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

Stochastic PDE model for spatial population growth in random environments

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.
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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