Destruction of the relaxation oscillations in the model of extreme dynamics of the population

A. Yu. Perevaryukha

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2017), pp. 55-65

Résumé

In this article within the context of the simulation of nontrivial changes in the development of population processes is offered equation with delay $\dot x = \lambda x(t)f(x(t-\tau)) \psi(x(t-\tau))$, where $\lambda, x>0,\ \psi(x) $—the function changes sign. In the new model of interpretation subthreshold carrying capacity differs from the asymptotic equilibrium of the balance $x(t) \rightarrow K$ from the equation Verhulst–Pearl. Computational investigation of loss of stability of a singular point in addition to the well-known scenario of the formation of the global cycle of orbital stability in the logistic equation with delay indicates the existence of another version of metamorphosis—the destruction occurred after the change of reproductive parameter transient relaxation oscillations and the advent of unlimited from the top pseudoperiodic solutions. With increasing amplitude of the relaxation oscillations original scenario for catastrophic completing of the growth phase of population size is realized, depending not on achieving a critical minimum and maximum of the situation in the case of exceeding the permissible unstable population supporting capacity of the environment. The model is applicable for describing outbreaks of mass reproduction of many species of insects, which strongly affect availability of its breeding environment.

Détail

Keywords: model of population fluctuations, equation with delay, cycles and outbreaks, occasional ecological scenarios, threshold situations.