Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

Weam Alharbi,Sergei Petrovskii

doi:10.3390/math6040059

Abstract

A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction–telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain’s critical size coincides with the critical size of the corresponding reaction–diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction–telegraph equation is larger than the critical domain size of the reaction–diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions.

Highlights

Habitat fragmentation due to the climate change and anthropogenic activities is regarded as a major threat to biodiversity worldwide [1]
Our main goal is to understand what the conditions are of species survival (i.e., when u( x, t) does not tend to zero uniformly over the domain in the large-time limit), how they may differ from the predictions of the corresponding reaction–diffusion model, and how they may depend on the type of density dependence in the population growth, e.g., the existence of the Allee effect
The main mathematical framework for modelling animal movement and dispersal was based on the simple isotropic random walk model (SRW) which is unbiased in the sense that, at any moment, the walker can move in any possible direction [13]

Summary

Introduction

Habitat fragmentation due to the climate change and anthropogenic activities is regarded as a major threat to biodiversity worldwide [1]. Whilst reaction–diffusion models have been used extensively in mathematical ecology for several decades [8,12,13,14,15] helping to make a number of important insights and inferences, they have their limits and their relevance may sometimes become questionable One point of their criticism is that diffusion (Brownian motion) as a baseline animal movement pattern is not entirely realistic as the diffusively moving ‘animal’ can change its movement direction with a high frequency Our main goal is to understand what the conditions are of species survival (i.e., when u( x, t) does not tend to zero uniformly over the domain in the large-time limit), how they may differ from the predictions of the corresponding reaction–diffusion model, and how they may depend on the type of density dependence in the population growth, e.g., the existence of the Allee effect

Non-Conservative Property of the Telegraph Equation

Telegraph Equation with Linear Growth

Telegraph Equation with Nonlinear Growth

Empirical Model

Discussion and Conclusions

Methods