A Simple Fitness Proxy for Structured Populations with Continuous Traits, with Case Studies on the Evolution of Haplodiploids and Genetic Dimorphisms

Metz, J.A.J. & Leimar, O. (2011). A Simple Fitness Proxy for Structured Populations with Continuous Traits, with Case Studies on the Evolution of Haplodiploids and Genetic Dimorphisms. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-11-014

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Abstract

For structured populations in equilibrium with everybody born equal ln(R_0) is a useful fitness proxy for ESS and most adaptive dynamics calculations, with R_0 the average lifetime number of offspring in the clonal and haploid cases, and half the average lifetime number of offspring fathered or mothered for Mendelian diploids. When individuals have variable birth states, as is e.g. the case in spatial models, R_0 is itself an eigenvalue, which usually cannot be expressed explicitly in the trait vectors under consideration. In that case Q(Y|X) := -det(I-L(Y|X)) can often be used as fitness proxy, with L the next-generation matrix for a potential mutant characterised by the trait vector Y in the (constant) environment engendered by a resident characterised by X. If the trait space is connected, global univadability can be determined from it. Moreover it can be used in all the usual local calculations like the determination of evolutionarily singular trait vectors and their local invadability and attractivity. We conclude with three extended case studies demonstrating the usefulness of Q: the calculation of ESSes under haplo-diploid genetics (I), of Evolutionarily Steady genetic Dimorphisms with a priori proportionality of macro- and micro-gametic outputs (an assumption that is generally made but the fulfillment of which is a priori highly exceptional) (II), and of ESDs without such proportionality (III). These case studies should also have some interest in their own right for the spelled out calculation recipes and their underlying modelling methodology.

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