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Oscillation and stability in a delay model of a perennial grass. (English) Zbl 0856.39012
The authors study the oscillation and stability of the delay model of a perennial grass, $x_{n + 1} = ax_n + (b + cx_{n - 1}) e^{- x_n}$, $n = 0, 1, \dots$, where $a,c \in (0,1)$ and $b \in (0, \infty)$, and where $x_{-1}$ and $x_0$ are arbitrary positive initial conditions. It is shown that every solution is bounded and persists, every nonoscillatory or positive solution converges to a positive equilibrium, and for certain values of $a$ and $b$ there exists an attractive $n$ cycle, for $n$ a positive integer which increases as $b$ increases.

39A12Discrete version of topics in analysis
92B99Mathematical biology
39A11Stability of difference equations (MSC2000)
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