On the Evolution of the Timing of Reproduction with Non-equilibrium Resident Dynamics

We study the evolution of an individual’s reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics. We find that, as in the equilibrium case, the presence or absence of inter-juvenile aggression affects the functional form of the evolutionarily stable reproductive strategy. If an ESS exists, it can have an absolutely continuous part only if inter-juvenile aggression is included in the model. If inter-juvenile aggression is not included in the model, an ESS can have no continuous parts, and only Dirac measures are possible.     

An Evolutionarily Stable Strategy (ESS) model of postcopulatory guarding in insects

A simple Evolutionarily Stable Strategy (ESS) model for promiscuous insect species is analyzed to obtain the optimal strategy for the duration of male guarding behavior after copulation with a female. Such guarding behavior prevents other males from copulating with that female. Predictions of the model are (i) that the ESS is either a non-guarding strategy, a perfect guarding strategy until oviposition, or a polymorphic equilibrium between the two types, and (ii) that the perfect guarding strategy has more advantages than the non-guarding strategy when (a) the ratio of males to females is large, (b) the searching efficiency is high, (c) the population density is high, and (d) the preoviposition period is short. Male guarding behavior in several species seems to agree with the predictions of the model.     

Examination of the λ equilibrium point hypothesis when applied to single degree of freedom movements performed with different inertial loads

 One of the theories of human motor control is the λ Equilibrium Point Hypothesis. It is an attractive theory since it offers an easy control scheme where the planned trajectory shifts monotionically from an initial to a final equilibrium state. The feasibility of this model was tested by reconstructing the virtual trajectory and the stiffness profiles for movements performed with different inertial loads and examining them. Three types of movements were tested: passive movements, targeted movements, and repetitive movements. Each of the movements was performed with five different inertial loads. Plausible virtual trajectories and stiffness profiles were reconstructed based on the λ Equilibrium Point Hypothesis for the three different types of movements performed with different inertial loads. However, the simple control strategy supported by the model, where the planned trajectory shifts monotonically from an initial to a final equilibrium state, could not be supported for targeted movements performed with added inertial load. To test the feasibility of the model further we must examine the probability that the human motor control system would choose a trajectory more complicated than the actual trajectory to control. Received: 20 June 1995 / Accepted in revised form: 6 August 1996     

Non-smooth plant disease models with economic thresholds

In order to control plant diseases and eventually maintain the number of infected plants below an economic threshold, a specific management strategy called the threshold policy is proposed, resulting in Filippov systems. These are a class of piecewise smooth systems of differential equations with a discontinuous right-hand side. The aim of this work is to investigate the global dynamic behavior including sliding dynamics of one Filippov plant disease model with cultural control strategy. We examine a Lotka–Volterra Filippov plant disease model with proportional planting rate, which is globally studied in terms of five types of equilibria. For one type of equilibrium, the global structure is discussed by the iterative equations for initial numbers of plants. For the other four types of equilibria, the bounded global attractor of each type is obtained by constructing appropriate Lyapunov functions. The ideas of constructing Lyapunov functions for Filippov systems, the methods of analyzing such systems and the main results presented here provide scientific support for completing control regimens on plant diseases in integrated disease management.     

Mistakes can stabilise the dynamics of rock-paper-scissors games

A game of rock-paper-scissors is an interesting example of an interaction where none of the pure strategies strictly dominates all others, leading to a cyclic pattern. In this work, we consider an unstable version of rock-paper-scissors dynamics and allow individuals to make behavioural mistakes during the strategy execution. We show that such an assumption can break a cyclic relationship leading to a stable equilibrium emerging with only one strategy surviving. We consider two cases: completely random mistakes when individuals have no bias towards any strategy and a general form of mistakes. Then, we determine conditions for a strategy to dominate all other strategies. However, given that individuals who adopt a dominating strategy are still prone to behavioural mistakes in the observed behaviour, we may still observe extinct strategies. That is, behavioural mistakes in strategy execution stabilise evolutionary dynamics leading to an evolutionary stable and, potentially, mixed co-existence equilibrium.     

Equilibrium structure and stability in a frequency-dependent,two-population diploid model

We investigate the equilibrium structure for an evolutionary genetic model in discrete time involving two monoecious populations subject to intraspecific and interspecific random pairwise interactions. A characterization for local stability of an equilibrium is found, related to the proximity of this equilibrium with evolutionarily stable strategies (ESS). This extends to a multi-population framework a principle initially proposed for single populations, which states that the mean population strategy at a locally stable equilibrium is as close as possible to an ESS.     

A Geometrical Perspective for the Bargaining Problem

A new treatment to determine the Pareto-optimal outcome for a non-zero-sum game is presented. An equilibrium point for any game is defined here as a set of strategy choices for the players, such that no change in the choice of any single player will increase the overall payoff of all the players. Determining equilibrium for multi-player games is a complex problem. An intuitive conceptual tool for reducing the complexity, via the idea of spatially representing strategy options in the bargaining problem is proposed. Based on this geometry, an equilibrium condition is established such that the product of their gains over what each receives is maximal. The geometrical analysis of a cooperative bargaining game provides an example for solving multi-player and non-zero-sum games efficiently.     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.     

Evolutionary game dynamics in diploid populations

Selection that influences behaviour can be studied using game theory if individual behavioural success depends on the frequencies of various behavioural types in the population. The evolutionarily stable strategy of J. Maynard Smith and G. R. Price (1973. Nature (London) 246, 15–18) is an equilibrium concept like the solution of a game. The dynamic model of Taylor and Jonker, studied in detail by Zeeman, goes beyond game theory using fitness to cause evolution, perhaps towards an equilibrium. A diploid version of their haploid model is considered and it is found that diploid evolution can be quite different. For example “catastrophic” bifurcations can occur between stable internal polymorphisms when the game matrix entries are changed slowly. A slight drop in food supply may cause extinction. Totally unfit altruistic genotypes can be maintained if they help the rest of the population. The relation of haploid game models to constant selection in diploids is also discussed.     

Dangerous nutrients: Evolution of phytoplankton resource uptake subject to virus attack

Phytoplankton need multiple resources to grow and reproduce (such as nitrogen, phosphorus, and iron), but the receptors through which they acquire resources are, in many cases, the same channels through which viruses attack. Therefore, phytoplankton can face a bottom-up vs. top-down tradeoff in receptor allocation: Optimize resource uptake or minimize virus attack? We investigate this top-down vs. bottom-up tradeoff using an evolutionary ecology model of multiple essential resources, specialist viruses that attack through the resource receptors, and a phytoplankton population that can evolve to alter the fraction of receptors used for each resource/virus type. Without viruses present the singular continuously stable strategy is to allocate receptors such that resources are co-limiting, which also minimizes the equilibrium concentrations of both resources. Only one virus type can be present at equilibrium (because phytoplankton, in this model, are a single resource for viruses), and when a virus type is present, it controls the equilibrium phytoplankton population size. Despite this top-down control on equilibrium densities, bottom-up control determines the evolutionary outcome. Regardless of which virus type is present, the allocation strategy that yields co-limitation between the two resources is continuously stable. This is true even when the virus type attacking through the limiting resource channel is present, even though selection for co-limitation in this case decreases the equilibrium phytoplankton population and does not decrease the equilibrium concentration of the limiting resource. Therefore, although moving toward co-limitation and decreasing the equilibrium concentration of the limiting resource often co-occur in models, it is co-limitation, and not necessarily the lowest equilibrium concentration of the limiting resource, that is the result of selection. This result adds to the growing body of literature suggesting that co-limitation at equilibrium is a winning strategy.     

Resistance management: the stable zone strategy

The different strategies of insecticide resistance management that have been formulated so far consist of delaying the appearance and spread of resistance genes. In this paper, we propose a strategy that can be used even if resistance genes are already present. This strategy consists of applying insecticides in an area smaller than a certain critical size, so that gene flow from the untreated area, combined with the fitness cost of the resistance genes, prevents its frequency reaching high equilibrium value. A two-locus model was analysed numerically to determine population densities at equilibrium as a function of selection coefficients (insecticide selection, fitness costs of resistance genes and dominances), gene flow and size of the treated area. This model indicates that there is an optimal size for the treated area where a minimal and stable density reach equilibrium, and where resistance genes cannot invade. This resistance management strategy seems applicable to a large variety of field situations, but eventually it may encounter obstacles due to a modifier which reduces the fitness costs of resistance genes.     

Population-level ecological effect assessment: estimating the effect of toxic chemicals on density-dependent populations

We examined the relationship between individual-level and population-level effects of toxic chemicals, employing the equilibrium population size as an index of population-level effects. We first analyzed two-stage matrix models considering four life-history types and four density-dependent models, and then we analyzed ecotoxicological and life-history data of the fathead minnow (Pimephales promelas) and brook trout (Salvelinus fontinalis) as real examples. Our elasticity analysis showed that toxic impacts on density-dependent populations depended largely on the differences in density-dependence and in life histories of the organisms. In particular, the importance of adult survivability was considerably increased in iteroparous organisms with density-dependent juvenile survivability or fertility. Our results also suggested that population-level effects, as indicated by the percentage reduction in equilibrium population size, were often greater than the percentage reductions in vital rates of individuals. Our analysis indicates that assessing population-level risk and developing a risk-reduction strategy without considering density-dependence can be risky.     

Evolutionarily stable strategy in a sex- and frequency-dependent selection model

In this paper, a sex-dependent matrix game haploid model is investigated. For this model, since the phenotypes of female and male individuals are determined by alleles located at a single locus and are sex dependent, any given genotype corresponds to a strategy pair. Thus, a strategy pair is an ESS if and only if the allele corresponding to this strategy pair cannot be invaded by any mutant allele. We show that an ESS equilibrium must be locally asymptotically stable if it exists.     

Is there a role for amplifiers in sexual selection?

The amplifier hypothesis states that selection could favour the evolution of traits in signallers that improve the ability of receivers to extract honest information from other signals or cues. We provide a formal definition of amplifiers based on the receiver's mechanisms of signal perception and we present a game-theoretical model in which males advertise their quality and females use sequential-sampling tactics to choose among prospective mates. The main effect of an amplifier on the female mating strategy is to increase her mating threshold, making the female more selective as the effectiveness of the amplifier increases. The effects of the amplifier on male advertising strategy depends both on the context and on the types of the amplifier involved. We consider two different contexts for the evolution of amplifiers (when the effect of amplifiers is on signals and when it is on cues) and two types of amplifiers (the ‘neutral amplifier’, when it improves quality assessment without altering male attractiveness, and the ‘attractive amplifier’, when it improves both quality assessment and male attractiveness). The game-theoretical model provides two main results. First, neutral and attractive amplifiers represent, respectively, a conditional and an unconditional signalling strategy. In fact, at the equilibrium, neutral amplifiers are displayed only by males whose advertising level lays above the female acceptance threshold, whereas attractive amplifiers are displayed by all signalling males, independent of their quality. Second, amplifiers of signals increase the differences in advertising levels between amplifying and not-amplifying males, but they decrease the differences within each group, so that the system converges towards an ‘all-or-nothing’ signalling strategy. By applying concepts from information theory, we show that the increase in information transfer at the perception level due to the amplifier of signals is contrasted by a decrease in information transfer at the emitter level due to the increased stereotypy of male advertising strategy.     

Differences in dynamics between discrete strategies and continuous strategies in a multi-player game with a linear payoff structure

A deductive analysis concerning replicator dynamics proved that a continuous strategy game (in which a player chooses an arbitrary real number between [0, 1] as a cooperative fraction) has the same equilibrium as a discrete strategy game (in which a player chooses only C or D), which has the same linear payoff structure as a continuous strategy game. The deduction is shown for two-player and multi-player games.     

Short- vs. long-range disperser: the evolutionarily stable allocation in a lattice-structured habitat

The population dynamics of two types of organisms in a lattice-structured habitat are studied and the evolutionarily stable allocation between short- and long-range disperser is calculated. Offsprings of short-range dispersal stay in the vicinity of their parent and cause local competition. Using pair approximation, I derive a closed system of ordinary differential equations of global and local densities (or mean crowding), and calculate the condition for one type to invade the population dominated by the other type. The evolutionarily stable strategy (ESS) of resource allocation is derived for the case in which there is a linear trade-off between short- and long-range dispersers. The maximum equilibrium abundance of the population may be achieved by a mixture of both types of dispersers, but it is in general different from the ESS resource allocation calculated from the invasibility condition. For the same parameter values, the ESS invests a larger fraction of resources to short-range disperser than the optimal allocation which maximizes the equilibrium population density. This difference can be explained by the fact that long-range disperser is more effective in the preoccupation of space than short-range disperser. The predictions are confirmed by the direct computer simulations of the lattice stochastic models. Copyright 1999 Academic Press.     

Haptoglobin and transferrin types in schizophrenia

Haptoglobin and transferrin types were studied in schizophrenic patients and controls. In the haptoglobin system a significant departure from the Hardy-Weinberg equilibrium with an excess of heterozygotes was found among the patients (p less than 0.01). The distribution of haptoglobin types in the schizophrenic patients was significantly different from that in the controls. The distribution of transferrin types showed a good agreement with the Hardy-Weinberg equilibrium. There was no significant difference between patients and controls with respect to transferrin types.     

The dynamics of plant disease models with continuous and impulsive cultural control strategies

Plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are proposed and investigated. This novel theoretical framework could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Firstly, continuous replanting of healthy plants and removing of infected plants is taken. The existence and stability of disease-free equilibrium and positive equilibrium are studied and continuous cultural control strategy is given. Secondly, plant disease model with impulsive replanting of healthy plants and removing of infected plants is also considered. Using Floquet's theorem and small amplitude perturbation, the sufficient conditions under which the infected plant free periodic solution is locally stable are obtained. Moreover, permanence of the system is investigated. Under certain parameter spaces, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations. The modeling methods and analytical analysis presented can serve as an integrating measure to identify and design appropriate plant disease control strategies.     

Plasma equilibrium in axisymmetric poloidal magnetic field configurations in flux coordinates

A simple derivation is given of equilibrium equations in flux coordinates in the general case of an anisotropic-pressure plasma. The issue of how to formulate the boundary conditions for these equations is discussed for two types of configurations—a straight system and a system with an internal conductor. Examples of numerical solutions to the equilibrium problem for these configurations are presented.     

Local mate competition with lethal male combat: effects of competitive asymmetry and information availability on a sex ratio game

We constructed a sex allocation model for local mate competition considering the asymmetry of competitive abilities among sons. This model assumes two females of a parasitoid wasp oviposit on the same host in sequential order. The evolutionarily stable strategy will be in either Stackelberg or Nash equilibrium, depending on whether the females can recognize their opponent's sex ratio or not, respectively. The Nash equilibrium predicts the second female produce more males than the first. If the second female is able to know and respond to the strategy of the first (a Stackelberg equilibrium), the first will decide an optimal sex ratio assuming that the second reply to it. Under such an assumption, our model predicts that not producing sons is adaptive for the second female when the sons she produces have low competitive ability. Males of parasitoid wasps Melittobia spp. are engaged in lethal male-male combat, indicating large asymmetry in mating success among sons. If females have the ability to recognize their opponent's sex ratio, our model suggests that the severe lethal male-male combat may be one factor explaining their extremely female-biased sex ratio that is unexplainable by pre-existent models.