Stochastic PDE model for spatial population growth in random environments

Pao-Liu Chow

doi:10.3934/dcdsb.2016.21.55

Article Contents

2016, Volume 21, Issue 1: 55-65. Doi: 10.3934/dcdsb.2016.21.55

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Stochastic PDE model for spatial population growth in random environments

Pao-Liu Chow^1,

1.
Department of Mathematics, Wayne State University, Detroit, MI 48202

Received: January 31, 2015

Revised: August 31, 2015

Published: December 2015

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Abstract

The paper is concerned with a class of stochastic reaction-diffusion equations arising from a spatial population growth model in random environments. Under some sufficient conditions, Theorem 3.1 shows that the equation has a unique positive global solution in space $H^1(D)$. Then it is proven in Theorem 4.1 that the solution, as the population size, is ultimately bounded in the mean $L^2-$norm as the time tends to infinity. An almost-sure upper bound is also obtained for the long run time-average of the exponential rate of the population growth in $L^2-$norm together with the $L^p-$moment of the population size with $p \geq 2.$ It is also shown in Theorem 4.3 that there is a unique invariant measure that leads to a stationary population distribution. For illustration, an example is given.

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Mathematics Subject Classification: Primary: 60H15; Secondary: 60H05.

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