MODELING HARVESTING PROCESSES FOR POPULATIONS WITH NON-OVERLAPPING GENERATIONS

Abstract

Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have the form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete logistic equation and Ricker's equation are most often used in practice. In the given paper, these equations are considered width taking into account an effect of harvesting, that is, the equations of the form below are studied $N_{t+1}=r N_t (1- N_t) - c$ and $N_{t+1}= N_t \exp (r(1 - N_t / K )) - c$, where the parameters $r$, $K>0$, $c>0$ are harvesting intensity. Positive equilibrium points and conditions for their stability for these equations were found. These kinds of states are often realized in nature. For practice, periodic solutions are also important, especially with periods $T=2 (N_{t+2} = N_t)$ and $T=3 (N_{t+3} = N_t)$, since, with their existence, by Sharkovskii's theorem, one can do conclusions about the existence of periodic solutions of other periods. For the discrete logistic equation in analytical form, the values that make up the periodic solution with period $T=2$ were found. We used numerical methods in order to find solutions with period $T=3$. For Ricker's model, the question of the existence of periodic solutions can be investigated by computer analysis only. In the paper, a number of computer experiments were conducted in which periodic solutions were found and their stability was studied. For Ricker's model with harvesting, chaotic solutions were also found. As we can see, the study of difference equations gives many unexpected results.