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.

The papers \cite{Matsenko3,Matsenko4} consider a generalization of the Skellam model. It is shown that in 
the case when $\displaystyle f\left(N_t\right)= \frac{a}{b+N_t^2}$ and 
$\displaystyle f\left(N_t\right)= \frac{aN_t}{b+N_t^2}$ the models only monotonic stabilization of the 
number to some stationary level is observed. If $\displaystyle f\left(N_t\right)= \frac{a}{b+N_t^3}$, then 
the model has periodic solutions of different periods.

This paper studies the behavior of solutions of the generalized Skellam model for a non-monotonic 
multiplication function of the form $\displaystyle f\left(N_t\right)= \frac{aN_t}{b+N_t^4}$. 
It is shown that the equation $N_{t+1}=f\left(N_t\right) N_t$ has stationary and periodic solutions of 
any period. The stability of these solutions is investigated. In the exponent of the quantity $N_t$ in 
the denominator or is reduced to three, than the Skellam model with the function 
$\displaystyle f\left(N_t\right)= \frac{aN_t}{b+N_t^3}$ no longer has periodic solution, but has only 
monotonic stabilization to stationary regimes.

The paper presents the results of a number of computer experiments with such generalizations of 
the Skellam model.

[1] Skellam J.G. Random dispersial in theoretical populations. Biometrica, 1951. 38. 196-218.
[2] Suba J., Kawata Y., Linden A. Properties and interpretation of the Skellam model. A discrete-time contest competition population model. Population Ecolody. Online Version, 2023. https://doi.org/10.1002/1438-390x.12169.
[3] Matsenko V.G. Analysis of Skellam models with a rigid harvesting strategy. Bukovinian Math. Journal. 12(1). 2024. 74-83. (in Ukrainian)
[4] Matsenko V.G. Analysis of Skellam-type models width with periodic regimes. Bukovinian Math. Journal. 12(2). 2024. 128–142. (in Ukrainian)
[5] Sharkovskii A. N. Coexistence of cycles of continuous transformation straight into itself. Ukrainian Mathematical Journal, 1964. XVI (1). P. 61-71. (in Russian)
[6] Li T., Yorke I. Period three implies chaos. The American Mathematical Monthly, 1975. 82 (10). Pp. 985-992.