The influence of density in population dynamics with strong and weak Allee effect

Md Kamrujjaman,Md Shafiqul Islam,Kamrun Nahar Keya

doi:10.1186/s42787-021-00114-x

Md Kamrujjaman, Md Shafiqul Islam + Show 1 more

Open Access

https://doi.org/10.1186/s42787-021-00114-x

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Abstract

In this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

Highlights

In population biology, a small or sparse population for some species is difficult to study because many factors such as mating may be difficult to analyze
Allee effects are broadly defined as a decline in individual fitness at low population size or low density, which can result in critical population thresholds below which populations crash to extinction
We show that negative sparsity implies weak Allee effect and the population survives in some domain and diverges otherwise

Summary

Introduction

A small or sparse population for some species is difficult to study because many factors such as mating may be difficult to analyze. Logistic model with Allee effects Consider the case when diffusion is ignorable; that is d → 0 (for historical interest, see [42]), the species birth (or death) is only dependent on growth rate (or death rate).

Results

Conclusion