Global Asymptotic Stability of Lotka–Volterra Diffusion Models with Continuous Time Delay
E. Beretta, Y. Takeuchi
Source abstract
We consider the global asymptotic stability of diffusion models with multiple species heterogeneous patches and incorporate the effect of continuous time delays for species in each patch. For two different patches connected by diffusion with multiple species, and with patch dynamics governed by Lotka–Volterra models with continuous time delays (and constant influxes), we give sufficient conditions for boundedness of solutions and the existence of a nonnegative equilibrium point, providing also a candidate for a Lyapunov function. By applying the homotopy function technique for different symbiotic patches with time delays, we provide a sufficient condition for the existence of a positive globally asymptotically stable equilibrium point. Further, we discuss the global asymptotic stability for the prey-predator model, where the prey has its shelter from predation and only the predator can diffuse between patches. We consider also the existence condition for the globally asymptotically stable equilibrium for the model with two identical patches and continuous time delays.
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