Stability of a stochastic predator-prey system (Q2726131)

The authors investigate the stochastic stability of a stochastic version of the Lotka-Volterra predator-prey model. The stochastic model is described by the two-dimensional stochastic differential equation system NEWLINE\[NEWLINE \begin{aligned} dx_t&=[rx_t(1-x_t)-q_0x_ty_t] dt-\sigma x_ty_t dw_t,\\ dy_t&=[cq_0x_ty_t-uy_t] dt-c\sigma x_ty_t dw_t,\end{aligned}NEWLINE\]NEWLINE with a one-dimensional driving Wiener process \(w\). The authors prove that a) the stationary solution \(E_0=(0,0)\) is always unstable; b) the stationary solution \(E_1=(1,0)\) is asymptotically stable if \(0<q_0<T_1:=u/c-c\sigma^2/2\) and unstable if \(q_0>T_2:=u/c+c\sigma^2/2\). Numerical simulations are also performed that allow the authors to conjecture that c)~for \(q_0\in(T_1,T_2)\), the solution is still stable; d)~for \(q_0>T_2\), the solution fluctuates around the deterministic equilibrium \(E^*=({u\over qc},{r\over q}(1-{u\over qc}))\) and has an invariant distribution.