Where and how do localized perturbations affect stream and coastal ocean populations with nonlinear growth dynamics?
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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.
KeywordsAsymmetrical dispersal Chaos Disturbance Perturbation Nonlinear dynamics Population model
The authors would like to thank two anonymous referees, ES Klein and MJ Fogarty for thoughtful discussion of this research. This paper is comprised of content from a thesis submitted to the Graduate School at the University of New Hampshire as part of the requirements for completion of a doctoral degree. Funding for the research was provided by the US National Science Foundation (NSF) award OCE 0961344.
Effects of perturbation as a function of the perturbation location (Ω = 0.01, a = 3.50, Ψ = 0.9). The graph overlaps 100 population distributions immediately following the perturbation. The dotted vertical line indicates the location of the local perturbation. We observe that when perturbations are applied to the upstream region (leftmost area of the domain), the population distribution is visibly affected, but as the perturbations are applied further downstream, they do not grow sufficiently over time to affect the population distribution (MP4 3473 kb)
Population at time (t-1) plotted against population at time (t) over every other time step (Ω = 0.01, a = 3.50, Ψ = 0.9). One time step is one generation of the model. The color bar indicates the location of the sampled point within the domain. Therefore, the gray markers are population locations in the upstream region of the domain, and the red markers are population locations in the downstream region of the domain. The black markers indicate the t = 1 solution. The perturbation is applied at time t = 1 to location x = 0.1. There is an initial ramp-up period where the perturbation is too small to alter the (t-1) vs (t) plots. After t ≈ 35, the perturbation is large enough to produce instability in the downstream region (red – pink markers) and temporarily destroys the periodic population behavior. After t ≈ 180, the downstream region slowly regains periodicity but with a shifted phase (MP4 223 kb)
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