The study of age structure in population models has had a long history, as this list of authors shows.
More recently, the use of age and stage structured models has been emphasized in applied contexts as well, including conservation areas, and in fisheries. In this lecture, we will only have time to give the background and not to cover some of the more specialized topics. For a nice discussion of some of the issues underlying choices about the inclusion of age or stage structure, see de Roos and Persson (2005).
Our goal in this lecture is quite limited. We will cover some of the basic classic modeling approaches, and get some of the basic results. More modern developments will be covered in detail later. The basic results will look at questions such as whether a population will increase, and what will the rate of increase be. Also, one might want to know about the dynamics of different age classes. Finally, what would the influence of density dependence be?
There are two basic ways to think about age structure. One can focus on births through time (the renewal approach) or on individuals of all different ages at a given time. One can also focus on discrete time or continuous time. In this lecture, I will primarily focus on the continuous time formulations, partly based on the assumption that you may be more familiar with discrete time approaches. Additionally, recent work focussing on causes of cycling in populations has tended to focus on either the continuous time model or extensions of it.
We will first consider the renewal approach, and develop Lotka's equation which is the basic equation describing this approach.
Here we will follow McKendrick's initial derivation, and then consdier extensions. In this case the derivation of the model gives hints as to its solution.
Here we will continue to focus on the the continuous time model, and emphasize the concept of reproductive value which is really related to the concept of discount rate from economics.
In this case, the best results are typically for discrete time models.
This list is very selective - the book by Caswell does a superb job of covering the literature for discrete time models.
* means required reading
Caswell, H. 2001. Matrix Population Models. Second Edition. Sinauer Associates, Sunderland, MA.
de Roos, A.M. and Persson, L.A. 2005. Unstructured population models: do population-level assumptions yield general theory? pp 31-62 in Cuddington, K. and Besiner, B. eds Ecological Paradigms Lost. ELsevier.
Higgins, K., Hastings, A., Botsford, L. 1997. Density dependence and age structure: nonlinear dynamics and population behavior. Amer. Nat. 149:247-269
Kooijman, S.A.L.M. 2000. Dynamic Energy and Mass Budgets in Biological Systems. Cambridge University Press
*Leslie, P.H. 1945. On the use of matrices in certain population mathematics.
Metz, J.A.J. and Diekmannm O. eds. 1986. The Dynamics of Physiologically Structured Populations. Springer-Verlag, Berlin.
*Sharpe, F.R. and Lotka, A.J. 1911. A problem in age distribution. Philosophical Magazine. 21:435-438.