Models of specific forms of insect outbreaks in modifications of Bazykin and Verhulst-Pearl equations.

The article discusses environmentally sound modifications for two popularpopulation models of Bazykin and Verhulst-Pearl for the task of describing particular and non- trivial changes in population processes. A variety of the extreme nature of the number dynamics of an invasive insect species — outbreak activity of pests is modeled. The problem of applied computational modeling of transient modes of oscillating and destructive invasions of alien pests is relevant for many cases of sporadic mass reproduction of insect pests without biological control. As a result of analyzing the properties of known ecological models, we propose a modification that combines the most ecologically relevant trajectory behaviors after the cycle birth Hopf bifurcation. In the new equation of insect population dynamics with the deviating argument $\dot{x} = rx(t − h)f(x^k (t − \tau))$, an alternative unimodal regulator function $f(x), \lim_{\,x \to \infty} f(x) > \epsilon$ instead of $r(x/K)^k$ and $rx \exp (−bx)$ or $rx \sqrt[3p]{\,(x − L)}$ — for threshold in population dynamic:
$$\frac{dx}{dt} = rx \ln\,\left(\frac{K}{x(t − \tau)}\right) \sqrt[3p]{\,(x − L)}.$$
$L$ — is a critical threshold. With a significant increase in the value of the delay in such equations, the behavior of the trajectory becomes more complicated. As a result of the modification, we were able to overcome and correct the lack of unrealistically low minima in the Hutchinson equation, as we see when a relaxation cycle of considerable amplitude occurs. We have proposed a new equation describing the effect of a critical minimum based on a modification of the Bazykin model with a regulator function $\ln K/N(t − h)$. In the extreme extension of the Ferhulst-Pearl equation with delay:
$$\frac{dx}{dt} = rx \left(1 − \frac{x(t − \tau)}{K}\right) (
\mathfrak{H} − x(t − \tau_1)).$$
In the new equation the destruction of relaxation oscillations is described as a transitional regime of a specific outbreak of the number of insects affecting the state of agrocenosis. The cycle is compressed and destroyed, there is a loss of dissipative properties with preservation of the pseudoperiodic component. The computational experiment is stopped by the internal message of the instrumental environment about overflow in floating-point calculations.
Keywords: equation with deviating argument, cycles, Hopf bifurcation, control functions, forms of delay in biosystems, outbreaks and insect invasion