Eur. Phys. J. Special Topics 205, 313-354 (2012)
https://doi.org/10.1140/epjst/e2012-01577-3

Regular Article

Extreme events in population dynamics with functional carrying capacity

¹ Department of Management, Technology and Economics, ETH Zürich, Swiss Federal Institute of Technology, 8032 Zürich, Switzerland
² Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia
³ Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna 141980, Russia
⁴ Swiss Finance Institute, c/o University of Geneva, 40 blvd. Du Pont d’Arve, 1211 Geneva 4, Switzerland

Received: 23 November 2011
Revised: 9 March 2012
Published online: 1 May 2012

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.