Adaptive Dynamics: Some Basic Theory and an Application

The theory of structured populations is a mathematical framework for developing and analyzing ecological models that can take account of relatively realistic detail at the level of individual organisms. This framework in turn has given rise to the theory of adaptive dynamics, a versatile framework for dealing with the evolution of the adaptable traits of individuals through repeated mutant substitutions directed by ecologically driven selection. The step from the former to the latter theory is possible thanks to effective procedures for calculating the expected rate of invasion of mutants with altered trait values into a community the dynamics of which has relaxed to an attractor. The mathematical underpinning is through a sequence of limit theorems starting from individual-based stochastic processes and culminating in (i) a differential equation for long-term trait evolution and (ii) various geometrical tools for classifying the evolutionary singular points such as Evolutionarily Steady Strategies, where evolution gets trapped, and branching points, where an initially quasi-monomorphic population starts to diversify. Traits that have been studied using adaptive dynamics tools are, among others, the virulence of infectious diseases and various other sorts of life-history parameters such as age at maturation. As one example, adaptive dynamics models of respiratory diseases tell that such diseases will evolve towards the upper air passages and hence towards lesser virulence, while at the same time diversifying as a result of limited cross-immunity. Since the upper airways offer the largest scope for disease persistence, they also allow for the largest disease diversification. Moreover, the upward evolution brings with it a tendency for vacating the lower reaches, which leads to the prediction that emerging respiratory diseases will tend to act low and therefore be both unusually virulent and not overly infective

Eight Personal Rules for Doing Science

Adaptive dynamics (AD) is not a scientific theory, but a mathematical framework for dealing with eco-evolutionary problems, based on a varied set of simplifying assumptions as a means of approaching problems of otherwise greater complexity. As such it may be compared with e.g. the theory of stochastic processes, or of differential equations. AD can make predictions only in a similar way to these theories: it lays bare consistent patterns in mathematical structures, some of which hopefully connect to the real world. Predictions largely come from specific models. AD studies the tools for analysing such models. Like in the theory of differential equations or bifurcation theory, a number of these tools already existed before the abstract theory took off. AD creates order on an abstract level, which in turn helps in constructing new tools. As far as the use of the newer tools is concerned, AD can be said to have contributed to predictions. Another class of predictions from AD arise from arguments on the frequency with which one may expect different situations to occur

Adaptive dynamics

Adaptive dynamics (AD) is a mathematical framework for dealing with eco-evolutionary problems, primarily based on the following simplifying assumptions: clonal reproduction, rare mutations, small mutational effects, smoothness of the demographic parameters in the traits, and well-behaved community attractors. However, often the results from AD models turn out to apply also under far less restrictive conditions. The main AD tools are its so-called canonical equation (CE) that captures how the trait value(s) currently present in the population should develop over evolutionary time, and graphical techniques for analyzing evolutionary progress for one-dimensional trait spaces like .pairwise invasibility plots. (PIPs) and .trait evolution plots. (TEPs). The equilibria of the CE, customarily referred to as evolutionarily singular strategies, or ess-es, comprise in addition to the evolutionary equilibria, or ESSes, also points in trait space where the population comes under a selective pressure to diversify. Such points mathematically capture the ecological conditions conducive to adaptive (Darwinian) speciation

Fitness

The fitness concept of evolutionary ecology differs from that of population genetics. The former is geared towards dealing with long term evolution through the repeated invasion of mutants for potentially complicated ecological scenarios, the latter with short term changes in relative frequencies of types for heavily simplified ecological scenarios. After a discussion of the conditions allowing for the definition of a general invasion fitness concept, among which that reproduction should be clonal, a framework is built within which the definition can be formalized. Recipes are given for calculating (proxies for) fitness in a large variety of instances. The main use of invasion fitness is in ESS calculations. Only under ecologically very special conditions ESSes can be calculated from optimization principles. These conditions are detailed, as well as the, even more special, conditions under which evolution maximizes r or Ro. The invasion fitness concept extends to any aggregates treatable as meta-individuals. Individual- and meta-individual-level invasion fitness coincide when the latter is larger than per capita within aggregate growth. Calculating invasion fitness through a meta-individual route often works beyond calculations based on inclusive fitness arguments, but provides less insight. Mendelian diploids are aggregates of clonally reproducing genes. Conditions are given for when predictions for virtual cloning diploids coincide with those from gene-based calculations

On the concept of individual in ecology and evolution

Part of the art of theory building is to construct effective basic concepts, with a large reach and yet powerful as tools for getting at conclusions. The most basic concept of population biology is that of individual. An appropriately reengineered form of this concept has become the basis for the theories of structured populations and adaptive dynamics. By appropriately delimiting individuals, followed by defining their states as well as their environment, it become possible to construct the general population equations that were introduced and studied by Odo Diekmann and his collaborators. In this essay I argue for taking the properties that led to these successes as the defining characteristics of the concept of individual, delegating the properties classically invoked by philosophers to the secondary role of possible empirical indicators for the presence of those characteristics. The essay starts with putting in place as rule for effective concept engineering that one should go for relations that can be used as basis for deductive structure building rather than for perceived ontological essence. By analysing how we want to use it in the mathematical arguments I then build up a concept of individual, first for use in population dynamical considerations and then for use in evolutionary ones. These two concepts do not coincide, and neither do they on all occasions agree with common intuition-based usage

Perspectives for Virulence Management: Relating Theory to Experiment

This paper reviews our current knowledge about the evolution of virulence in pathogen-host systems, with an emphasis on the interface between the theoretical and experimental literature. After giving a methodically oriented overview of the field, stressing restrictions and caveats, the paper attempts to summarize the main results on virulence evolution gleaned from the literature. From that perspective the authors identify what they see as gaps in our current knowledge that need to be filled to transform the study of virulence evolution and management into a mature science

On Diploid versus Clonal ESSes in Metapopulations

Most studies of evolutionary stable strategies (ESSes) assume clonal reproduction. At least in the simplest cases, more realistic genetic models yield results compatible with the clonal results. In this paper we study a case where the diploid and clonal results are not expected to be similar: evolution in a metapopulation with small local population sizes. It turns out, that although there are differences between the clonal and diploid ESS dispersal rates, the trait under consideration, the discrepancy is irrelevant for all practical purposes (less than 2%)

The adaptive dynamics of life histories: From fitness-returns to selection gradients and Pontryagin's maximum principle

Using a fitness-returns argument we derive an expression for the selection gradient for the age dependent allocation strategy in a common class of state variable based life-history models. By setting the selection gradient equal to zero as part of the calculation of the ESS-es for such models, we get a marginal value argument and through this recover the local version of Pontryagin's maximum principle. This fills in a minor gap in a recent paper by Parvinen, Heino and Dieckmann (2012; DOI10.1007/s00285-012-0549-2), who treated the calculation of the selection gradient and of the ESS-es as separate issues. As bonuses we (i) provide an extension of the framework of these authors that can handle also the more complicated evolutionary dynamics of the life histories that we consider, and (ii) derive also the full Pontryagin's maximum principle from a fitness-returns argument