Oscillation and stability in models of a perennial grass.

*(English)*Zbl 0812.39003Ladde, 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 $\alpha ,\beta ,\gamma \in (0,\infty )$ 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 $a\in (0,1)$ and $b\in (0,\infty )$.

Reviewer: A.D.Mednykh (Novosibirsk)

##### MSC:

39A10 | Additive difference equations |

39A11 | Stability of difference equations (MSC2000) |

92D25 | Population dynamics (general) |