Revisiting the logistic map: A closer look at the dynamics of a classic chaotic population model with ecologically realistic spatial structure and dispersal.

There is an ongoing debate about the applicability of chaotic and nonlinear models to ecological systems. Initial introduction of chaotic population models to the ecological literature was largely theoretical in nature and difficult to apply to real-world systems. Here, we build upon and expand prior work by performing an in-depth examination of the dynamical complexities of a spatially explicit chaotic population, within an ecologically applicable modeling framework. We pair a classic chaotic growth model (the logistic map) with explicit dispersal length scale and shape via a Gaussian dispersal kernel. Spatio-temporal heterogeneity is incorporated by applying stochastic perturbations throughout the spatial domain. We witness a variety of population dynamics dependent on the growth rate, dispersal distance, and domain size. Dispersal serves to eliminate chaotic population behavior for many of the parameter combinations tested. The model displays extreme sensitivity to changes in growth rate, dispersal distance, or domain size, but is robust to low-level stochastic population perturbations. Large and temporally consistent perturbations can lead to a change in population dynamics. Frequent switching occurs between chaotic/non-chaotic behaviors as dispersal distance, domain size, or growth rate increases. Small changes in these parameters are easy to imagine in real populations, and understanding or anticipating the abrupt resulting shifts in population dynamics is important for population management and conservation.