Biological invasions are devastating ecosystems and economies world-wide, while many native species' survival depends on their ability to track climate change. Characterising the spread of biological populations is therefore of utmost importance, and can be studied with spatially explicit, discrete-time integro-difference equations (IDEs), which reflect numerous species' processes of demography and dispersal. While spatial variation has often been ignored when implementing IDE models, real landscapes are rarely spatially uniform and environmental variation is crucial in determining biological spread. To address this, we use novel methods to characterise population spread in heterogeneous landscapes. Asymptotic analysis is used for highly fragmented landscapes, where habitat patches are isolated and smaller than the dispersal scale, and in landscapes with low environmental variation, where the ecological parameters vary by no more than a small factor from their mean values. We find that the choice of dispersal kernel determines the effect of landscape structure on spreading speed, indicating that accurately fitting a kernel to data is important in accurately predicting speed. For the low-variation case, the spreading speeds in the heterogeneous and homogeneous landscapes differ by ϵ², where ϵ governs the degree of variation, suggesting that in many cases, a simpler homogeneous model gives similar spread rates. For irregular landscapes, analytical methods become intractable and numerical simulation is needed to predict spread. Accurate simulation requires high spatial resolution, which, using existing techniques, requires prohibitive amounts of computational resources (RAM, CPU etc). We overcome this by developing and implementing a novel algorithm that uses adaptive mesh refinement. The approximations and simulation algorithm produce accurate results, with the adaptive algorithm providing large improvements in efficiency without significant losses of accuracy compared to non-adaptive simulations. Hence, the adaptive algorithm enables faster simulation at previously unfeasible scales and resolutions, permitting novel areas of scientific research in species spread modelling.

Spatial segregation among life cycle stages has been observed in many stage-structured species, including species of the flour beetle Tribolium. Patterns have been observed both in homogeneous and heterogeneous environments. We investigate density dependent dispersal of life cycle stages as a mechanism responsible for this separation. By means of mathematical analysis and numerical simulations, we explore this hypothesis using stage-structured, integrodifference equation (IDE) models that incorporate density dependent dispersal kernels.In Chapter 2 we develop a bifurcation theory approach to the existence and stability of (non-extinction) equilibria for a general class of structured integrodifference equation models on finite spatial domains with density dependent kernels. We show that a continuum of such equilibria bifurcates from the extinction equilibrium when it loses stability as the net reproductive number n increases through 1. We give several examples to illustrate the theory.In Chapter 3 we investigate mechanisms that can lead to spatial patterns in two dimensional Juvenile-Adult IDE models. The bifurcation theory shows that such patterns do not arise for n near 1. For larger values of n we show, via numerical simulation, that density dependent dispersal can lead to the segregation of life cycle stages in the sense that each stage peaks in a different spatial location.Finally, in Chapter 4, we construct spatial models to describe the population dynamics of T. castaneum, T. confusum and T. brevicornis and use them to assess density dependent dispersal mechanisms that are able to explain spatial patterns that have been observed in these species.

Marine populations worldwide are in decline due to anthropogenic effects. Spatial management via marine reserves may be an effective conservation method for many species, but the requisite theory is still underdeveloped. Integrodifference equation (IDE) models can be used to determine the critical domain size required for persistence and provide a modelling framework suitable for many marine populations. Here, we develop a novel spatially implicit approximation for the proportion of individuals lost outside the reserve areas which consistently outperforms the most common approximation. We examine how results using this approximation compare to the existing IDE results on the critical domain size for populations in a single reserve, in a network of reserves, in the presence of alongshore currents, and in structured populations. We find that the approximation consistently provides results which are in close agreement with those of an IDE model with the advantage of being simpler to convey to a biological audience while providing insights into the significance of certain model components. We also design a stochastic individual based model (IBM) to explore the probability of extinction for a population within a reserve area. We use our spatially implicit approximation to estimate the proportion of individuals which disperse outside the reserve area. We then use this approximation to obtain results on extinction using two different approaches, which we can compare to the baseline IBM; the first approach is based on the Central Limit Theorem and provides efficient simulation results, and the second modifies a simple Galton-Watson branching process to include loss outside the reserve area. We find that this spatially implicit approximation is also effective in obtaining results similar to those produced by the IBM in the presence of both demographic and environmental variability. Overall, this provides a set of complimentary methods for predicting the reserve area required to sustain a population in the presence of strong fishing pressure in the surrounding waters.

In this dissertation, we study integrodifference equations in patchy landscapes. Specifically, we provide a framework for linking individual dispersal behavior with population-level dynamics in patchy landscapes by integrating recent advances in modeling dispersal into an integrodifference equation. First, we formulate a random-walk model in a patchy landscape with patch-dependent diffusion, settling, and mortality rates. We incorporate mechanisms for individual behavior at an interface which, in general, results in the probability-density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator and illustrate the kind of (discontinuous) dispersal kernels that arise from our approach. We examine the dependence of obtained kernels on model parameters. Secondly, we analyze integrodifference equations in patchy landscapes equipped with discontinuous kernels. We obtain explicit formulae for the critical-domain-size problem, as well as, explicit formulae for the analogous critical size of good patches on an infinite, periodic, patchy landscape. We examine the dependence of obtained formulae on individual behavior at an interface. Through numerical simulations, we observe that, if the population can persist on an infinite, periodic, patchy landscape, its spatial profile can evolve into a discontinuous traveling periodic wave. We derive a dispersion relation for the speed of the wave and illustrate how interface behavior affects invasion speeds. Lastly, we develop a strategic model for the spread of the emerald ash borer and its interaction with host trees. A thorough literature search provides point estimates and interval ranges for model parameters. Numerical simulations show that the spatial profile of an emerald ash borer invasion evolves into a pulse-like solution that moves with constant speed. We employ Latin hypercube sampling to obtain a plausible collection of parameter values and use a sensitivity analysis technique, partial rank correlation coefficients, to identify model parameters that have the greatest influence on obtained speeds. We illustrate the applicability of our framework by exploring the effectiveness of barrier zones on slowing the spread of the emerald ash borer invasion.

In this dissertation, we study integrodifference equations in patchy landscapes. Specifically, we provide a framework for linking individual dispersal behavior with population-level dynamics in patchy landscapes by integrating recent advances in modeling dispersal into an integrodifference equation. First, we formulate a random-walk model in a patchy landscape with patch-dependent diffusion, settling, and mortality rates. We incorporate mechanisms for individual behavior at an interface which, in general, results in the probability-density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator and illustrate the kind of (discontinuous) dispersal kernels that arise from our approach. We examine the dependence of obtained kernels on model parameters. Secondly, we analyze integrodifference equations in patchy landscapes equipped with discontinuous kernels. We obtain explicit formulae for the critical-domain-size problem, as well as, explicit formulae for the analogous critical size of good patches on an infinite, periodic, patchy landscape. We examine the dependence of obtained formulae on individual behavior at an interface. Through numerical simulations, we observe that, if the population can persist on an infinite, periodic, patchy landscape, its spatial profile can evolve into a discontinuous traveling periodic wave. We derive a dispersion relation for the speed of the wave and illustrate how interface behavior affects invasion speeds. Lastly, we develop a strategic model for the spread of the emerald ash borer and its interaction with host trees. A thorough literature search provides point estimates and interval ranges for model parameters. Numerical simulations show that the spatial profile of an emerald ash borer invasion evolves into a pulse-like solution that moves with constant speed. We employ Latin hypercube sampling to obtain a plausible collection of parameter values and use a sensitivity analysis technique, partial rank correlation coefficients, to identify model parameters that have the greatest influence on obtained speeds. We illustrate the applicability of our framework by exploring the effectiveness of barrier zones on slowing the spread of the emerald ash borer invasion.