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Browsing Pure Mathematics by Author "Apaloo, Joseph."

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    Evolutionary dynamics of coexisting species.
    (2000) Muir, Peter William.; Apaloo, Joseph.; Hearne, John W.
    Ever since Maynard-Smith and Price first introduced the concept of an evolutionary stable strategy (ESS) in 1973, there has been a growing amount of work in and around this field. Many new concepts have been introduced, quite often several times over, with different acronyms by different authors. This led to other authors trying to collect and collate the various terms (for example Lessard, 1990 & Eshel, 1996) in order to promote better understanding ofthe topic. It has been noticed that dynamic selection did not always lead to the establishment of an ESS. This led to the development ofthe concept ofa continuously stable strategy (CSS), and the claim that dynamic selection leads to the establishment of an ESSif it is a CSS. It has since been proved that this is not always the case, as a CSS may not be able to displace its near neighbours in pairwise ecological competitions. The concept of a neighbourhood invader strategy (NIS) was introduced, and when used in conjunction with the concept of an ESS, produced the evolutionary stable neighbourhood invader strategy (ESNIS) which is an unbeatable strategy. This work has tried to extend what has already been done in this field by investigating the dynamics of coexisting species, concentrating on systems whose dynamics are governed by Lotka-Volterra competition models. It is proved that an ESNIS coalition is an optimal strategy which will displace any size and composition of incumbent populations, and which will be immune to invasions by any other mutant populations, because the ESNIS coalition, when it exists, is unique. It has also been shown that an ESNIS coalition cannot exist in an ecologically stable state with any finite number of strategies in its neighbourhood. The equilibrium population when the ESNIS coalition is the only population present is globally stable in a n-dimensional system (for finite n), where the ESNIS coalition interacts with n - 2 other strategies in its neighbourhood. The dynamical behaviour of coexisting species was examined when the incumbent species interacted with various invading species. The different behaviour ofthe incumbent population when invaded by a coalition using either an ESNIS or an NIS phenotype underlines the difference in the various strategies. Similar simulations were intended for invaders who were using an ESS phenotype, but unfortunately the ESS coalition could not be found. If the invading coalition use NIS phenotypes then the outcome is not certain. Some, but not all of the incumbents might become extinct, and the degree to which the invaders flourish is very dependent on the nature ofthe incumbents. However, if the invading species form an ESNIS coalition, one is certain of the outcome. The invaders will eliminate the incumbents, and stabilise at their equilibrium populations. This will occur regardless of the composition and number of incumbent species, as the ESNIS coalition forms a globally stable equilibrium point when it is at its equilibrium populations, with no other species present. The only unknown fact about the outcome in this case is the number ofgenerations that will pass before the system reaches the globally stable equilibrium consisting ofjust the ESNIS. For systems whose dynamics are not given by Lotka-Volterra equations, the existence ofa unique, globally stable ESNIS coalition has not been proved. Moreover, simulations of a non Lotka-Volterra system designed to determine the applicability ofthe proof were inconclusive, due to the ESS coalition not having unique population sizes. Whether or not the proof presented in this work can be extended to non Lotka-Volterra systems remains to be determined.