PubMed Health⌕ Search

Biomedical subjects

G T Vickers

Publications and source records attributed to G T Vickers.

13 recordsLinked to original sources
PubMed2003

The evolution of dispersal.

A non-local model for dispersal with continuous time and space is carefully justified and discussed. The necessary mathematical background is developed and we point out some interesting and challenging problems. While the basic model is not new, a 'spread' parameter (effectively the width of the dispersal kernel) has been introduced along with a conventional rate paramter, and we compare their competitive advantages and disadvantages in a spatially heterogeneous environment. We show that, as in the case of reaction-diffusion models, for fixed spread slower rates of diffusion are always optimal. However, fixing the dispersal rate and varying the spread while assuming a constant cost of dispersal leads to more complicated results. For example, in a fairly general setting given two phenotypes with different, but small spread, the smaller spread is selected while in the case of large spread the larger spread is selected.

Animals↗
PubMed2000

A sequential-arrivals model of territory acquisition II.

Birds arrive sequentially at their breeding ground where the nest sites vary in value (measured by reproductive success). Each bird may choose a vacant site or challenge an occupier for its site. In the latter case, the occupier is presumed to be the more-likely winner; the loser incurs a cost and must go to a vacant site. In a previous paper (Broom et al., 1997, J. theor. Biol.189, 257-272), we considered the optimal strategy. However, that optimal strategy was complex and perhaps could not be realized in real bird populations, making possibly costly demands both perceptually and at the coding level. With this in mind we introduce certain restricted classes of strategy, and consider how populations might evolve. Computer simulations of various populations have been performed to model the competition amongst several strategies in the presence of recurrent mutations. Certain combinations of strategies persisted and corresponded approximately to the ESSs found in Broom et al. (1997).

Animals↗
PubMed2000

Evolution in knockout conflicts: the fixed strategy case.

A group of individuals resolve their disputes by a knockout tournament. In each round of the tournament, the remaining contestants form pairs which compete, the winners progressing to the next round and the losers being eliminated. The payoff received depends upon how far the player has progressed and a cost is incurred only when it is defeated. We only consider strategies in which individuals are constrained to adopt a fixed play throughout the successive rounds. The case where individuals can vary their choice of behaviour from round to round will be treated elsewhere. The complexity of the system is investigated and illustrated both by special cases and numerical examples.

Agonistic Behavior↗
PubMed1999

Sperm competition I: basic model, ESS and dynamics.

We examine models of sperm competition to determine which strategies are evolutionarily stable according to game theory. Games are considered in which the males of a species must divide a fixed amount of sperm between a fixed number of rounds in competition over fertilization of a given set of eggs. Sperm success with a single female is allocated using the raffle principle". A two round model is formulated and we show that the evolutionarily stable strategy (ESS) is a pure strategy in which a male should use at least half his sperm in the first round if given the opportunity to mate. The ESS is unique and globally stable, in contrast to most classical ESSs which are only locally stable. The model is extended to include the effects of sperm replenishment and egg oviposition between rounds. Both serve to increase the ESS amount of sperm inseminated in round one.

Animals↗
PubMed1997

Spatial and density effects in evolutionary game theory.

Two models are considered for the study of game dynamics in a spatial domain. Both models are continuous in space and time and give rise to reaction-diffusion equations. The spatial domain is homogeneous but the mobility of the individuals is allowed to depend upon the strategy. The models are analysed for spatial patterns (via a Turing instability) and also for the direction of the travelling wave that replaces one strategy by another. It is shown that the qualitative behaviour of the two models is quite different. When considering the existence of spatial patterns and deciding whether increased mobility is helpful or not, it is shown that the answers depend crucially upon the model equations. Since both models (in the absence of spatial variation) are quite standard, it is clear that considerable care has to be exercised in the formulation of spatial models and in their interpretation.

Animals↗
PubMed1996

ESS patterns: adding pairs to an ESS.

The notion of a pattern of evolutionarily stable strategies was introduced by Cannings and Vickers in 1988 (J. Theor. Biol. 132:387-420). In this paper a specific class of patterns is considered. Suppose that there is an evolutionarily stable strategy (ESS) on some set of n strategies {1,2,...,n} and that new strategies {n + 1,n + 2,...,n + k} are added. Supposing that for this new enlarged conflict there is still an ESS on {1,2,...,n} and also that there are ESSs on {n + i,j} for 1 < or = i < or = k and j epsilon Si [symbol: see text] {1,2,...,n}, the authors investigate the restrictions on the Si. These restrictions are related to certain properties of strong tournaments introduced by Reid and Beineke. We also specify, given the Si, what ESSs of the form {n + i,n + j} can be added.

Animals↗
PubMed1995

The spatial struggle of tit-for-tat and defect.

The pioneering work by Trivers (1971), Axelrod (1984) and Axelrod & Hamilton (1981) has stimulated continuing interest in explaining the evolution of cooperation by game theory, in particular, the iterated prisoner's dilemma and the strategy of tit-for-tat. However these models suffer from a lack of biological reality, most seriously because it is assumed that players meet opponents at random from the population and, unless the population is very small, this excludes the repeated encounters necessary for tit-for-tat to prosper. To meet some of the objections, we consider a model with two types of players, defectors (D) and tit-for-tat players (T), in a spatially homogeneous environment with player densities varying continuously in space and time. Players only encounter neighbours but move at random in space. The analysis demonstrates major new conclusions, the three most important being as follows. First, stable coexistence with constant densities of both players is possible. Second, stable coexistence in a pattern (a spatially inhomogeneous stationary state) may be possible when it is impossible for constant distributions (even unstable ones) to exist. Third, invasion by a very small number of T-players is sometimes possible (in contrast with the usual predictions) and so a mutation to tit-for-tat may lead to a population of defectors being displaced by the T-players.

Animals↗
PubMed1993

Routes to polymorphism.

It is well known that for a single, autosomal locus with differential viabilities convergence to a stable equilibrium is assured. However if new alleles are introduced sequentially by mutation the final equilibrium may depend on the particular sequence. Thus the equilibrium to which the population moves will be randomly determined. When there is an internal, stable polymorphism there is no such randomness in the final equilibrium as there is only the one stable point. However the time taken to reach that equilibrium will vary with the order of occurrence of mutations, and with the specific set of viabilities. The object here is to discuss the possible routes by which an internal polymorphism can be reached, and to begin the elucidation of which sets of alleles can be stable in their own space.

Animals↗
PubMed1991

Spatial patterns and travelling waves in population genetics.

We consider a reaction-diffusion equation to model a multi-allelic, single locus problem. The population can migrate in a homogeneous region and the diffusion rates depend upon the genotype. It is shown that if there is an equilibrium point with all alleles present and if this polymorphism is stable for the classical reaction system then it is also stable for the reaction-diffusion equation. Also a simplified model is used to investigate which allele will spread in the two-allele case. Alleles which are associated with large fitness and small dispersion do best.

Alleles↗
PubMed1989

Spatial patterns and ESS's.

The classical idea of an evolutionarily stable strategy (ESS) does not involve any spatial dependence. An evolution equation for analyzing games in a region is suggested and the possibility of spatial patterns is investigated. It is shown that an ESS is so stable that it forbids any spatial dependence but that other equilibria may have patterns associated with them if the dispersion rates are suitably chosen.

Biological Evolution↗
PubMed1988

Patterns of ESS's. I.

A matrix may have several evolutionarily stable strategies (ESS's). It is thus possible for different populations of a species to adopt a different ESS even when the pay-offs for the populations are the same. The occurrence of different strategies does not imply different circumstances. However, there are constraints upon the collection of supports of the ESS's (i.e. pattern) that any matrix can have. The best-known of these is that the support of one ESS cannot be contained in that of another and this gives bounds on the number of different patterns possible for n x n matrices. Other general constraints are presented here. The enumeration of the patterns for 3 x 3 and 4 x 4 matrices is completed and considerable progress made on 5 x 5 matrices where the number of (permutationally distinct, maximal) patterns exceeds 16.

Biological Evolution↗
PubMed1988

Patterns of ESS's. II.

For symmetric matrix conflicts with aij = +/- 1 and aii = 0, an ESS corresponds to a clique in an associated graph. This result is proved and exploited to yield results on the attainable patterns for this class of conflicts and bounds for the number of ESS's which may coexist. Randomly generated matrices in this class are considered, and some results on the size of the support of a typical ESS given.

Biological Evolution↗