Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviors will be driven to extinction by mutant cheats that absorb common benefits but do not themselves contribute. Here we show that when demographic stochasticity is accounted for, selection can in fact act in the reverse direction to that predicted deterministically, instead favoring cooperative behaviors that appreciably increase the carrying capacity of the population. Populations that exist in larger numbers experience a selective advantage by being more stochastically robust to invasions than smaller populations, and this advantage can persist even in the presence of reproductive costs. We investigate this general effect in the specific context of public goods production and find conditions for stochastic selection reversal leading to the success of public good producers. This insight, developed here analytically, is missed by the deterministic analysis as well as by standard game theoretic models that enforce a fixed population size. The effect is found to be amplified by space; in this scenario we find that selection reversal occurs within biologically reasonable parameter regimes for microbial populations. Beyond the public good problem, we formulate a general mathematical framework for models that may exhibit stochastic selection reversal. In this context, we describe a stochastic analog to r − K theory, by which small populations can evolve to higher densities in the absence of disturbance. O ver the past century, mathematical biology has provided a framework with which to begin to understand the complexities of evolution. Historically, development has focused on deterministic models (1). However, when it comes to questions of invasion and migration in ecological systems, it is widely acknowledged that stochastic effects may be paramount, because the incoming number of individuals is typically small. The importance of demographic (intrinsic) noise has long been argued for in population genetics; it is the driver of genetic drift and can undermine the effect of selection in small populations (2, 3). This concept has also found favor in game theoretic models of evolution that seek to understand how apparently altruistic traits can invade and establish in populations (4). However, the past decade has seen an increase in the awareness of some of the more exotic and counterintuitive aspects of demographic noise: It has the capacity to induce cycling of species (5), pattern formation (6, 7), speciation (8), and spontaneous organization in systems that do not display such behavior deterministically.Here we explore the impact of demographic noise on the direction of selection in interactions between multiple phenotypes or species. Historically, a key obstacle to progress in this area has been the analytical intractability of multidimensional stochastic models. This is particularly apparent when trying to investigate problems related to invasion, where systems are typically far from equilibrium. A promising avenue of ...

The relationship between the Moran model and stochastic Lotka-Volterra competition (SLVC) model is explored via time scale separation arguments. For neutral systems the two are found to be equivalent at long times. For systems with selective pressure, their behavior differs. It is argued that the SLVC is preferable to the Moran model since in the SLVC population size is regulated by competition, rather than arbitrarily fixed as in the Moran model. As a consequence, ambiguities found in the Moran model associated with the introduction of more complex processes, such as selection, are avoided.

We analyse a model consisting of a population of individuals which is subdivided into a finite set of demes, each of which has a fixed but differing number of individuals. The individuals can reproduce, die and migrate between the demes according to an arbitrary migration network. They are haploid, with two alleles present in the population; frequency-independent selection is also incorporated, where the strength and direction of selection can vary from deme to deme. The system is formulated as an individual-based model and the diffusion approximation systematically applied to express it as a set of nonlinear coupled stochastic differential equations. These can be made amenable to analysis through the elimination of fast-time variables. The resulting reduced model is analysed in a number of situations, including migration-selection balance leading to a polymorphic equilibrium of the two alleles and an illustration of how the subdivision of the population can lead to non-trivial behaviour in the case where the network is a simple hub. The method we develop is systematic, may be applied to any network, and agrees well with the results of simulations in all cases studied and across a wide range of parameter values.

We investigate the stochastic dynamics of entities which are confined to a set of islands, between which they migrate. They are assumed to be one of two types, and in addition to migration, they also reproduce and die. Birth and death events are later moderated by weak selection. Systems which fall into this class are common in biology and social science, occurring in ecology, population genetics, epidemiology, biochemistry, linguistics, opinion dynamics, and other areas. In all these cases the governing equations are intractable, consisting as they do of multidimensional Fokker-Planck equations or, equivalently, coupled nonlinear stochastic differential equations with multiplicative noise. We develop a methodology which exploits a separation in time scales between fast and slow variables to reduce these equations so that they resemble those for a single island, which are amenable to analysis. The technique is generally applicable, but we choose to discuss it in the context of population genetics, in part because of the extra features that appear due to selection. The idea behind the method is simple, its application is systematic, and the results are in very good agreement with simulations of the full model for a range of parameter values.

It is unclear why sexually reproducing isogamous species frequently contain just two self-incompatible mating types. Deterministic theory suggests that since rare novel mating types experience a selective advantage (by virtue of their many potential partners), the number of mating types should consistently grow. However, in nature, species with thousands of mating types are exceedingly rare. Several competing theories for the predominance of species with two mating types exist, yet they lack an explanation for how many are possible and in which species to expect high numbers. Here, we present a theoretical null model that explains the distribution of mating type numbers using just three biological parameters: mutation rate, population size and the rate of sex. If the number of mating types results from a mutation-extinction balance, the rate of sexual reproduction plays a crucial role. If sex is facultative and rare (a very common combination in isogamous species), mating type diversity will remain low. In this rare sex regime, small fitness differences between the mating types lead to more frequent extinctions, further lowering mating type diversity. We also show that the empirical literature supports the role of drift and facultativeness of sex as a determinant of mating type dynamics.

The theory of slow manifolds is an important tool in the study of deterministic dynamical systems, giving a practical method by which to reduce the number of relevant degrees of freedom in a model, thereby often resulting in a considerable simplification. In this article we demonstrate how the same basic methodology may also be applied to stochastic dynamical systems, by examining the behaviour of trajectories conditioned on the event that they no not depart the slow manifold. We apply the method to a pair of example models from ecology and epidemiology, achieving a reduction in model dimension and gaining excellent analytical approximations. PACS numbers:

Evolutionary transitions between male and female heterogamety are common in both vertebrates and invertebrates. Theoretical studies of these transitions have found that, when all genotypes are equally fit, continuous paths of intermediate equilibria link the two sex chromosome systems. This observation has led to a belief that neutral evolution along these paths can drive transitions, and that arbitrarily small fitness differences among sex chromosome genotypes can determine the system to which evolution leads. Here, we study stochastic evolutionary dynamics along these equilibrium paths. We find non-neutrality, both in transitions retaining the ancestral pair of sex chromosomes, and in those creating a new pair. In fact, substitution rates are biased in favor of dominant sex determining chromosomes, which fix with higher probabilities than mutations of no effect. Using diffusion approximations, we show that this non-neutrality is a result of "drift-induced selection" operating at every point along the equilibrium paths: stochastic jumps off the paths return with, on average, a directional bias in favor of the dominant segregating sex chromosome. Our results offer a novel explanation for the observed preponderance of dominant sex determining genes, and hint that drift-induced selection may be a common force in standard population genetic systems.

Evolutionary transitions between male and female heterogamety are common in both vertebrates and invertebrates. Theoretical studies of these transitions have found that, when all genotypes are equally fit, continuous paths of intermediate equilibria link the two sex chromosome systems. This observation has led to a belief that neutral evolution along these paths can drive transitions, and that arbitrarily small fitness differences among sex chromosome genotypes can determine the system to which evolution leads. Here, we study stochastic evolutionary dynamics along these equilibrium paths. We find non-neutrality, both in transitions retaining the ancestral pair of sex chromosomes and in those creating a new pair. In fact, substitution rates are strongly biased in favor of dominant sex determining chromosomes, which fix with higher probabilities than mutations of no effect. Using diffusion approximations, we show that this non-neutrality is a result of 'drift-induced selection' operating at every point along the equilibrium paths: stochastic jumps off the paths return, on average, with a directional bias in favor of the dominant segregating sex chromosome. Our results offer novel explanations for the ubiquity of transitions between male and female heterogamety and the preponderance of dominant major sex determining genes.