Ruihua WU, Ke WANG

Population dynamical behaviors of stochastic logistic system with jumps

This paper is concerned with a stochastic logistic model driven by martingales with jumps. In the model, generalized noise and jump noise are taken into account. This model is new and more feasible. The explicit global positive solution of the system is presented, and then sufficient conditions for extinction and persistence are established. The critical value of extinction, nonpersistence in the mean, and weak persistence in the mean are obtained. The path-wise and moment properties are also investigated. Finally, some simulation figures are introduced to illustrate the main results.

Anahtar Kelimeler:

Logistic equation, martingale, extinction, persistence

Population dynamical behaviors of stochastic logistic system with jumps

Keywords:

Logistic equation, martingale, extinction, persistence,

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Applebaum D. L´evy Process and Stochastic Calculus. 2nd ed. Cambridge, UK: Cambridge University Press, 2009.
Bandyopadhyay M, Chattopadhyay J. Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity 2005; 10: 913–936.
Bao J, Mao X, Yin G, Yuan C. Competitive Lotka-Volterra population dynamics with jumps. Nonlinear Anal 2011; 74: 6601–6616.
Bao J, Yuan C. Stochastic population dynamics driven by L ´e vy noise. J Math Anal Appl 2012; 391: 363–375.
Freedman HI, Wu J. Periodic solutions of single-species models with periodic delay. SIAM J Math Anal 1992; 23: 689–701.
Gard T. Persistence in stochastic food web models. Bull Math Biol 1984; 46: 357–370.
Gard T. Stability for multispecies population models in random environments. Nonlinear Anal 1986; 10: 1411–1419.
Golpalsamy K. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Dordrecht, the Netherlands: Kluwer Academic, 1992.
Gong G. The Introduction to Stochastic Differential Equations. Beijing, China: Peking University press, 1995 (in Chinese).
Huang Z. The Introduction to Stochastic Analysis. Beijing, China: Science Press, 2001 (in Chinese).
Jiang D, Shi N. A note on nonautonomous logistic equation with random perturbation. J Math Anal Appl 2005; 303: 164–172.
Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston, MA, USA: Academic Press, 1993.
Li X, Gray A, Jiang D, Mao X. Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. J Math Anal Appl 2011; 376: 11–28.
Li X, Jiang D, Mao X. Population dynamical behavior of Lotka-Volterra system under regime switching. J Comput Appl Math 2009; 232: 427–448.
Lipster R. A strong law of large numbers for local martingales. Stochastics 1980; 3: 217–228.
Lisena B. Global attractivity in nonautonomous logistic equations with delay. Nonlinear Anal Real World Appl 2008; 9: 53–63.
Liu M, Wang K. Persistence and extinction in stochastic non-autonomous logistic systems. J Math Anal Appl 2011; 375: 443–457.
Liu M, Wang K. Asymptotic properties and simulations of a stochastic logistic model under regime switching. Math Comput Modelling 2011; 54: 2139–2154.
Liu M., Wang K. On a stochastic logistic equation with impulsive perturbations. Comput Math Appl 2012; 63: 871–886.
Liu M, Wang K. Asymptotic properties and simulations of a stochastic logistic model under regime switching II . Math Comput Modelling 2012; 55: 405–418.
Liu M, Wang K. Analysis of a stochastic autonomous mutualism model. J Math Anal Appl 2013; 402: 392403.
Mao X. Stochastic Differential Equations and Applications. Chichester, UK: Horwood Publishing, 1997.
Mao X, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stoch Process Appl 2002; 97: 95–110.
May R.M. Stability and Complexity in Model Ecosystems. Princeton, NJ, USA: Princeton University, 1973.
Situ R. Theory of Stochastic Differential Equation with Jumps and Applications. New York, NY, USA: Springer, 20 Zhu C, Yin G. On competitive Lotka-Volterra model in random environments. J Math Anal Appl 2009; 357: 154– 1
Zou X, Wang K. Numerical simulations and modeling for stochastic biological systems with jumps. Commun Nonlinear Sci Numer Simul 2014; 19: 1557–1568.