zbMATH — the first resource for mathematics

Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system. (English) Zbl 1261.60057
This paper studies the long term dynamics of the generalized logistic stochastic differential equation \[ dx_t=x_t (r-a x^{\theta})dt + \sum_{i=1}^{n}a_i x dB_i(t)+ \sum_{i=1}^{n}\beta_{i} x^{1+\theta}dB_{i}(t), \] where \(B_i\) are independent Brownian motions and \(a>0\), \(\theta>0\). In the absence of noise, this system is known to have a positive equilibrium which is globally asymptotically stable for \(r>0\). The paper examines the validity of this result in the presence of noise, where now the statement has to be modified in terms of the stationary distribution and its ergodicity properties. It is proved that, for small enough values of the noise which depend also on the values of the parameters of the deterministic system, there exists a stationary invariant distribution which is ergodic. Furthermore, conditions under which extinction is possible are provided. The results are based essentially on appropriately constructed Lyapunov functions and are supported by simulation.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX Cite
Full Text: DOI
[1] May, R.M., Stability and complexity in model ecosystems, (1973), Princeton Univ. Press
[2] Jiang, D.; Shi, N.; Li, X., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. anal. appl., 340, 588-597, (2006) · Zbl 1140.60032
[3] Liu, M.; Wang, K., Extinction and permanence in a stochastic nonautonomous population system, Appl. math. lett., 23, 1464-1467, (2010) · Zbl 1206.34079
[4] Liu, M.; Wang, K.; Wu, Q., Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. math. biol., 73, 1969-2012, (2011) · Zbl 1225.92059
[5] Liu, M.; Wang, K., Persistence and extinction in stochastic non-autonomous logistic systems, J. math. anal. appl., 375, 443-457, (2011) · Zbl 1214.34045
[6] 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., 376, 11-28, (2011) · Zbl 1205.92058
[7] Ji, C.; JIang, D.; Shi, N., A note on a predator – prey model with modified leslie – gower and Holling-type II schemes with stochastic perturbation, J. math. anal. appl., 377, 435-440, (2011) · Zbl 1216.34040
[8] Atar, R.; Budhiraja, A.; Dupuis, P., On positive recurrence of constrained diffusion processes, Ann. probab., 29, 979-1000, (2001) · Zbl 1018.60081
[9] Hasminskii, R.Z., ()
[10] T.C. Gard, Introduction to Stochastic Differential Equations, New York, 1988. · Zbl 0628.60064
[11] Strang, G., Linear algebra and its applications, (1988), Thomson Learning, Inc.
[12] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. control optim., 46, 1155-1179, (2007) · Zbl 1140.93045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.