A Boundary Value Problem for Integrodifference Population Models with Cyclic Kernels

The population dynamics of species with separate growth and dispersal stages can be modeled by a discrete-time, continuous-space integrodifference equation. Many authors have considered the case where the model parameters remain fixed over time, however real environments are constantly in flux. We develop a framework for analyzing the population dynamics when the dispersal parameters change over time in a cyclic fashion. In particular, for the case of N cyclic dispersal kernels modeling movement in the presence of unidirectional flow, we derive a 2Nth-order boundary value problem that can be used to study the linear stability of the associated integrodifference model. 1. Motivation and background. Humans depend on biodiversity for trade, sustenance, pharmaceuticals, and entertainment, not to mention the ethical imperative of wildlife protection [13]. However, Earth’s biodiversity is under serious threat. The International Union for Conservation of Nature lists 21,285 species as threatened, which is likely an underestimate due to insufficient information [2]. In fact, extinction rates are projected to increase as the human population continues to grow and put increased demand on Earth’s resources [2]. Many sources agree that the primary cause of threat is habitat loss and degradation, followed by direct exploitation and competition with invasive non-native species [2, 15]. How can we ensure that threatened populations are able to survive in a limited habitat, or that non-native species will not overwhelm native populations? Population models are invaluable to conservation ecology because they help us predict the likely impact of environmental change on a given population. In this paper, we consider a population model for species that have separate growth and dispersal stages. Many plants demonstrate this pattern of growth and dispersal, as well as a variety of arthropods (insects, arachnids, and crustaceans) [7, 5]. We restrict our attention to populations living in a bounded habitat surrounded by 2010 Mathematics Subject Classification. Primary: 45C05, 92B05; Secondary: 34B05.