Dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries

Dawei Zhang,Beiping Duan,Binxiang Dai

doi:10.1051/mmnp/2020034

Dawei Zhang, Beiping Duan + Show 1 more

Open Access

https://doi.org/10.1051/mmnp/2020034

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Abstract

This paper focuses on the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species. The existence, uniqueness and estimates of the global solution are discussed firstly. Then we prove a spreading–vanishing dichotomy, specifically, the top predator species either successfully spreads to the entire space as time t goes to infinity and survives in the new environment, or fails to establish and dies out in the long run. The long time behavior of the three species and criteria for spreading and vanishing are also obtained. Besides, our simulations illustrate the impacts of initial occupying area and expanding capability on the dynamics of top predator for free boundaries.

Highlights

In ecosystems, Prey–predator interaction is one of the basic interspecies relations, and it is the basic block of more complicated food chain, food web, and biophysical network structures (e.g. [16])
The larger the expanding capability μ is, the faster the free boundaries x = g(t) and x = h(t) increase, and Z(t, x) stabilizes to a positive solution Z3∗. This means that the top predator Z(t, x) will eventually survive in the new environment
We have studied the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species

Summary

Introduction

Prey–predator interaction is one of the basic interspecies relations, and it is the basic block of more complicated food chain, food web, and biophysical network structures (e.g. [16]). Du and Lin [6] firstly discussed a free boundary problem for the diffusive one-species Logistic model They investigated the existence and uniqueness, regularity and uniform estimates, and long-time behavior of global solution. We prove the following local existence and uniqueness result by applying the contraction mapping theorem and the upper and lower solutions method, and we show the global existence via some suitable estimates. Following the proof of Theorem 2.1 again, there exists τ > 0 independent of Tmax such that the solution of (1.4) with initial time. This means that the solution of can be extended as long as N , P and Z remain bounded, which is in contradiction with our hypothesis.

Let l be as in Proposition

The criteria governing spreading and vanishing

Numerical simulations

Conclusion and discussion