We have discussed so far the diffusion equation and its application on relatively small spatial and temporal scales. What we need to do next is to understand how robust the model is to the assumptions that are made, and what are some of the issues that arise.
We will discuss the application of the model to a variety of different cases, and examine the robgustness of the model to changes in the assumptions -- particularly assumptions about the age structure of the system under consideration, and about the uniformity of the population. This examination will naturally lead into a discussion of assumptions that will segue into the next lecture on integrodifference equations.
Last time we talked about some modifications of the basic model for spatial spread, but kept within the framework of the diffusion equation approach. Here, we ask ourselves what happens if we change our basic model and make it discrete in time -- can we get different results? We look at the spatial spread question first, picking up from the Shigesada et al. example of last time. We then look at the implications of this approach for other questions.
*Persistence
of Transients in Spatially Structured Ecological Models (in Reports) Alan
Hastings, Kevin Higgins Science, New Series, Vol. 263, No. 5150. (Feb.
25, 1994), pp. 1133-1136.
VanKirk, RW; Lewis, MA. Integrodifference models
for persistence in fragmented habitats.
BULLETIN OF MATHEMATICAL BIOLOGY, JAN, 1997, V59(N1):107-137.
* means required reading