The ESS in an asymmetric generalized war of attrition with mistakes in role perception

We derive the ESS for the generalized asymmetric war of attrition, where payoffs to contestants may vary in time and may depend on some characteristic, called the "role" of an individual. We use the same approach as Hammerstein & Parker (1982), who examined an asymmetric war of attrition. We consider two roles, A and B. Role A is assumed to be favoured with respect to payoffs. It is assumed that there is always a true asymmetry, so in each contest one individual has role A and the other has role B. It is assumed that roles are assigned to contestants at random and that they can make mistakes in role perception. It is shown that, under certain assumptions about shapes of payoff functions and probabilities of making mistakes, there is an ESS which can be characterized by two probability distributions with non-overlapping support. Individuals who perceive their role as A should choose larger persistence times. This ESS structure is similar to that in the asymmetric war of attrition. In that model, the resource values and the cost rates are constant. We consider situations where all these values may change in time and where rewards and costs may be equal after some finite time. Such shapes of payoff functions arise naturally in competitive patch depletion (Sjerps & Haccou, 1994a,b). As a result, the probability density functions that specify the conditional strategies are no longer necessarily negative exponentials (as in the war of attrition), but may have very different shapes. Furthermore, under some conditions there is a maximum persistence time, at which there can be an atom of probability. We give explicit expressions as well as numerical approximations for the ESS.     

Patch leaving strategies and superparasitism: an asymmetric generalized war of attrition

When several competitors deplete a patch, it can be advantageous for each of them to stay provided that others leave, whereas, on the other hand, staying longer decreases the expected payoff for everyone. This situation can be considered as a generalized war of attrition. Previous studies have shown that optimal patch leaving strategies become stochastic and the expected leaving time is much larger than predicted by the marginal value theorem when competitors interfere. The possibility of superparasitism, as occurs for example in parasitoids, induces such interference. In addition, it gives several complications. First, the payoff of females that have left the patch is affected by the ovipositions of the remaining individuals. Second, differences in the arrival time of females cause payoff-relevant asymmetries, since females that arrived early on have parasitized more hosts in a patch at the moment superparasitism starts than those that arrived later. We show that this can be modelled as an asymmetric generalized war of attrition, and derive global characteristics of the ESS for simultaneous decisions on when to start superparasitism and when to leave a patch.     

The war of attrition with random rewards

This paper describes a type of fitness indicating trait which is particularly likely to be exaggerated because of sexual selection on fitness-faking modifier genes, describes the selection process which tends to exaggerate such traits, and indicates by citing examples from different taxa the wide range of traits which can be explained as resulting from this type of selection.     

A generalized war of attrition

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.     

Evolution of kleptoparasitism as a war of attrition

Previous models of kleptoparasitism (resource stealing) assume that contests over resource items are of fixed duration. Here we suggest that such contests will often be well represented as a war of attrition, with the winner being the individual who is prepared to contest for the longer time. Given that time spent in contests cannot be used to search for other resource items, we provide an analytical expression for the evolutionarily stable distribution of contest times. This can be used to investigate the circumstances under which we would expect kleptoparasitism to evolve. In particular, we focus on situations where searching for conspecifics to kleptoparasitize can only be achieved at a cost of reduced resource discovery by other means; under such circumstances we show that kleptoparasitism is not evolutionarily stable.     

Strong evolutionary equilibrium and the war of attrition

In developing the concept of an evolutionarily stable strategy, Maynard Smith proposed formal conditions for stability. These conditions have since been shown to be neither necessary nor sufficient for evolutionary stability in finite populations. This paper provides a strong stability condition which is sensitive to the population size. It is then demonstrated that in the war of attrition with uncertain rewards there is a unique “strong evolutionary equilibrium” strategy. As the population becomes large this is shown to approach the solution strategy proposed by Bishop, Cannings and Maynard Smith.The analysis is then extended to wars of attrition between different populations. It is concluded that for such contests there is a whole family of potential strong evolutionary equilibria.     

Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration μ and the boundary concentration of the diffusing ion c ₀. Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k → ∞, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute μ and c ₀. The position of the advancing front ξ then provides, together with μ, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), μ and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.     

The evolving war on cancer

Building on years of basic scientific discovery, recent advances in the fields of cancer genetics and medicinal chemistry are now converging to revolutionize the treatment of cancer. Starting with serendipitous observations in rare subsets of cancer, a paradigm shift in clinical research is poised to ensure that new molecular insights are rapidly applied to shape emerging cancer therapies. Could this mark a turning point in the "War on Cancer"?     

The logic of asymmetric contests

A theoretical analysis is made of the evolution of behavioural strategies in contest situations. It is assumed that behaviour will evolve so as to maximize individual fitness. If so, a population will evolve an ‘evolutionarily stable strategy’, or ESS, which can be defined as a strategy such that, if all members of a population adopt it, no ‘mutant’ strategy can do better. A number of simple models of contest situations are analysed from this point of view. It is concluded that in ‘symmetric’ contests the ESS is likely to be a ‘mixed’ strategy; that is, either the population will be genetically polymorphic or individuals will be behaviourally variable. Most real contests are probably asymmetric, either in pay-off to the contestants, or in size or weapons, or in some ‘uncorrelated’ fashion; i.e. in a fashion which does not substantially bias either the pay-offs or the likely outcome of an escalated contest. An example of an uncorrelated asymmetry is that between the ‘discoverer’ of a resource and a ‘late-comer’. It is shown that the ESS in asymmetric contests will usually be to permit the asymmetric cue to settle the contest without escalation. Escalated contests will, however, occur if information to the contestants about the asymmetry is imperfect.     

The cold war of the social amoebae

When confronted with starvation, the amoebae of Dictyostelium discoideum initiate a developmental process that begins with cell aggregation and ends with a ball of spores supported on a stalk. Spores live and stalk cells die. Because the multicellular organism is produced by cell aggregation and not by growth and division of a single cell, genetically diverse amoebae may enter an aggregate and, if one lineage has a capacity to avoid the stalk cell fate, it may have a selective advantage. Such cheater mutants have been found among wild isolates and created in laboratory strains. The mutants raise a number of questions--how did such a cooperative system evolve in the face of cheating? What is the basis of self recognition? What genes are involved? How is cheating constrained? This review summarizes the results of studies on the social behavior of Dictyostelium and its relatives, including the familiar asexual developmental cycle and the lesser known, but puzzling, sexual cycle.