We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.
Newman, T. J., Antonovics, J., & Wilbur, H. M. (2002). Population dynamics with a refuge: fractal basins and the suppression of chaos. Theoretical Population Biology, 62(2), 121-128. https://doi.org/10.1006/tpbi.2002.1584