Stability analysis of a stochastic Gilpin-Ayala model driven by Lévy noise

Abstract A stochastic one-dimensional Gilpin–Ayala model driven by Levy noise is presented in this paper. Firstly, we show that this model has a unique global positive solution under certain conditions. Then sufficient conditions for the almost sure exponential stability and moment exponential stability of the trivial solution are established. Results show that the jump noise can make the trivial solution stable under some conditions. Numerical example is introduced to illustrate the results.

[1]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .

[2]  David Applebaum,et al.  STOCHASTIC STABILIZATION OF DYNAMICAL SYSTEMS USING LÉVY NOISE , 2010 .

[3]  D. Applebaum,et al.  Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise , 2009, Journal of Applied Probability.

[4]  Miljana Jovanovic,et al.  Stochastic Gilpin-Ayala competition model with infinite delay , 2011, Appl. Math. Comput..

[5]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[6]  X. Mao,et al.  Competitive Lotka–Volterra population dynamics with jumps , 2011, 1102.2163.

[7]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[8]  M. Fan,et al.  Optimal harvesting policy for single population with periodic coefficients. , 1998, Mathematical biosciences.

[9]  Daqing Jiang,et al.  A note on nonautonomous logistic equation with random perturbation , 2005 .

[10]  M. Gilpin,et al.  Global models of growth and competition. , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Ke Wang,et al.  Global periodic solutions of a generalized n-species Gilpin-Ayala competition model☆ , 2000 .

[12]  Ke Wang,et al.  A note on stability of stochastic logistic equation , 2013, Appl. Math. Lett..

[13]  Daqing Jiang,et al.  Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation☆ , 2008 .

[14]  Ke Wang,et al.  Stochastic Lotka–Volterra systems with Lévy noise , 2014 .

[15]  Maja Vasilova,et al.  Dynamics of Gilpin-Ayala competition model with random perturbation , 2010 .

[16]  Hong Qiu,et al.  Stability of a stochastic logistic model with distributed delay , 2013, Math. Comput. Model..

[17]  R. Situ.  Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

[18]  Eric Renshaw,et al.  Asymptotic behaviour of the stochastic Lotka-Volterra model , 2003 .

[19]  Shigeng Hu,et al.  Asymptotic behaviour of the stochastic Gilpin–Ayala competition models , 2008 .

[20]  R. Liptser.  A strong law of large numbers for local martingales , 1980 .

[21]  C. Yuan,et al.  Stochastic Population Dynamics Driven by Levy Noise , 2011, 1105.1174.

[22]  Jia Li,et al.  Stability in Gilpin-Ayala competition models with diffusion , 1997 .

[23]  H. Kunita.  Itô's stochastic calculus: Its surprising power for applications , 2010 .

[24]  David Applebaum,et al.  Lévy Processes and Stochastic Calculus by David Applebaum , 2009 .

[25]  Fengde Chen,et al.  Some new results on the permanence and extinction of nonautonomous Gilpin–Ayala type competition model with delays , 2006 .