Abstract
This paper deals with an almost global convergence result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global convergence result, which provides sufficient conditions to rule out oscillatory or more complicated behavior that is often observed in predator-prey systems.
Highlights
Predator-prey systems have been -and still are- attracting a lot of attention [8, 18, 11, 5] since the early work of Lotka and Volterra
The phase portrait consists of an infinite number of periodic solutions centered around an equilibrium point
As in example 1, (1, 1, 1) is always an equilibrium point and it can be shown that a supercritical Hopf bifurcation occurs at k = kc where kc = 105/2 = 52.5, again illustrating oscillatory behavior, see figure 1
Summary
Predator-prey systems have been -and still are- attracting a lot of attention [8, 18, 11, 5] since the early work of Lotka and Volterra. Oscillatory behavior can be found within the class of Lotka-Volterra predator-prey systems, but the number of predator and prey species must be larger than two. As in example 1, (1, 1, 1) is always an equilibrium point and it can be shown that a supercritical Hopf bifurcation occurs at k = kc where kc = 105/2 = 52.5, again illustrating (stable) oscillatory behavior, see figure 1. Note that almost global attractivity of an equilibrium point is in a sense the strongest achievable stability property for a Lotka-Volterra system. These systems typically possess multiple equilibrium points. These systems typically possess multiple equilibrium points. (for instance, zero is always an equilibrium point; it is usually an uninteresting one from biological point of view)
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