Global behavior of a plant-herbivore model

Advances in Difference Equations

Qamar Din¹ 

[1] Department of Mathematics, Faculty of Basic and Applied Sciences, University of Poonch Rawalakot, Rawalakot, Pakistan

关键词: plant-herbivore system;    steady-states;    local stability;    global behavior;    rate of convergence;    39A10;    40A05;   

DOI  :  10.1186/s13662-015-0458-y

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

The present work deals with an analysis of the local asymptotic stability and global behavior of the unique positive equilibrium point of the following discrete-time plant-herbivore model:xn+1=αxnβxn+eyn$x_{n+1}=\frac{\alpha x_{n}}{\beta x_{n}+e^{y_{n}}}$,yn+1=γ(xn+1)yn$y_{n+1}=\gamma(x_{n}+1)y_{n}$, whereα∈(1,∞)$\alpha\in\mathbb{(}1,\infty)$,β∈(0,∞)$\beta\in\mathbb{(}0,\infty)$, andγ∈(0,1)$\gamma\in\mathbb{(}0,1)$withα+β>1+βγ$\alpha+\beta>1+\frac{\beta}{\gamma}$and initial conditionsx0$x_{0}$,y0$y_{0}$are positive real numbers. Moreover, the rate of convergence of positive solutions that converge to the unique positive equilibrium point of this model is also discussed. In particular, our results solve an open problem and a conjecture proposed by Kulenović and Ladas in their monograph (Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, 2002). Some numerical examples are given to verify our theoretical results.

【 授权许可】

CC BY