Global stability of discrete models of Lotka-Volterra type.

*(English)*Zbl 0919.92030Global stability of population models governed by difference equations has been studied by a number of papers. But most of them focus their attention on one-species models with or without time delays. For many-species difference models, some papers studied the permanence and extinction of populations. Few papers concern with the global stability of positive equilibria of these models. In this paper we will consider the following Lotka-Volterra model
\[
x_i(k+1) =x_i(k)\exp \left\{r_i(k)-\sum^n_{j=1} a_{ij}(k) x_j(k) \right\}, \quad i=1, \dots,n, \tag{1}
\]
where \(x_i(k)\) is the density of population \(i\) at \(k\)th generation, \(r_i(k)\) is the growth rate of population \(i\) at \(k\)th generation, \(a_{ij}(k)\) measures the intensity of intraspecific competition or interspecific action of species. It is assumed that \(\{r_i(k)\}\) and \(\{a_{ij} (k)\} (i,j=1, \dots,n)\) are bounded.

System (1) is a counterpart of continuous Lotka-Volterra models. It is shown in the literature that a continuous Lotka-Volterra system is globally asymptotically stable if the interaction matrix is diagonally dominant. The purpose of this paper is to find further conditions to ensure that the discrete system (1) is globally asymptotically stable.

System (1) is a counterpart of continuous Lotka-Volterra models. It is shown in the literature that a continuous Lotka-Volterra system is globally asymptotically stable if the interaction matrix is diagonally dominant. The purpose of this paper is to find further conditions to ensure that the discrete system (1) is globally asymptotically stable.

##### MSC:

92D25 | Population dynamics (general) |

39A11 | Stability of difference equations (MSC2000) |