Matrix models of population dynamics:
straightforward applications and inspiring mathematics
Dmitrii O. Logofet
Laboratory of Mathematical Ecology, IAP RAS, and
M.V. Lomonosov Moscow State University, Russia
danilal@postman.ru
Hal Caswell
Biology Department, Woods Hole Oceanographic Institution,
Woods Hole MA 02543, USA
hcaswell@whoi.edu http://www.whoi.edu/WHOI/SciTechDir/hal_caswell.html
The concepts of age or size classes and of life-cycle stages are among the simplest and most natural in population biology. Mathematical object such as matrices, in contrast, have been until recently considered esoteric outside mathematics, appalling even by the fact A´B¹B´A, unlike with usual numbers. But matrices formalize the very natural ideas that individuals at different stages are biologically different, and that the age (respectively size or stage) structure of a population causes the population dynamics, thus giving rise to the population projection matrix – perhaps, the most powerful tool of population modelling. The limitation of the Leslie matrix formalism, published more than 50 years ago, to age classification discouraged straightforward applications until the mathematics made the next step of principal importance: introducing additional non-zero entries into the matrix, expanding the strict age class to a less rigid stage group and thus expanding the horizons of application. Those expanding horizons continue to pose challenging mathematical problems in branches such as (but not limited to) matrix theory, graph theory, stochastic chains, bifurcations, and computational aspects. Advances in modern ITs (information technologies) induce progress also in model testing and visualization techniques. A special session at the Alcala-2 Conference on Mathematical Ecology, focusing on the above aspects of matrix population models, would encourage a better interface between the “straightforward applications” and the relevant mathematics, and it would, perhaps, inspire younger mathematical ecologists to learn and use this powerful tool of ecological modelling.