一类具有时滞与Lévy跳的随机捕食者-食饵模型
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摘要
研究了一类具有时滞与Lévy跳的随机捕食者-食饵模型.首先利用Lyapunov方法和It■公式,给出了模型全局正解的存在唯一性.然后根据切比雪夫不等式和指数鞅不等式以及BorelCantelli引理等,得到了解的随机最终有界性以及灭绝性.最后,运用数值模拟验证了理论结果.
This research focuses on a class of the stochastic predator-prey model with delay and Lévy jump. Firstly, the Lyapunov method and Ito^formula are used to give the existence and uniqueness of the global positive solution of this model. Then according to Chebyshev's inequality, exponential martingale inequality and Borel-Cantelli lemma, etc., the stochastic ultimate boundedness and extinction are obtained. Finally, the theoretical results are verified by numerical simulations.
This research focuses on a class of the stochastic predator-prey model with delay and Lévy jump. Firstly, the Lyapunov method and Ito^formula are used to give the existence and uniqueness of the global positive solution of this model. Then according to Chebyshev's inequality, exponential martingale inequality and Borel-Cantelli lemma, etc., the stochastic ultimate boundedness and extinction are obtained. Finally, the theoretical results are verified by numerical simulations.
引文
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[7] MAO Xue-rong,RENSHAW E,MARION G.Environmental Brownian noise suppresses explosions in population dynamics[J].Stochastic Processes & Their Applications,2002,97(1):95-110.
[8] DU N H,SAM V H.Dynamics of a stochastic Lotka–Volterra model perturbed by white noise[J].Journal of Mathematical Analysis and Applications,2006,324(1):82-97.
[9] JI Chun-yan,JIANG Da-qing,SHI Ning-zhong.Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation[J].Journal of Mathematical Analysis and Applications,2009,359(2):482-498.
[10] MAO Xue-rong.Stationary distribution of stochastic population systems[J].Systems & Control Letters,2011,60(6):398-405.
[11] GENG Jing,LIU Meng,ZHANG You-qiang.Stability of a stochastic one-predator-two-prey population model with time delays[J].Communications in Nonlinear Science and Numerical Simulation,2017:S1007570417301284.
[12] MAO Xue-rong.Delay population dynamics and environmental noise[J].Stochastics and Dynamics,2005,05(02):149-162.
[13] LIPTSER R S.A strong law of large numbers for local martingales[J].Stochastics,1980,3(1/2/3/4):217-228.
[14] MAO Xue-rong.Stochastic differential equations and applications[M].Chichester:Horwood Publishing,2007.
[15] HIGHAM D.An algorithmic introduction to numerical simulation of stochastic differential equations[J].SIAM Review,2001,43(2):525–546.
[2] TAKEUCHI Y,ADACHI N.Existence and bifurcation of stable equilibrium in two-prey,one-predator communities[J].Bulletin of Mathematical Biology,1983,45(6):877-900.
[3] HUTSON V,VICKERS G T.A criterion for permanent coexistence of species,with an application to a two-prey one-predator system[J].Mathematical Biosciences,1983,63(2):0-269.
[4] WEI FENG.Coexistence,stability,and limiting behavior in a one-predator-two-prey model[J].Journal of Mathematical Analysis and Applications,1993,179(2):592-609.
[5] AHMAD S,STAMOVA I M.Almost necessary and sufficient conditions for survival of species[J].Nonlinear Analysis:Real World Application,2004,5(1):219-229.
[6] SEIFERT G.A Lotka-Volterra predator-prey system involving two predators[J].Methods and Applications of Analysis,1995,2(2).
[7] MAO Xue-rong,RENSHAW E,MARION G.Environmental Brownian noise suppresses explosions in population dynamics[J].Stochastic Processes & Their Applications,2002,97(1):95-110.
[8] DU N H,SAM V H.Dynamics of a stochastic Lotka–Volterra model perturbed by white noise[J].Journal of Mathematical Analysis and Applications,2006,324(1):82-97.
[9] JI Chun-yan,JIANG Da-qing,SHI Ning-zhong.Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation[J].Journal of Mathematical Analysis and Applications,2009,359(2):482-498.
[10] MAO Xue-rong.Stationary distribution of stochastic population systems[J].Systems & Control Letters,2011,60(6):398-405.
[11] GENG Jing,LIU Meng,ZHANG You-qiang.Stability of a stochastic one-predator-two-prey population model with time delays[J].Communications in Nonlinear Science and Numerical Simulation,2017:S1007570417301284.
[12] MAO Xue-rong.Delay population dynamics and environmental noise[J].Stochastics and Dynamics,2005,05(02):149-162.
[13] LIPTSER R S.A strong law of large numbers for local martingales[J].Stochastics,1980,3(1/2/3/4):217-228.
[14] MAO Xue-rong.Stochastic differential equations and applications[M].Chichester:Horwood Publishing,2007.
[15] HIGHAM D.An algorithmic introduction to numerical simulation of stochastic differential equations[J].SIAM Review,2001,43(2):525–546.