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Oscillation and stability in models of a perennial grass. (English) Zbl 0812.39003
Ladde, G. S. (ed.) et al., Dynamic systems and applications. Vol. 1. Proceedings of the 1st international conference, held at Morehouse College, Atlanta, GA, USA, May 26-29, 1993. Atlanta, GA: Dynamic Publishers, Inc. 87-93 (1994).

The permanence and oscillation of a model of a perennial grass are studied. The model was introduced by Tilman and Wedin and can be expressed by the equation

${x}_{n+1}=\frac{{x}_{n}^{2}}{1+{x}_{n}}+\frac{\alpha }{1+\beta {e}^{\gamma {x}_{n}}},\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,$

where $\alpha ,\beta ,\gamma \in \left(0,\infty \right)$ and ${x}_{0}$ is an arbitrary positive initial condition.

The study of the global stability of the positive equilibrium of the above equation is very difficult. For this reason the authors also investigate the model

${x}_{n+1}=a{x}_{n}+b{e}^{-{x}_{n}},\phantom{\rule{1.em}{0ex}}n=0,1,\cdots ,$

where $a\in \left(0,1\right)$ and $b\in \left(0,\infty \right)$.

##### MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000) 92D25 Population dynamics (general)