Predator-prey System with the Effect of Environmental Fluctuation
Main Article Content
Abstract
Abstract: In this paper we study the trajectory behavior of Lotka - Volterra predator - prey systems with periodic coefficients under telegraph noises. We describe the - limit set of the solution, give sufficient conditions for the persistence and prove the existence of a Markov periodic solution.
Keywords: Key words and phrases, Lotka-Volterra Equation, Predator - Prey, Telegraph noise, - limit set, Markov periodic solution.References
[1] K. Gopalsamy. Global asymptotic stability in a periodic Lotka-Volterra system, J.Austral. Math. Soc. Ser , B 27 (1985), pp. 66-72.
[2] A. Tineo. Permanence of a large class periodic predator-prey systems, J.Math.Anal.Appl, 241 (2000), pp. 83-91.
[3] P.Yang, R. Xu. Global attractivity of the periodic Lotka-Volterra system, J. Math. Anal. Appl, 233 (1999), no. 1, pp. 221-232.
[4] J. Zhao, W. Chen. Global asymptotic stability of a periodic ecological model, Appl.Math.Comput, 147 (2004), pp 881-892.
[5] Z. Amine, R. Ortega. A periodic prey-predator system, J.Math.Anal.Appl, 185 (1994), pp. 477-489.
[6] M.Bardi. Predator-prey models in periodically uctuating environments, J.Math.Biology, 12 (1981), pp. 127-140.
[7] J. M. Cushing. Periodic time-dependent predator-prey system, Siam J. Appl. Math, Vol. 32, No. 1 (January 1977), pp. 82-95.
[8] J. Lpez-Gmez, R. Ortega, A. Tineo. The periodic predator-prey Lotka-Volterra model. Adv.Differential Equations, 1 (1996), no. 3, pp. 403-423.
[9] A. Tineo. On the asymptotic behavior of some population models, J.Math.Anal.Appl, 167 (1992), pp. 516-529.
[10] F. Zanolin, T. Ding, H. Huang. A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations, DCDS, Vol. 1, No 1 (January 1995), pp. 103-117.
[11] H. Kesten, Y. Ogura. Recurrence properties of Lotka-Volterra models with random uctuations. J. Math. Soc. Japan, 33 (1981), no. 2, pp. 335-366.
[12] M. Liu, K. Wang. Persistence, extinction and global asymptotical stability of a nonautonomous predator-prey model with random perturbation. Appl. Math. Model, 36 (2012), no. 11, pp. 5344-5353.
[13] S. S. De. Random predator-prey interactions in a varying environment: extinction or survival Bull. Math. Biol, 46 (1984), no. 1, pp. 175-184.
[14] P. Auger, N. H. Du, N. T. Hieu. Evolution of Lotka-Volterra predator-prey systems under telegraph noise. Math. Biosci. Eng, 6 (2009), no. 4, 683-700.
[15] I.I Gihman and A.V. Skorohod. The Theory of Stochastic Processes. Springer -Verlag Berlin Heidelberg New York 1979.
[16] R.S. Lipshter and Shyriaev. Statistics of Stochastic Processes. Nauka, Moscow 1974 (in Russian).
[2] A. Tineo. Permanence of a large class periodic predator-prey systems, J.Math.Anal.Appl, 241 (2000), pp. 83-91.
[3] P.Yang, R. Xu. Global attractivity of the periodic Lotka-Volterra system, J. Math. Anal. Appl, 233 (1999), no. 1, pp. 221-232.
[4] J. Zhao, W. Chen. Global asymptotic stability of a periodic ecological model, Appl.Math.Comput, 147 (2004), pp 881-892.
[5] Z. Amine, R. Ortega. A periodic prey-predator system, J.Math.Anal.Appl, 185 (1994), pp. 477-489.
[6] M.Bardi. Predator-prey models in periodically uctuating environments, J.Math.Biology, 12 (1981), pp. 127-140.
[7] J. M. Cushing. Periodic time-dependent predator-prey system, Siam J. Appl. Math, Vol. 32, No. 1 (January 1977), pp. 82-95.
[8] J. Lpez-Gmez, R. Ortega, A. Tineo. The periodic predator-prey Lotka-Volterra model. Adv.Differential Equations, 1 (1996), no. 3, pp. 403-423.
[9] A. Tineo. On the asymptotic behavior of some population models, J.Math.Anal.Appl, 167 (1992), pp. 516-529.
[10] F. Zanolin, T. Ding, H. Huang. A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations, DCDS, Vol. 1, No 1 (January 1995), pp. 103-117.
[11] H. Kesten, Y. Ogura. Recurrence properties of Lotka-Volterra models with random uctuations. J. Math. Soc. Japan, 33 (1981), no. 2, pp. 335-366.
[12] M. Liu, K. Wang. Persistence, extinction and global asymptotical stability of a nonautonomous predator-prey model with random perturbation. Appl. Math. Model, 36 (2012), no. 11, pp. 5344-5353.
[13] S. S. De. Random predator-prey interactions in a varying environment: extinction or survival Bull. Math. Biol, 46 (1984), no. 1, pp. 175-184.
[14] P. Auger, N. H. Du, N. T. Hieu. Evolution of Lotka-Volterra predator-prey systems under telegraph noise. Math. Biosci. Eng, 6 (2009), no. 4, 683-700.
[15] I.I Gihman and A.V. Skorohod. The Theory of Stochastic Processes. Springer -Verlag Berlin Heidelberg New York 1979.
[16] R.S. Lipshter and Shyriaev. Statistics of Stochastic Processes. Nauka, Moscow 1974 (in Russian).