ESS modelling of diploid populations II: stability analysis of possible equilibria

1994 ◽  
Vol 26 (2) ◽  
pp. 361-376 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Stability Analysis ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Stable Strategy ◽  
Local Attractors ◽  
The Mean ◽  
Arbitrary Frequency ◽  
Probability Simplex ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus.The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.

ESS modelling of diploid populations II: stability analysis of possible equilibria

1994 ◽  
Vol 26 (02) ◽  
pp. 361-376
Author(s):  
W. G. S. Hines
Keyword(s):  
Stability Analysis ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Stable Strategy ◽  
Local Attractors ◽  
The Mean ◽  
Arbitrary Frequency ◽  
Probability Simplex ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus. The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.

ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices

1994 ◽  
Vol 26 (02) ◽  
pp. 341-360 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Genetic Variability ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Allelic Frequency ◽  
Stable Strategy ◽  
The Mean ◽  
Probability Simplex ◽  
Weaker Conditions ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, relevant features of the probability simplex of allelic frequencies for a population with genetic variability at a single locus are explored. Singularities and related properties of mappings from the space of allele frequencies to the space of strategy frequencies are examined, and related to a certain covariance measure of variability present in the population.A companion paper builds on this characterization to establish that previous claims of stability in fact hold under slightly weaker conditions than initially indicated. The pair of papers also determines conditions under which unstable equilibria can occur, and establishes that these conditions are exceptional in practice.

ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices

1994 ◽  
Vol 26 (2) ◽  
pp. 341-360 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Genetic Variability ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Allelic Frequency ◽  
Stable Strategy ◽  
The Mean ◽  
Probability Simplex ◽  
Weaker Conditions ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, relevant features of the probability simplex of allelic frequencies for a population with genetic variability at a single locus are explored. Singularities and related properties of mappings from the space of allele frequencies to the space of strategy frequencies are examined, and related to a certain covariance measure of variability present in the population.A companion paper builds on this characterization to establish that previous claims of stability in fact hold under slightly weaker conditions than initially indicated. The pair of papers also determines conditions under which unstable equilibria can occur, and establishes that these conditions are exceptional in practice.

On the local stability of an evolutionarily stable strategy in a diploid population

1984 ◽  
Vol 21 (02) ◽  
pp. 215-224 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Local Stability ◽  
Evolutionarily Stable Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Diploid Population ◽  
Haploid Population ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.

Evolutionarily stable strategies in diploid populations with general inheritance patterns

1983 ◽  
Vol 20 (2) ◽  
pp. 395-399 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Sexual Selection ◽  
Evolutionarily Stable Strategy ◽  
Evolutionarily Stable Strategies ◽  
Stable Strategy ◽  
Simple Argument ◽  
Sexual Population ◽  
Inheritance Patterns ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (ESS) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided that the ESS is both attainable and maintainable.

On the local stability of an evolutionarily stable strategy in a diploid population

1984 ◽  
Vol 21 (2) ◽  
pp. 215-224 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Local Stability ◽  
Evolutionarily Stable Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Diploid Population ◽  
Haploid Population ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.

Evolutionarily stable strategies in diploid populations with general inheritance patterns

1983 ◽  
Vol 20 (02) ◽  
pp. 395-399 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Sexual Selection ◽  
Evolutionarily Stable Strategy ◽  
Evolutionarily Stable Strategies ◽  
Stable Strategy ◽  
Simple Argument ◽  
Sexual Population ◽  
Inheritance Patterns ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (ESS) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided that the ESS is both attainable and maintainable.

Strategy stability in complex randomly mating diploid populations

1982 ◽  
Vol 19 (03) ◽  
pp. 653-659 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Convex Hull ◽  
Lyapunov Functions ◽  
Evolutionarily Stable Strategy ◽  
Stable Strategy ◽  
Convergence In Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations ◽  
Better Than

A class of Lyapunov functions is used to demonstrate that strategy stability occurs in complex randomly mating diploid populations. Strategies close to the evolutionarily stable strategy tend to fare better than more remote strategies. If convergence in mean strategy to an evolutionarily stable strategy is not possible, evolution will continue until all strategies in use lie on a unique face of the convex hull of available strategies. The results obtained are also relevant to the haploid parthenogenetic case.

Strategy stability in complex randomly mating diploid populations

1982 ◽  
Vol 19 (3) ◽  
pp. 653-659 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Convex Hull ◽  
Lyapunov Functions ◽  
Evolutionarily Stable Strategy ◽  
Stable Strategy ◽  
Convergence In Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations ◽  
Better Than

A class of Lyapunov functions is used to demonstrate that strategy stability occurs in complex randomly mating diploid populations. Strategies close to the evolutionarily stable strategy tend to fare better than more remote strategies. If convergence in mean strategy to an evolutionarily stable strategy is not possible, evolution will continue until all strategies in use lie on a unique face of the convex hull of available strategies.The results obtained are also relevant to the haploid parthenogenetic case.

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