Extreme events in population dynamics with functional carrying capacity

A class of models is introduced describing the evolution of population species whose carrying capacities are functionals of these populations. The functional dependence of the carrying capacities reflects the fact that the correlations between populations can be realized not merely through direct interactions, as in the usual predator-prey Lotka-Volterra model, but also through the influence of species on the carrying capacities of each other. This includes the self-influence of each kind of species on its own carrying capacity with delays. Several examples of such evolution equations with functional carrying capacities are analyzed. The emphasis is given on the conditions under which the solutions to the equations display extreme events, such as finite-time death and finite-time singularity. Any destructive action of populations, whether on their own carrying capacity or on the carrying capacities of co-existing species, can lead to the instability of the whole population that is revealed in the form of the appearance of extreme events, finite-time extinctions or booms followed by crashes. 1 1 Brief survey of population models Evolution equations, describing population dynamics, are widely employed in various branches of biology, ecology, and sociology. The main forms of such equations are given by the variants of the predator-prey Lotka-Volterra models. In this paper, we introduce a novel class of models whose principal feature, making them different from other models, is the functional dependence of the population carrying capacities on the population species. This general class of models allows for different particular realizations characterizing specific correlations between coexisting species. The functional dependence of the carrying capacities describes the mutual influence of species on the carrying capacities of each other, including the self-influence of each kind of species on its own capacity. Such a dependence is, both mathematically and biologically, principally different from the direct interactions typical of the predator-prey models. Before formulating the general approach, we give in this section a brief survey of the main known models of population dynamics. This will allow us to stress the basic difference of our approach from other models used for describing the population dynamics in biology, ecology, and sociology. The first model, describing interacting species, one of which is a predator with population N 1 , and another is a prey with population N 2 , has been the Lotka-Volterra [1, 2] model dN 1 dt = −γ 1 N 1 + A 12 N 2 N 1 , dN 2 …