Scenario of involuntary destruction of a population in a modified Hutchinson equation

A. Yu. Perevarukha

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 4, pp. 58-69

Résumé

The problem of simulating abrupt changes in the mode of self-oscillations inherent in species that are capable of affecting their habitat is considered. The relevance of this work is the need to improve methods of mathematical biology to study non-stationary and extreme types of population dynamics, which often occur in practice. Rapid transitions to sharp fluctuations in the number of infestations occur during invasions of actively breeding pest species as Ostrinia nubilalis. Modification of the Hutchinson equation suggested in the article, taking into account a significant role of achieving the subthreshold point number that is less than the limiting capacity of the ecological niche $K$ in Verhulst equation, but a significantly higher number of lower threshold $L$ in Bazykin equation: $L\ll H<K$. In our equation, the atypical scenario of the development of a dangerous outbreak of insects is described with the change in the acting delay of regulation $\tau$ value. As follows from ecological examples, population cycles with large amplitude are often unstable. Often the cycle is a transitional mode. The smooth damping of the oscillations $\overline{N_*(r, t)}\rightarrow K$ does not always occur. In the new model, after the Hopf bifurcation, with the value $\hat\tau = \tau_* + \xi$ and the appearance of auto-oscillations of the nonharmonic form with increasing amplitude, the loss of the dissipative property of the trajectory sharply occurs. The computational scenario with the sudden output of the transient cycle $N_*(\hat\tau r, t)$ from the range of admissible values of abundance is interpreted as a specific disturbance of the functioning of the habitat. The loss of a compact attracting set leads to the destruction of the biosystem in the locus of an outbreak of insects or irretrievable death in the case of an island population of mammals.

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