Population Dynamics of Central Place Foragers - Wei, Hsin-Hua (University of Ottawa (Canada), 2010)
In this thesis we use analysis and simulation to study the qualititative behavior of several mathematical models for consumer-resource interaction. While such models have been studied for several decades now, our approach to include a random-walk model for the foraging behavior of the consumer is novel. With this approach we are able to link individual movement rules to population-level patterns. For every species, acquisition and assimilation of food is an important process that enables survival, growth, and reproduction. Different species have developed different strategies for this process. Individuals who depart from one location on foraging trips and return to the same location are called central place foragers. Examples include cave crickets, beavers, and colonial seabirds. Resources around central place foragers' habitats are the major source of food. In this thesis, we consider discrete-time consumer-resource models and examine the following questions: (1) Under what conditions can consumers and resources stabily coexist? (2) How does the distribution of consumers affect resource abundance spatially? (3) How does the abundance and distribution of resource affect the foraging behavior of consumers? First, we study a non-spatial model by considering that consumers disperse evenly over the resource patch. We apply several different functions for resource growth (compensatory and overcompensatory), and we consider various foraging behaviors (random versus clumped). We analyze the stability behavior of steady states in each of these models and we summarize our results in bifurcation diagrams. Next, we study the spatial model by assuming that consumers tend to concentrate closer to the central place. We apply the Laplace distribution to describe consumers' dispersal. We focus on the compensatory dynamics for resource growth and consider both random search and clumped search for resources. In this section, we use a combination of analytical and numerical techniques to study the qualitative behavior of the system. Finally, we apply a random-walk framework to derive the distribution of consumers based on individual movement rules and the abundance of resources. That is, we have different consumer distributions depending on local resource abundance at different times. The analysis of this complex model is exclusively based on numerical simulation. For all consumer-resource models, we present one-parameter bifurcation diagrams for each parameter in the model to illustrate the qualitative behavior. We investigate the effects of parameters on the stability of steady states and limit cycles. For models with fixed consumer distributions, we find that the clumped search for resources stabilizes the system. The resource and consumer populations can reach stable steady states if consumers aggregate intensively at the place with more abundant resources. We find that a resource-dependent distribution of consumers also stabilize the system as consumers settle at a more resource-rich location to forage. All parameters in our models have impacts on the population dynamics of resource and consumer. For the compensatory resource growth, we find that consumers can invade the system and both populations can reach a stable steady state for a higher value of resource growth rate or a lower value of consumer conversion efficiency. Other parameters can have different impacts on stability depending on different model structures.
Keywords: Mathematics.