Difference equations are used in order to model the dynamics of population with overlapping generations. In the symple case such equations have the form $N_{t+1}=f\left(N_t\right)N_t$, where $N_t>0$, in the population size at a moment of time $t$, $f\left(N_t\right)$ is a coefficient of natural reproductions.

The paper [3] consider a generalization of Skellam models when $\displaystyle f\left(N_t\right)= \frac{a}{b+N_t^2}$ and $\displaystyle f\left(N_t\right)= \frac{a N_t}{b+N_t^2}$ with a tough harvesting strategy. Functions are monotonic. But as ecological observations show, functions $f\left(N_t\right)$ are not always monotonic, at small $N_t>0$ they increase and at large $N_t$ they decrease. Such model where studied in [4].

Since Humans use various natural resources in their activities, it is important that the exploitation of populations does not lead to their destruction. Therefore, it is important to study models with harvesting.

This paper studies behavior of solution of the generalized Skellam model for a non-monotonic multiplication function of the form (3). This show that the equation (5) has stationary and periodic solution of any period. It is shown that with an incase in the harvest intensity coefficient, stationary and periodic solutions lose their stability and may disappear altogether.

For practice, it is extremely important to know the critical values of the parameter $k$, the fragments of which will lead to a decrease in the population.

[1] Matsenko V.G. Mathematical modelling of ecological processes : study guide. Chernivtsi : Yuriy Fedkovych Chernivtsi National University, 2019. 376p. (in Ukrainian)

[2] Skellam J.G. Random dispersial in theoretical populations. Biometrica. 1951. 38. 196-218.

[3] Matsenko V.G. Analysis of Skellam-type models with non-monotonic reproduction function. Bukovinian Math. Journal. 2025. 13(1). 52–65. (in Ukrainian)

[4] Matsenko V.G. Analysis of Skellam models with a rigid harvesting strategy. Bukovinian Math. Journal. 2024. 12(1). 74-83.