Peter D. Taylor

Evolutionarily Stable Strategies and Game Dynamics

We consider a class of matrix games in which successful strategies are rewarded by high reproductive rates, so become more likely to participate in subsequent playings of the game. Thus, over time, the strategy mix should evolve to some type of optimal or stable state. Maynard Smith and Price (1973) have introduced the concept of ESS (evolutionarily stable strategy) to describe a stable state of the game. We attempt to model the dynamics of the game both in the continuous case, with a system of non-linear first-order differential equations, and in the discrete case, with a system of non-linear difference equations. Using this model, we look at the notions of stability and asymptotic behavior. Our notion of stable equilibrium for the continuous dynamic includes, but is somewhat more general than, the notion of ESS.

Cryptographic solution to a problem of access control in a hierarchy

Assume t h a t the users of a c o m p u t e r (or c o m m u n i c a t i o n ) s y s t e m are d iv ided in to a n u m b e r of d i s jo in t sets, /_}1, U2 . . . . . Un. T h e t e r m security class (or class, for short) is used to des igna te each of the Ui. A s s u m e fu r the r t h a t a b i n a r y r e l a t ion _< par t i a l ly orders the set S = ( U1, U 2 , . . . , /_7,} of classes. T h e m e a n i n g of Ui <Uj in the pa r t i a l ly ordered set (poset) (S, ~) is t h a t users in Ui have a security clearance lower t h a n or equa l to those in Uj. S i m p l y put , th is m e a n s t h a t users in Uj can have access to i n f o r m a t i o n he ld by (or de s t i ned to) users in Ui, while the opposi te is no t allowed. Le t xm be a piece of in fo rmat ion , or object, t h a t a cen t r a l a u t h o r i t y (CA) desires to s tore in (or b roadcas t over) the sys tem. T h e m e a n i n g of the subsc r ip t m is t h a t ob jec t x is accessible to users in class Urn. T h e par t i a l o rder on S impl ies t h a t xm is also accessible to users in all classes Ui such t h a t Um Ui. I t is r equ i r ed to design a sy s t em which, in add i t i on to sa t i s fy ing the above condi t ions , e n su r e s t h a t access to i n f o r m a t i o n is as decen t ra l i zed as possible. T h i s m e a n s t h a t au thor ized users shou ld be able to retrieve xm independent ly as soon as it is stored or broadcast by CA.

Altruism in viscous populations- an inclusive fitness model

SummaryA viscous population (Hamilton, 1964) is one in which the movement of organisms from their place of birth is relatively slow. This viscosity has two important effects: one is that local interactions tend to be among relatives, and the other is that competition for resources tends to be among relatives. The first effect tends to promote and the second to oppose the evolution of altruistic behaviour. In a simulation model of Wilsonet al. (1992) these two factors appear to exactly balance one another, thus opposing the evolution of local altruistic behaviour. Here I show, with an inclusive fitness model, that the same result holds in a patchstructured population.

PARENTAL CARE AND THE EVOLUTION OF EGG SIZE IN FISHES

The quality of parental care appears to correlate positively with egg size, both among and within species of fishes. Past models of the trade-off between quantity and quality of offspring have been inadequate in explaining this correlation. Using features of models by Smith and Fretwell (1974), Shine (1978), and Taylor and Williams (1984), we constructed a model to explain continuous covariation between the quality of parental care and egg size. Our model contains three major assumptions about the dependence of offspring survival on egg size: offspring from larger eggs develop more slowly and take longer to resorb their yolk sacs and become juveniles; egg size determines initial juvenile size; and larger juveniles, which hatch from larger eggs, have lower mortalities, experience faster growth, and take less time to become adults. Under these assumptions, as parental care reduces instantaneous egg mortality, the optimal egg size increases. This increase is expected both among and within populations. Thus, the general conclusion that each population should have a single optimal egg size (see, e.g., Smith and Fretwell 1974; Maynard Smith 1978) may be incorrect.

Evolutionary stability in one-parameter models under weak selection

A general notion of evolutionary stability is formulated in models in which the possible behaviours are parameterized by a continuous variable, and selection is assumed to be weak. Two local stability conditions are formulated, m-stability and (δ-stability, the former being first-order and the latter second-order in the mutant behavioural deviation. The conditions are interpreted in two standard formulations of a one-locus genetic model: a covariance approach and a structured population approach. A weak selection theorem is proved which says that m-stability can be calculated using the neutral covariances. These in turn can be calculated as relatedness coefficients; hence an inclusive fitness formulation is capable of checking m-stability. But (δ-stability, being second-order, is more difficult to handle.

Evolution of cooperation in a finite homogeneous graph

Recent theoretical studies of selection in finite structured populations have worked with one of two measures of selective advantage of an allele: fixation probability and inclusive fitness. Each approach has its own analytical strengths, but given certain assumptions they provide equivalent results. In most instances the structure of the population can be specified by a network of nodes connected by edges (that is, a graph), and much of the work here has focused on a continuous-time model of evolution, first described by ref. 11. Working in this context, we provide an inclusive fitness analysis to derive a surprisingly simple analytical condition for the selective advantage of a cooperative allele in any graph for which the structure satisfies a general symmetry condition (‘bi-transitivity’). Our results hold for a broad class of population structures, including most of those analysed previously, as well as some for which a direct calculation of fixation probability has appeared intractable. Notably, under some forms of population regulation, the ability of a cooperative allele to invade is seen to be independent of the nature of population structure (and in particular of how game partnerships are specified) and is identical to that for an unstructured population. For other types of population regulation our results reveal that cooperation can invade if players choose partners along relatively ‘high-weight’ edges.

Confidence of Paternity and Male Parental Care

We present a theoretical framework in order to understand how the relationship between male parental care and paternity is dependent on the relationship between male parental care and offspring recruitment. When there is an S-shaped relationship between offspring recruitment and the parental care of a single male, we predict a threshold relationship between male parental care and paternity. Tree swallows, monogamous dunnocks, and red-winged black-birds may follow this pattern. Alternatively, when there is a concave-down relationship between offspring recruitment and male parental care, we predict a continuous (gradual) decline in male parental care in response to decreasing paternity. Studies of noisy miners and polyandrous dunnocks suggest this pattern. Our model illustrates that the relationship between male parental care and paternity is more complex than assumed previously. A predicted adjustment of male parental care in response to paternity must be examined in relation to the effect of male parental care on offspring recruitment and the effect of alternative activities on male fitness

An inclusive fitness model for dispersal of offspring

A series of papers, Hamilton & May (1977) , Motro, 1982 , Motro, 1983 and Frank (1986) have developed and generalized a model of dispersal of offspring to random sites in a stable environment. I present an inclusive fitness model for such dispersal behaviour, applicable in diploid or haplodiploid populations with dispersal under offspring or maternal control. This model provides an illustration of the general discussion of inclusive fitness models found in Taylor, 1988a , Taylor, 1988b . In particular, it provides a good example of the way in which relatedness coefficients can depend, not only on the genetic structure of the population, but on the mechanism of control of the behaviour under study.

Local mate competition and the sex ratio

Abstract Hamilton (1967) pointed out that Fishers (1930) argument predicting an equality of the sex ratio may break down when there is local competition for mates. He considered in particular a model in which the environment consists of a number of isolated patches, each of which is colonized by a number of inseminated females; the offspring breed within the patch before dispersal. The present paper provides a careful derivation of the equilibrium sex ratio under this model in both diploid and haplo-diploid populations, and extends the model to consider the effects of having a finite number of patches. We suggest that the equilibrium sex ratio is not simply a function of the amount of inbreeding or sib-mating, as suggested by Maynard Smith (1978) , but that the detailed breeding structure of the population must be taken into account.

Inclusive fitness in a homogeneous environment.

A general formulation of inclusive fitness is proposed which specifically accounts for competitive effects between relatives. As an example, for an asexual population in a homogeneous inelastic environment, such as is found in a stepping-stone model of dispersal, the inclusive fitness of a breeding female, under weak selection, is independent of her direct effect on the fitness of other individuals in the population. More precisely, suppose a female acts in a way that changes not only her own fitness but the fitness of several other females in the population who may be relatives (i. e. she changes the number o f their offspring). I call these changes the direct effects of the action. There will also be indirect effects on the fitness of these and other females which come from the competitive impact of the extra offspring in the next generation. The principal result is that these indirect effects exactly cancel all the direct effects except the direct effect of the actor on herself.