Difference equations are widely used as models of population dynamics with non-overlapping generations. In the simplest case, they have the form $N_{t+1}=f\left(N_t\right) N_t$, where $N_t$ is the population size at a given time $t$, $f\left(N_t\right)$ is the natural reproduction rate. This function, in particular in the Skellam model, is monotonically decreasing. But as ecological observations show, $f\left(N_t\right)$ is not always monotonic; for small values of $N_t$, the function $f\left(N_t\right)$ is increasing, and for large values, it is decreasing.

Therefore, in paper \cite{Matsenko5}, for $f\left(N_t\right)$, a generalization of the Skellam model for a nonmonotonic multiplication function of the form $\displaystyle  f\left(N_t\right)= \frac{aN_t}{b+N_t^4}$ is considered. It is shown that such a model has stationary and periodic modes of any period.

If we lower the exponent of the value of en in the denominator to 3, then the model with $\displaystyle  f\left(N_t\right)= \frac{aN_t}{b+N_t^3}$ allows only stationary solutions, and periodic modes no longer exist.

This paper considers a generalization of these models with an exponent for $N_t$ ranging from 3 to 4, i.e. a model of the form $\displaystyle  N_{t+1}= \frac{aN_t}{b+N_t^{3+\alpha}}$, $a,b>0$, $\alpha\in(0,1]$. It is shown how, as the parameter a increases from 0 to 1, the behavior of the solutions of this model changes, namely, a bifurcation of the doubling of the cycle lengths occurs. In computational experiments, such solutions were found and their stability was studied. The existence of periodic solutions with period 3 has also been established, which means that this model has periodic solutions of any period and solutions with chaotic behavior.

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