Models with Multiple Attractors

The main focus of this chapter is on the introduction of models that exhibit multiple attractors. Most single-species density-dependent population models without structure that are widely studied, such as the discrete logistic (Beverton-Holt) equation ( 1.25 ), the Leslie-Gower competition model ( 3.34 ), and the Ricker (exponential) competition model ( 3.55 ), are, in general, not capable of supporting multiple attractors [233]. However, we showed in Chapter 3 that the Leslie-Gower competition model can have multiple attractors on the boundary, in the case where there are no stable interior equilibrium points ( 3.34 ). In this chapter, we consider some single species and multiple species population models (with and without structure) that support multiple attractors. When multiple attractors occur, the long-term fate of the population depends on initial conditions. In such systems, one obviously cannot determine all possibilities for a population’s ultimate fate from a single time series data set or initial condition.