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.Comment: Latex file, 60 pages, 24 figure

The effects of time delay on the decline and propagation processes of population in the Malthus-Verhulst model with cross-correlated noises

The effects of time delay on the decline and propagation processes of population in the Malthus-Verhulst model with cross-correlated noises are investigated separately. Through numerically computing and stochastically simulating, we find that: (i) inclusion of time delay in the decline process, increasing the delay time τ weakens the stability of population with short delay and strengthens it with long delay. The stability of population reduces monotonically as the cross-correlated intensity λ increasing. The population of a species goes to extinction with increasing τ and increasing λ; (ii) inclusion of time delay in the propagation process, the increasing τ strengthens the stability of population and the increasing λ weakens it. The increasing τ slows down the growth process of a species while the increasing λ speeds it up. That is, the increasing delay time does not affect roughly the stability of population with short delay but strengthens it with long delay, and the population of species is restricted in the lower level by the larger delay time. The stability of population is weakened and the replacement of old individuals with young ones is accelerated by the increasing cross-correlation intensity between two noises