The effectiveness of control measures against the diffusion of the COVID-19 pandemic is grounded on the assumption that people are prepared and disposed to cooperate. From a strategic decision point of view, cooperation is the unreachable strategy of the prisoner\'s dilemma game, where the temptation to exploit the others and the fear to be betrayed by them drives the people behavior, which eventually results fully defective. In this work, we integrate the SIRS epidemic model with the replicator equation of evolutionary games in order to study the interplay between the infection spreading and the propensity of people to become cooperative under the pressure of the epidemic. We find that the developed model possesses several steady states, including fully or partially cooperative ones and that the presence of such states allows to take the disease under control. Moreover, assuming a seasonal variation of the infection rate, the system presents rich dynamics, including chaotic behavior and epidemic extinction.

Understanding the conditions for maintaining cooperation in groups of unrelated individuals despite the presence of non-cooperative members is a major research topic in contemporary biological, sociological, and economic theory. The N-person snowdrift game models the type of social dilemma where cooperative actions are costly, but there is a reward for performing them. We study this game in a scenario where players move between play groups following the casual group dynamics, where groups grow by recruiting isolates and shrink by losing individuals who then become isolates. This describes the size distribution of spontaneous human groups and also the formation of sleeping groups in monkeys. We consider three scenarios according to the probability of isolates joining a group. We find that for appropriate choices of the cost-benefit ratio of cooperation and the aggregation-disaggregation ratio in the formation of casual groups, free-riders can be completely eliminated from the population. If individuals are more attracted to large groups, we find that cooperators persist in the population even when the mean group size diverges. We also point out the remarkable similarity between the replicator equation approach to public goods games and the trait group formulation of structured demes.

Multispecies community composition and dynamics are key to health and disease across biological systems, a prominent example being microbial ecosystems. Explaining the forces that govern diversity and resilience in the microbial consortia making up our body\'s defences remains a challenge. In this, theoretical models are crucial, to bridge the gap between species dynamics and underlying mechanisms and to develop analytic insight. Here we propose a replicator equation framework to model multispecies dynamics where an explicit notion of invasion resistance of a system emerges and can be studied explicitly. For illustration, we derive the conceptual link between such replicator equation and N microbial species\' growth and interaction traits, stemming from micro-scale environmental modification. Within this replicator framework, mean invasion fitness arises, evolves dynamically, and may undergo critical predictable shifts with global environmental changes. This mathematical approach clarifies the key role of this resident system trait for invader success, and highlights interaction principles among N species that optimize their collective resistance to invasion. We propose this model based on the replicator equation as a powerful new avenue to study, test and validate mechanisms of invasion resistance and colonization in multispecies microbial ecosystems and beyond.

We investigate the evolutionary dynamics of an age-structured population subject to weak frequency-dependent selection. It turns out that the weak selection is affected in a non-trivial way by the life-history trait. We disentangle the dynamics, based on the appearance of different time scales. These time scales, which seem to form a universal structure in the interplay of weak selection and life-history traits, allow us to reduce the infinite dimensional model to a one-dimensional modified replicator equation. The modified replicator equation is then used to investigate cooperation (the prisoner\'s dilemma) by means of adaptive dynamics. We identify conditions under which age structure is able to promote cooperation. At the end we discuss the relevance of our findings.

Understanding the interplay of different traits in a co-infection system with multiple strains has many applications in ecology and epidemiology. Because of high dimensionality and complex feedback between traits manifested in infection and co-infection, the study of such systems remains a challenge. In the case where strains are similar (quasi-neutrality assumption), we can model trait variation as perturbations in parameters, which simplifies analysis. Here, we apply singular perturbation theory to many strain parameters simultaneously and advance analytically to obtain their explicit collective dynamics. We consider and study such a quasi-neutral model of susceptible-infected-susceptible (SIS) dynamics among N strains, which vary in 5 fitness dimensions: transmissibility, clearance rate of single- and co-infection, transmission probability from mixed coinfection, and co-colonization vulnerability factors encompassing cooperation and competition. This quasi-neutral system is analyzed with a singular perturbation method through an appropriate slow-fast decomposition. The fast dynamics correspond to the embedded neutral system, while the slow dynamics are governed by an N-dimensional replicator equation, describing the time evolution of strain frequencies. The coefficients of this replicator system are pairwise invasion fitnesses between strains, which, in our model, are an explicit weighted sum of pairwise asymmetries along all trait dimensions. Remarkably these weights depend only on the parameters of the neutral system. Such model reduction highlights the centrality of the neutral system for dynamics at the edge of neutrality and exposes critical features for the maintenance of diversity.

The Calogero-Leyvraz Lagrangian framework, associated with the dynamics of a charged particle moving in a plane under the combined influence of a magnetic field as well as a frictional force, proposed by Calogero and Leyvraz, has some special features. It is endowed with a Shannon \"entropic\" type kinetic energy term. In this paper, we carry out the constructions of the 2D Lotka-Volterra replicator equations and the N=2 Relativistic Toda lattice systems using this class of Lagrangians. We take advantage of the special structure of the kinetic term and deform the kinetic energy term of the Calogero-Leyvraz Lagrangians using the κ-deformed logarithm as proposed by Kaniadakis and Tsallis. This method yields the new construction of the κ-deformed Lotka-Volterra replicator and relativistic Toda lattice equations.

Plant-soil feedbacks (PSFs) are considered a key mechanism generating frequency-dependent dynamics in plant communities. Negative feedbacks, in particular, are often invoked to explain coexistence and the maintenance of diversity in species-rich communities. However, the primary modelling framework used to study PSFs considers only two plant species, and we lack clear theoretical expectations for how these complex interactions play out in communities with natural levels of diversity. Here, we extend this canonical model of PSFs to include an arbitrary number of plant species and analyse the dynamics. Surprisingly, we find that coexistence of more than two species is virtually impossible, suggesting that alternative theoretical frameworks are needed to describe feedbacks observed in diverse natural communities. Drawing on our analysis, we discuss future directions for PSF models and implications for experimental study of PSF-mediated coexistence in the field.

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.

A general theory for competitive dynamics among many strains at the epidemiological level is required to understand polymorphisms in virulence, transmissibility, antibiotic resistance and other biological traits of infectious agents. Mathematical coinfection models have addressed specific systems, focusing on the criteria leading to stable coexistence or competitive exclusion, however, due to their complexity and nonlinearity, analytical solutions in coinfection models remain rare. Here we study a 2-strain Susceptible-Infected-Susceptible (SIS) compartmental model with co-infection/co-colonization, incorporating five strain fitness dimensions under the same framework: variation in transmissibility, duration of carriage, pairwise susceptibilities to coinfection, coinfection duration, and transmission priority effects from mixed coinfection. Taking advantage of a singular perturbation approach, under the assumption of strain similarity, we expose how strain dynamics on a slow timescale are explicitly governed by a replicator equation which encapsulates all traits and their interplay. This allows to predict explicitly not only the final epidemiological outcome of a given 2-player competition, but moreover, their entire frequency dynamics as a direct function of their relative variation and of strain-transcending global parameters. Based on mutual invasion fitnesses, we analyze and report rigorous results on transition phenomena in the 2-strain system, strongly mediated via endemic coinfection prevalence. We show that coinfection is not always a promoter of coexistence; instead, its effect to favour or prevent polymorphism is non-monotonic and depends on the type and level of phenotypic differentiation between strains. This framework offers a deeper analytical understanding of 2-strain competitive games in coinfection, with theoretical and practical applications in epidemiology, ecology and evolution.

Suppose we have n different types of self-replicating entity, with the population Pi of the ith type changing at a rate equal to Pi times the fitness fi of that type. Suppose the fitness fi is any continuous function of all the populations P1,…,Pn. Let pi be the fraction of replicators that are of the ith type. Then p=(p1,…,pn) is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher\'s fundamental theorem of natural selection. We compare it to Fisher\'s original result as interpreted by Price, Ewens and Edwards.