, Moreover, the system (1.3) is ergodic and its transition probility P((x, y), t, .) satisfies P((x 0 , y 0 ), t, ?) ? µ(?) when t ? ? for each

, The ecologically less interesting case when (x, y) stays in one of the coordinate axes has similar features, since, by [21, Theorem 3.2], the stochastic logistic equation admits a unique invariant ergodic distribution when the diffusion coefficient is positive but not too large

, Denote by A(x) the m × m matrix g(x) g(x) T. Assume that ]0, +?[ d is invariant by (3.14) and that there exists a bounded open subset U of ]0, +?[ d such that the following conditions are satisfied: (B.1) In a neighborhood of U , the smallest eigenvalue of A(x) is bounded away from 0, (B.2) If x ? R d \U , the expectation of the hitting time ? U at which the solution to (3.14) starting from x reaches the set U is finite, R d ? R d and g : R d ? R m×d are locally Lipschitz functions with locally sublinear growth, and W is a standard Brownian motion on R m

, 14) is ergodic, its transition probility P(x, t, .) satisfies (3.15) P(x, t, ?) ? µ(f ) when t ? ? for each x ? R d and each bounded continuous ?, Internat. J. Bifur. Chaos Appl. Sci. Engrg, vol.28, issue.3, p.1850089, 2018.

W. Abid, R. Yafia, M. A. Aziz-alaoui, H. Bouhafa, and A. Abichou, Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain, Appl. Math. Comput, vol.260, pp.292-313, 2015.

M. A. Aziz-alaoui and M. Daher-okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett, vol.16, pp.90096-90102, 2003.

M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, vol.18, pp.913-936, 2005.

N. P. Bhatia and G. P. Szegö, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, vol.161, 1970.

B. I. Camara, Waves analysis and spatiotemporal pattern formation of an ecosystem model, Nonlinear Anal. Real World Appl, vol.12, pp.2511-2528, 2011.

F. Chen, L. Chen, and X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl, vol.10, pp.2905-2908, 2009.

G. Da-prato and H. Frankowska, Stochastic viability of convex sets, J. Math. Anal. Appl, vol.333, pp.151-163, 2007.

M. Daher-okiye and M. A. Aziz-alaoui, On the dynamics of a predatorprey model with the Holling-Tanner functional response, Mathematical modelling & computing in biology and medicine, vol.1, pp.270-278, 2003.

N. Dalal, D. Greenhalgh, and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl, vol.341, pp.1084-1101, 2008.

F. Dumortier, J. Llibre, and J. C. Artés, Qualitative theory of planar differential systems, 2006.

G. Ferreyra and P. Sundar, Comparison of solutions of stochastic equations and applications, Stochastic Anal. Appl, vol.18, pp.211-229, 2000.

J. Fu, D. Jiang, N. Shi, T. Hayat, and A. Alsaedi, Qualitative analysis of a stochastic ratio-dependent Holling-Tanner system, Acta Math. Sci. Ser, issue.18, pp.30758-30764, 2018.

F. R. Gantmacher, The theory of matrices, Vols, vol.1, 1959.

D. H. Gottlieb, A de Moivre like formula for fixed point theory, Fixed point theory and its applications, vol.72, pp.99-105, 1986.

J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol.42, 1983.

C. Ji, D. Jiang, and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl, vol.359, pp.482-498, 2009.

R. Khasminskii, Stochastic stability of differential equations, vol.66, 2012.

P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, vol.23, 1992.

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, vol.47, pp.219-234, 1960.

L. Liu and Y. Shen, Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system, Adv. Difference Equ, vol.13, issue.10, 2015.

Z. Liu, Stochastic dynamics for the solutions of a modified Holling-Tanner model with random perturbation, Internat. J. Math, vol.25, p.1450105, 2014.

J. Llibre and J. Villadelprat, A Poincaré index formula for surfaces with boundary, Differential Integral Equations, vol.11, pp.191-199, 1998.

J. Lv and K. Wang, Analysis on a stochastic predator-prey model with modified Leslie-Gower response, Abstr. Appl. Anal., Art, vol.518719

J. Lv and K. Wang, Asymptotic properties of a stochastic predator-prey system with Holling II functional response, Commun. Nonlinear Sci. Numer. Simul, vol.16, pp.4037-4048, 2011.

T. Ma and S. Wang, A generalized Poincaré-Hopf index formula and its applications to 2-D incompressible flows, Nonlinear Anal. Real World Appl, vol.2, pp.4-9, 2001.

P. S. Mandal and M. Banerjee, Stochastic persistence and stability analysis of a modified Holling-Tanner model, Math. Methods Appl. Sci, vol.36, 2013.

R. M. May, Stability and Complexity in Model Ecosystems, 1973.

A. F. Nindjin, M. A. Aziz-alaoui, and M. Cadivel, Analysis of a predatorprey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl, vol.7, pp.1104-1118, 2006.

E. C. Pielou, Mathematical ecology, 1977.

C. C. Pugh, A generalized Poincaré index formula, Topology, vol.7, pp.217-226, 1968.

J. Tong, b 2 ? 4ac and b 2 ? 3ac, Math. Gaz, vol.88, pp.511-513, 2004.

R. Yafia and M. A. Aziz-alaoui, Existence of periodic travelling waves solutions in predator prey model with diffusion, Appl. Math. Model, vol.37, pp.3635-3644, 2013.

R. Yafia, F. E. Adnani, and H. T. Alaoui, Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Nonlinear Anal. Real World Appl, vol.9, pp.2055-2067, 2008.

R. Yafia, F. E. Adnani, and H. Talibi-alaoui, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Appl. Math. Sci, pp.119-131, 2007.