An example of my current research interests is the interaction between population dynamics and dispersal. Theoretical ecologists have traditionally formulated continuous-time partial differential equations to describe the growth of a population in time and its distribution in space. These models have been used in order to answer such ecological questions as:
Unfortunately, continuous-time models are inappropriate for species
(such as annual plants) that have discrete generations and well-defined
growth and dispersal stages. In such cases we can model the population
dynamics using integrodifference equations such as
Here, nt(x) represents the population density in the tth generation at location x and the nonlinear function f describes density dependent processes at a given position in space. The redistribution kernel k is a probability density function that describes how new individuals (seeds in the case of annual plants) are dispersed throughout the habitat. k is often called a "redistribution kernel" or "dispersal kernel". Dispersal kernels have been measured for corals, grasses, trees, wasps, and birds to name but a few examples.
My own work on these models has to date focussed on their applicability as models of ecological invasions. Integrodifference equation models are becoming popular, and are currently being used to understand the growth and spread of species as diverse as fruit flies, House Finches, Pine Trees, and Scotch Broom.
I am also interested in the ability of integrodifference equation models to produce spatial pattens. The combined effects of discrete time and a generalized description of dispersal promote pattern formation in these models as compared with their continuous-time counterparts.
An important yet under-studied class of theoretical problems involves the effects of dispersal on demographic processes and vice-versa. Integrodifference equations coupled with matrix population models provide an ideal model framework for studying this interaction. In preliminary analyses, I have determined how critical patch sizes, travelling wave speeds, and pattern formation are modified by age structure. Hal Caswell and I are currently being funded by the National Science Foundation to extend these and other results to more general forms of population structure (e.g. size or stage structure). Our work is also being
A population typically consists of a variety of individuals. Imagine that we classify individuals by age using a simple system: juveniles and adults. Adults are characterized by the ability to reproduce. Also imagine that between each generation time there are probabilities governing the transition between each class. Juveniles survive to become adults with a certain probability; adults survive with a certain probability; and adults reproduce with a certain probability. These probabilities probably depend upon the local population density in a nonlinear way. (I.e. local popualtion density effects local mortality and reproductive rates.) Finally between each transition some dispersal occurs. The way in which an individual moves depends upon the transition that individual is making. Juveniles move differently than adults.
The combination of these processes can produce spatial patterns in population density. The image below shows the asymptotic state of a model based on the one just described. The color gives the local population density of the juvenile population. (The adult population looks qualitatively the same.) Initially, the population was randomly distributed. However, interaction of nonlinear population dynamics with differential dispersal among population classes eventually produces a spatial pattern. The pattern is more or less the same for any positive initial condition. I am currently developing a theory which predicts the size and shape of these patterns.
Click on the image below to see a movie of pattern formation in action.
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