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Report of visit by Mark A. Lewis

Mark A. Lewis (Department of Mathematics, University of Utah Salt Lake City, UT 84112, USA) visited Utrecht and Leiden, December 2-8, 1998, reported by Hans Metz.

The main purpose of the visit was to let Mark interact and collaborate with those researchers in the Netherlands that specialize in the modelling and mathematical analysis of spatial ecological processes and in particular invasion waves, to wit Hans Metz (Leiden), Odo Diekmann (Utrecht) and Frank van den Bosch (Wageningen) and their collaborators. The Dutch researchers and Mark have at some points quite different approaches, which meant that considerable time was spent comparing the approaches and discussing how the differences could be put to the test. All parties found the contacts most enjoyable, and some good amount of information was exchanged.

During the visit Mark gave three talks:

On the asymptotic speed of a stochastic invasion (Utrecht, December 3, and Leiden, December 4, 1998): Despite the recognized importance of stochastic factors, models for ecological invasions are almost exclusively formulated using deterministic equations. Stochastic factors relevant to invasions can be either extrinsic (quantities such as temperature or habitat quality which vary randomly in time and space and are external to the population itself) or intrinsic (arising from a finite population of individuals each reproducing, dying, and interacting with other individuals in a probabilistic manner). It has been long conjectured that intrinsic stochastic factors associated with interacting individuals can slow the spread of a population or disease, even in a uniform environment.

The Leiden talk gave an introduction to this problem from an overall perspective. The Utrecht talk was devoted to a mathematical analysis of the conjecture for the case where individuals interact locally over small spatial neighborhoods. A set of equations describing the dynamics of spatial moments of the population is formulated. Although the full equations cannot be expressed in closed form, a mixture of a moment closure and comparison methods can be used to derive upper and lower bounds for the expected density of individuals. Analysis of upper and solution gives an bound on the rate of spread of the stochastic invasion process. The stochastic process spreads more slowly than the corresponding deterministic model would predict. The slow spread situation corresponds to invaders occurring in widely spaced high density foci. In these cases spatial correlations between individuals mean that density dependent effects are significant even when expected population densities are low. Finally a heuristic formula for estimating the rate of spread was proposed, based on a scaling argument for moments.

Wolf home ranges and prey survival (Leiden December 7, 1998): Wolves are now recolonizing much of North America. It is not yet clear how they will impact the ecosystems to which they return. Data from established wolf populations indicate that they when wolves are close to prey they interact strongly with them. However, much of the time wolves are far from their prey: radio tracking data shows wolves moving around their large home ranges in complex spatial patterns, using scent marks, trails and other features as movement cues.

With new spatial modeling methods it is possible to describe wolf movement, behavior and impact on prey. Here the home ranges arise through the interactions between scent-marking and detailed spatial movements. When incorporated into a mechanistic model, these interactions result in distinctive territories with `walls' of scent marks defining the edges the territory, and the `buffer zones' found between territories. Such mechanistic models are an improvement on traditional statistical models for home ranges and can also be used to predict the effects of changing conditions on the wolf home ranges. Finally, the model results can be compared with radio-tracking data. For example, data from northeastern Minnesota that indicates buffer zones between wolf territories act as refuges for their primary prey species, white-tailed deer.

next up previous contents
Next: Third Winter school on Up: nls-98-4.html Previous: Summer school Spatio-Temporal Patterns

Bob Kooi
Tue Dec 29 15:54:50 MET 1998