he Extinction Equation: What Happens When Advection Goes Too Far? | HackerNoon

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Abstract and 1. Introduction

2. The well-posed global solution

3. Nontrivial stationary solution

   3.1 Spectral theory of integro-differential operator

   3.2 The existence, uniqueness and stability of nontrivial stationary solution

4. The sharp criteria for persistence or extinction

5. The limiting behaviors of solutions with respect to advection

6. Numerical simulations

7. Discussion, Statements and Declarations, Acknowledgement, and References

7 Discussion

The paper introduces a nonlocal reaction-diffusion-advection problem in a bounded region with Dirichlet/Neumann boundary condition. Our objective is to investigate its spatial dynamics, namely, the sharp criteria for persistence or extinction and the limiting behaviors of solutions with respect to advection rate. The main conclusions can be summarized as follows:

(1) The existence/nonexistence, uniqueness and stability of nontrivial stationary solutions are obtained. To be specific:

(3) The limiting behaviors of solutions with respect to advection rate are considered. It tells us that a sufficiently large directional motion will cause the species extinction in any situations.

(4) The numerical simulations verify our theoretical proof and show that the advection rate has a great impact on the dynamic behaviors of species.

Statements and Declarations

Competing Interests We declare that the authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Acknowledgement

This work is supported in part by the National Natural Science Foundation of China (No. 11871475, 12271525) and the Fundamental Research Funds for the Central Universities of Central South University (No. CX20230218).

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Authors:

(1) Yaobin Tang, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China;

(2) Binxiang Dai, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China ([email protected]).

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This paper is available on arxiv under CC BY 4.0 DEED license.

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