Replicator dynamics for the game theoretic selection models based on state.

The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describeenergy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game,with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed.

Correction to: From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies.

Unfortunately, part of the article title was updated as subtitle which in turn resulted with complete title not appearing on website and in the bibliographic data. The complete version of title is updated here.

From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies.

The idea that selection works in different ways during free population growth and at the equilibrium population size has been present in theoretical biology for a long time. It was first expressed as an r and K selection concept and later clarified in the debate on fitness measures in life history theory. The latest discussion related to this topic is focused on the nest site lottery mechanism and the resulting new population growth model. In this mechanistic biphasic model, the suppression of growth is induced by a shortage of free nest sites for newborns. Before it occurs, the population can grow exponentially. In this paper, the continuous version of the model and its selective properties are analysed. We show a continuous smooth transition between different fitness measures operating during the exponential growth and suppressed growth phase and at the equilibrium population size. Then, the model is extended to the case of a population of parasites, where a constant number of nest sites is replaced by the dynamics of a population of their hosts, in the role of the limiting supply. Parasite strategies are selected under exponential and suppressed growth phases of the population of hosts. Transitions between different fitness measures and conditions for extinction of hosts by parasites are analysed. An interesting result is the possibility of a continuum of fitness measures of parasites for the unsuppressed exponential growth of the host population.

Extrinsic Mortality Can Shape Life-History Traits, Including Senescence.

The Williams' hypothesis is one of the most widely known ideas in life history evolution. It states that higher adult mortality should lead to faster and/or earlier senescence. Theoretically derived gradients, however, do not support this prediction. Increased awareness of this fact has caused a crisis of misinformation among theorists and empirical ecologists. We resolve this crisis by outlining key issues in the measurement of fitness, assumptions of density dependence, and their effect on extrinsic mortality. The classic gradients apply only to a narrow range of ecological contexts where density-dependence is either absent or present but with unrealistic stipulations. Re-deriving the classic gradients, using a more appropriate measure of fitness and incorporating density, shows that broad ecological contexts exist where Williams' hypothesis is supported.

Nest site lottery revisited: towards a mechanistic model of population growth suppressed by the availability of nest sites.

In the "nest site lottery" mechanism, newborns form a pool of candidates and randomly drawn candidates replace the dead adults in their nest sites. However, the selection process has only been analyzed under the assumption of an equilibrium population size. In this study, we extend this model to cases where the population size is not at an equilibrium, which yields a simplified (but fully mechanistic) biphasic population growth model, where the suppression of growth is driven only by the availability of free nest sites for newborns. This new model is free of the inconsistency found in the classical single phase models (such as the logistic model), where the number of recruited newborns can exceed the number of free nest sites. We analyzed the stability of the stationary density surfaces and the selection mechanisms for individual strategies described by different vital rates, which are implied by the new model.

The dynamics of sex ratio evolution: from the gene perspective to multilevel selection.

The new dynamical game theoretic model of sex ratio evolution emphasizes the role of males as passive carriers of sex ratio genes. This shows inconsistency between population genetic models of sex ratio evolution and classical strategic models. In this work a novel technique of change of coordinates will be applied to the new model. This will reveal new aspects of the modelled phenomenon which cannot be shown or proven in the original formulation. The underlying goal is to describe the dynamics of selection of particular genes in the entire population, instead of in the same sex subpopulation, as in the previous paper and earlier population genetics approaches. This allows for analytical derivation of the unbiased strategic model from the model with rigorous non-simplified genetics. In effect, an alternative system of replicator equations is derived. It contains two subsystems: the first describes changes in gene frequencies (this is an alternative unbiased formalization of the Fisher-Dusing argument), whereas the second describes changes in the sex ratios in subpopulations of carriers of genes for each strategy. An intriguing analytical result of this work is that the fitness of a gene depends on the current sex ratio in the subpopulation of its carriers, not on the encoded individual strategy. Thus, the argument of the gene fitness function is not constant but is determined by the trajectory of the sex ratio among carriers of that gene. This aspect of the modelled phenomenon cannot be revealed by the static analysis. Dynamics of the sex ratio among gene carriers is driven by a dynamic "tug of war" between female carriers expressing the encoded strategic trait value and random partners of male carriers expressing the average population strategy (a primary sex ratio). This mechanism can be called "double-level selection". Therefore, gene interest perspective leads to multi-level selection.

The toxicokinetics cell demography model to explain metal kinetics in terrestrial invertebrates.

Metal toxicokinetics in invertebrates are usually described by one-compartment first-order kinetic model. Although the model gives an adequate description of the toxicokinetics in certain cases, it has been shown to fail in some situations. It also does not seem acceptable on purely theoretical grounds as accumulation and excretion rates may change depending on instantaneous toxicant concentration in the gut. We postulate that the mechanism behind such changes is connected with the toxic effect of metals on gut epithelial cells. Based on published data, we have constructed a mechanistic model assuming a dynamic rate of replacement of epithelial cells with increasing contamination. We use a population-type modeling, with a population of gut epithelial cells characterized by specific death and birth rates, which may change depending on the metal concentration in food. The model shows that the equilibrium concentration of a toxicant in an organism is the net result of gut cell death and replacement rates. At low constant toxicant concentrations in food, the model predicts that toxicant-driven cell mortality is moderate and the total amount of toxicant in the intestine increases slowly up to the level resulting from the gradual increase of the cell replacement rate. At high constant concentration, total toxicant amount in the gut increases very fast, what is accompanied by massive cell death. The increased cell death rate results in reduced toxicant absorption, which in turn brings its body load down. The resulting pattern of toxicokinetic trajectory for high metal concentration closely resemble that found in empirical studies, indicating that the model probably describes the actual phenomenon.

The dynamics of sex ratio evolution dynamics of global population parameters.

Classical formalizations of the Fisherian theory of sex ratio evolution are based on the assumption that the number of grand offspring of a female serves as a measure of fitness. However, the classical population genetics approach also considers the contribution of male individuals to gene proliferation. The difference between the predictions of phenotypic and genetic models is that the phenotypic approach describes the primary sex ratio of 0.5 as the ESS value, while genetic models describe the stable state of a population by a combination of the stable states of the male and female subpopulations. In this paper, we formulate an alternative model of sex ratio evolution that is focused on the dynamics and quantitative properties of this process and that combine a rigorous genetic approach with a game theoretic strategic analysis. In the new model, females are the strategic agents and males are the passive carriers on unexpressed genes. Fitness functions in the new model are derived with respect to a "fitness exchange" effect, i.e. the contribution of male individuals to female fitness and vice versa. This new model shows that the dynamics of this system are complex and consist of two phases. The first, rapid, phase converges the system to a stable manifold (termed the male subpopulation equilibrium-MSE) where the male subpopulation state is in equilibrium, conditional on the current state of the female subpopulation. Double phase dynamics occur when the population state is not compatible with the current strategic composition of the population (determined by the value of the primary sex ratio) which can be caused by ecological factors. The trajectory of convergence to the MSE can be very complicated and may contain a dramatic change in the primary sex ratio. Thus, the primary sex ratio of 0.5 is unstable for perturbations of gene frequencies among male carriers. Therefore, the new model supports predictions of genetic models that the evolutionary stability of the sex ratio should be characterized by a combination of a stable value of the primary sex ratio and the male subpopulation equilibrium.

How can we model selectively neutral density dependence in evolutionary games.

The problem of density dependence appears in all approaches to the modelling of population dynamics. It is pertinent to classic models (i.e., Lotka-Volterra's), and also population genetics and game theoretical models related to the replicator dynamics. There is no density dependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the question arises: How is unlimited population growth suppressed in frequency-dependent models? Two categories of solutions can be found in the literature. In the first, replicator dynamics is independent of background fitness. In the second type of solution, a multiplicative suppression coefficient is used, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory and basic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reaches the maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitly consider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium value of population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectories is the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and then remains constant and dependent on the rate of turn-over of individuals.

Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept.

This paper contains the basic extensions of classical evolutionary games (multipopulation and density dependent models). It is shown that classical bimatrix approach is inconsistent with other approaches because it does not depend on proportion between populations. The main conclusion is that interspecific proportion parameter is important and must be considered in multipopulation models. The paper provides a synthesis of both extensions (a metasimplex concept) which solves the problem intrinsic in the bimatrix model. It allows us to model interactions among any number of subpopulations including density dependence effects. We prove that all modern approaches to evolutionary games are closely related. All evolutionary models (except classical bimatrix approaches) can be reduced to a single population general model by a simple change of variables. Differences between classic bimatrix evolutionary games and a new model which is dependent on interspecific proportion are shown by examples.