On the relationship between continuous and discrete models for size-structured population dynamics.
Committee ChairCushing, Jim M.
MetadataShow full item record
PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractWe address the problem of the consistency between discrete and continuous models for density-dependent size-structured populations. Some earlier works have discussed the consistency of density independent age and size-structured models. Although the issue of consistency between these models has raised interest in recent years, it has not been discussed in depth, perhaps because of the non-linear nature of the equations involved. We construct a numerical scheme of the continuous model and show that the transition matrix of this scheme has the form of the standard discrete model. The construction is based on the theory of Upwind Numerical Schemes for non-Linear Hyperbolic Conservation Laws with one important difference, that we do have a non-linear source at the boundary; interestingly, this case has not been explored in depth from the purely mathematical point of view. We prove the consistency, non-linear stability and hence convergence of the numerical scheme which guarantee that both models yield results that are completely consistent with each other. Several examples are worked out: a simple linear age-structured problem, a density-independent size-structured problem and a non-linear size-structured problem. These examples confirm the convergence just proven theoretically. An ample revision of relevant biological and computational literature is also presented and used to establish realistic restrictions on the objects under consideration and to prepare significant examples to illustrate our points.
Degree ProgramApplied Mathematics