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The complex dynamics of a stochastic predator-prey model. (English) Zbl 1253.93142

Summary: A modified stochastic ratio-dependent Leslie-Gower predator-prey model is formulated and analyzed. For the deterministic model, we focus on the existence of equilibria, local, and global stability; for the stochastic model, by applying Itô formula and constructing Lyapunov functions, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. Based on these results, we perform a series of numerical simulations and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

MSC:

93E20 Optimal stochastic control
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
60H30 Applications of stochastic analysis (to PDEs, etc.)
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[1] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0539.03004
[2] P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, no. 3-4, pp. 213-245, 1948. · Zbl 0034.23303 · doi:10.1093/biomet/35.3-4.213
[3] P. H. Leslie, “A stochastic model for studying the properties of certain biological systems by numerical methods,” Biometrika, vol. 45, pp. 16-31, 1958. · Zbl 0089.15803 · doi:10.1093/biomet/45.1-2.16
[4] H.-F. Huo and W.-T. Li, “Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model,” Mathematical and Computer Modelling, vol. 40, no. 3-4, pp. 261-269, 2004. · Zbl 1067.39008 · doi:10.1016/j.mcm.2004.02.026
[5] H.-F. Huo and W.-T. Li, “Periodic solutions of delayed Leslie-Gower predator-prey models,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 591-605, 2004. · Zbl 1060.34039 · doi:10.1016/S0096-3003(03)00801-4
[6] F. Chen, “Permanence in nonautonomous multi-species predator-prey system with feedback controls,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 694-709, 2006. · Zbl 1087.92059 · doi:10.1016/j.amc.2005.04.047
[7] F. Chen and X. Cao, “Existence of almost periodic solution in a ratio-dependent Leslie system with feedback controls,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1399-1412, 2008. · Zbl 1145.34026 · doi:10.1016/j.jmaa.2007.09.075
[8] C. Ji, D. Jiang, and N. Shi, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 482-498, 2009. · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[9] M. Liu and K. Wang, “Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1114-1121, 2011. · Zbl 1221.34152 · doi:10.1016/j.cnsns.2010.06.015
[10] T. C. Gard, “Persistence in stochastic food web models,” Bulletin of Mathematical Biology, vol. 46, no. 3, pp. 357-370, 1984. · Zbl 0533.92028 · doi:10.1007/BF02462011
[11] T. C. Gard, “Stability for multispecies population models in random environments,” Nonlinear Analysis. Theory, Methods & Applications, vol. 10, no. 12, pp. 1411-1419, 1986. · Zbl 0598.92017 · doi:10.1016/0362-546X(86)90111-2
[12] R. M. May, Stability and Complexity in Model Ecosystems, vol. 6, Princeton University Press, 2001.
[13] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited, 1997. · Zbl 0892.60057
[14] Q. Luo and X. Mao, “Stochastic population dynamics under regime switching,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 69-84, 2007. · Zbl 1113.92052 · doi:10.1016/j.jmaa.2006.12.032
[15] X. Mao, C. Yuan, and J. Zou, “Stochastic differential delay equations of population dynamics,” Journal of Mathematical Analysis and Applications, vol. 304, no. 1, pp. 296-320, 2005. · Zbl 1062.92055 · doi:10.1016/j.jmaa.2004.09.027
[16] S. Pang, F. Deng, and X. Mao, “Asymptotic properties of stochastic population dynamics,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 15, no. 5, pp. 603-620, 2008. · Zbl 1171.34038
[17] D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525-546, 2001. · Zbl 0979.65007 · doi:10.1137/S0036144500378302
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