Local Stability in a Seed Provisioning Model

The "'calculus" of inclusive fitness (Hamilton, 1964, t970, 1972) provides a powerful method, both conceptually and computationally, o f modeling social behavior. An important question concerns the extent to which inclusive fitness models are able to give the same results as genetic models. There is quite a general result, various versions of which have been discussed by Hamil ton (1970, 1979), Charlesworth (1980), Seger (1981), Grafen (1985), and Taylor (submitted), which provides a partial equivalence between inclusive fitness conditions and one-locus genetic models when selection is weak. The purpose of this paper is to illustrate this equivalence result with a seed provisioning model of Queller (1983, 1984), paying particular attention to the extent to which the inclusive fitness model is able to diagnose the local stability of the equilibrium. The model will also serve as a good example of the relationship between two different types of local stability. I begin with a simple description of an ESS model o f behavior, general enough to embrace both an inclusive fitness model, and a one-locus genetic model. I assume that the range of possible behaviors is described by a continuous parameter m which can be regarded as the probabili ty of engaging in a certain activity, or the proport ion of resources invested in one activity instead of another. I let m stand for "normal" behavior, and consider a rare "deviant" behavior m + 8. The objective of the model is to produce an expression W(m, 8) for the incremental fitness of the deviant behavior. I f 8 = 0, deviant behavior is the same as normal behavior, and W will be zero. A "weak-select ion" model is interested in the form of W for values of 8 close to zero, and looks at the Taylor series expansion of W about 8 = 0: