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Long-time behaviour of a stochastic prey–predator model. (English) Zbl 1075.60539
A stochastic version $dX_ {t} = (\alpha X_ {t} -\beta X_ {t}Y_ {t} - \mu X^ 2_ {t})\,dt + \sigma X_ {t}\,dW_ {t}, \quad dY_ {t} = (-\gamma Y_ {t} + \delta X_ {t}Y_ {t} - \nu Y^ 2_ {t})\,dt + \rho Y_ {t}\,dW_ {t}\tag{1}$ of the Lotka-Volterra system is studied, where $$\alpha$$, $$\beta$$, $$\gamma$$, $$\delta$$, $$\mu$$, $$\nu$$, $$\rho$$ and $$\sigma$$ are positive constants, and $$W$$ is a standard Wiener process. By setting $$X_ {t} = \exp (\xi _ {t})$$ and $$Y_ {t} = \exp (\eta _ {t})$$ the equations (1) are transformed to $d\xi _ {t} = (\alpha - \sigma ^ 2/2 - \mu e^ {\xi _ {t}} -\beta e^ {\eta _ {t}})\,dt + \sigma \,dW_ {t},\quad d\eta _ {t} = (-\gamma -\rho ^ 2/2 + \delta e^ {\xi _ {t}} - \nu e^ {\eta _ {t}})\,dt + \rho \,dW_ {t}.\tag{2}$ Let us set $$c_ 1 = \alpha - \sigma ^ 2/2$$, $$c_ 2 = \gamma + \rho ^ 2 /2$$. Let $$(\xi ,\eta )$$ be an arbitrary solution to (2). It is proven that if $$c_ 1>0$$ and $$\mu c_ 2 <\delta c_ 1$$, then there exists a unique invariant probability measure $$m^ *$$ for (2) and the distribution of $$(\xi _ {t},\eta _ {t})$$ converges to $$m^ *$$ as $$t\to \infty$$ in the total variation norm. If $$c_ 1>0$$ and $$\mu c_ 2 >\delta c_ 1$$, then $$\lim _ {t\to \infty } \eta _ {t} = -\infty$$ almost surely, while the law of $$\xi _ {t}$$ converges weakly to a measure having density $$C\exp (2c_ 1\sigma ^ {-2}x - 2\mu \sigma ^ {-2}e^ {x})$$. Finally, if $$c_ 1<0$$, then both $$\xi _ {t}$$ and $$\eta _ {t}$$ converge to $$-\infty$$ as $$t\to \infty$$ almost surely. In the course of proofs, it is shown that the laws of both $$(\xi _ {t},\eta _ {t})$$ and $$m^ *$$ have density with respect to two-dimensional Lebesgue measure, hence recent results on long-time behaviour of integral Markov semigroups [see e.g.K.Pichór and R.Rudnicki, J. Math. Anal. Appl. 249, 668–685 (2000; Zbl 0965.47026)] may be applied.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 47D07 Markov semigroups and applications to diffusion processes 60J60 Diffusion processes 92D25 Population dynamics (general)
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##### References:
 [1] Aida, S., Kusuoka, S., Strook, D., 1993. On the support of Wiener functionals. In: Elworthy, K.D., Ikeda, N. (Eds.), Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotic. Pitman Research Notes in Mathematical Series, Vol. 284, Longman Scient. Tech., pp. 3-34. · Zbl 0790.60047 [2] Arnold, L.; Horsthemke, W.; Stucki, J.W., The influence of external real and white noise on the lotka – volterra model, Biomedical J., 21, 451-471, (1979) · Zbl 0433.92019 [3] Ben Arous, G.; Léandre, R., Décroissance exponentielle du noyau de la chaleur sur la diagonale (II), Probab. theory related fields, 90, 377-402, (1991) · Zbl 0734.60027 [4] Chessa, S.; Fujita Yashima, H., The stochastic equation of predator – prey population dynamics (Italian), Boll. unione mat. ital. sez. B artic. ric. mat., 5, 8, 789-804, (2002) · Zbl 1177.60057 [5] Foguel, S.R., Harris operators, Israel J. math., 33, 281-309, (1979) · Zbl 0434.60012 [6] Khasminskiı̆, R.Z.; Klebaner, F.C., Long term behaviour of solutions of the lotka – volterra system under small random perturbations, Ann. appl. probab., 11, 953-963, (2001) · Zbl 1061.34513 [7] Komorowski, T.; Tyrcha, J., Asymptotic properties of some Markov operators, Bull. Polish acad. sci. math., 37, 221-228, (1989) · Zbl 0767.47012 [8] Lasota, A.; Mackey, M.C., Chaos, fractals and noise. stochastic aspects of dynamics, Springer applied mathematical sciences, Vol. 97, (1994), Springer New York · Zbl 0784.58005 [9] Malliavin, P., Stochastic calculus of variations and hypoelliptic operators, (), 195-263 [10] Malliavin, P., Ck-hypoellipticity with degeneracy, (), 199-214 [11] Norris, J., 1986. Simplified Malliavin calculus. In: Séminaire de probabilitiés XX, Lecture Notes in Mathematics, Vol. 1204. Springer, New York, pp. 101-130. [12] Pichór, K.; Rudnicki, R., Stability of Markov semigroups and applications to parabolic systems, J. math. anal. appl., 215, 56-74, (1997) · Zbl 0892.35072 [13] Pichór, K.; Rudnicki, R., Continuous Markov semigroups and stability of transport equations, J. math. anal. appl., 249, 668-685, (2000) · Zbl 0965.47026 [14] Rudnicki, R., On asymptotic stability and sweeping for Markov operators, Bull. Polish acad.: math., 43, 245-262, (1995) · Zbl 0838.47040 [15] Rudnicki, R., Pichór, K., Tyran-Kamińska, M., 2002. Markov semigroups and their applications, Markov semigroups and their applications. In: Garbaczewski, P., Olkiewicz, R. (Eds.), Dynamics of Dissipation. Lecture Notes in Physics, Vol. 597. Springer, Berlin, pp. 215-238. [16] Stroock, D.W., Varadhan, S.R.S., 1972. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III. Univ. Cal. Press, Berkeley, pp. 333-360. · Zbl 0255.60056 [17] Volterra, V., Leçons sur la théorie mathematique de la lutte pour la vie, (1931), Gauthier-Villar Paris · JFM 57.0466.02
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