Allee effects and density-dependent dispersal interact to generate variable invasion speed
The speed at which a population spreads as it invades new habitat, the invasion speed, is a fundamental quantity in both theoretical and applied population biology. Mathematical theory predicts that the invasion speed often is asymptotically constant and is determined by the vital rates and dispersal behavior of individuals at the low-density leading edge of the invasion wave. These results rest on two important assumptions: (1) the dispersal behavior of individuals does not depend upon density, and (2) the absence of Allee effects (i.e., positive density dependence in the vital rates). One consequence of relaxing these assumptions is that the added complexity makes difficult (if not impossible) to write down a formula for the invasion speed in terms of the model’s parameters. Nevertheless, numerical simulations indicate that for realistic dispersal models the population asymptotically spreads at a constant speed when either dispersal is density dependent or there are Allee effects. To see if this result holds in the face of both Allee effects and density-dependent dispersal we formulated an integrodifference equation that includes both.
We simulated our model for a wide range of parameter values and found that, in contrast with simpler models, there are cases in which the population does not spread at a constant rate. In these cases the invasion speed oscillates, sometimes with large amplitude. In the extreme, when Allee effects are strong and the propensity for individuals to disperse increases with density, the invasion speed can alternate sign. This indicates alternation between population spread and contraction. Such oscillations have been documented previously in models with environmental variability. Our results establish that oscillating rates of population spread can also be generated by the interaction between Allee effects and density-dependent dispersal in a constant environment.