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In order to examine the spatial dynamics of freshwater species in river habitats, this study creates a nonlocal reaction-diffusion-advection model. Using spectral analysis of an integro-differential operator, we incorporate directional flow (advection) and define conditions for species persistence or extinction. The findings show that advection rate has a significant impact on long-term survival, with considerable directional motion always resulting in extinction. With the help of numerical simulations, we further investigate the existence, uniqueness, and stability of stationary solutions. These results offer a robust mathematical framework for forecasting how ecosystems will react to river currents and comprehending the drift dilemma.In order to examine the spatial dynamics of freshwater species in river habitats, this study creates a nonlocal reaction-diffusion-advection model. Using spectral analysis of an integro-differential operator, we incorporate directional flow (advection) and define conditions for species persistence or extinction. The findings show that advection rate has a significant impact on long-term survival, with considerable directional motion always resulting in extinction. With the help of numerical simulations, we further investigate the existence, uniqueness, and stability of stationary solutions. These results offer a robust mathematical framework for forecasting how ecosystems will react to river currents and comprehending the drift dilemma.

The Mathematics of Life and Death

Author: Hackernoon

Source: Hackernoon

2025/09/21 15:39

2 min read

Table of Links

Abstract and 1. Introduction

The well-posed global solution
Nontrivial stationary solution

3.1 Spectral theory of integro-differential operator

3.2 The existence, uniqueness and stability of nontrivial stationary solution
The sharp criteria for persistence or extinction
The limiting behaviors of solutions with respect to advection
Numerical simulations
Discussion, Statements and Declarations, Acknowledgement, and References

3 Nontrivial stationary solution

3.1 Spectral theory of integro-differential operator

The eigenvalue problem is an important tool to study the longtime behaviors of elliptic and parabolic problems, especially the sign of principal eigenvalue of the linearized operator can determine the existence and stability of nontrivial stationary solution, even the persistence of species, see [20, 32, 44, 45].

\ And some standard properties have shown in [32].

3.2 The existence, uniqueness and stability of nontrivial stationary solution

Firstly, we shall the strong maximal principle for the stationary problem.

\ Notice that (3.5) (or (3.6)) has only one trivial solution u = 0.In the following, we still only consider the details for (3.5).

\ Hence, we obtain a contradiction, and so the theorem holds.

\ In summary, the results about the existence/nonexistence, uniqueness of nontrivial solutions of (3.5) and the stability of nontrivial/trivial solution are as follows.

4 The sharp criteria for persistence or extinction

:::info 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 (bxdai@csu.edu.cn).

:::

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

:::

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