Global stability results for a generalized Lotka-Volterra system with distributed delays. Applications to predator-prey and epidemic systems. (English) Zbl 0716.92020

Beretta, E.; Capasso, V.; Rinaldi, F.

J. Math. Biol. 26, No. 6, 661-688 (1988).

Summary: The paper contains an extension of the general ODE system proposed in previous papers by the same authors [see e.g. the first two authors’ paper, Comput. Math. Appl., Part A 12, 677-694 (1986; Zbl 0622.92016)], to include distributed time delays in the interaction terms. The new system describes a large class of Lotka-Volterra like population models and epidemic models with continuous time delays. Sufficient conditions for the boundedness of solutions and for the global asymptotic stability of nontrivial equilibrium solutions are given. A detailed analysis of the epidemic system is given with respect to the conditions for global stability. For a relevant subclass of these systems an existence criterion for steady states is also given.

MSC:

92D30	Epidemiology
92D25	Population dynamics (general)
34C11	Growth and boundedness of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations
34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20	Stability theory of functional-differential equations
34K25	Asymptotic theory of functional-differential equations

Keywords:

predator-prey systems; distributed time delays; epidemic models; continuous time delays; Sufficient conditions; boundedness of solutions; global asymptotic stability of nontrivial equilibrium solutions; global stability; existence criterion for steady states

Citations:

Zbl 0622.92016

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