On the global behavior of solutions of a biological model.

*(English)* Zbl 1110.39300
Introduction: In this paper, our purpose is to study the stability and the semicycles of solutions of the delay difference equation $$x_{n+1}= ax_n+ bx_{n-1} e^{-x_n},\quad n= 0,1,\dots,$$ where $a\in (0,1)\text{ and }b\in (0,\infty)$ and where $x_{-1}$ and $x_0$ are positive initial conditions. This equation arises from models for the amount of litter in a perennial grassland. We use the amount of litter because it is observable above ground, while most of the biomass is in subterranean roots. Each year the grass grows a new from the underground roots, and at the end of the season dies and falls as litter.
The total amount of litter on the ground at the end of the season in a given year is equal to the amount of litter remaining from the previous year plus new litter from this year’s growth. Thus the amount of litter $x_{n+1}$ at the end of year $n+ 1$ consists of two components. The first component is the fraction $ax_n$ of litter remaining from year $n$. The second component $bx_{n-1} e^{-x_n}$ represents new litter produced in year $n+ 1$. We assume that new growth emerges only through bare soil and that litter currently on the ground supresses growth. We also assume that the nutrients come only from the decay of the previous year’s litter. Therefore the production of new litter is inhibited by the litter $x_n$ currently on the ground, but increased by the recycling of the previous year’s litter $x_{n-1}$. This approximates a tropical environment where much of the nutrient is stored in the biomass.

##### MSC:

39A10 | Additive difference equations |

39A11 | Stability of difference equations (MSC2000) |

92D25 | Population dynamics (general) |