Modelling biological invasions : population cycles, waves and time delays

Biological invasions are rapidly gaining importance due to the ever-increasing number of introduced species. Alongside the plenitude of empirical data on invasive species there exists an equally broad range of mathematical models that might be of use in understanding biological invasions. This thesis aims to address several issues related to modelling invasive species and provide insight into their dynamics. Part I (Chapter 2) documents a case study of the gypsy moth, Lymantria dispar, invasion in the US. We propose an alternative hypothesis to explain the patchiness of gypsy moth spread entailing the interplay between dispersal, predation or a viral infection and the Allee effect. Using a reaction-diffusion framework we test the two models (prey-predator and susceptible-infected) and predict qualitatively similar patterns as are observed in natural populations. As high density gypsy moth populations cause the most damage, estimating the spread rate would be of help in any suppression strategy. Correspondingly, using a diffusive SI model we are able to obtain estimates of the rate of spread comparable to historical data. Part II (Chapters 3, 4 and 5) is more methodological in nature, and in a single species context we examine the effect of an ubiquitous phenomenon influencing population dynamics time delay. In Chapter 3 we show that contrary to the general opinion, time delays are not always destabilising, using a delay differential equation with discrete time delay. The concept of distributed delay is introduced in Chapter 4 and studied through an integrodifferential model. Both Chapters 3 and 4 focus on temporal dynamics of populations, so we further this consideration to include spatial effects in Chapter 5. Using two different representations of movement, we show that the onset of spatiotemporal chaos in the wake of population fronts is possible in a single species model.