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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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