Dynamics in a Delayed Competition System on a Weighted Network

Yaohua Tong,Xiaoling Wang,Stefan Balint

doi:10.1155/2020/6625060

Yaohua Tong, Xiaoling Wang + Show 1 more

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https://doi.org/10.1155/2020/6625060

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Abstract

In this paper, we study the stability of positive steady states in a delayed competition system on a weighted network, which does not satisfy the comparison principle appealing to classical competitive systems. By introducing some auxiliary equations and constructing proper contracting rectangles, we present some sufficient conditions on the stability of the unique positive steady state. Moreover, some numerical examples are given to explore the complex dynamics of this nonmonotone model, which implies the nontrivial roles of weights and time delays.

Highlights

Competition is one of the most universal phenomena in the natural world due to the limit resources including water and sunshine
⎪⎪⎪⎪⎪⎩ dN2 dt in which all the parameters are positive, and N1, N2 present the densities of two competitive species; we may refer to Murray [1], section 3.5, for the dynamics and biology background of (1)
We studied a delayed model on a weighted network, which may be regarded as a delayed competition model on patches

Summary

Introduction

Competition is one of the most universal phenomena in the natural world due to the limit resources including water and sunshine. When the amount of individuals is concerned in population dynamics, some differential systems modeling both interspecific and intraspecific competitions have been proposed and studied. The controllability of many mathematical models in finite time is important, which partly depends on the stability conditions of some states; see a number of examples about switched nonlinear systems [22,23,24]. Liu et al [21] and Sun and Mai [18] studied the stability of different steady states of (4) and (5) by using the comparison principle and other techniques, which include some applications in population dynamics. We show some numerical results to support our theoretical conclusions and present the complex dynamics of this system when our stability conditions do not hold, which implies the nontrivial role of time delays and partial degenerate weights.

Preliminaries

Main Results

Numerical Simulation

Discussion