Where and how do localized perturbations affect stream and coastal ocean populations with nonlinear growth dynamics?

Abstract

Chaotic systems are sensitive to small changes in parameters and initial conditions, but are spatially distributed chaotic populations sensitive to the location of perturbations within their domain? Here, we examine the transient responses to perturbation in a density-dependent population model with asymmetrical dispersal, where offspring are moved some distance downstream of the parent population and diffusively spread, an appropriate approximation for species in streams, rivers, and ocean currents. We find that over a portion of the studied parameter space, our system displays extreme convective instability; that is, an arbitrarily small point-source perturbation to the upstream region grows and propagates downstream (in our finite one-dimensional domain, the current moves offspring preferentially in the downstream direction). We can then divide our domain into distinct upstream and downstream regions based on the response to perturbation. A perturbation to the upstream region will create transient downstream instability, and a perturbation to the downstream region has no qualitative effect on the population. We investigate the factors affecting the existence/size of the sensitive upstream region, discuss the ecologically relevant time scales of the transient dynamics following a perturbation, and consider the ecological implications of managing and predicting a population with transient amplification of perturbations.