Eur. Phys. J. B 16, 183-187 (2000)
https://doi.org/10.1007/s100510070263

Instabilities in population dynamics

Institute of Theoretical Physics, University of Wroclaw, pl. M. Borna 9, 50-204 Wrocław, Poland

Corresponding author: ^a kweron@ift.uni.wroc.pl

Received: 15 February 2000
Published online: 15 July 2000

Abstract

Biologists have long known that the smaller the population, the more susceptible it is to extinction from various causes. Biologists define minimum viable population size (MVP), which is the critical population size, below which the population has a very small chance to survive. There are several theoretical models for predicting the probability that a small population will become extinct. But these models either embody unrealistic assumptions or lead to currently unresolved mathematical problems. In other popular models of population dynamics, like the logistic model, MVP does not exist. In this paper we find the existence of such a critical concentration in a simple model of evolution. We solve this model by a mean field theory and show, in one and two dimensions, the existence of the critical adaptation and concentration below which a population dies out. We also show that, like in the logistic model, above the critical value a population reaches its carrying capacity. Moreover, in the two-dimensional case we find – the so common in biological models – periodic solutions and their biffurcations.