Stochastic PDE model for spatial population growth in random environments

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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

Abstract
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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 and 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, Chapman & Hall/CRC, Boca Raton-London-New York, 2015.
[2]	P. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Comm. Stoch. Analy., 3 (2009), 211-222.
[3]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[4]	L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, 1988.
[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), 153-173.
[6]	R. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146.
[7]	Y. Lou, T. Nagylaki and W. Ni, An introduction to migration, selection PDE models, Discr. Conti. Dyn. Syst., 33 (2013), 4349-4373. doi: 10.3934/dcds.2013.33.4349.
[8]	X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavier of the sochastic Lotka-Volterra model, J. Math. Analy. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.
[9]	W. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Series in Appl. Math., SIAM 82, 2011. doi: 10.1137/1.9781611971972.
[10]	B. Øksendal, Stochastic Differential Equations, Springer, New York, 2003.
[11]	E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. doi: 10.1080/17442507908833142.
[12]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin/New York, 1983.
[13]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Walter de Gruyter, Berlin/New York, 1988. doi: 10.1007/978-1-4684-0313-8.
[14]	C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.

show all references

References:

[1]	P. Chow, Stochastic Partial Differential Equations, $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton-London-New York, 2015.
[2]	P. Chow, Unbounded positive solutions of nonlinear parabolic Itô equations, Comm. Stoch. Analy., 3 (2009), 211-222.
[3]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
[4]	L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, 1988.
[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), 153-173.
[6]	R. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146.
[7]	Y. Lou, T. Nagylaki and W. Ni, An introduction to migration, selection PDE models, Discr. Conti. Dyn. Syst., 33 (2013), 4349-4373. doi: 10.3934/dcds.2013.33.4349.
[8]	X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavier of the sochastic Lotka-Volterra model, J. Math. Analy. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.
[9]	W. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Series in Appl. Math., SIAM 82, 2011. doi: 10.1137/1.9781611971972.
[10]	B. Øksendal, Stochastic Differential Equations, Springer, New York, 2003.
[11]	E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. doi: 10.1080/17442507908833142.
[12]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin/New York, 1983.
[13]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Walter de Gruyter, Berlin/New York, 1988. doi: 10.1007/978-1-4684-0313-8.
[14]	C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.

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