Dynamics of the stochastic equation of cooperative population (Q712283)

The author investigates solutions \((X_t,Y_t)\) of a Lotka-Volterra equation with stochastic perturbations which are assumed to be proportional to the number of individuals (cf., e.g., \textit{R. Rudnicki} and \textit{K. Pichór}, Influence of stochastic perturbation on prey-predator systems. Math. Biosci. 206, No. 1, 108--119 (2007; Zbl 1124.92055), and the references given there). This leads to stochastic differential equations of the form \[ dX_t =(a_1 + b_{12} Y_t - b_{11}X_t) X_t d t + X_t(\sigma_1 d W_t^1 + \sigma_2 d W_t^2), \] \[ dY_t =(a_2 + b_{21} X_t - b_{22}Y_t) Y_t d t + Y_t(\rho_1 d W_t^1 + \rho_2 d W_t^2), \] where \(a_i, b_{ij}\) are positive coefficients, \(\sigma_i, \rho_j\geq 0\), not all \(=0\), and \(W_t^i,\;i=1,2\), are independent Brownian motions. Substituting \(X_t = e^{\xi_t}, Y_t = e^{\eta_t}\) and replacing the coefficients \(a_i\) by \(c_1 = a_1-(\sigma_1^2+\sigma_2^2)/2\), resp., \(c_2 = a_2-(\rho_1^2+\rho_2^2)/2\), leads to equations of the type: \[ d\xi_t =(c_1 + b_{12} e^{\eta_t} - b_{11}e^{\xi_t}) d t + (\sigma_1 d W_t^1 + \sigma_2 d W_t^2), \] \[ d\eta_t =(c_2 + b_{21} e^{\xi_t} - b_{22}e^{\eta_t}) d t + (\rho_1 d W_t^1 + \rho_2 d W_t^2). \] The solution \((\xi_t,\eta_t)\) defines a Markov diffusion semigroup with transition kernels \(P(t)\), which, if smooth densities exist, is related to the Fokker-Planck equation. Under suitable conditions on the coefficients it is shown that \(P(t)\) is asymptotically stable (on \(\mathbb{R}^2\), resp., on a subspace \(E\subseteq \mathbb{R}^2\)), i.e., if \(u(t, \cdot, \cdot)\) denote the densities of the distributions of \((\xi_t,\eta_t)\) at time \(t>0\), then there exists a unique stationary density \(u^*(\cdot, \cdot)\) such that \[ \int\int|u(t,x,y)-u^*(x,y)|dxdy\to 0,\;t\to\infty. \] In fact, the author distinguishes six different cases depending on the values of the coefficients of the equation, which lead to asymptotic stability or allow to reduce the equation to a more simple form.