Interaction rates, vital rates, background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

Interaction intensity in strategic fitness: A quantifying yardstick of selection optimization for evolutionary game

The notion of the fitness of a strategy has been assimilated as the reproductive success in the evolutionary game. Initially, this fitness was tied to the game's pay-off and the strategy's relative frequency. However, density dependence becomes exigent in order to make ecologically reliable fitness. However, the contributions of each different type of interaction to the species's overall growth process were surprisingly under-explored. This oversight has occasionally led to either more or less prediction of strategy selection compared to the actual possibility. Moreover, density regulation of the population has always been analysed in a general way compared to strategy selection. In this context, our study introduces the concept of mean relative death payoff, which helps in assessing interaction intensity coefficients and integrates them into strategic fitness. Based on this fitness function, we develop the frequency-density replicator dynamics, which eventually provides distinguishing criteria for directional and balancing selection. Our optimized, evolutionarily stable strategy emerges as a superior alternative to the conventional trade-off between selection forces and ecological processes. More significantly, mean relative death pay-off has both conditional and quantitative roles in getting a stable population size. As a case study, we have extensively analysed the evolution of aggression using the Hawk-Dove game. We have shown that pure Dove selection is always beneficial for species growth rather than pure Hawk selection, and the condition of selection is dependent on external mortality pressure. However, the condition of coexistence is independent of external mortality pressure, representing a strong evolutionary selection that optimizes population density governed by interaction intensity.

Life-History traits and the replicator equation

Due to the relevance for conservation biology, there is an increasing interest to extend evolutionary genomics models to plant, animal or microbial species. However, this requires to understand the effect of life-history traits absent in humans on genomic evolution. In this context, it is fundamentally of interest to generalize the replicator equation, which is at the heart of most population genomics models. However, as the inclusion of life-history traits generates models with a large state space, the analysis becomes involving. We focus, here, on quiescence and seed banks, two features common to many plant, invertebrate and microbial species. We develop a method to obtain a low-dimensional replicator equation in the context of evolutionary game theory, based on two assumptions: (1) the life-history traits are per se neutral, and (2) frequency-dependent selection is weak. We use the results to investigate the evolution and maintenance of cooperation based on the Prisoner's dilemma and the snowdrift game. We first consider the generalized replicator equation, and then refine the investigation using adaptive dynamics. It turns out that, depending on the structure and timing of the quiescence/dormancy life-history trait, cooperation in a homogeneous population can be stabilized. We finally discuss and highlight the relevance of these results for plant, invertebrate and microbial communities.

Towards a replicator dynamics model of age structured populations

We present a new modelling framework combining replicator dynamics, the standard model of frequency dependent selection, with an age-structured population model. The new framework allows for the modelling of populations consisting of competing strategies carried by individuals who change across their life cycle. Firstly the discretization of the McKendrick von Foerster model is derived. We show that the Euler–Lotka equation is satisfied when the new model reaches a steady state (i.e. stable frequencies between the age classes). This discretization consists of unit age classes where the timescale is chosen so that only a fraction of individuals play a single game round. This implies a linear dynamics and individuals not killed during the round are moved to the next age class; linearity means that the system is equivalent to a large Bernadelli–Lewis–Leslie matrix. Then we use the methodology of multipopulation games to derive two, mutually equivalent systems of equations. The first contains equations describing the evolution of the strategy frequencies in the whole population, completed by subsystems of equations describing the evolution of the age structure for each strategy. The second contains equations describing the changes of the general population’s age structure, completed with subsystems of equations describing the selection of the strategies within each age class. We then present the obtained system of replicator dynamics in the form of the mixed ODE-PDE system which is independent of the chosen timescale, and much simpler. The obtained results are illustrated by the example of the sex ratio model which shows that when different mortalities of the sexes are assumed, the sex ratio of 0.5 is obtained but that Fisher’s mechanism, driven by the reproductive value of the different sexes, is not in equilibrium.

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.

Are the better cooperators dormant or quiescent?

Despite the wealth of empirical and theoretical studies, the origin and maintenance of cooperation is still an evolutionary riddle. In this context, ecological life-history traits which affect the efficiency of selection may play a role despite being often ignored. We consider here species such as bacteria, fungi, invertebrates and plants which exhibit resting stages in the form of a quiescent state or a seed bank. When quiescent, individuals are inactive and reproduce upon activation, while under seed bank parents produce offspring remaining dormant for different amount of time. We assume weak frequency-dependent selection modeled using game-theory and the prisoner's dilemma (cooperation/defect) as payoff matrix. The cooperators and defectors are allowed to evolve different quiescence or dormancy times. By means of singular perturbation theory we reduce the model to a one-dimensional equation resembling the well known replicator equation, in which the gain functions are scaled with lumped parameters reflecting the time scale of the resting state of the cooperators and defectors. If both time scales are identical cooperation cannot persist in a homogeneous population. If, however, the time scale of the cooperator is distinctively different from that of the defector, cooperation may become a locally asymptotically stable strategy. Interestingly enough, in the seed bank case the cooperator needs to become active faster than the defector, while in the quiescent case the cooperator has to be slower. We use adaptive dynamics to identify situations where cooperation may evolve and form a convergent stable ESS. We conclude by highlighting the relevance of these results for many non-model species and the maintenance of cooperation in microbial, invertebrate or plant populations.

A stochastic differential game approach toward animal migration

A stochastic differential game model for animal migration between two habitats under uncertain environment, a new population dynamics model, is formulated. Its novelty is the use of an impulse control formalism to naturally describe migrations with different timings and magnitudes that the conventional models could not handle. Uncertainty of the environment that the population faces with is formulated in the context of the multiplier robust control. The optimal migration strategy to give the maximized minimal profit is found through a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). A key message from HJBQVI is that its free boundary determines the optimal migration strategy. Solving the HJBQVI is carried out with a specialized stable and convergent finite difference scheme. This paper theoretically suggests that the sub-additivity of the performance index, the index to be optimized through the migration, critically affects the resulting strategy. The computational results with the established scheme are consistent with the theoretical predictions and support importance of the sub-additivity property. Social interaction to reduce the net mortality rate is also quantified, suggesting a linkage between the present and existing population dynamics models.

Beyond replicator dynamics: From frequency to density dependent models of evolutionary games

Game theoretic models of evolution such as the Hawk-Dove game assume that individuals gain fitness (which is a proxy of the per capita population growth rate) in pair-wise contests only. These models assume that the equilibrium distribution of phenotypes involved (e.g., Hawks and Doves) in the population is given by the Hardy-Weinberg law, which is based on instantaneous, random pair formation. On the other hand, models of population dynamics do not consider pairs, newborns are produced by singles, and interactions between phenotypes or species are described by the mass action principle. This article links game theoretic and population approaches. It shows that combining distribution dynamics with population dynamics can lead to stable coexistence of Hawk and Dove population numbers in models that do not assume a priori that fitness is negative density dependent. Our analysis shows clearly that the interior Nash equilibrium of the Hawk and Dove model depends both on population size and on interaction times between different phenotypes in the population. This raises the question of the applicability of classic evolutionary game theory that requires all interactions take the same amount of time and that all single individuals have the same payoff per unit of time, to real populations. Furthermore, by separating individual fitness into birth and death effects on singles and pairs, it is shown that stable coexistence in these models depends on the time-scale of the distribution dynamics relative to the population dynamics. When explicit density-dependent fitness is included through competition over a limited resource, the combined dynamics of the Hawk-Dove model often lead to Dove extinction no matter how costly fighting is for Hawk pairs.

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.