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A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. (English) Zbl 1216.34040
The authors prove that there is a stationary distribution of a predator-prey model with modified Leslie-Gower and Holling-type schemes with stochastic perturbations and it has the ergodic property.

34C60Qualitative investigation and simulation of models (ODE)
34F05ODE with randomness
92D25Population dynamics (general)
Full Text: DOI
[1] Atar, R.; Budhiraja, A.; Dupuis, P.: On positive recurrence of constrained diffusion processes, Ann. probab. 29, 979-1000 (2001) · Zbl 1018.60081 · doi:10.1214/aop/1008956699
[2] Aziz-Alaoui, M. A.; Okiye, M. Daher: Boundedness and global stability for a predator-prey model with modified Leslie-gower and Holling-type II schemes, Appl. math. Lett. 16, 1069-1075 (2003) · Zbl 1063.34044 · doi:10.1016/S0893-9659(03)90096-6
[3] Gard, T. C.: Introduction to stochastic differential equations, (1988) · Zbl 0628.60064
[4] Hasminskii, R. Z.: Stochastic stability of differential equations, Monogr. textb. Mech. solids fluids 7 (1980)
[5] Ji, C. Y.; Jiang, D. Q.; Shi, N. Z.: Analysis of a predator-prey model with modified Leslie-gower and Holling-type II schemes with stochastic perturbation, J. math. Anal. appl. 359, 482-498 (2009) · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[6] Kliemann, W.: Recurrence and invariant measures for degenerate diffusions, Ann. probab. 15, 690-707 (1987) · Zbl 0625.60091 · doi:10.1214/aop/1176992166
[7] Mao, X.; Yuan, C.; Zou, J.: Stochastic differential delay equations of population dynamics, J. math. Anal. appl. 304, 296-320 (2005) · Zbl 1062.92055 · doi:10.1016/j.jmaa.2004.09.027
[8] Strang, G.: Linear algebra and its applications, (1988) · Zbl 0338.15001
[9] Zhu, C.; Yin, G.: Asymptotic properties of hybrid diffusion systems, SIAM J. Control optim. 46, 1155-1179 (2007) · Zbl 1140.93045 · doi:10.1137/060649343