Ecological Archives E095-086-A1

James M. Pringle, James E. Byers, Paula Pappalardo, John P. Wares, Dustin Marshall. 2014. Circulation constrains the evolution of larval development modes and life histories in the coastal ocean. Ecology 95:1022–1032. http://dx.doi.org/10.1890/13-0970.1

Appendix A. Sensitivity to dispersal kernel shape.

The dispersal kernel of marine organisms with planktonic dispersal is poorly understood, and is likely to vary with the details of spawning timing, local and regional ocean circulation, and larval behavior (Siegel et al. 2003, 2008, Mitarai et al. 2008). Some insight into the effect of various dispersal kernel shapes is provided by Lutscher (2007) and Kot et al. (1996), who found that the tails of the dispersal kernel drive the speed of dispersal of an organism spreading upstream, and thus the ability of an organism to persist when undergoing asymmetric dispersal (Pachepsky et al. 2005, Pringle et al. 2009). These results suggest that the shape of the dispersal kernel has two important effects on marine organisms: first, if the dispersal kernel is finite (i.e., there is a maximum upstream and downstream distance the larvae can spread), there is a maximum downstream transport of larvae that can allow a species to persist, regardless of fecundity. Essentially, if none of the larvae can return to the location of their parents, the population will go locally extinct at that location. Thus for finite dispersal kernels, there is a maximum time in plankton (and thus downstream transport) that can be favored. Beyond that limit, any further increase is strongly selected against. Secondly, for kernels with infinite extent, the greater the kurtosis of the dispersal kernel (essentially, the fatter the tails of the spatial larval distribution) the less the fecundity needed to allow a species to persist for a given mean and standard deviation of larval dispersal. This strongly suggests that behavior that increases the kurtosis of the larval distribution will be selected for, and increased kurtosis lessens the penalty for downstream transport driven by long larval durations.

To test these ideas, Fig. 2(A) was recalculated for a broad range of ∏1 and a number of different dispersal kernels. To do so, for each value of ∏1 shown, a modified Brent root finding algorithm (Press et al. 2007) was used to find the value of ∏2 for which a change in time in plankton was selectively neutral; this point is the dividing line between where longer and shorter time in plankton is favored.

The “Tent” kernel, with a finite spatial extent and low kurtosis, has a lower threshold in ∏2 for evolutionarily stable long planktonic durations than the Gaussian dispersal kernel with its infinite extent (Fig. A1). Furthermore, as the time in plankton and thus ∏1 increases past some threshold, the maximum mean current, and thus ∏2, that favors longer planktonic duration decreases. Thus for this finite kernel, there is an evolutionary steady state for long time in plankton, instead of (as with the Gaussian kernel) an evolutionary pressure that always drives evolution towards longer times in plankton. This is seen in Fig. A1 where the neutral stability curve turns back towards ∏1=0 as ∏1 increases. This will occur for any kernel with finite extent.

At the opposite extreme, a Laplacian kernel has a large kurtosis, and can be shown to lead much larger upstream invasion speeds (Powell et al. 2005). This, in the numerical model, leads to there always being a value of ∏1 that allows the evolution of longer planktonic duration; this threshold increases as ∏2 increases (Fig. A1). With this kernel, there could always be long planktonic durations, or a ∏2 threshold that depends on the maximum TPLD that is otherwise possible in the system. However, the authors believe this result is implausible, for it depends on the infinite extent of the fat tails of the Laplacian distribution, which depend on infinite currents being available in the ocean to transport the larvae infinitely far in a finite time. In Fig. A1, we show results for the same kernel, but truncated with a short linear taper to a finite extent of 3 and 4*Ldiff from the center of the distribution. In these more realistic finite kernels, there is again a finite threshold value of ∏2 consistent with the persistence of long planktonic durations. While the behavior of the tails of larval dispersal kernels is poorly understood, assuming an infinite extent to the kernel in an ocean with finite extent would not seem to be the best null hypothesis (of course, anthropogenic transport of larvae can lead to some very long tail transport events, e.g., Pringle et al. (2011)– but that is a discussion for another paper).

All of the truncated kernels allow for an evolutionary steady state at finite but long planktonic duration, as explained in the paper. It has not escaped our notice that this could be used to devise a theory for the evolutionarily optimal and stable magnitude of TPLD and larval size in planktonic dispersers. While this is mathematically straightforward (e.g., Lutscher (2007)), our understanding of the tails of larval dispersal kernels is so poor that we feel doing so is premature.

 

Literature cited

Kot, M., M. A. Lewis, and P. van den Driessche. 1996. Dispersal Data and the Spread of Invading Organisms. Ecology 77:2027–2042.

Lutscher, F. 2007. A short note on short dispersal events. Bulletin of Mathematical Biology 69:1615–1630.

Mitarai, S., D. A. Siegel, and K. B. Winters. 2008. A numerical study of stochastic larval settlement in the California Current system. Journal of Marine Systems 69:295–309.

Pachepsky, E., F. Lutscher, R. M. Nisbet, and M. A. Lewis. 2005. Persistence, spread and the drift paradox. Theoretical Population Biology 67:61–73.

Powell, J. A., I. Slapničar, and W. van der Werf. 2005. Epidemic spread of a lesion-forming plant pathogen—analysis of a mechanistic model with infinite age structure. Linear Algebra and its Applications 398:117–140.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. 2007. Numerical Recipes 3rd Edition: The Art of Scientific Computing, Third edition. Cambridge University Press.

Pringle, J. M., A. M. H. Blakeslee, J. E. Byers, and J. Roman. 2011. Asymmetric dispersal allows an upstream region to control population structure throughout a species’ range. Proceedings of the National Academy of Sciences 108(37):15288–15293.

Pringle, J. M., F. Lutscher, and E. Glick. 2009. Going against the flow: effects of non-Gaussian dispersal kernels and reproduction over multiple generations. Marine Ecology Progress Series 377:13–17.

Siegel, D. A., B. P. Kinlan, B. Gaylord, and S. D. Gaines. 2003. Lagrangian descriptions of marine larval dispersion. Marine Ecology Progress Series 260:83–96.

Siegel, D. A., S. Mitarai, C. J. Costello, S. D. Gaines, B. E. Kendall, R. R. Warner, and K. B. Winters. 2008. The stochastic nature of larval connectivity among nearshore marine populations. Proceedings of the National Academy of Sciences 105:8974 –8979.

FigA1

Fig. A1. (A) The shape of the different dispersal kernels evaluated. (B) The threshold between increased fitness for longer or shorter planktonic durations for Gaussian, Tent, Laplacian, or Laplacian kernels linearly tapered to zero at 3 or 4 standard deviations from the center of the kernel.


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