Stable Positive Equilibrium Research Articles

Equilibria in structured populations.

The existence of a stable positive equilibrium state for the density rho of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of of nontrivial equilibrium pairs (n, rho) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.

Global stability and persistence of simple food chains

The main purpose of this paper is to develop criteria for which a simple food-chain model of intermediate type and of arbitrary length has a globally stable positive equilibrium and to develop criteria under which such a food chain exhibits uniform persistence. The same techniques are used to obtain conditions for a model of a predator-prey system with mutual interference of the predator to possess a globally stable positive equilibrium.

Predator prey interactions with time delays.

A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example.