Parameter estimation techniques for interaction and redistribution models: a predator-prey example.

The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.

Local movement in herbivorous insects: applying a passive diffusion model to mark-recapture field experiments.

A simple passive diffusion model is used to analyze the local within-habitat dispersal of twelve species of herbivorous insects. The data comprise field mark-recapture studies in relatively homogeneous habitats. For eight of the species, the cumulative frequency distributions of dispersal distances are consistent with a model of movement by passive diffusion. The observed departures from passive diffusion indicate the directions in which we need to modify our mathematical descriptions of movement if we are to develop realistic models of population dynamics and dispersal. The analyses also synthesize in a standard way the relative dispersal rates of several ecologically similar species. The variation both within and between species in diffusion coefficients is striking-certainly sufficient to generate significant consequences for population dynamics and interactions.

Analyzing insect movement as a correlated random walk.

This paper develops a procedure for quantifying movement sequences in terms of move length and turning angle probability distributions. By assuming that movement is a correlated random walk, we derive a formula that relates expected square displacements to the number of consecutive moves. We show this displacement formula can be used to highlight the consequences of different searching behaviors (i.e. different probability distributions of turning angles or move lengths). Observations of Pieris rapae (cabbage white butterfly) flight and Battus philenor (pipe-vine swallowtail) crawling are analyzed as a correlated random walk. The formula that we derive aptly predicts that net displacements of ovipositing cabbage white butterflies. In other circumstances, however, net displacements are not well-described by our correlated random walk formula; in these examples movement must represent a more complicated process than a simple correlated random walk. We suggest that progress might be made by analyzing these more complicated cases in terms of higher order markov processes.