In this work we suggest and consider an original, simple mathematical modelof a "quasi-rapid" extinction population dynamics. It describes a decrease andfinal extinction of the population of one prey species by a "quasi-rapid"interaction with one predator species with increasing population. This"quasi-rapid" interaction means ecologically that prey species behavespractically quite passively (since there is no time for any reaction, i.e.defense), like an appropriate environment, in respect to "quasi-rapid" activityof the predator species that can have different "quasi-rapid" huntingabilities. Mathematically, our model is based on a non-Lotka-Volterraian systemof two differential equations of the first order, first of which is linearwhile second, depending of a parameter that characterizes hunting ability isnonlinear. We compare suggested "quasi-rapid" extinction population dynamicsand the global model of the overkill of the prehistoric megafauna (mammoths).We demonstrate that our "quasi-rapid" extinction population dynamics is able torestitute successfully correlations between empirical (archeological) data andoverkill theory in North America as well as Australia. For this reason, weconclude that global overkill theory, completely mathematically modelable by"quasi-rapid" extinction population dynamics can consistently explain thePleistocene extinctions of the megafauna.
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