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Dynamics of a two-prey one-predator system in random environments. (English) Zbl 1279.92088
A two prey one predator model perturbed by a Brownian motion is considered. The main results are: characterization of parameter values that lead to the extinction of none, one or more species, sufficient conditions for ergodicity (i.e., a unique stationary distribution), and conditions for ‘global stability’ or ‘asymptotic flatness’, i.e., two solutions initiated with different initial conditions asymptotically approach each other. Examples and numerical simulations are also included.

MSC:
92D40 Ecology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92-08 Computational methods for problems pertaining to biology
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[1] Ahmad, S.; Lazer, A.C., Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40, 37-49, (2000) · Zbl 0955.34041
[2] Ahmad, S.; Stamova, I.M., Almost necessary and sufficient conditions for survival of species, Nonlinear Anal., 5, 219-229, (2004) · Zbl 1080.34035
[3] Bahar, A.; Mao, X., Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11, 377-400, (2004) · Zbl 1043.92028
[4] Barbalat, I.: Systems d’equations differentielles d’oscillations nonlineaires. Rev. Roum. Math. Pures Appl. 4, 267-270 (1959) · Zbl 0090.06601
[5] Beddington, J.R.; May, R.M., Harvesting natural populations in a randomly fluctuating environment, Science, 197, 463-465, (1977)
[6] Braumann, C.A., Itô versus Stratonovich calculus in random population growth, Math. Biosci., 206, 81-107, (2007) · Zbl 1124.92039
[7] Chen, L., Chen, J.: Nonlinear Biological Dynamical System. Science Press, Beijing (1993)
[8] Cheng, S., Stochastic population systems, Stoch. Anal. Appl., 27, 854-874, (2009) · Zbl 1180.92071
[9] Feng, W., Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179, 592-609, (1993) · Zbl 0846.35067
[10] Freedman, H.; Waltman, P., Mathematical analysis of some three-species food-chain models, Math. Biosci., 33, 257-276, (1977) · Zbl 0363.92022
[11] Freedman, H.; Waltman, P., Persistence in models of three interacting predator-prey populations, Math. Biosci., 68, 213-231, (1984) · Zbl 0534.92026
[12] Gard, T.: Introduction to Stochastic Differential Equations. Dekker, New York (1988) · Zbl 0628.60064
[13] Has’minskii, R.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)
[14] Higham, D., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546, (2001) · Zbl 0979.65007
[15] Hu, G.; Wang, K., The estimation of probability distribution of SDE by only one sample trajectory, Comput. Math. Appl., 62, 1798-1806, (2011) · Zbl 1231.60051
[16] Hutson, V.; Vickers, G., A criterion for permanent co-existence of species, with an application to a two-prey one-predator system, Math. Biosci., 63, 253-269, (1983) · Zbl 0524.92023
[17] Jiang, D.; Shi, N.; Li, X., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340, 588-597, (2008) · Zbl 1140.60032
[18] Jiang, D.; Ji, C.; Li, X.; O’Regan, D., Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390, 582-595, (2012) · Zbl 1258.34099
[19] Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991) · Zbl 0734.60060
[20] Li, X.; Mao, X., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24, 523-545, (2009) · Zbl 1161.92048
[21] Li, X.; Gray, A.; Jiang, D.; Mao, X., Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376, 11-28, (2011) · Zbl 1205.92058
[22] Liu, M.; Wang, K., Survival analysis of a stochastic cooperation system in a polluted environment, J. Biol. Syst., 19, 183-204, (2011) · Zbl 1228.92074
[23] Liu, M.; Wang, K.; Wu, Q., Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73, 1969-2012, (2011) · Zbl 1225.92059
[24] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching II, J. Math. Anal. Appl., 355, 577-593, (2009) · Zbl 1162.92032
[25] Øsendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2003)
[26] Sridhara, R.; Watson, R., Stochastic three species systems, J. Math. Biol., 28, 595-607, (1990) · Zbl 0737.92027
[27] Strang, G.: Linear Algebra and Its Applications. Thomson Learning, New York (1988) · Zbl 0338.15001
[28] Takeuchi, Y.; Adachi, N., Existence of bifurcation of stable equilibrium in two-prey, one-predator communities, Bull. Math. Biol., 45, 877-900, (1983) · Zbl 0524.92025
[29] Ton, T.V., Survival of three species in a nonautonomous Lotka-Volterra system, J. Math. Anal. Appl., 362, 427-437, (2010) · Zbl 1184.34057
[30] Vance, R., Predation and resource partitioning in one predator-two prey model communities, Am. Nat., 112, 797-813, (1978)
[31] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46, 1155-1179, (2007) · Zbl 1140.93045
[32] Zhu, C.; Yin, G., On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71, e1370-e1379, (2009) · Zbl 1238.34059
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