Adaptive dynamics for an age-structured population model with a Shepherd recruitment function

Abstract

English: In this study the evolution of the genetic composition of certain species will be replaced by the evolution of the traits that represent these genetic compositions. Depending on the nature of the trait of interest, a scalar valued parameter called the strategy parameter will be assigned to this trait making the simulation of strategy evolution possible. The trait of interest, and therefore the strategy associated, will be the ability of a population to keep its densities within the carrying capacity of the environment they find themselves in. The Shepherd function, on account of its wide use in population simulations as well as composing of exactly such a density parameter, will be the density curbing mechanism of choice in the age-structured population model designed here. An algorithm will be designed to simulate strategy evolution towards an evolutionary stable strategy or ESS that will ensure not only an optimal fit for this environment but also render the population immune against future invasion by other members of the population practising slight variations of this strategy. There are two ways to come by such an optimal strategy without directly involving genetics. The first is game theory, allowing strategists to compete for this position, and the second is with the use of adaptive dynamics, converting winning and loosing instead into tangible mathematics. Combining these two classics will show that the quest is an exercise in strategy optimization, not only from the point of view of an already established population but also from the point of view of an initially small one. It will be interesting!

Afrikaans: In hierdie studie word die evolusie van die genetiese samestellings verantwoordelik vir sekere karakteristieke van ’n spesies vervang deur die strategie wat hierdie karakteristieke verteenwoordig. As die strategie hom daartoe leen, kan dit as ’n skalaar verteenwoordig word wat die simulasie van strategie evolusie moontlik maak. In hierdie studie is die strategie van belang die vermo¨e van die spesie om sy digtheid te reguleer binne die draagkrag van sy omgewing. As gevolg van die wye toepassing van die Shepherd funksie in populasie simulering en die teenwoordigheid van ’n strategie parameter, is dit die funksie van keuse in hierdie studie se ouderdoms gestruktureerde populasie model. ’n Algoritme word spesiaal vir die optimerings proses ontwerp wat die optimale strategie, ook genoem die evolusionere stabiele strategie of ESS, bereken deur die evolusie proses van die strategie te simuleer. Die implimentering van die ESS sal die populasie die voordeel bo ander gee wat ’n klein variasie van hierdie strategie beoefen. Beide spelteorie, waar populasies teen mekaar meding, en aanpassings-dinamika, wat wen en verloor wiskundig verteenwoordig, kan begruik word om die ESS te bepaal. Die kombinasie van die twee metodes wys dat die optimerings proses beide vanaf die oogpunt van ’n hoë digtheid asook ’n lae digtheid populasie kom.