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 = alpha X-n/beta X-n+e(yn), y(n+1) = gamma(x(n) + 1)y(n), wehre alpha is an element of(1, infinity), beta is an element of(0, infinity), and gamma is an element of(0, 1) with alpha + beta > 1 + beta/gamma and inital conditions x(0),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 Kulenovic 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.