Abstract
A stochastic generalization of the Ricker discrete population model is studied under the assumption that noise impacts the population reproduction rate. The obtained results demonstrate that the demographic-type stochastic noise increases the risk of the population extinction. In particular, the paper establishes conditions on the noise intensity under which the population will extinct even if the corresponding population with no noise survives.
Highlights
The stability of stochastic difference equations has been investigated in numerous papers
A stochastic generalization of the Ricker discrete population model is studied under the assumption that noise impacts the population reproduction rate
From the biological point of view, the paper analyzes and clarifies the fundamental fact that certain types of stochastic noise cannot improve the population growth. It increases the risk of population extinction
Summary
The stability of stochastic difference equations has been investigated in numerous papers (see, e.g., [8, 13,14,15,16,17,18,19]). In this paper a martingale-based technique [13] is applied to a stochastic version of the well-known Ricker population model [12]: xn+1 = ae−bxn xn, n = 0, 1, 2,. The Ricker model (1.1) provides a classic description of an isolated single-species population in the inhibiting environment, which only produces offspring at a specific time each year. It is very popular in biological literature because of its remarkable dynamics and good correspondence with various experimental data (especially, for fish populations) [4, 11].
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