Does advection make my population chaotic? Introducing ecological realism to the logistic map

Background/Question/Methods

Chaotic behavior has been directly detected in ecological systems, e.g. fisheries, yet chaotic models are underutilized in the ecological literature.  Existing chaotic models from other fields of research are often difficult to apply to real-world ecological systems due to idealizations and ecologically unrealistic boundary conditions.  Here, we seek to provide easily applicable chaotic models by introducing both (1-dimensional) diffusive and advective-diffusive population models with explicit length scale, explicit dispersal shape, and absorbing boundaries.  The growth is dictated by the logistic map, a discrete-time, point-source, density-dependent growth function that exhibits chaotic behavior for growth rates larger than ≈3.56.  The dispersal distance is dictated by a discrete Gaussian kernel.  We test a range of growth rates and dispersal distances, which are normalized by the domain size so only the ratio of (dispersal distance/domain) is important.   

Results/Conclusions

Diffusive dispersal paired with logistic growth leads to many spatiotemporally periodic population behaviors in the chaotic growth range of the regular logistic map. We find that small changes in parameters can change a periodically behaving population to a chaotically behaving population, or vice versa.  With the addition of advection we find prevalence of spatiotemporally periodic behavior is significantly reduced in comparison to the diffusive model, and chaotic behavior is far more common.  Such reduction in periodic behavior is due to the fact that the dynamics of the advective system are driven by the dynamics of the upstream population.  Therefore, the dynamics are driven by a small portion of the total domain and thus the diffusive dispersal has less of a smoothing effect versus the diffusion-only model.  Additionally we find external population perturbations at the upstream edge of the domain have a more pronounced effect on the qualitative population behavior than perturbations at the downstream edge of the domain.  Presence of chaotic dynamics in an ecological system has myriad ecological implications, and we have listed several above. Identifying chaotic behavior in ecological systems is therefore crucial for monitoring and management.