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A stochastic model of symbiosis. (English) Zbl 1204.47049
The author studies the solution of the following system of SDEs: \begin{aligned} dX(t) &= [(a_1+ b_1 Y(t)- c_1 X(t))dt+\rho_{11} dW_1(t)+ \rho_{12}dW_2(t)] X(t),\\ dY(t) &= [(a_2+ b_2 X(t)- c_2 Y(t))dt+ \rho_{21} dW_1(t)+ \rho_{22}dW_2(t)] Y(t),\end{aligned} where $$\text{det\,}\rho\neq 0$$, and $$W_1$$, $$W_2$$ are independent Wiener processes. The system may be viewed as a stochastic analogue of the deterministic system proposed 75 years ago by G. F. Gause and A. A. Witt [“Behaviour of Mixed Populations and the Problem of Natural Selection”, Amer. Naturalist 69, 596–609 (1935; per bibl.)] as a model for the development of two populations living in symbiosis. If $$b_1 b_2< c_1 c_2$$, then for any initial state in the first quadrant of the $$xy$$-plane, there is a unique solution staying in that quadrant for all $$t> 0$$. If, in addition, $$2a_1-({\rho_{11}}^2+{\rho_{12}}^2)> 0$$ and $$2a_2- ({\rho_{21}}^2+ {\rho_{22}}^2)> 0$$ or $$2a_1- ({\rho_{11}}^2+ {\rho_{12}}^2)> 0$$, $$2a_2- ({\rho_{21}}^2+ {\rho_{22}}^2)< 0$$, and $$[2a_1-({\rho_{11}}^2+{\rho_{12}}^2)]b_2+ [2a_2-({\rho_{21}}^2+ {\rho_{22}}^2)]c_1> 0$$, then the transition semigroup of $$(\log X(t),\log Y(t))$$ is asymptotically stable.

##### MSC:
 47D07 Markov semigroups and applications to diffusion processes 35K15 Initial value problems for second-order parabolic equations 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 92D25 Population dynamics (general)
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