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

  • Laura S. StorchEmail author
  • James M. Pringle


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.


Asymmetrical 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.

Supplementary material

Movie 1S.

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)

Movie 2S.

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)

12080_2019_446_MOESM3_ESM.mp4 (14.3 mb)
Movie 3S. Spread of perturbation as a function of population density (Ω = 0.01, a = 7.00, Ψ = 1.5). The leftmost graph plots the magnitude of the response to perturbation versus the location of perturbation (where the magnitude of response is Eq. 8, summed divergence between perturbed and unperturbed trajectories over 100 iterations immediately following the perturbation). The area under the curve is filled in to easily distinguish regions where the perturbation response is zero (or near zero). The second graph is the population distribution overlapped for 100 generations immediately following the perturbation. When the population distributions are colored red, it indicates that the trajectories of the perturbed and unperturbed populations diverge over time, while gray coloring indicates that the population distribution is periodic over time and the perturbed versus unperturbed trajectories do not diverge. The thick black curve shows the population distribution that the perturbation was applied to. The dotted vertical line (graphs one and two) indicates the location of the perturbation, which moves downstream as the movie progresses. The third plot illustrates the spread of the local perturbation over time (y-axis) and space (x-axis). We observe that when a perturbation is applied to a region of local high population density, the perturbation has no effect on the population distribution because the local population fails to successfully reproduce due to density-dependent effects (MP4 14615 kb)


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Copyright information

© Springer Nature B.V. 2020

Authors and Affiliations

  1. 1.Department of MathematicsWilliam & MaryWilliamsburgUSA
  2. 2.Coastal Oregon Marine Experiment StationOregon State UniversityNewportUSA
  3. 3.Department of Earth Sciences and the Institute for Earth, Ocean, and SpaceUniversity of New HampshireDurhamUSA

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