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Global stability in \(n\)-dimensional discrete Lotka-Volterra predator-prey models. (English) Zbl 1417.34039

Summary: There are few theoretical works on global stability of Euler difference schemes for two-dimensional Lotka-Volterra predator-prey models. Furthermore no attempt is made to show that the Euler schemes have positive solutions. In this paper, we consider Euler difference schemes for both the two-dimensional models and \(n\)-dimensional models that are a generalization of the two-dimensional models. It is first shown that the difference schemes have positive solutions and equilibrium points which are globally asymptotically stable in the two-dimensional cases. The approaches used in the two-dimensional models are extended to the \(n\)-dimensional models for obtaining the positivity and the global stability. Numerical examples are presented to verify the results.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34D23 Global stability of solutions to ordinary differential equations
39A10 Additive difference equations
40A05 Convergence and divergence of series and sequences
37N05 Dynamical systems in classical and celestial mechanics
92D25 Population dynamics (general)
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