Department of Mathematics and Statistics, University of Ottawa Ottawa, ON, Canada
Background/Question/Methods
Individual variability in dispersal and reproduction abilities can lead to evolutionary processes that may have significant effects on the speed and shape of biological invasions. A fundamental evolutionary mechanism for increased invasion speeds is spatial sorting: individuals with the highest dispersal ability are located at the leading edge of an invasion front. A famous example are cane toads in Australia, whose evolution led to an accelerating invasion front.
Most mathematical models for these processes are based on reaction-diffusion equations, i.e., continuous time and Gaussian dispersal. We develop novel theory for how evolution shapes biological invasions with integrodifference equations, i.e., for organisms that show distinct non-overlapping growth and dispersal phases during their life cycle and that may have non-Gaussian dispersal patterns.
Our model tracks how the distribution of growth rates and dispersal ability in the population changes from one generation to the next in continuous space. We include mutation between types and a potential trade-off between dispersal ability and growth rate. We determine the invasion speed of the population and give a formula for the distribution of types at the leading edge of the invasion. In particular, we observe conditions for when spatial sorting emerges and when it does not.
Results/Conclusions
Without trade-offs, the leading edge is composed of a high density of the most mobile individuals, i.e., spatial sorting arises, for all levels of mutation rates. The corresponding speeds of invasion exceed the ones expected by the median dispersal trait, i.e, evolution can increase invasion speed.
With weak trade-offs, at low mutation rates, the distributions tend to center at the traits that maximize speed, shifting towards having the most mobile at the leading edge as mutation rate increases.
For strong trade-offs, population distributions at the leading edge range from having a high density of best reproducers when mutations are rare, bi-modal distributions between best reproducers and good dispersers for intermediate values of mutation rates, and finally having the most mobile ones at higher densities for elevated mutation rates.
Under a strong trade-off, an anomalous spreading speed can arise, i.e., the speed of invasion of the entire population is faster than the fastest single trait could attain if alone.
Our results contribute to the theory of spatial sorting in IDE models with very general dispersal patterns, and show that under some trade-off and mutation settings, spatial sorting may not arise.
