American Institute of Mathematical Sciences

Advanced
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

[1]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[2]

Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
[3]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[4]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[5]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033
[6]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182
[7]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
[8]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65
[9]

Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200
[10]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[11]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[12]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
[13]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105
[14]

Benedetto Bozzini, Deborah Lacitignola, Ivonne Sgura. Morphological spatial patterns in a reaction diffusion model for metal growth. Mathematical Biosciences & Engineering, 2010, 7 (2) : 237-258. doi: 10.3934/mbe.2010.7.237
[15]

Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81
[16]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385
[17]

Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102
[18]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[19]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
[20]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences