Chris Cannings

Probability functions on complex pedigrees

The calculation of probabilities on pedigrees of arbitrary complexity is discussed for a basic model of transmission and penetrance (encompassing Mendelian inheritance, and certain environmental influences). The structure of pedigrees, and the types of loops occurring, is discussed. Some results in graph theory are obtained and, using these, a recurrence relation derived for certain probabilities. The recursive procedure enables the successive peeling off of certain members of the pedigree, and the condensation of the information on those individuals into a function on a subset of those remaining. The underlying theory is set out, and examples given of the utilization of the resulting algorithm. PEDIGREE; PROBABILITY; LOOPS; PEELING; GRAPH; OUSIOTYPE; GENETICS

A generalized war of attrition

Abstract In the “War of Attrition” model of animal conflict, introduced by Maynard Smith, a reward is obtained by whichever of two opponents displays longer, each individual incurring a cost associated with the length of the contest. This model is generalized to allow more general reward and cost functions, and restrictions on the length of contest permitted. This permits unification of the “War of Attrition” model and the “Graduated Risks” model, and also the extension to models in which contests may end either due to injury, or to retreat. In each case it is demonstrated that either (i) there is no evolutionary stable strategy (ESS) or (ii) there is a unique ESS, which is fully specified. In the case where only a finite number of pure strategies are available, global convergence to the ESS is shown. A variety of interesting conclusions of biological relevance emerge, perhaps the most striking being the occurrence of a dichotomous behavioural pattern in an essentially continuous conflict.

The war of attrition with random rewards.

The War of Attrition model of animal conflict was introduced by Maynard Smith (1974), who derived the unique Evolutionarily Stable Strategy (ESS). A generalisation of this model, incorporating the Graduated Risk model (Maynard Smith & Parker, 1976) and a variety of new models, was analysed by Bishop & Cannings (1978), who also found a unique ESS. In essence, the War of Attrition supposes that two animals are competing for an indivisible prize of value V, and that their only choice of strategy is of the length of time for which they will continue, and hence of the price they are prepared to pay. The ESS is to continue for a variable length of time t, with a probability density

Multi-player matrix games

Game theory has had remarkable success as a framework for the discussion of animal behaviour and evolution. It suggested new interpretations and prompted new observational studies. Most of this work has been done with 2-player games. That is the individuals of a population compete in pairwise interactions. While this is often the case in nature, it is not exclusively so. Here we introduce a class of models for situations in which more than two (possibly very many) individuals compete simultaneously. It is shown that the solutions (i.e. the behaviour which may be expected to be observable for long periods) are more complex than for 2-player games. The concluding section lists some of the new phenomena which can occur.

The coevolution of predator—prey interactions : ESSS and Red Queen dynamics

A model for the coevolution of body size of predators and their prey is described. Body sizes are assumed to affect the interactions between individuals, and the Lotka-Volterra population dynamics arising from these interactions provide the driving force for evolutionary change. The space of phenotypes of predator and prey contains a region, oval in shape, in which the predator and prey species coexist. Within this region, evolutionarily stable strategies (ESSS) and evolutionary saddles may be found, and coevolution may tend to an ESS, develop a Red Queen dynamic, or move to predator extinction. Ten qualitatively distinct kinds of phenotype space are described, depending mainly on the number of ESSS and evolutionary saddles. These varied outcomes are in part due to the range of ways in which density-dependent selection within the prey interacts with density- and frequency-dependent selection on the prey due to the predator. The results point to a ‘loser wins’ principle, in which the evolution leads to a weakening of the interaction between predator and prey. The results also illustrate the deterioration of the environment associated with each evolutionary step of the species and the lack of a net improvement in their mean fitness.

On the definition of an evolutionarily stable strategy

An ESS must be able to withstand invasion by a small group. It is shown that there are (at least) two possible mathematical interpretations of this statement. In some important applications the two definitions of an ESS are equivalent, but this is not generally the case and a simple example is given to illustrate this. In the weaker form, not favoured here, an ESS may not withstand some infinitely small invasions, but in the stronger form the ESS is certain to survive all invasions up to a fixed fraction of the population. Also, when a pay-off matrix is used to define a dynamic, the stronger definition is the more convenient.

The n-Person War of Attrition

The War of Attrition (WA) was one of the earliest examples studied in the use of the theory of games to understand animal behavior (see Maynard Smith (1974)). The setup is that two contestants compete for a prize worth V(V > 0), and the one who is prepared to wait longer collects the prize; both contestants incur a cost equal to the length of time taken to resolve the contest. Symbolically, if E(x,y) denotes the amount gained by a contestant prepared to wait time x when the opponent is prepared to wait y,

Patterns of ESS's. II

E\left( {x,y} \right) = \left\{ {\begin{array}{*{20}{c}} {V - y\quad if\;x > y} \\ { - x\quad if\;x < y} \\ \end{array} } \right.