Chaotic dynamics of a nonlinear density dependent population model

We study the dynamics of an overcompensatory Leslie population model where the fertility rates decay exponentially with population size. We find a plethora of complicated dynamical behaviour, some of which has not been previously observed in population models and which may give rise to new paradigms in population biology and demography. We study the two- and three-dimensional models and find a large variety of complicated behaviour: all codimension 1 local bifurcations, period doubling cascades, attracting closed curves that bifurcate into strange attractors, multiple coexisting strange attractors with large basins (which cause an intrinsic lack of 'ergodicity'), crises that can cause a discontinuous large population swing, merging of attractors, phase locking and transient chaos. We find (and explain) two different bifurcation cascades transforming an attracting invariant closed curve into a strange attractor. We also find one-parameter families that exhibit most of these phenomena. We show that some of the more exotic phenomena arise from homoclinic tangencies.