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Posts from the ‘ecology’ Category

17
Feb

Scientists track nighttime bird migration using weather radar


Discovery
Scientists track nighttime bird migration using weather radar

Weather radar offers new view of nocturnal bird flight

 sandhill cranes flying

Researchers studied sandhill cranes, among the largest migrating birds in North America.
Credit and Larger Version

February 16, 2016

Using recently developed techniques for analyzing Doppler weather radar data, researchers looked at the impediments — crosswinds and oceans — facing nighttime-migrating birds in eastern North America.

The migrants drifted sideways on crosswinds, the scientists found, but compensated for that drift near the Atlantic coast.

Coastal migrating birds’ ability to compensate for wind drift increased through the night, but no strong changes were observed at inland sites. The behavior suggests that birds adapt in flight and compensate for wind drift near coastal areas.

Weather radar tracks bird migration

“The research has taken an innovative approach in showing how existing weather radar systems can be used to investigate the behavior of migrating birds,” said Liz Blood, program director in the National Science Foundation (NSF) Division of Environmental Biology, which funded the research.

Blood said the ability to use the U.S. weather radar network to track migrating birds “opens new opportunities to study — in real-time — billions of birds during their migrations.”

Kyle Horton and Phillip Stepanian of the University of Oklahoma developed an application for observing migrating birds during nighttime flight.

Jeffrey Kelly of the Oklahoma Biological Survey and the University of Oklahoma and Cornell University’s Benjamin Van Doren, Wesley Hochachka and Andrew Farnsworth were also involved in the research.

The results are published in the current issue of the journal Scientific Reports.

“Until now, no studies have documented this large-scale phenomenon using weather radars,” Horton said. “Our analyses are based on detection of millions of migrating birds, as many as five million on a single night.”

The researchers looked at strategies of nocturnally migrating birds using Doppler polarimetric radars at three coastal and three inland sites in the eastern U.S. during the autumns of 2013 and 2014.

Radars collected data every five to 10 minutes, yielding approximately 1.6 million samples from 55 nights.

Birds drift sideways, then compensate

The results show a greater propensity of birds to drift sideways at inland sites; birds flying near the Atlantic coast increasingly oriented and tracked westward away from the coast.

The prediction that migrating birds compensate more for drift when encountering a migration barrier is consistent with the results, the scientists said.

The research indicates that at a regional scale, in a regularly and heavily traveled airspace of the bird migration system, birds routinely fly in crosswind conditions and have managed to compensate for such conditions, according to Kelly.

Migrants likely know their location relative to migration barriers while in flight, he said, and actively assess the degree to which they need to compensate for wind.

“The increasing automation of radar analysis will enable further exploration of U.S. weather radar data to achieve real-time monitoring of billions of birds during their migrations,” Kelly said.

The U.S. weather radar network offers the largest sensor array worldwide for monitoring animal migrations, the scientists said, including birds, bats and insects.

Investigators

Eli Bridge

Le Gruenwald

Jeffrey Kelly

Phillip Chilson

Valliappa Lakshmanan

Related Institutions/Organizations

University of Oklahoma Norman Campus

Related Programs

MacroSystems Biology and Early NEON Science

Related Awards

#1340921 EAGER: Advancing Biological Interpretations of Radar Data

Total Grants

$301,641

17
Jun

Big dinosaurs steered clear of the tropics


Press Release 15-067
Big dinosaurs steered clear of the tropics

Climate swings lasting millions of years too much for dinos

Some 212 million years ago, landscapes weren’t all dinosaur-friendly: dry, hot, with wildfires.
Credit and Larger Version

June 15, 2015

For more than 30 million years after dinosaurs first appeared, they remained inexplicably rare near the equator, where only a few small-bodied meat-eating dinosaurs made a living.

The long absence at low latitudes has been one of the great, unanswered questions about the rise of the dinosaurs.

Now the mystery has a solution, according to scientists who pieced together a detailed picture of the climate and ecology more than 200 million years ago at Ghost Ranch in northern New Mexico, a site rich with fossils.

The findings, reported today in the journal Proceedings of the National Academy of Sciences (PNAS), show that the tropical climate swung wildly with extremes of drought and intense heat.

Wildfires swept the landscape during arid regimes and reshaped the vegetation available for plant-eating animals.

“Our data suggest it was not a fun place,” says scientist Randall Irmis of the University of Utah.

“It was a time of climate extremes that went back and forth unpredictably. Large, warm-blooded dinosaurian herbivores weren’t able to exist close to the equator–there was not enough dependable plant food.”

The study, led by geochemist Jessica Whiteside, now of the University of Southampton, is the first to provide a detailed look at climate and ecology during the emergence of the dinosaurs.

Atmospheric carbon dioxide levels then were four to six times current levels. “If we continue along our present course, similar conditions in a high-CO2 world may develop, and suppress low-latitude ecosystems,” Irmis says.

“These scientists have developed a new explanation for the perplexing near-absence of dinosaurs in late Triassic [the Triassic was between 252 million and 201 million years ago] equatorial settings,” says Rich Lane, program director in the National Science Foundation’s (NSF) Division of Earth Sciences, which funded the research.

“That includes rapid vegetation changes related to climate fluctuations between arid and moist climates and the resulting extensive wildfires of the time.”

Reconstructing the deep past

The earliest known dinosaur fossils, found in Argentina, date from around 230 million years ago.

Within 15 million years, species with different diets and body sizes had evolved and were abundant except in tropical latitudes. There the only dinosaurs were small carnivores. The pattern persisted for 30 million years after the first dinosaurs appeared.

The scientists focused on Chinle Formation rocks, which were deposited by rivers and streams between 205 and 215 million years ago at Ghost Ranch (perhaps better known as the place where artist Georgia O’Keeffe lived and painted for much of her career).

The multi-colored rocks of the Chinle Formation are a common sight on the Colorado Plateau at places such as the Painted Desert at Petrified Forest National Park in Arizona.

In ancient times, North America and other land masses were bound together in the supercontinent Pangea. The Ghost Ranch site stood close to the equator, at roughly the same latitude as present-day southern India.

The researchers reconstructed the deep past by analyzing several kinds of data: from fossils, charcoal left by ancient wildfires, stable isotopes from organic matter, and carbonate nodules that formed in ancient soils.

Fossilized bones, pollen grains and fern spores revealed the types of animals and plants living at different times, marked by layers of sediment.

Dinosaurs remained rare among the fossils, accounting for less than 15 percent of vertebrate animal remains.

They were outnumbered in diversity, abundance and body size by reptiles known as pseudosuchian archosaurs, the lineage that gave rise to crocodiles and alligators.

The sparse dinosaurs consisted mostly of small, carnivorous theropods.

Big, long-necked dinosaurs, or sauropodomorphs–already the dominant plant-eaters at higher latitudes–did not exist at the study site nor any other low-latitude site in the Pangaea of that time, as far as the fossil record shows.

Abrupt changes in climate left a record in the abundance of different types of pollen and fern spores between sediment layers.

Fossilized organic matter from decaying plants provided another window on climate shifts. Changes in the ratio of stable isotopes of carbon in the organic matter bookmarked times when plant productivity declined during extended droughts.

Drought and fire

Wildfire temperatures varied drastically, the researchers found, consistent with a fluctuating environment in which the amount of combustible plant matter rose and fell over time.

The researchers estimated the intensity of wildfires using bits of charcoal recovered in sediment layers.

The overall picture is that of a climate punctuated by extreme shifts in precipitation and in which plant die-offs fueled hotter fires. That in turn killed more plants, damaged soils and increased erosion.

Atmospheric carbon dioxide levels, calculated from stable isotope analyses of soil carbonate and preserved organic matter, rose from about 1,200 parts per million (ppm) at the base of the section, to about 2,400 ppm near the top.

At these high CO2 concentrations, climate models predict more frequent and more extreme weather fluctuations consistent with the fossil and charcoal evidence.

Continuing shifts between extremes of dry and wet likely prevented the establishment of the dinosaur-dominated communities found in the fossil record at higher latitudes across South America, Europe, and southern Africa, where aridity and temperatures were less extreme and humidity was consistently higher.

Resource-limited conditions could not support a diverse community of fast-growing, warm-blooded, large dinosaurs, which require a productive and stable environment to thrive.

“The conditions would have been something similar to the arid western United States today, although there would have been trees and smaller plants near streams and rivers, and forests during humid times,” says Whiteside.

“The fluctuating and harsh climate with widespread wildfires meant that only small two-legged carnivorous dinosaurs could survive.”

-NSF-

Media Contacts

Cheryl Dybas, NSF, (703) 292-7734, cdybas@nsf.gov

Joe Rojas-Burke, University of Utah, (801) 585-6861, joe.rojas@utah.edu

Related Websites
NSF Grant: Collaborative Research: An Exceptional Window into Late Triassic Terrestrial Ecosystems from the Western United States: http://www.nsf.gov/awardsearch/showAward?AWD_ID=1349650HistoricalAwards=false

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2015, its budget is $7.3 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 48,000 competitive proposals for funding, and makes about 11,000 new funding awards. NSF also awards about $626 million in professional and service contracts yearly.

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12
Jan

A free boundary problem arising in the ecological models with N -species

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M228','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M228View MathML/a be given and let (2.5) hold for some a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M229','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M229View MathML/a. The aim of this section is to prove Theorem 2.1.

3.1 An approximation problem

We will construct an approximation problem of (1.1a)-(1.3). We first construct some approximation functions.

For an arbitrary a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M230','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M230View MathML/a, choose a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M231','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M231View MathML/a such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M232','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M232View MathML/a in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M233','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M233View MathML/a as a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M234','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M234View MathML/a. Then, for small enough ε,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M235','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M235View MathML/a

(3.1)

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M236','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M236View MathML/a be a smooth function with values between 0 and 1 such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M237','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M237View MathML/a for all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M238','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M238View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M239','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M239View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M240','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M240View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M241','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M241View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M242','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M242View MathML/a, and let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M243','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M243View MathML/a

Define

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M244','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M244View MathML/a

(3.2)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M245','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M245View MathML/a

(3.3)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M246','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M246View MathML/a

Then according to hypothesis (H)(ii) and (iii), it follows that the vector function
a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M247','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M247View MathML/a is mixed quasimonotone in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M248','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M248View MathML/a with index vector a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M249','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M249View MathML/a, and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M250','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M250View MathML/a

(3.4)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M251','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M251View MathML/a

(3.5)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M252','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M252View MathML/a

(3.6)

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M253','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M253View MathML/a. The definition of function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M254','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M254View MathML/a implies that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M255','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M255View MathML/a

(3.7)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M256','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M256View MathML/a

(3.8)

In addition, it is obvious from (2.5) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M257','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M257View MathML/a

(3.9)

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M258','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M258View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M259','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M259View MathML/a are the closure of a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M260','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M260View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M261','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M261View MathML/a, respectively. Thus by (3.9) and the definition of functions a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M262','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M262View MathML/a, an argument similar to the one used in 16], Lemma 3.2] shows that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M263','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M263View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M264','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M264View MathML/a

(3.10)

We next construct the approximation functions of a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M265','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M265View MathML/a. Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M266','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M266View MathML/a be a sufficiently smooth nonnegative function such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M267','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M267View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M268','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M268View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M269','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M269View MathML/a, and let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M270','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M270View MathML/a be a sufficiently smooth nonnegative function taking values in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M271','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M271View MathML/a such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M272','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M272View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M273','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M273View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M274','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M274View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M275','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M275View MathML/a or a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M276','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M276View MathML/a, and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M277','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M277View MathML/a for all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M278','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M278View MathML/a. Define

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M279','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M279View MathML/a

Then hypothesis (H)(i) and 17], Chapter II] imply that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M280','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M280View MathML/a

(3.11)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M281','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M281View MathML/a

and (2.1) and (2.3) imply that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M282','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M282View MathML/a

(3.12)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M283','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M283View MathML/a

(3.13)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M284','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M284View MathML/a

(3.14)

In addition, for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M285','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M285View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M286','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M286View MathML/a

(3.15)

Employing the above approximation functions, we consider the following approximation
problem:

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M287','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M287View MathML/a

(3.16)

Lemma 3.1

Problem (3.16) has a unique classical solutiona onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M288','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M288View MathML/aina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M289','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M289View MathML/a, and the following estimates hold:

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M290','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M290View MathML/a

(3.17)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M291','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M291View MathML/a

(3.18)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M292','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M292View MathML/a

(3.19)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M293','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M293View MathML/a

(3.20)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M294','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M294View MathML/a

(3.21)

where constantsa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M295','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M295View MathML/aandCare independent ofε.

Proof

In 15], by using the method of upper and lower solutions, together with the associated monotone
iterations and various estimates, we investigated the existence and uniqueness of
the global piecewise classical solutions of the quasilinear parabolic system with
discontinuous coefficients and continuous delays under various conditions including
mixed quasimonotone property of reaction functions. The same problem was also discussed
for the system with continuous coefficients without time-delay.

It is obvious that problem (3.16) is the special case of 15], problem (1.1)] without discontinuous coefficients and time delays. Hypothesis (H)(ii)
shows that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M296','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M296View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M297','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M297View MathML/a are the coupled weak upper and lower solutions of (3.16) in the sense of 15], Definition 2.2]. By (3.4)-(3.6) and (3.11)-(3.14), we conclude from 15], Theorem 4.1] that problem (3.16) has a unique classical solution a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M298','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M298View MathML/a in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M299','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M299View MathML/a. Furthermore, using (3.1), (3.5), (3.6) and (3.12)-(3.14), the proof similar to that of 16], Lemma 3.3] shows that estimates (3.17) and (3.18) hold.

To prove (3.19), we first fix a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M300','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M300View MathML/a. In view of (3.7), (3.8), (3.10) and (3.15), we find that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M301','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M301View MathML/a is the solution of the following problems for single equation:

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M302','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M302View MathML/a

(3.22)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M303','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M303View MathML/a

(3.23)

By (3.22), (3.23), (3.12) and (3.14), the proof similar to that of 17], Chapter VI, Lemma 3.1] gives (3.19) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M304','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M304View MathML/a. The similar argument shows that (3.19) holds for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M305','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M305View MathML/a.

We next prove (3.20). For any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M306','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M306View MathML/a, let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M307','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M307View MathML/a

and let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M308','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M308View MathML/a

By a direct computation we have

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M309','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M309View MathML/a

(3.24)

Then (3.16), (3.2), (3.3) and (3.10) imply that the function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M310','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M310View MathML/a satisfies

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M311','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M311View MathML/a

(3.25)

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M312','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M312View MathML/a.

A double integration by parts gives

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M313','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M313View MathML/a

(3.26)

Thus multiplying the equation in (3.25) by a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M314','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M314View MathML/a, integrating it on a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M315','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M315View MathML/a and using (3.5), (3.6), (3.24) and (3.26), we find that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M316','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M316View MathML/a

Furthermore, by (3.12)-(3.14), (3.19) and Cauchy’s inequality, we deduce that for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M317','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M317View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M318','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M318View MathML/a

Choosing a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M319','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M319View MathML/a, we get

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M320','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M320View MathML/a

(3.27)

Consequently,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M321','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M321View MathML/a

This, together with Gronwall’s inequality (see 17], Chapter II, Lemma 5.5]), implies that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M322','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M322View MathML/a

Hence we deduce from (3.25), (3.27), (3.5) and (3.6) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M323','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M323View MathML/a

which, together with (3.24), yields (3.20) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M324','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M324View MathML/a.

For any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M325','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M325View MathML/a, we consider the equality

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M326','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M326View MathML/a

A similar argument gives (3.20) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M327','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M327View MathML/a. Therefore, (3.20) holds for all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M328','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M328View MathML/a.

It remains to prove (3.21). For each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M329','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M329View MathML/a, since a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M330','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M330View MathML/a is in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M331','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M331View MathML/a, then (3.20) and 17], Chapter 2, formula (3.8)] show that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M332','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M332View MathML/a

Thus (3.21) holds. □

Lemma 3.2

Leta onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M333','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M333View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M334','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M334View MathML/abe the solutions ina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M335','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M335View MathML/afor problem (3.16) corresponding toa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M336','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M336View MathML/aanda onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M337','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M337View MathML/a, respectively. Then

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M338','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M338View MathML/a

(3.28)

wherea onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M339','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M339View MathML/a.

Proof

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M340','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M340View MathML/a, and let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M341','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M341View MathML/a be fixed. We see from (3.16) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M342','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M342View MathML/a

(3.29)

In view of (3.15), we find a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M343','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M343View MathML/a. Multiplying the equation in (3.29) by a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M344','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M344View MathML/a and integrating by parts on a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M345','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M345View MathML/a, we deduce that, for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M346','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M346View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M347','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M347View MathML/a

(3.30)

Let us estimate a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M348','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M348View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M349','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M349View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M350','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M350View MathML/a. Since (3.2), (3.3) and (3.10) imply that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M351','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M351View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M352','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M352View MathML/a

then it follows from Cauchy’s inequality that, for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M353','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M353View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M354','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M354View MathML/a

(3.31)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M355','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M355View MathML/a

(3.32)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M356','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M356View MathML/a

(3.33)

where

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M357','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M357View MathML/a

According to the definition of function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M358','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M358View MathML/a, we see that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M359','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M359View MathML/a if a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M360','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M360View MathML/a or a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M361','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M361View MathML/a. Thus by (3.21) we have

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M362','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M362View MathML/a

(3.34)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M363','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M363View MathML/a

(3.35)

Summing equality (3.30) with respect to l from a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M364','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M364View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M365','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M365View MathML/a, using (3.31)-(3.35) and Minkowski’s inequality, and choosing a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M366','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M366View MathML/a, we then conclude that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M367','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M367View MathML/a

This, together with Gronwall’s inequality, yields

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M368','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M368View MathML/a

Combining the two inequalities above and (3.21) leads us to estimate (3.28). □

3.2 The solutions of the diffraction problem

Proof of Theorem 2.1

We divide the proof into three steps.

Step 1. We prove the global existence of the solutions. Let us discuss the behavior
of the solution a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M369','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M369View MathML/a associated with a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M370','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M370View MathML/a by Theorem 2.1 as a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M371','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M371View MathML/a.

We first see from (3.16) that for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M372','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M372View MathML/a and any vector function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M373','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M373View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M374','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M374View MathML/a

(3.36)

Furthermore, according to estimates (3.17), (3.20), (3.28) and the Arzela-Ascoli theorem, we conclude that there exists a subsequence (we retain
the same notation for it) a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M375','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M375View MathML/a such that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M376','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M376View MathML/a

Thus u is in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M377','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M377View MathML/a, u satisfies the parabolic condition (1.2), and estimates (2.6) and (2.7) hold.

We next show that for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M378','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M378View MathML/a, the sequences a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M379','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M379View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M380','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M380View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M381','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M381View MathML/a converge in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M382','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M382View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M383','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M383View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M384','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M384View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M385','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M385View MathML/a, respectively. Since

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M386','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M386View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M387','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M387View MathML/a

then it follows from (2.2), (3.3) and Lebesgue dominated convergence theorem that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M388','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M388View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M389','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M389View MathML/a

and from (3.10) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M390','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M390View MathML/a

The similar argument shows that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M391','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M391View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M392','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M392View MathML/a.

Based on the above arguments for sequences a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M393','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M393View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M394','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M394View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M395','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M395View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M396','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M396View MathML/a, by letting a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M397','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M397View MathML/a, we conclude from (3.36) that (2.4) holds.

We also see from (3.20) that there exists a subsequence a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M398','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M398View MathML/a (denoted by a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M399','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M399View MathML/a still) such that for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M400','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M400View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M401','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M401View MathML/a converge weakly in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M402','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M402View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M403','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M403View MathML/a. Recalling that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M404','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M404View MathML/a in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M405','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M405View MathML/a, we deduce a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M406','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M406View MathML/a. These, together with (3.20), imply that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M407','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M407View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M408','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M408View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M409','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M409View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M410','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M410View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M411','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M411View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M412','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M412View MathML/a, and (2.8) holds. Thus (2.4) implies that u satisfies the equations in (1.1a) and (1.1b) for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M413','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M413View MathML/a and the inner boundary condition (1.3) for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M414','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M414View MathML/a (see 17], Chapter 3, Section 13]). As we have done in the derivation of (3.21), estimate (2.8) yields (2.9).

For fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M415','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M415View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M416','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M416View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M417','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M417View MathML/a satisfies the linear equation

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M418','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M418View MathML/a

where

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M419','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M419View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M420','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M420View MathML/a

Then for any subdomains a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M421','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M421View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M422','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M422View MathML/a satisfying a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M423','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M423View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M424','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M424View MathML/a, we have a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M425','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M425View MathML/a. The parabolic regularity theory shows that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M426','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M426View MathML/a. Hence u satisfies pointwise the equations in (1.1a) and (1.1b) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M427','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M427View MathML/a. Consequently, u is a solution in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M428','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M428View MathML/a of problem (1.1a)-(1.3) and estimates (2.6)-(2.9) hold.

Step 2. In what follows, we will show that the solution in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M429','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M429View MathML/a for problem (1.1a)-(1.3) is unique and estimates (2.10) and (2.11) hold.

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M430','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M430View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M431','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M431View MathML/a be the solutions in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M432','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M432View MathML/a corresponding to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M433','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M433View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M434','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M434View MathML/a, respectively. Set a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M435','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M435View MathML/a. Then a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M436','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M436View MathML/a. We choose a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M437','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M437View MathML/a in (2.4) to find

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M438','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M438View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M439','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M439View MathML/a

For each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M440','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M440View MathML/a, by a subtraction of the above equations for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M441','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M441View MathML/a, we conclude that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M442','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M442View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M443','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M443View MathML/a

Then

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M444','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M444View MathML/a

Setting a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M445','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M445View MathML/a and summing the above inequalities with respect to l from a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M446','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M446View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M447','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M447View MathML/a, we have

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M448','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M448View MathML/a

Again by Gronwall’s inequality we deduce (2.10), which, together with 17], Chapter 2, formula (3.8)], gives (2.11). Therefore the solution in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M449','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M449View MathML/a for problem (1.1a)-(1.3) associated with a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M450','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M450View MathML/a is unique.

Step 3. For a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M451','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M451View MathML/a, we discuss the regularity of u.

For any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M452','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M452View MathML/a, let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M453','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M453View MathML/a

Then v satisfies

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M454','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M454View MathML/a

We will use the result of 18] to obtain the regularity of v. To do this, we need the estimate of a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M455','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M455View MathML/a for any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M456','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M456View MathML/a. Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M457','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M457View MathML/a be a smooth function with values between 0 and 1 such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M458','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M458View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M459','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M459View MathML/a or a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M460','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M460View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M461','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M461View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M462','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M462View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M463','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M463View MathML/a, and let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M464','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M464View MathML/a

For any small enough Δt, consider the equality a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M465','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M465View MathML/a. By employing the formula of integration by parts and 17], Chapter II, formula (4.7)], we get

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M466','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M466View MathML/a

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M467','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M467View MathML/a. Some tedious computation and Cauchy’s inequality yield

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M468','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M468View MathML/a

We choose a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M469','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M469View MathML/a and employ (2.8) to find

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M470','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M470View MathML/a

As we have done in the derivation of (3.20), by Gronwall’s inequality we get

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M471','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M471View MathML/a

Consequently,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M472','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M472View MathML/a

By 17], Chapter II, Lemma 4.11], this inequality implies that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M473','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M473View MathML/a

(3.37)

Hence a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M474','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M474View MathML/a. Using (3.37), hypothesis (H)(iii) and 18], Theorem 1.1], we deduce that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M475','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M475View MathML/a is continuous with respect to y in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M476','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M476View MathML/a and in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M477','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M477View MathML/a for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M478','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M478View MathML/a, and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M479','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M479View MathML/a is continuous in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M480','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M480View MathML/a. Since a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M481','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M481View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M482','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M482View MathML/a, then for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M483','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M483View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M484','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M484View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M485','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M485View MathML/a are continuous with respect to x in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M486','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M486View MathML/a and in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M487','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M487View MathML/a. □

The following corollary follows directly from Theorem 2.1.

Corollary 3.3

Assume thata onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M488','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M488View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M489','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M489View MathML/a , satisfy (2.5) and the sequencea onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M490','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M490View MathML/aconverges ina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M491','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M491View MathML/atoa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M492','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M492View MathML/a. Ifa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M493','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M493View MathML/a, uare the solutions ina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M494','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M494View MathML/aof (1.1a)-(1.3) corresponding toa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M495','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M495View MathML/aandφ , respectively, then there exists a subsequence (we retain the same notation for it) a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M496','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M496View MathML/asuch that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M497','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M497View MathML/a

11
Jun

West Nile Virus Notes

 

2014 budget is $30,097,170 than last year.  Failing to solve the problem means more money and resources (personnel) involved the next time there’s a spike in temperatures and drop in precipitation.

Current drought map

Drought monitor archive

CDC West Nile Virus stats (2002-2012) PDF archives

Mother Jones (2012) interactive maps

The middle class suburban areas appeared to support the appropriate combination of vegetation, open space, and potential vector habitat favoring WNV transmission. Wealthier neighborhoods had more vegetation, more diverse land use, and less habitat fragmentation likely resulting in higher biological diversity potentially protective against the WNV human transmission, e.g. the avian host “dilution effect” [45].

CDC WNV stats 2002-2012 by state

 

 TIME 2/28/2014

The biggest indicator of whether West Nile virus will occur is the maximum temperature of the warmest month of the year, which is why the virus has caused the most damage in hot southern states like Texas.

The UCLA model indicates that higher temperatures and lower precipitation will generally lead to more cases of West Nile

 

 

2012 Scientific American

A nearly frost-free winter followed by the summer’s drought has worsened the epidemic

 

West Nile Virus outbreak map

west nile virus

21
Dec

Pigging out at TRAC

I’ve been volunteering at Trinity River Audubon Center since 2009; long enough to be allowed to do “special projects” for them; projects that will start with research and end with publications in magazines and (hopefully) journals or conference presentations.

TRAC is 200 acres of a “blackfield remediation” site — an illegal dump that had polluted the Trinity and the neighborhood for over 30 years which was reclaimed and turned into an Audubon center.  A recent decision by the City of Dallas to turn a part of the Trinity River corridor area next to the center into a golf course has chased the feral hogs from there onto our property, with the result that we’re seeing a lot of landscape damage from these animals.

IMG_20131220_141155

 

This is a section of the trail near the building, where the hogs have been rooting all along the gravel walkways.  The damaged landscape left by their actions is vulnerable to erosion, and any native plants in this area that are destroyed are often replaced by invasives, which represents a step backwards in the effort to recreate a “pristine prairie environment” similar to what would have been here fifty or a hundred years ago.   The idea that you can take a damaged piece of land and magically return it to a pristine state is a bit of a pipe dream — we have been fighting a constant war with invasive species since the center’s opening.

So, this is my new research — find out about the hogs on THIS piece of property and see if we can manage them — because they are also destroying the juniper forest, as you can see in the two photos below.

IMG_20131220_141628

IMG_20131220_142132I’m going to focus on “manage them to minimize damage” rather than “eliminate” since much of the research indicates that they’re difficult to eradicate and that habitats free of hogs are just an invitation for other hogs to move in.

Here’s what I know:

* There are two to four different herds.  I’ve set up a cheap game camera (and am hoping it works) to start to get pictures of the pigs so we can identify them.

* the pigs are mainly going for areas with Johnson grass and areas with junipers (cedars).  There’s minor damage in other areas, but the Johnson grass places and juniper forest are the places that are most heavily damaged.

Here’s what I’m thinking:

* that it might be possible to landscape the area (brush piles and so forth) to make it less convenient habitat for the pigs.

* that when they tear up the invasive Johnson grass, they’re doing us a favor.  We can plant over those areas with native grasses and the hogs have done a lot of the removal and soil tilling work for us.

* would a maintained and controlled herd keep other herds from entering the land?

* Are there certain types of landscapes that the pigs don’t like?  In other words, do they avoid walking over cobblestone-sized rocks or do they avoid brush piles — what do they avoid and what do they prefer?

My research assistant, John Snodgrass, is hunting up web pages for me to look at on wild hogs, but so far the information seems to come down to: they breed rapidly, they’re destructive, trapping and killing are the best ways to get rid of them, and Wild Hogs Are Tasty.

So — I’m also soliciting thoughts and observations here — if you have a thought or an observation or an idea about hogs (remembering that this will be done by One Small Woman… so don’t advocate putting up 30 miles of barbed wire fence, ’cause it just ain’t happening), let me know and I’ll add it to my list.

Alternatively, if you have a game camera to loan me or want to help me come map the trails on the property with a GPS or help map the damage), let me know and you can be part of the team.

 

30
Nov

National Science Foundation: Emerging Vector-Borne Diseases Create New Public Health Challenge


Press Release 12-218
Emerging Vector-Borne Diseases Create New Public Health Challenge

Land-use change, globalization of trade and travel, and social upheaval drive emergence of diseases

Close-up photo of a tick.

Vector-borne diseases are transmitted by ticks, mosquitoes and fleas.
Credit and Larger Version

November 30, 2012

Human activities are advancing the spread of vector-borne, zoonotic diseases such as West Nile virus, Lyme disease and dengue fever, report scientists publishing a series of papers today in the journal The Lancet.

Vector-borne zoonotic diseases result from disease-causing agents or pathogens that naturally infect wildlife, and are transmitted to humans by carriers such as mosquitoes and ticks. In short, they’re diseases transmitted between animals and humans.

Widespread land-use change, globalization of trade and travel, and social upheaval are driving the emergence of zoonotic diseases around the world, said biologist Marm Kilpatrick, who studies the ecology of infectious diseases at the University of California, Santa Cruz.

Kilpatrick co-authored one of several papers in The Lancet, along with Sarah Randolph of the University of Oxford. The Lancet papers are part of a special series in the journal focused on emerging zoonotic diseases.

“Increasing human population, and the urbanization and agricultural intensification of landscapes, put strong selective pressure on vector-borne pathogens to infect humans–and to be transmitted by vectors and hosts that live around humans,” Kilpatrick said.

“Humans are altering the environment and moving ourselves and other organisms around the globe at an ever-increasing pace,” said Sam Scheiner, a program director for the Ecology and Evolution of Infectious Diseases (EEID) program at the National Science Foundation. “Our fast-track has led to a growing disease threat.”

EEID is a joint effort with NSF and the National Institutes of Health. At NSF, the Directorate for Biological Sciences and Directorate for Geosciences fund the program.

EEID funded much of the research discussed in The Lancet papers. “These papers show how and why zoonotic diseases are emerging, and what we need to know to ease the disease burden,” said Scheiner.

The papers “offer a bridge between ecologists and clinicians whose combined efforts are needed to address the ongoing challenges of emerging zoonotic diseases,” said Kilpatrick.

Added scientist Peter Daszak, president of the EcoHealth Alliance in New York City and author of a paper in the series, “Pandemic zoonoses such as SARS, Ebola and HIV/AIDS are devastating when they emerge. What this series shows is that we have new ways of predicting their origins, of discovering them even before they reach our population–truly a brave new world for pandemic prevention.”

There are roughly two types of emerging infectious diseases: introduced and locally emerging.

Introduced diseases arise from the spread of a pathogen to a new location, as when West Nile virus arrived in New York in 1999 and subsequently spread across North America.

Locally emerging diseases increase in importance in areas where they are endemic, as with Lyme disease in the United States during the past three decades.

These two types of emerging diseases can differ markedly with respect to infection dynamics or the number of cases over time, Kilpatrick said.

“Introduced diseases often cause a big spike in infections, and then decrease substantially. Locally emerging diseases often show a steady, sustained rise.”

The movement of pathogens by global trade and travel results in the emergence of diseases in new regions.

Once established, introduced pathogens often evolve to take advantage of their new environments, including new hosts and vectors.

With much of the landscape shaped by human activities, pathogens may thrive by infecting hosts and vectors that do well in man-made environments.

Emergence of endemic vector-borne diseases can result from changes in land use, such as movement of people into new habitats, or environmental changes that affect wild animals that serve as natural hosts–and the insect vectors that spread the disease to humans.

Although vector-borne diseases are sensitive to climate, climate change does not appear to be a major driving force behind emerging diseases.

“So far, climate change has been a relatively minor player compared to land use and socioeconomic factors in the emergence of vector-borne disease,” Kilpatrick said.

Social and economic changes, ranging from economic downturns to displacement of populations by armed conflict, frequently precipitate disease outbreaks through their effects on public health systems, sanitation systems, behavioral patterns and uses of natural environmental resources.

The incidence of any vector-borne disease involves a complex interplay of multiple factors affecting animal hosts, vectors and people.

Kilpatrick and Randolph emphasize that control of these diseases requires combined efforts by clinicians and public health officials to treat patients; promote behavior likely to minimize the risk of infection; and advise on efforts to reverse the ecological drivers of transmission through vector control, urban planning and ecological restoration.

The Lancet papers are published ahead of a special 20th anniversary symposium to be held on Dec. 11 and 12, 2012, in Washington, D.C.

The symposium is hosted by the National Academies’ Institute of Medicine’s Forum on Microbial Threats. The symposium will take a retrospective look at the Institute of Medicine’s 1992 report on Emerging Infections and its 2003 report on Microbial Threats to Health, as well as its creation of the forum in 1996.

-NSF-

Media Contacts

Cheryl Dybas, NSF (703) 292-7734 cdybas@nsf.gov

Tim Stephens, UCSC (831) 459-2495 stephens@ucsc.edu

Anthony Ramos, EcoHealth Alliance (212) 380-4469 ramos@ecohealthalliance.org

Related Websites
NSF-NIH Special Report: Ecology and Evolution of Infectious Diseases: http://www.nsf.gov/news/special_reports/ecoinf/index.jsp
West Nile Virus Transmission Linked with Land-Use Patterns and “Super-spreaders”: http://www.nsf.gov/news/news_summ.jsp?cntn_id=122007
Social Bats Pay a Price: Fungal Disease, White-Nose Syndrome … Extinction?: http://www.nsf.gov/news/news_summ.jsp?cntn_id=124679
Controlling the Spread of Diseases Among Humans, Other Animals and the Environment: http://www.nsf.gov/news/news_summ.jsp?cntn_id=125496
Snails in the Waters, Disease in the Villages: http://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=126031

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2012, its budget is $7.0 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives over 50,000 competitive requests for funding, and makes about 11,000 new funding awards. NSF also awards nearly $420 million in professional and service contracts yearly.

 

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

14
Nov

Why We Need Insects–Even "Pesky" Ones


Press Release 12-189
Why We Need Insects–Even “Pesky” Ones

Hard evidence of evolution: a five-year study shows that plants may quickly lose important traits through evolution soon after insects are removed from the environment

Photo of yellow flowers of evening primrose in Ithaca, New York.

A large natural population of evening primrose (yellow flowers) in Ithaca, New York.
Credit and Larger Version

October 4, 2012

View a video interview with Anurag Agrawal of Cornell University.

At first blush, many people would probably love to get rid of insects, such as pesky mosquitoes, ants and roaches. But a new study indicates that getting rid of insects could trigger some unwelcome ecological consequences, such as the rapid loss of desired traits in plants, including their good taste and high yields.

Specifically, the study–described in the Oct. 5, 2012 issue of Science and funded by the National Science Foundation showed that evening primroses grown in insecticide-treated plots quickly lost, through evolution, defensive traits that helped protect them from plant-eating moths. The protective traits lost included the production of insect-deterring chemicals and later blooms that gave evening primroses temporal distance from plant-eating larvae that peak early in the growing season.

These results indicate that once the plants no longer needed their anti-insect defenses, they lost those defenses. What’s more, they did so quickly–in only three or four generations.

Anurag Agrawal, the leader of the study and a professor of ecology and evolutionary biology at Cornell University, explains, “We demonstrated that when you take moths out of the environment, certain varieties of evening primrose were particularly successful. These successful varieties have genes that produce less defenses against moths.”

In the absence of insects, the evening primroses apparently stopped investing energy in their anti-insect defenses, and so these defenses disappeared through natural selection. Agrawal says that he was “very surprised” by how quickly this process occurred, and that such surprises, “tell us something about the potential speed and complexities of evolution. In addition, experiments like ours that follow evolutionary change in real-time provide definitive evidence of evolution.”

Agrawal believes that his team’s study results are applicable to many other insect-plant interactions beyond evening primroses and moths.  Here’s why: The ubiquitous consumption of plants by insects represents one of the dominant species interactions on Earth. With insect-plant relationships so important, it is widely believed that many plant traits originally evolved solely as defenses against insects. Some of these anti-insect plant defenses, such as the bitter taste of some fruits, are desirable.

“This experimental demonstration of how rapid evolution can shape ecological interactions supports the idea that we need to understand feedbacks between evolutionary and ecological processes in order to be able to predict how communities and ecosystems will respond to change,” said Alan Tessier, a program director in NSF’s Directorate for Biological Sciences.

“One of the things farmers are trying to do is breed agricultural crops to be more resistant to pests,” said Agrawal. “Our study indicates that various genetic tradeoffs may make it difficult or impossible to maintain certain desired traits in plants that are bred for pest resistance.”

In addition, oils produced by evening primroses have been used medicinally for hundreds of years and are beginning to be used as herbal remedies. Agrawal’s insights about pests that attack these plants and about chemical compounds produced by these plants may ultimately be useful to the herbal and pharmaceutical industries.

Agrawal says that most previous real-time experiments on evolution have been conducted with bacteria in test tubes in laboratories. “One of things we were excited about is that we were able to repeat that kind of experiment in nature. You can expect to see a lot more of this kind of thing in future. We will keep our experiment running as a long-term living laboratory. ”

More information about this study is available from a Cornell University press release.

-NSF-

Media Contacts

John Carberry, Cornell University (607) 255-5353 johncarberry@cornell.edu

Lily Whiteman, National Science Foundation (703) 292-8310 lwhitema@nsf.gov

Program Contacts

Alan Tessier, National Science Foundation (703) 292-7198 atessier@nsf.gov

Principal Investigators

Anurag Agrawal, Cornell University (607) 254-4255 aa337@cornell.edu

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2012, its budget is $7.0 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives over 50,000 competitive requests for funding, and makes about 11,000 new funding awards. NSF also awards nearly $420 million in professional and service contracts yearly.

 

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

24
Aug

Tarrant County’s guide to Gambusia (mosquito fish)

This humble little Texas minnow is one of the primary resources used by cities to fight mosquitoes — and they do a very good job of it, too.   They don’t come in pretty colors and they never get very large (up to 2 inches for females) but each fish can eat over 100 mosquito larvae in a day.  This makes them an ideal method of controlling mosquitoes, since when they’re in the larval or egg stage, mosquitoes don’t move about much and they float on top of the water.  They’re easier for predators such as the gambusia to spot… and eat.  Once they hatch out and are flying around, they’re much harder to catch, even for quick chimney swifts and other birds that catch flying insects. And they are the mosquito control method preferred by many Texas cities.

 

But this year in North Texas, the much-needed fish were in very short supply due to problems with the hatchery, leaving cities scrambling to find other sources — or resorting to spraying techniques (which residents don’t like.)

http://dfw.cbslocal.com/2012/08/02/mosquito-controlling-fish-in-short-supply/

 

For people interested in these fish, Tarrant County has a local stocking guide for city governments that can be applied on a smaller scale.

http://www.tarrantcounty.com/ehealth/lib/ehealth/Gambusia_Affinis_Stocking_Guide_for_Local_Governments.pdf

Homeowners with ponds can get these fish from Keller Fish Farms: http://www.kellerfish.com/

24
Aug

Native Plants in Urban Yards Offer Birds "Mini-Refuges"


Press Release 12-155
Native Plants in Urban Yards Offer Birds “Mini-Refuges”

Landscaping with native vegetation helps local bird species

Photo of a xeric, or desert, yard in Phoenix.

A xeric, or desert, yard in Phoenix: This yard with native vegetation is a mini-refuge for birds.
Credit and Larger Version

August 22, 2012

Yards with plants that mimic native vegetation offer birds “mini-refuges” and help to offset losses of biodiversity in cities, according to results of a study published today in the journal PLOS ONE.

“Native” yards support birds better than those with traditional grass lawns and non-native plantings.

Researchers conducted the study through the National Science Foundation’s (NSF) Central Arizona-Phoenix Long-Term Ecological Research (LTER) site, one of 26 such sites around the globe in ecosystems from coral reefs to deserts, from forests to grasslands.

“To a desert bird, what’s green is not necessarily good,” says Doug Levey, program director in NSF’s Division of Environmental Biology. “Arizona birds don’t view lush urban landscapes as desert oases. The foraging behavior of birds in greener yards suggests that there’s less food for them there than in yards with more natural vegetation.”

The research, led by scientists Susannah Lerman and Paige Warren of the University of Massachusetts-Amherst, and Hilary Gan and Eyal Shochat of Arizona State University, looked at residential landscape types and native bird communities in Phoenix, Ariz.

It’s among the first to use quantitative measures and a systematic approach–including 24-hour video monitoring–in yards to assess and compare foraging behavior of common backyard birds.

The scientists found that desert-like, or xeric, yards had a more even bird community and superior habitat compared with moist, or mesic, grass lawns.

“We already know that bird communities differ, and that there are more desert birds found in a desert-type yard,” says Lerman.

“With this study, we’re starting to look at how different yards function–whether birds behave differently by yard type. We’re doing that using behavioral indicators, especially foraging, as a way of assessing birds’ perceptions of habitat quality between differing yard designs.”

Lerman and colleagues conducted the experiment in 20 residential yards at least 1.8 miles apart, making it unlikely that the same birds would visit more than one study yard.

Half the yards were desert-like, while the others had green lawns.

From February through April 2010, homeowners removed bird feeders before and during a 24-hour experimental data collection period.

The researchers set up feeding stations–seed trays–in each yard to simulate resource patches similar to ones where birds feed in the wild. Plastic trays contained 0.70 ounces of millet seed mixed into six pounds of sand. The trays were placed on low stools and left out for 24 hours.

Later, Lerman removed the trays, sifted out and weighed uneaten seed to the nearest 0.01 gram. The amount of seed remaining quantified the giving-up densities (GUD), or the foraging decision and quitting point for the last bird visiting a seed tray.

Trays were videotaped for the entire 24-hour experiment.

The experiment assumed that an animal behaving optimally would stop foraging from a seed tray when its energy gains equal the “costs” of foraging, Lerman says.

Costs include predation risk, digestion and missed opportunities to find food elsewhere.

As time spent foraging at a seed tray increases, so do the costs associated with foraging. When a bird first arrives at the tray, seeds are easy to find, but that gets harder as the tray becomes depleted.

Each bird makes a decision about whether to spend time searching in the tray or to move on to a new patch in the yard.

The “giving up” point will be different for different species and in different environmental conditions. Birds visiting seed trays in yards with more natural food available will quit a tray sooner than birds in resource-poor yards.

Since the method only measures the foraging decisions for the last species visiting the seed tray, the researchers devised a mathematical model for estimating the foraging decisions for all visiting species.

Using the videotapes, they counted every peck by every bird for each tray to calculate the relationship between the number of pecks and grams of seed consumed for each seed tray. This was the GUD-peck ratio for the last species visiting the seed tray.

They then estimated the seed consumption–GUD ratio for all other species visiting the seed tray based on the number of pecks per tray when each species quit.

“We know how many pecks each species had and can put that number into the model and calculate the number of grams at that point,” Lerman says. This greatly enhances the GUD method by expanding the ability to assess foraging decisions for all species visiting trays.

In all, 14 species visited the trays, 11 of which visited both yard types. Abert’s towhee, curve-billed thrasher (a species unique to the Sonoran desert), house finch and house sparrow were the most widespread tray visitors.

Species that visited trays in both yard designs consumed more seed from trays placed in mesic yards, indicating lower habitat quality compared with xeric yards.

Similarly, foragers in the desert-like yards quit the seed trays earlier due to greater abundance of alternative food resources in those yards, spending more time foraging in the natural yards and less at the seed trays.

Lerman says that by videotaping the trays, counting pecks and measuring giving-up points by species, the research also advanced the GUD method, allowing researchers to disentangle some of the effects of bird community composition and density of competitors, and how these factors affect foraging decisions between two different landscape designs.

The results build upon evidence that native landscaping can help mitigate the effects of urbanization on common songbirds, she says.

-NSF-

Media Contacts

Cheryl Dybas, NSF (703) 292-7734 cdybas@nsf.gov

Janet Lathrop, UMass-Amherst (413) 545-0444 jlathrop@admin.umass.edu

Related Websites
NSF Central Arizona-Phoenix LTER Site: http://caplter.asu.edu/about/site-description/
NSF LTER Network: http://www.lternet.edu

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2012, its budget is $7.0 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives over 50,000 competitive requests for funding, and makes about 11,000 new funding awards. NSF also awards nearly $420 million in professional and service contracts yearly.

 Get News Updates by Email 

Useful NSF Web Sites:

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14
Aug

Current Community Science Projects

CURRENT AND ONGOING:

The Arlington Archosaur site is closing for the remainder of the summer (as is usual) — in August it’s just Too Darned Hot to dig.  They will return later in the year.

Trinity River Audubon Center’s Third Thursday will have Frogwatch, Amphibian Watch, and National Phenological Database expedition as well as a Chimney Swift watch and possibly an Owl Prowl (whew!)  This time we’ll also be doing a “transect survey” for dragonflies.

Botannical Research Institute of Texas (BRIT) has an ongoing need for volunteers to help with their databases (indoor work!) and do some field research.  Contact them for details.

 

GENERAL EDUCATION, public welcome

Connemara Conservacy has an astronomy walk (August 18th) and an evening Open House (September 23.)  See website for details.

The John Bunker Sands Wetland Center has a First Saturday Walk & Talk Bird Tour.    9:00 – 11:00 am; $10 includes admission if you’re not a member; $5 for members.

Botannical Research Institute of Texas (BRIT)’s Science Saturday – open plant ID, exhibits, demonstrations, tours, etc First Saturday of every month at  10:00 – 2:00.  This is a fun place to visit, even if you’re not really into plants.

 

PLANNED:

The John Bunker Sands Wetland Center is planning to start some Citizen Science initiatives in the near future.  I will be in a planning meeting about this effort at the end of August.