Demographic noise can reverse the direction of deterministic selection.

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 [Formula: see text] theory, by which small populations can evolve to higher densities in the absence of disturbance.

Models of genetic drift as limiting forms of the Lotka-Volterra competition model.

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.

Noise-induced bistable states and their mean switching time in foraging colonies.

We investigate a type of bistability occurring in population systems where noise not only causes transitions between stable states, but also constructs the states themselves. We focus on the experimentally well-studied system of ants choosing between two food sources to illustrate the essential points, but the ideas are more general. The mean time for switching between the two bistable states of the system is calculated. This suggests a procedure for estimating, in a real system, the critical population size above which bistability ceases to occur.

Population genetics on islands connected by an arbitrary network: an analytic approach.

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.

Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond.

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

Are there species smaller than 1 mm?

The rapid advance in genetic sequencing technologies has provided an unprecedented amount of data on the biodiversity of meiofauna. It was hoped that these data would allow the identification and counting of species, distinguished as tight clusters of similar genomes. Surprisingly, this appears not to be the case. Here, we begin a theoretical discussion of this phenomenon, drawing on an individual-based ecological model to inform our arguments. The determining factor in the emergence (or not) of distinguishable genetic clusters in the model is the product of population size with mutation rate-a measure of the adaptability of the population as a whole. This result suggests that indeed one should not expect to observe clearly distinguishable species groupings in data gathered from ultrasequencing of meiofauna.

The mode of host-parasite interaction shapes coevolutionary dynamics and the fate of host cooperation.

Antagonistic coevolution between hosts and parasites can have a major impact on host population structures, and hence on the evolution of social traits. Using stochastic modelling techniques in the context of bacteria-virus interactions, we investigate the impact of coevolution across a continuum of host-parasite genetic specificity (specifically, where genotypes have the same infectivity/resistance ranges (matching alleles, MA) to highly variable ranges (gene-for-gene, GFG)) on population genetic structure, and on the social behaviour of the host. We find that host cooperation is more likely to be maintained towards the MA end of the continuum, as the more frequent bottlenecks associated with an MA-like interaction can prevent defector invasion, and can even allow migrant cooperators to invade populations of defectors.

Spontaneous genetic clustering in populations of competing organisms.

We introduce and analyse an individual-based evolutionary model, in which a population of genetically diverse organisms compete with each other for limited resources. Through theoretical analysis and stochastic simulations, we show that the model exhibits a pattern-forming instability which is highly amplified by the effects of demographic noise, leading to the spontaneous formation of genotypic clusters. This mechanism supports the thesis that stochasticity has a central role in the formation and coherence of species.

Stochastic amplification in an epidemic model with seasonal forcing.

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.

Enhanced stochastic oscillations in autocatalytic reactions.

We study a simplified scheme of k coupled autocatalytic reactions, previously introduced by Togashi and Kaneko. The role of stochastic fluctuations is elucidated through the use of the van Kampen system-size expansion and the results compared with direct stochastic simulations. Regular temporal oscillations are predicted to occur for the concentration of the various chemical constituents, with an enhanced amplitude resulting from a resonance which is induced by the intrinsic graininess of the system. The associated power spectra are determined and have a different form depending on the number of chemical constituents k . We make detailed comparisons in the two cases k=4 and k=8 . Agreement between the theoretical and numerical results for the power spectrum is good in both cases. The resulting spectrum is especially interesting in the k=8 system, since it has two peaks, which the system-size expansion is still able to reproduce accurately.

Limit cycles, complex Floquet multipliers, and intrinsic noise.

We study the effects of intrinsic noise on chemical reaction systems, which in the deterministic limit approach a limit cycle in an oscillatory manner. Previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude. We show that a similar effect occurs when the stable attractors are limit cycles. We compute the correlation functions and spectral properties of the fluctuations in suitably comoving Frenet frames for several model systems including driven and coupled Brusselators, and the Willamowski-Rössler system. Analytical results are confirmed convincingly in numerical simulations. The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.

Stochastic fluctuations in the susceptible-infective-recovered model with distributed infectious periods.

We investigate a stochastic model of infection dynamics based on the Susceptible-Infective-Recovered (SIR) model, where the distribution of the recovery times can be tuned, interpolating between exponentially distributed recovery times, as in the standard SIR model, and recovery after a fixed infectious period. This is achieved by introducing L infective classes, as compared to 1 in the standard model. For large populations, the spectrum of fluctuations around the deterministic limit of the model can be computed analytically. The demographic stochasticity has the effect of transforming the decaying oscillations of the deterministic model into sustained oscillations in the stochastic formulation. We find that the amplification of these stochastic oscillations increases with L , as well as their coherence in frequency. For large values of L (of the order of 10 and greater), the height and position of the peak of the power spectra changes little and is described well by the model with fixed recovery period (L-->infinity) . In this limit we give a closed-form expression for the power spectrum of fluctuations of infective individuals.

The maintenance of heritable variation through social competition.

The paradoxical persistence of heritable variation for fitness-related traits is an evolutionary conundrum that remains a preeminent problem in evolutionary biology. Here we describe a simple mechanism in which social competition results in the evolutionary maintenance of heritable variation for fitness related traits. We demonstrate this mechanism using a genetic model with two primary assumptions: the expression of a trait depends upon success in social competition for limited resources; and competitive success of a genotype depends on the genotypes that it competes against. We find that such social competition generates heritable (additive) genetic variation for "competition-dependent" traits. This heritable variation is not eroded by continuous directional selection because, rather than leading to fixation of favored alleles, selection leads instead to allele frequency cycling due to the concerted coevolution of the social environment with the effects of alleles. Our results provide a mechanism for the maintenance of heritable variation in natural populations and suggest an area for research into the importance of competition in the genetic architecture of fitness related traits.

Predicting the species abundance distribution using a model food web.

A large number of models of the species abundance distribution (SAD) have been proposed, many of which are generically similar to the log-normal distribution, from which they are often indistinguishable when describing a given data set. Ecological data sets are necessarily incomplete samples of an ecosystem, subject to statistical noise, and cannot readily be combined to yield a closer approximation to the underlying distribution. In this paper, we adopt the Webworld ecosystem model to study the predicted SAD in detail. The Webworld model is complex, and does not allow analytic examination of such features; rather, we use simulation data and an approach similar to that of ecologists analysing empirical data. By examining large sets of fully described data we are able to resolve features which can distinguish between models but which have not been investigated in detail in field data. We find that the power-law normal distribution is superior to both the log-normal and logit-normal distributions, and that the data can improve on even this at the high-population cut-off.

The characteristics of species in an evolutionary food web model.

We explore the consequences of modifying the way in which species are defined in an evolutionary food web model. In the original version of the model, the species were defined in terms of a fixed number of features, chosen from a large number of possibilities. These features represented phenotypic and behavioural characteristics of the species. Speciation consisted in occasionally replacing one of the features by another. Here we modify this scheme by firstly allowing for a richer structure and secondly by testing whether we are able to eliminate the need for an explicit choice of features altogether. In the first case we allow for changing the number of features which define a species, as well as their nature, and find that in the resulting webs the higher trophic levels typically contain species with the greatest number of features. In the second case, by a simplification of the mechanisms for inter and intra-species competition, we construct a model without any explicit features and find that we are still able to grow model food webs. We assess the quality of the food webs produced and discuss the consequences of our findings for the future modelling of food webs.

Abundance-body size relationships: the roles of metabolism and population dynamics.

1. Species' abundance scales approximately as an inverse power of body mass. This property has been explained on the basis of metabolic rates of organisms of different sizes. 2. This paper considers the additional effect of population dynamics on the abundance-body size relationship, on the grounds that mass flow through food webs also depends on interactions between predators and their prey. To do this, an analysis of simple dynamical food-chain models was carried out, using rate parameters which scaled with body mass according to empirically based rules. 3. The analysis shows that a function for the abundance-body size relationship derived from metabolic theory is a good first approximation to a function derived for food chains at dynamic equilibrium, although the mechanistic interpretation of terms in the functions is not the same. 4. The results are sensitive to assumptions about the scaling of the self-limitation of basal species with respect to body size. Depending on the assumption made, the abundance-body size relationship may have a power parameter -1 at all trophic levels, or be described by different functions at different trophic levels.

Quasicycles in a spatial predator-prey model.

We use spatial models of simple predator-prey interactions to predict that predator and prey numbers oscillate in time and space. These oscillations are not seen in the deterministic versions of the models, but are due to stochastic fluctuations about the time-independent solutions of the deterministic equations which are amplified due to the existence of a resonance. We calculate the power spectra of the fluctuations analytically and show that they agree well with results obtained from stochastic simulations. This work extends the analysis of these quasicycles from that previously developed for well-mixed systems to spatial systems, and shows that the ideas and methods used for nonspatial models naturally generalize to the spatial case.

Thermodynamics of vesicle growth and instability.

We describe the growth of vesicles, due to the accretion of lipid molecules to their surface, in terms of linear irreversible thermodynamics. Our treatment differs from those previously put forward by consistently including the energy of the membrane in the thermodynamic description. We calculate the critical radius at which the spherical vesicle becomes unstable to a change of shape in terms of the parameters of the model. The analysis is carried out for the case both when the increase in volume is due to the absorption of water and when a solute is also absorbed through the walls of the vesicle.

Goldilocks Forgetting in Cross-Situational Learning.

Given that there is referential uncertainty (noise) when learning words, to what extent can forgetting filter some of that noise out, and be an aid to learning? Using a Cross Situational Learning model we find a U-shaped function of errors indicative of a "Goldilocks" zone of forgetting: an optimum store-loss ratio that is neither too aggressive nor too weak, but just the right amount to produce better learning outcomes. Forgetting acts as a high-pass filter that actively deletes (part of) the referential ambiguity noise, retains intended referents, and effectively amplifies the signal. The model achieves this performance without incorporating any specific cognitive biases of the type proposed in the constraints and principles account, and without any prescribed developmental changes in the underlying learning mechanism. Instead we interpret the model performance as more of a by-product of exposure to input, where the associative strengths in the lexicon grow as a function of linguistic experience in combination with memory limitations. The result adds a mechanistic explanation for the experimental evidence on spaced learning and, more generally, advocates integrating domain-general aspects of cognition, such as memory, into the language acquisition process.