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 "symmetric dispersal" 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.