Extreme events in population dynamics with functional carrying capacity

TL;DR

This paper introduces models of population dynamics where carrying capacities depend on populations functionally, revealing conditions that lead to extreme events like finite-time extinctions or singularities, highlighting potential instability in ecological systems.

Contribution

It develops new models with functional carrying capacities that incorporate delayed and indirect influences among species, extending traditional predator-prey models.

Findings

Extreme events such as finite-time death can occur in these models.

Population actions on carrying capacities can destabilize the system.

Conditions for the emergence of singularities are identified.

Abstract

A class of models is introduced describing the evolution of population species whose carrying capacities are functionals of these populations. The functional dependence of the carrying capacities reflects the fact that the correlations between populations can be realized not merely through direct interactions, as in the usual predator-prey Lotka-Volterra model, but also through the influence of species on the carrying capacities of each other. This includes the self-influence of each kind of species on its own carrying capacity with delays. Several examples of such evolution equations with functional carrying capacities are analyzed. The emphasis is given on the conditions under which the solutions to the equations display extreme events, such as finite-time death and finite-time singularity. Any destructive action of populations, whether on their own carrying capacity or on the carrying capacities of co-existing species, can lead to the instability of the whole population that is revealed in the form of the appearance of extreme events, finite-time extinctions or booms followed by crashes.