Structural and Dynamic Stability of Model Predator-Prey Systems

Abstract

A modified set of Volterra's differential equations for dynamics of prey and predator populations is analyzed. This modification takes three effects into consideration: (1) Satiation of predator resulting in the inability of either predation rate or predator reproduction rate to increase infinitely with growth of prey numbers; (2) Limited resources of prey, as a result of which prey populations cannot increase infinitely even in the absence of predators; 3) Limited external resources (unrelated to prey) of predators, as a result of which predator populations cannot grow infinitely even when there is an excess of prey. Analysis of this set of equations gives many different behavioral regimes depending on the values of parameters. 
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\nThis model as a whole can be used to demonstrate a number of situations: situations in which the behavior of a predator-prey system is adequately described by Volterra's equations; situations in which these equations cannot describe the dynamics of prey-predator interactions; situations in which the system behaves similarly to Volterra's under certain initial conditions but not under other conditions.