Stability of ecosystem: global properties of a general predator-prey model. (English) Zbl 1178.92053

Korobeinikov, Andrei

Math. Med. Biol. 26, No. 4, 309-321 (2009).

Summary: Establishing the conditions for the stability of ecosystems and for stable coexistence of interacting populations is a problem of highest priority in mathematical biology. This problem is usually considered under specific assumptions made regarding the functional forms of nonlinear feedbacks. However, there is growing understanding that this approach has a number of major deficiencies. The most important of these is that the precise forms of the functional responses involved in the model are unknown in detail, and we can hardly expect that these will be known in feasible future.
We consider the dynamics of two species with interaction of consumer-supplier (prey-predator) type. This model generalizes a variety of models of population dynamics, including a range of prey-predator models, \(SIR\) and \(SIRS\) epidemic models, chemostat models, etc. We assume that the functional responses that are usually included in such models are given by unspecified functions. Using the direct Lyapunov method, we derive conditions which ensure global asymptotic stability of this general model. It is remarkable that these conditions impose much weaker constraints on the system properties than are usually assumed. We also identify the parameter that allows us to distinguish between existence and non-existence of the coexisting steady state.

MSC:

92D40	Ecology
34D23	Global stability of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations
34C60	Qualitative investigation and simulation of ordinary differential equation models

Keywords:

ecosystem; coexistence; global stability; nonlinear attack rate; nonlinear incidence rate; infectious disease; mass action; endemic equilibrium state; direct Lyapunov method; Lyapunov function; SIR model; chemostat model; compartment model; consumer-supplier model; prey-predator model; Lotka-Volterra model