Non-unique population dynamics: basic patterns

We review the basic patterns of complex non-uniqueness in simple discrete-t ime population dynamics models. We begin by studying a population dynamics model of a single species with a two-stage, two-habitat life cycle. We then explore in greater detail two ecological models describing host-macroparas ite and host-parasitoid interspecific interactions. In general, several typ es of attractors, e.g. point equilibria vs. chaotic, periodic vs. quasiperi odic and quasiperiodic vs, chaotic attractors, may coexist in the same mapp ing. This non-uniqueness also indicates that the bifurcation diagrams, or t he routes to chaos, depend on initial conditions and are therefore nbn-uniq ue. The basins of attraction, defining the initial conditions leading to a certain attractor, may be fractal sets. The fractal structure may be reveal ed by fractal basin boundaries or by the patterns of self-similarity. The f ractal basin boundaries make it more difficult to predict the final state o f the system, because the initial values can be known only up to some preci sion. We conclude that non-unique dynamics, associated with extremely compl ex structures of the basin boundaries, can have a profound effect on our un derstanding of the dynamical processes of nature. (C) 2000 Elsevier Science B.V. All rights reserved.