Allee Dynamics and the Spread of Invading Organisms

We examine how an Allee effect in local population dynamics (reduced reproductive success at low densities) influences the spatio-temporal dynamics of ecological invasions. Our approach is to use a partial differential equation model of dispersal and population growth, and then ask whether we can identify "rates of spread" for an invading organism subject to an Allee effect. Results indicate that an Allee effect may substantially reduce the rate at which the invader moves into a new environment. Analysis of spread in two spatial dimensions entails application of a singular perturbation theory approach. Here the two-dimensional spread velocity is given in terms of the one-dimensional asymptotic spread rate and the curvature of a boundary between invaded and non-invaded regions. Using this result, we show that invasions cannot propagate unless they initially exceed a critical area. This prediction is verified by numerically solving the original model. Numerical solutions are used throughout in demonstrating the nature of the two-dimensional spread.