On stochastic Gilpin-Ayala population model with Markovian switching

In this paper, we analyze a stochastic Gilpin-Ayala population model with Markovian switching and white noise. The Gilpin-Ayala parameter is also allowed to switch. We establish the global stability of the trivial equilibrium state of the model. Verifiable sufficient conditions which guarantee the extinction and persistence are provided. Furthermore, we show the existence of a stationary distribution. The analytical results are illustrated by computer simulations.

[1]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

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

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

[4]  Shigeng Hu,et al.  STOCHASTIC DELAY GILPIN–AYALA COMPETITION MODELS , 2006 .

[5]  Daqing Jiang,et al.  Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System , 2013 .

[6]  Adel Settati,et al.  Stationary distribution of stochastic population systems under regime switching , 2014, Appl. Math. Comput..

[7]  Y. Takeuchi,et al.  Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise , 2004 .

[8]  G. Yin,et al.  Stability of regime-switching diffusions , 2007 .

[9]  W. J. Anderson.  Continuous-Time Markov Chains , 1991 .

[10]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[11]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[12]  Xuerong Mao,et al.  Population dynamical behavior of Lotka-Volterra system under regime switching , 2009, J. Comput. Appl. Math..

[13]  M. Begon,et al.  Ecology: From Individuals to Ecosystems , 2005 .

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

[15]  Dingshi Li,et al.  The stationary distribution and ergodicity of a stochastic generalized logistic system , 2013 .

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

[17]  Ke Wang,et al.  Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system , 2012, Appl. Math. Lett..

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

[19]  G. Yin,et al.  On competitive Lotka-Volterra model in random environments , 2009 .

[20]  Xuerong Mao,et al.  Stochastic population dynamics under regime switching II , 2007 .

[21]  Miljana Jovanovic,et al.  Dynamics of non-autonomous stochastic Gilpin-Ayala competition model with time-varying delays , 2013, Appl. Math. Comput..

[22]  A. Shwartz,et al.  The Poisson Equation for Countable Markov Chains: Probabilistic Methods and Interpretations , 2002 .

[23]  Ke Wang,et al.  Stability in distribution of competitive Lotka–Volterra system with Markovian switching , 2011 .

[24]  K. Sato,et al.  Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment , 2006 .

[25]  H. B. Wilson,et al.  Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[26]  Gang George Yin,et al.  Asymptotic Properties of Hybrid Diffusion Systems , 2007, SIAM J. Control. Optim..

[27]  Meng Liu,et al.  Survival Analysis of Stochastic Competitive Models in a Polluted Environment and Stochastic Competitive Exclusion Principle , 2010, Bulletin of mathematical biology.

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

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

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

[31]  Ke Wang,et al.  Asymptotic properties and simulations of a stochastic logistic model under regime switching II , 2011, Math. Comput. Model..

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

[33]  Maja Vasilova.  Asymptotic behavior of a stochastic Gilpin-Ayala predator-prey system with time-dependent delay , 2013, Math. Comput. Model..