Non-unique population dynamics: basic patterns

We review the basic patterns of complex non-uniqueness in simple discrete-time 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–macroparasite and host–parasitoid interspecific interactions. In general, several types of attractors, e.g. point equilibria vs. chaotic, periodic vs. quasiperiodic and quasiperiodic vs. chaotic attractors, may coexist in the same mapping. This non-uniqueness also indicates that the bifurcation diagrams, or the routes to chaos, depend on initial conditions and are therefore non-unique. The basins of attraction, defining the initial conditions leading to a certain attractor, may be fractal sets. The fractal structure may be revealed by fractal basin boundaries or by the patterns of self-similarity. The fractal basin boundaries make it more difficult to predict the final state of the system, because the initial values can be known only up to some precision. We conclude that non-unique dynamics, associated with extremely complex structures of the basin boundaries, can have a profound effect on our understanding of the dynamical processes of nature.