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January  2018, 17(1): 67-84. doi: 10.3934/cpaa.2018005

Stochastic spatiotemporal diffusive predator-prey systems

 Yuan Yung Tseng Functinal Analysis Resarch Center, Harbin Normal University, Harbin, 150025, China, School of mathematics and Statistics, Northeast Normal Unversity, Changchun, 130024, China

* Corresponding author

Received  December 2016 Revised  June 2017 Published  September 2017

Fund Project: This work is supported by the National Natural Science Foundation of China grant 11471091,11571086

In this paper, a spatiotemporal diffusive predator-prey system with Holling type-Ⅲ is considered. By using a Lyapunov-like function, it is proved that the unique local solution of the system must be a a global one if the interaction intensity is small enough. A comparison theorem is used to show that the system can be extinction or stability in mean square under some additional conditions. Finally, an unique invariant measure for the system is obtained.

Citation: Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure & Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005
References:
 [1] P. Balasubramaniam and C. Vidhlya, Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms, J. Comput. App. Math., 234 (2010), 3458-3466. doi: 10.1016/j.cam.2010.05.007. Google Scholar [2] T. Caraballo and L. Shaikhet, Stability of delay evolution equations with stochastic perturbations, Commun. Pur. Appl. Anal., 13 (2014), 2095-2113. doi: 10.3934/cpaa.2014.13.2095. Google Scholar [3] P. Chow, Stochastic Partial Differential Equations, 2$^{nd}$ edition, CRC Press, New York, 2014. Google Scholar [4] H. I. Freedman, Deterministic Mathematical Models in Population Ecoloty, Marcel Dekker, New York, 1980. Google Scholar [5] S. L. Hollis, R. H. Martin and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761. doi: 10.1137/0518057. Google Scholar [6] C. Y. Ji, D. Q. Jiang and N. Z. Shi, Analysis of a predator-prey model with modified Leslie-Goweer and Holling-type Ⅱ schemes with sotchastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039. Google Scholar [7] Y. G. Kao and C. H. Wang, Global stability analysis for stochastic coupled reaction-diffusion systems on networks, Nonlinear Anal-real., 14 (2013), 1457-1465. doi: 10.1016/j.nonrwa.2012.10.008. Google Scholar [8] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discret Contin. Dyn. S., 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. Google Scholar [9] K. Liu, Stability of Infinite Diemnsional Stochastic Differential Equations with Applications, Chapmen & Hall\CRC, New York, 2006. Google Scholar [10] A. J. Lotka, Elements of Physical Biology, Dover Publications, 1925.Google Scholar [11] Q. Luo, F. Deng, J. Bao and Y. Fu, Stabilization of stochastic Hopfield neural network with distributed parameters, Science in China, Ser. F Information Sciences, 47 (2004), 752-762. doi: 10.1360/03yf0332. Google Scholar [12] Q. Luo and X. Mao, Sochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84. doi: 10.1016/j.jmaa.2006.12.032. Google Scholar [13] X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Chichester, UK, 1997. doi: 10.1533/9780857099402. Google Scholar [14] X. Mao, G. Marion and E. Renshaw, Enviromental Brownian noise suppresses explosions in population dynamis, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar [15] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.Google Scholar [16] O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theor. Probab., 29 (2016), 996-1026. doi: 10.1007/s10959-015-0606-z. Google Scholar [17] G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar [18] G. D. Prato and J. Zabczyk, Ergodicity for Infinite, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. Google Scholar [19] Y. Takeuchi, N. H. Dub, N.T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka--Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009. Google Scholar [20] J. Wang, Spatiotemporal Patterns of a Homogeneous Diffusive Predator-Prey System with Holling Type Ⅲ Functional Response, J. Dyn. Diff. Equ., (2016), 1-27. doi: 10.1007/s10884-016-9517-7. Google Scholar [21] J. Wang and H. Fan, Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence, Discret Contin. Dyn. S., 21 (2016), 909-918. doi: 10.3934/dcdsb.2016.21.909. Google Scholar [22] Y. Yang and D. Jiang, Dynamics of the stochastic low concentration trimolecular chemical reaction model, J. Math. Chem., 52 (2014), 2532-2545. doi: 10.1007/s10910-014-0398-x. Google Scholar [23] Y. Zhang and L. Li, Stability of numerical method for semi-linear stochastic pantograph differential equations, J. Inequal. Appl. , 1 (2016), 30. doi: 10.1186/s13660-016-0971-x. Google Scholar

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References:
 [1] P. Balasubramaniam and C. Vidhlya, Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms, J. Comput. App. Math., 234 (2010), 3458-3466. doi: 10.1016/j.cam.2010.05.007. Google Scholar [2] T. Caraballo and L. Shaikhet, Stability of delay evolution equations with stochastic perturbations, Commun. Pur. Appl. Anal., 13 (2014), 2095-2113. doi: 10.3934/cpaa.2014.13.2095. Google Scholar [3] P. Chow, Stochastic Partial Differential Equations, 2$^{nd}$ edition, CRC Press, New York, 2014. Google Scholar [4] H. I. Freedman, Deterministic Mathematical Models in Population Ecoloty, Marcel Dekker, New York, 1980. Google Scholar [5] S. L. Hollis, R. H. Martin and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761. doi: 10.1137/0518057. Google Scholar [6] C. Y. Ji, D. Q. Jiang and N. Z. Shi, Analysis of a predator-prey model with modified Leslie-Goweer and Holling-type Ⅱ schemes with sotchastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039. Google Scholar [7] Y. G. Kao and C. H. Wang, Global stability analysis for stochastic coupled reaction-diffusion systems on networks, Nonlinear Anal-real., 14 (2013), 1457-1465. doi: 10.1016/j.nonrwa.2012.10.008. Google Scholar [8] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discret Contin. Dyn. S., 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. Google Scholar [9] K. Liu, Stability of Infinite Diemnsional Stochastic Differential Equations with Applications, Chapmen & Hall\CRC, New York, 2006. Google Scholar [10] A. J. Lotka, Elements of Physical Biology, Dover Publications, 1925.Google Scholar [11] Q. Luo, F. Deng, J. Bao and Y. Fu, Stabilization of stochastic Hopfield neural network with distributed parameters, Science in China, Ser. F Information Sciences, 47 (2004), 752-762. doi: 10.1360/03yf0332. Google Scholar [12] Q. Luo and X. Mao, Sochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84. doi: 10.1016/j.jmaa.2006.12.032. Google Scholar [13] X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Chichester, UK, 1997. doi: 10.1533/9780857099402. Google Scholar [14] X. Mao, G. Marion and E. Renshaw, Enviromental Brownian noise suppresses explosions in population dynamis, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar [15] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.Google Scholar [16] O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, J. Theor. Probab., 29 (2016), 996-1026. doi: 10.1007/s10959-015-0606-z. Google Scholar [17] G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar [18] G. D. Prato and J. Zabczyk, Ergodicity for Infinite, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. Google Scholar [19] Y. Takeuchi, N. H. Dub, N.T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka--Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009. Google Scholar [20] J. Wang, Spatiotemporal Patterns of a Homogeneous Diffusive Predator-Prey System with Holling Type Ⅲ Functional Response, J. Dyn. Diff. Equ., (2016), 1-27. doi: 10.1007/s10884-016-9517-7. Google Scholar [21] J. Wang and H. Fan, Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence, Discret Contin. Dyn. S., 21 (2016), 909-918. doi: 10.3934/dcdsb.2016.21.909. Google Scholar [22] Y. Yang and D. Jiang, Dynamics of the stochastic low concentration trimolecular chemical reaction model, J. Math. Chem., 52 (2014), 2532-2545. doi: 10.1007/s10910-014-0398-x. Google Scholar [23] Y. Zhang and L. Li, Stability of numerical method for semi-linear stochastic pantograph differential equations, J. Inequal. Appl. , 1 (2016), 30. doi: 10.1186/s13660-016-0971-x. Google Scholar
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