Persistence and nonpersistence of a predator prey system with stochastic perturbation. (English) Zbl 1474.92080

Li, Haihong; Jiang, Daqing; Cong, Fuzhong; Li, Haixia

Abstr. Appl. Anal. 2014, Article ID 720283, 10 p. (2014).

Summary: We analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.

MSC:

92D25	Population dynamics (general)
34C60	Qualitative investigation and simulation of ordinary differential equation models
34F05	Ordinary differential equations and systems with randomness

References:

[1]	Gard, T. C., Stability for multispecies population models in random environments, Nonlinear Analysis: Theory, Methods & Applications, 10, 12, 1411-1419 (1986) · Zbl 0598.92017 · doi:10.1016/0362-546X(86)90111-2
[2]	Goh, B. S., Global stability in many species system, American Naturalist, 111, 135-143 (1997)
[3]	Lv, J. L.; Wang, K., Asymptotic properties of a stochastic predator-prey system with Holling II functional response, Communications in Nonlinear Science and Numerical Simulation, 16, 10, 4037-4048 (2011) · Zbl 1218.92072 · doi:10.1016/j.cnsns.2011.01.015
[4]	Chen, L. S.; Jing, Z. J., The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions, Chinese Science Bulletin, 9, 521-523 (1984)
[5]	Mao, X. R.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes and Their Applications, 97, 1, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[6]	Mao, X., Delay population dynamics and environmental noise, Stochastics and Dynamics, 5, 2, article 149 (2005) · Zbl 1093.60033 · doi:10.1142/S021949370500133X
[7]	Polansky, P., Invariant distributions for multipopulation models in random environments, Theoretical Population Biology, 16, 1, 25-34 (1979) · Zbl 0417.92019 · doi:10.1016/0040-5809(79)90004-2
[8]	Gard, T. C., Introduction to Stochastic Differential Equations (1988), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0628.60064
[9]	Kuang, Y.; Smith, H. L., Global stability for infinite delay Lotka-Volterra type systems, Journal of Differential Equations, 103, 2, 221-246 (1993) · Zbl 0786.34077 · doi:10.1006/jdeq.1993.1048
[10]	Hu, Y. Z.; Wu, F. K.; Huang, C., Stochastic Lotka-Volterra models with multiple delays, Journal of Mathematical Analysis and Applications, 375, 1, 42-57 (2011) · Zbl 1245.92063 · doi:10.1016/j.jmaa.2010.08.017
[11]	Wantanabe, N. I., Stochastic Differential Equations and Diffusion Processes (1981), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0495.60005
[12]	Ji, C. Y.; Jiang, D. Q., Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, Journal of Mathematical Analysis and Applications, 381, 1, 441-453 (2011) · Zbl 1232.34072 · doi:10.1016/j.jmaa.2011.02.037
[13]	May, R. M., Stability and Complexity in Model Ecosystem (2001), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 1044.92047
[14]	Ji, C. Y.; Jiang, D. Q.; Shi, N. Z., Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, Journal of Mathematical Analysis and Applications, 359, 2, 482-498 (2009) · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[15]	Ji, C. Y.; Jiang, D. Q.; Li, X. Y., Qualitative analysis of a stochastic ratio-dependent predator-prey system, Journal of Computational and Applied Mathematics, 235, 5, 1326-1341 (2011) · Zbl 1229.92076 · doi:10.1016/j.cam.2010.08.021
[16]	Liu, X. N.; Chen, L. S., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals, 16, 2, 311-320 (2003) · Zbl 1085.34529 · doi:10.1016/S0960-0779(02)00408-3
[17]	Mao, X. R., Stochastic Differential Equations and Applications (1997), Chichester, UK: Horwood, Chichester, UK · Zbl 0892.60057
[18]	Jiang, D.-Q.; Zhang, B.-X.; Wang, D.-H.; Shi, N.-Z., Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation, Science in China A: Mathematics, 50, 7, 977-986 (2007) · Zbl 1136.34324 · doi:10.1007/s11425-007-0071-y
[19]	Chen, L. S.; Chen, J., Nonlinear Biological Dynamical System (1993), Beijing, China: Science Press, Beijing, China
[20]	Xia, P. Y.; Zheng, X. K.; Jiang, D. Q., Persistence and nonpersistence of a nonautonomous stochastic mutualism system, Abstract and Applied Analysis, 2013 (2013) · Zbl 1402.92375 · doi:10.1155/2013/256249
[21]	Jiang, D. Q.; Ji, C. Y.; Li, X. Y.; O’Regan, D., Analysis of autonomous Lotka-Volterra competition systems with random perturbation, Journal of Mathematical Analysis and Applications, 390, 2, 582-595 (2012) · Zbl 1258.34099 · doi:10.1016/j.jmaa.2011.12.049
[22]	Feller, W., An Introduction to Probability Theory and Its Application, 2 (1971), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0219.60003
[23]	Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J., A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71, 3, 876-902 (2011) · Zbl 1263.34068 · doi:10.1137/10081856X
[24]	Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43, 3, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302