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January  2016, 21(1): 55-65. doi: 10.3934/dcdsb.2016.21.55

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  February 2015 Revised  September 2015 Published  November 2015

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.
Keywords: positive solutions, ultimate bound, Stochastic reaction-diffusion equation, spatial population growth, invariant measure..
Mathematics Subject Classification: Primary: 60H15; Secondary: 60H0.
Citation: Pao-Liu Chow. Stochastic PDE model for spatial population growth in random environments. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 55-65. doi: 10.3934/dcdsb.2016.21.55
References:
[1]

P. Chow, Stochastic Partial Differential Equations,, $2^{nd}$ edition, (2015). Google Scholar

[2]

P. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations,, Comm. Stoch. Analy., 3 (2009), 211. Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[4]

L. Edelstein-Keshet, Mathematical Models in Biology,, Random House, (1988). Google Scholar

[5]

I. Gyöngy, N. V. Krylov and B. L. Rozovskii, On stochastic equations with respect to semimartingales II: Itô formula in Banach spacses,, Stochastics, 6 (1982), 153. Google Scholar

[6]

R. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146. Google Scholar

[7]

Y. Lou, T. Nagylaki and W. Ni, An introduction to migration, selection PDE models,, Discr. Conti. Dyn. Syst., 33 (2013), 4349. doi: 10.3934/dcds.2013.33.4349. Google Scholar

[8]

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavier of the sochastic Lotka-Volterra model,, J. Math. Analy. Appl., 287 (2003), 141. doi: 10.1016/S0022-247X(03)00539-0. Google Scholar

[9]

W. Ni, The Mathematics of Diffusion,, CBMS-NSF Regional Conf. Series in Appl. Math., 82 (2011). doi: 10.1137/1.9781611971972. Google Scholar

[10]

B. Øksendal, Stochastic Differential Equations,, Springer, (2003). Google Scholar

[11]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes,, Stochastics, 3 (1979), 127. doi: 10.1080/17442507908833142. Google Scholar

[12]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1983). Google Scholar

[13]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Walter de Gruyter, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[14]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154. doi: 10.1016/j.jmaa.2009.03.066. Google Scholar

show all references

References:
[1]

P. Chow, Stochastic Partial Differential Equations,, $2^{nd}$ edition, (2015). Google Scholar

[2]

P. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations,, Comm. Stoch. Analy., 3 (2009), 211. Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[4]

L. Edelstein-Keshet, Mathematical Models in Biology,, Random House, (1988). Google Scholar

[5]

I. Gyöngy, N. V. Krylov and B. L. Rozovskii, On stochastic equations with respect to semimartingales II: Itô formula in Banach spacses,, Stochastics, 6 (1982), 153. Google Scholar

[6]

R. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146. Google Scholar

[7]

Y. Lou, T. Nagylaki and W. Ni, An introduction to migration, selection PDE models,, Discr. Conti. Dyn. Syst., 33 (2013), 4349. doi: 10.3934/dcds.2013.33.4349. Google Scholar

[8]

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavier of the sochastic Lotka-Volterra model,, J. Math. Analy. Appl., 287 (2003), 141. doi: 10.1016/S0022-247X(03)00539-0. Google Scholar

[9]

W. Ni, The Mathematics of Diffusion,, CBMS-NSF Regional Conf. Series in Appl. Math., 82 (2011). doi: 10.1137/1.9781611971972. Google Scholar

[10]

B. Øksendal, Stochastic Differential Equations,, Springer, (2003). Google Scholar

[11]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes,, Stochastics, 3 (1979), 127. doi: 10.1080/17442507908833142. Google Scholar

[12]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1983). Google Scholar

[13]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Walter de Gruyter, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[14]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154. doi: 10.1016/j.jmaa.2009.03.066. Google Scholar

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