Models Of Species Interactions Research Articles

Periodic solutions of two species interaction models with lags

The classical Volterra-Lotka system of differential equations modeling the interaction of two species is considered under the assumption that the interaction and/or density inhibition terms may possess time lags (of a very general nature). Under the assumption that there exists a positive equilibrium, sufficient conditions are given which guarantee the existence of nonconstant, positive periodic solution. These conditions are algebraic in nature and are easily and straightforwardly applied. A few specific examples (namely, some predator-prey and competing species models) are discussed and are used to illustrate how lags are able to alter significantly the general dynamics of the classical equations. The mathematical approach is to construct periodic solutions which bifurcate from the equilibrium (using the net birth rates as free parameters) by making use of Liapunov-Schmidt type expansions, an approach which requires the proof of a Fredholm-type alternative for Stieltjes integrodifferential systems.

Mathematical Models of Species Interactions in Time and Space

Three widely used species interaction models, the Nicholson-Bailey and Hassell-Varley host-parasite models and the Lotka-Volterra predator-prey model, are extended into space as well as time. One type of equilibrium having the same number of individuals at all locations is shown to exist analytically for the case. Another kind of equilibrium having different numbers of individuals at different locations in two-dimensional space is demonstrated by simulation. Stability analysis suggests that space-time models having symmetric dispersal are no more stable near equilibrium than the simple time models upon which they are based. In many situations the space-time models are less stable than the simple time models. This result can be reconciled with the experimental literature if one assumes a stochastic persistence effect which predicts that expected extinction time will increase with the number of independent subpopulations.