Multiple attractors in stage-structured population models with birth pulses

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.