Asymptotic behavior of stochastic Gilpin-Ayala mutualism model with jumps

This article concerns the study of stochastic Gilpin-Ayala mutualism models with white noise and Poisson jumps. Firstly, an explicit solution for one-dimensional Gilpin-Ayala mutualism model with jumps is obtained and the asymptotic pathwise behavior is analyzed. Then, sufficient conditions for the existence of global positive solutions, stochastically ultimate boundedness and stochastic permanence are established for the n-dimensional model. Asymptotic pathwise behavior of n-dimensional Gilpin-Ayala mutualism model with jumps is also discussed. Finally numerical examples are introduced to illustrate the results developed.

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

[2]  Hal L. Smith.  On the asymptotic behavior of a class of deterministic models of cooperating species , 1986 .

[3]  Shuenn-Ren Cheng,et al.  Stochastic Population Systems , 2009 .

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

[5]  Li-Chu Hung,et al.  Stochastic delay population systems , 2009 .

[6]  Ke Wang,et al.  Analysis of a stochastic autonomous mutualism model , 2013 .

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

[8]  Yasuhiro Takeuchi,et al.  Permanence and global stability for cooperative Lotka-Volterra diffusion systems , 1992 .

[9]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

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

[11]  Xinyuan Liao,et al.  On permanence and global stability in a general Gilpin-Ayala competition predator-prey discrete system , 2007, Appl. Math. Comput..

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

[13]  A. Tineo.  On the asymptotic behavior of some population models , 1992 .

[14]  M. Zhien,et al.  Thresholds of Survival for an n-Dimensional Volterra Mutualistic System in a Polluted Environment , 2000 .

[15]  Xuerong Mao,et al.  Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation , 2009 .

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

[17]  Fengde Chen,et al.  Global attractivity in an almost periodic multi-species nonlinear ecological model , 2006, Appl. Math. Comput..

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

[19]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

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

[21]  Shige Peng,et al.  Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations , 2006 .

[22]  Ke Wang,et al.  Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations , 2012 .

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

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

[25]  X. Mao,et al.  Environmental Brownian noise suppresses explosions in population dynamics , 2002 .

[26]  H. I. Freedman,et al.  On the structure and stability of mutualistic systems: Analysis of predator-prey and competition models as modified by the action of a slow-growing mutualist , 1984 .

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

[28]  B. S. Goh,et al.  Stability in Models of Mutualism , 1979, The American Naturalist.

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