Global behavior of a plant-herbivore model

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=α xₙβ xₙ+e^yₙ , yₙ₊₁=γ(xₙ+1)yₙ , where α∈(1,∞) , β∈(0,∞) , and γ∈(0,1) with α+β>1+β/γ and initial conditions x₀ , y₀ 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.