Class-structured Populations Research Articles

Direct fitness or inclusive fitness: how shall we model kin selection?

Two standard mathematical formulations of kin-selection models can be found. Inclusive fitness is an actor-centred approach, which calculates the fitness effect on a number of recipients of the behaviour of a single actor. Direct fitness is a recipient-centred approach, which calculates the fitness effect on the recipient of the behaviour of a number of actors. Inclusive fitness offers us a powerful heuristic, of choosing behaviour to maximize fitness, but direct fitness can be mathematically easier to work with and has recently emerged as the preferred approach of theoreticians. In this paper, we explore the fundamental connection between these two approaches in both homogeneous and class-structured populations, and we show that under simple assumptions (mainly fair meiosis and weak selection) they provide equivalent formulations, which correspond to the predictions of Price's equation for allele frequency change. We use a couple of examples to highlight differences in their conception and formulation, and we briefly discuss a two-species example in which we have a class of 'actor' that is never a 'recipient', which the standard direct fitness method can handle but the usual inclusive fitness cannot.

The economics of altruism and cooperation in class-structured populations: what's in a cost? What's in a benefit?

Lehmann & Keller (2006, and hereafter referred to as either ‘the authors’ or L&K) have identified the cost-tobenefit ratio, C/B as being a principal factor in the evolution of altruism and cooperation. The authors argue that this ratio (or some equivalent version) cannot exceed a given threshold if a selective advantage of altruism or cooperation is to exist. Of course, many biologists will readily recognize Hamilton’s (1964) rule as a special case of this result. An actor is willing to pay a cost, C, to give a benefit, B, to a recipient whenever

Dispersal, Kin Competition, and the Ideal Free Distribution in a Spatially Heterogeneous Population

In this paper, we reanalyze simple models of the evolution of dispersal in a heterogeneous landscape. Previous analyses concluded that without temporal variability, dispersal can evolve only if it is not costly and if it is conditional on the habitat. If both conditions hold, these models predict that selection on dispersal should lead to balanced dispersal between habitats (the number of immigrants equals the number of emigrants in each habitat). To evaluate the generality of these conclusions, we extended the analysis of these models to finite populations. This requires us to establish fitness measures for finite class-structured populations. These fitness measures allow us to take kin competition into account. Our analysis shows that even without temporal variability, conditional dispersal and the absence of a dispersal cost are not necessary conditions for dispersal to evolve. In the absence of a dispersal cost, we predict that selection on conditional dispersal will always lead to panmixia and not simply to balanced dispersal. When dispersal is costly, we show that the ideal free distribution (IFD) and balanced dispersal do not occur. Our results show that the deviations from IFD are of the order of the dispersal cost. We propose an approach to test our predictions.

Population dynamics of mangrove Avicennia bicolor on the Pacific coast of Costa Rica

Deterministic and stochastic class structured population models were used to simulate the life cycle of Avicennia bicolor of the Pacific coast of Costa Rica. The models were based on an extensive data set collected during a 6 year period in a 0.52 ha plot of monospecific A. bicolor. This data set included density, growth, mortality and transition rates of seedlings, saplings and trees of eight different diameter classes, as well as propagule production for the reproductive tree classes. Model simulations carried out over a 100 year period indicated a stable size class structure of the forest. Sensitivity analysis showed a significantly greater sensitivity of the model population to simulated changes in the mortality of seedlings, in comparison with the mortality of saplings and trees. An increase of 1% in the mortality of seedlings, for example, was sufficient to cause significant changes in the density of individual size classes. In contrast, neither a 10% increase in the mortality of saplings and trees nor a 20% decrease in the propagule production of fecund trees significantly affected the overall forest structure.

How to Make a Kin Selection Model

Kin selection arguments, based on Hamilton's (1964) concept of inclusive fitness, provide a powerful heuristic and can therefore give us valuable insights into the different pathways through which natural selection acts. Buttheir formulation can be quite tricky, requiring as they do, a close accounting of all the fitness effects of a particular item of behaviour. Here we propose a “direct fitness” formulation of inclusive fitness which often has a more straightforward derivation. Our method finds ESS trait values by the standard optimization techniques of simple differentiation plus two additional steps. First, slopes of group phenotype on individual genotype arise naturally during differentiation, and these slopes are replaced by coefficients by relatedness. Second, when behaviours influence different classes such as age, sex of recipient, or other life history components of fitness, the fitness effects on each component are weighted by reproductive value. We illustrate this technique first in a homogeneous population, with examples of group competition and partial dispersal behaviour, and then in a class-structured population, with examples of sex allocation and altruism between age classes.

Inclusive fitness arguments in genetic models of behaviour.

My purpose here is to provide a coherent account of inclusive fitness techniques, accessible to a mathematically literate graduate student in evolutionary biology, and to relate these to standard one-locus genetic models. I begin in Sect. 2 with a general formulation of evolutionary stability; in Sect. 3 and Sect. 4 I interpret the basic stability conditions within genetic and inclusive fitness models. In Sect. 5 I extend these concepts to the case of a class-structured population, and in Sect. 6 I illustrate these notions with a sex ratio example. In Sect. 7 I give a proof of the result that under additive gene action and weak selection, an inclusive fitness argument is able to verify an important stability condition (2.5) for one-locus genetic models. Most of these results have been published.

Allele-Frequency Change in a Class-Structured Population

Suppose that in a population with a class structure, we wish to calculate the change in frequency of an allele under the action of selection. If the allele affects the behavior of individuals in different classes differently and if parents of different classes have different class distributions of offspring, then the allele frequencies in different classes may be different, and any overall calculation of allele-frequency change must decide how to weight the different class frequencies. I show that the correct weights to use are the class reproductive values, and, using these, I formulate a covariance equation for allele-frequency change. This equation also provides the classical inclusive-fitness formulation if benefits (increased numbers of offspring) are calculated by weighting each offspring by the individual reproductive value that pertains to his class.