ANALYSIS OF SKELLAM-TYPE MODELS WITH PERIODIC REGIMES

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Difference equations are used in order to model the dynamics of population with non-overlapping generations. In the simplest case such equations have the form $N_{t+1}=f\left(N_t\right)N_t$, where $N_t>0$ is the population size at a moment of time $t$, $\displaystyle f\left(N_t\right)= \frac{N_{t+1}}{N_t}$ is a coefficient of natural reproduction. In Skellam's model this coefficient has the form of a decreasing hyperbolic function: $\displaystyle f\left(N_t\right)= \frac{a}{b+N_t}$, $a,b>0$. Parameter $a$ here plays the role of the largest value of the reproduction coefficient, and $b$ describes the influence of self-regulating mechanisms on population dynamics. For the Skellam's model, both without harvesting and with harvesting, only regimes with monotonic stabilization of the population size are observed. At the same time, as in other discrete models, there are periodic and even chaotic solutions. In this work, the following generalization of the Skellam model is proposed, which allows the existence of periodic regimes. Namely, a function is taken for $\displaystyle f\left(N_t\right)= \frac{a}{b+N_t^3}$. This shows that at certain values of $a$ and $b$ there are stable stationary states, that later lose stability, whereas with a corresponding change in $a$ and $b$, cycles of lengths 2, then 4, 8 appear. That is, there is a bifurcation of the doubling of the cycle. Periodic solutions with period 3 where not found, although the existence of chaotic solutions was established. It has been established that stable periodic regimes during harvesting can lose their stability.