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A beginner’s guide to adaptive dynamics. (English) Zbl 1051.92032
Rudnicki, Ryszard (ed.), Mathematical modelling of population dynamics. Collection of papers from the conference, Bȩdlewo, Poland, June 24–28, 2002. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 63, 47-86 (2004).
Summary: The aim of these notes is to illustrate, largely by way of examples, how standard ecological models can be put into an evolutionary perspective in order to gain insight in the role of natural selection in shaping life history characteristics. We limit ourselves to phenotypic evolution under clonal reproduction (that is, we simply ignore the importance of genes and sex). Another basic assumption is that mutation occurs on a time scale which is long relative to the time scale of convergence to an ecological attractor.
We begin by illustrating the idea of interaction via environmental variables through the example of competition for substrates in the chemostat. In this context we explain the trait/strategy substitution sequence, capturing how successful invaders/mutants outcompete the resident and then become the new resident. We also introduce the PIP, the pairwise invasibility plot, as a convenient graphical tool to study the adaptive dynamics of a one-dimensional trait. We high-light the pessimization principle: if the environmental condition is one-dimensional, mutation and natural selection inevitably lead to deterioration/Verelendung. We illuminate the Tragedy of the Commons as well as evolutionary suicide and, while we’re about it, adaptive dynamics as an added feature to a bifurcation diagram (with, possibly, AD-induced branch switching). Then we rewrite the invasion exponent as a function of resident and invader trait, define the selection gradient and turn to the core of the theory: the classification of singular points (where the selection gradient vanishes) in terms of ESS, CSS, mutual invasibility, converging and diverging dimorphisms and branching points. An extensive collection of allied examples focuses on the timing of reproduction of semelparous organisms.
We show how to analyse steady states of structured population models, how fitness measures relate to the dimension of the environmental conditions and to the specific form of density dependence, and we establish the central role of the ideal free distributions (i.e., the principle of indifference). We also describe the “resident strikes back” phenomenon. In the final section we very concisely sketch the wider perspective, alternative theories and the agenda of AD. An almost identical earlier version of these notes appeared in: A. Margheri, C. Rebelo, F. Zanolin (eds.), Summer School on Mathematical Biology, CIM, Lisboa, Portugal, 2002.
For the entire collection see [Zbl 1051.92034].

##### MSC:
 92D15 Problems related to evolution 92D40 Ecology 37N25 Dynamical systems in biology