  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.
