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.

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