Path integral calculation for emergence of rapid evolution from demographic stochasticity
Hong-Yan Shih, Nigel Goldenfeld
备注：
5 pages and 2 figures + 2 pages of supplementary material; v2 figures and supplementary material updated, other minor changes; v3 very minor changes; v4 further minor changes in supplementary material and figures; v5 acknowledgement updated; very minor changes
Genetic variation in a population can sometimes arise so fast as to modify ecosystem dynamics. Such phenomena have been observed in natural predator-prey systems, and characterized in the laboratory as showing unusual phase relationships in population dynamics, including a $π$ phase shift between predator and prey (evolutionary cycles) and even undetectable prey oscillations compared to those of the predator (cryptic cycles). Here we present a generic individual-level stochastic model of interacting populations that includes a subpopulation of low nutritional value to the predator. Using a master equation formalism, and by mapping to a coherent state path integral solved by a system-size expansion, we show that evolutionary and cryptic quasicycles can emerge generically from the combination of intrinsic demographic fluctuations and clonal mutations alone, without additional biological mechanisms.
Recent theoretical studies have shown that demographic stochasticity can greatly increase the tendency of asexually reproducing phenotypically diverse organisms to spontaneously evolve into localised clusters, suggesting a simple mechanism for sympatric speciation. Here we study the role of demographic stochasticity in a model of competing organisms subject to assortative mating. We find that in models with sexual reproduction, noise can also lead to the formation of phenotypic clusters in parameter ranges where deterministic models would lead to a homogeneous distribution. In some cases, nois…
We study the effects of demographic stochasticity on the long-term dynamics of biological coevolution models of community assembly. The noise is induced in order to check the validity of deterministic population dynamics. While mutualistic communities show little dependence on the stochastic population fluctuations, predator-prey models show strong dependence on the stochasticity, indicating the relevance of the finiteness of the populations. For a predator-prey model, the noise causes drastic decreases in diversity and total population size. The communities that emerge under influence of the …
We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean-time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dyna…
Biological populations are subject to two types of noise: demographic stochasticity due to fluctuations in the reproductive success of individuals, and environmental variations that affect coherently the relative fitness of entire populations. The rate in which the average fitness of a community increases has been considered so far using models with pure demographic stochasticity; here we present some theoretical considerations and numerical results for the general case where environmental variations are taken into account. When the competition is pairwise, fitness fluctuations are shown to re…
Demographic stochasticity is important in determining extinction risks of small populations, but it is largely unknown how its effect depends on the life histories of species. We modeled effects of demographic stochasticity on extinction risk in a broad range of generalized life histories, using matrix models and branching processes. Extinction risks of life histories varied greatly in their sensitivity to demographic stochasticity. Comparing life histories, extinction risk generally increased with increasing fecundity and decreased with higher ages of maturation. Effects of adult survival dep…
Classical models of biological invasions assess species spread in homogeneous landscapes by assuming constant growth rates and random local movement. Mounting evidence suggests, however, that demographic stochasticity, environmental heterogeneity and non-random movement of individuals affect considerably the spread dynamics. Here, we show that the dynamics of biological invasions are controlled by the spatial heterogeneity of the resource distribution. We show theoretically that increasing the landscape resource autocorrelation length causes a reduction in the average speed of species spread. …
In this paper we present the simplest individual level model of predator-prey dynamics and show, via direct calculation, that it exhibits cycling behavior. The deterministic analogue of our model, recovered when the number of individuals is infinitely large, is the Volterra system (with density-dependent prey reproduction) which is well-known to fail to predict cycles. This difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite. Our results indicate that additional biological mechani…
Recent collapses of many fisheries across the globe have challenged the mathematical approach to these systems through classic bioeconomic models. Decimated populations did not recover as fast as predicted by these models and depensatory effects were introduced to better fit the dynamics at low population abundances. Alternative to depensation, modeling captures by non-linear harvesting functions produces equivalent outcomes at small abundances, and the dynamics undergoes a bifurcation leading to population collapse and recovery once catching efforts are above or below certain thresholds, resp…
The affect of demographic stochasticity of a system of globally coupled chaotic maps is considered. A two-step model is studied, where the intra-patch chaotic dynamics is followed by a migration step that coupled all patches; the equilibrium number of agents on each site, $N$, controls the strength of the discreteness-induced fluctuations. For small $N$ (large fluctuations) a period-doubling cascade appears as the coupling (migration) increases. As $N$ grows an extremely slow dynamic emerges, leading to a flow along a one-dimensional family of almost period 2 solutions. This manifold become a …
It has been shown that differences in fecundity variance can influence the probability of invasion of a genotype in a population, i.e. a genotype with lower variance in offspring number can be favored in finite populations even if it has a somewhat lower mean fitness than a competitor. In this paper, Gillespie's results are extended to population genetic systems with explicit age structure, where the demographic variance (variance in growth rate) calculated in the work of Engen and colleagues is used as a generalization of "variance in offspring number" to predict the interaction between deter…
Life-history tradeoffs among species are major drivers of community assembly. Most studies investigate how tradeoffs promote deterministic coexistence of species. It remains unclear how tradeoffs may instead promote historically contingent exclusion of species, where species dominance is affected by initial abundances, causing alternative community states. Focusing on the establishment-longevity tradeoff, we study the transient dynamics and equilibrium outcomes of competitive interactions in a simulation model of plant community assembly. We show that, in this model, the establishment-longevit…
The dynamics of two competing species in a finite size community is one of the most studied problems in population genetics and community ecology. Stochastic fluctuations lead, inevitably, to the extinction of one of the species, but the relevant timescale depends on the underlying dynamics. The persistence time of the community has been calculated for neutral models, where the only drive of the system is drift (demographic stochasticity) and for models with strong selection. Following recent analyses that stress the importance of environmental stochasticity in empirical systems, we present he…
Logistic dynamics of small populations with demographic stochasticity
Consejo Nacional de Investigaciones Científicas y Técnicas（国家科学研究委员会）；Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energía Atómica and Universidad Nacional de Cuyo（巴里洛切原子中心与巴尔塞罗研究所，国家原子能委员会与门多萨国立大学）
Methods for predicting the probability and timing of a species' extinction are typically based on a combination of theoretical models and empirical data, and focus on single species population dynamics. Of course, species also interact with each other, forming more or less complex networks of interactions. Models to assess extinction risk often lack explicit incorporation of these interspecific interactions. We study a birth and death process in which the death rate includes an effect from predation. This predation rate is included via a general nonlinear expression for the functional response…
We present an explicit unified stochastic model of fluctuations in population size due to random birth, death, density-dependent competition and environmental fluctuations. Stochastic dynamics provide insight into small populations, including processes such as extinction, that cannot be correctly treated by deterministic methods. We present exact analytical and simulation-based results for extinction times of our stochastic model and compare the different effects of environmental stochasticity and intrinsic demographic stochasticity. We use both the discrete master equation approach and an exa…
May and Leonard (SIAM J. Appl. Math 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic E. coli), in the immune system, in neural information processing models ("winnerless competition"), and in mo…
Recreational fishing is a highly socio-ecological process. Although recreational fisheries are self-regulating and resilient, changing anthropogenic pressure drives these fisheries to overharvest and collapse. Here, we evaluate the effect of demographic and environmental stochasticity for a social-ecological two-species fish model. In the presence of noise, we find that an increase in harvesting rate drives a critical transition from high-yield/low-price fisheries to low-yield/high-price fisheries. To calculate stochastic trajectories for demographic noise, we derive the master equation corres…
Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only -at most- local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of t…
Neutral dynamics, where taxa are assumed to be demographically equivalent and their abundance is governed solely by the stochasticity of the underlying birth-death process, has proved itself as an important minimal model that accounts for many empirical datasets in genetics and ecology. However, the restriction of the model to demographic [${\cal{O}} ({\sqrt N})$] noise yields relatively slow dynamics that appears to be in conflict with both short-term and long-term characteristics of the observed systems. Here we analyze two of these problems - age size relationships and species extinction ti…
Classical stochastic demography predicts that environmental stochasticity reduces population growth rates and, thereby, can increase extinction risk. In contrast, the SAS-CFF model demonstrates that environmental stochasticity can promote genetic diversity. Extending the SAS-CFF to account for demography, I examine the simultaneous effects of environmental stochasticity on genetic diversity and population persistence. Consistent with Gillespie's analysis, if the log-fitness function is concave and allelic responses to the environment are not perfectly correlated, then per-capita growth rates o…
Frequency dependent selection and demographic fluctuations play important roles in evolutionary and ecological processes. Under frequency dependent selection, the average fitness of the population may increase or decrease based on interactions between individuals within the population. This should be reflected in fluctuations of the population size even in constant environments. Here, we propose a stochastic model, which naturally combines these two evolutionary ingredients by assuming frequency dependent competition between different types in an individual-based model. In contrast to previous…
Extinction is the ultimate absorbing state of any stochastic birth-death process, hence the time to extinction is an important characteristic of any natural population. Here we consider logistic and logistic-like systems under the combined effect of demographic and bounded environmental stochasticity. Three phases are identified: an inactive phase where the mean time to extinction $T$ increases logarithmically with the initial population size, an active phase where $T$ grows exponentially with the carrying capacity $N$, and temporal Griffiths phase, with power-law relationship between $T$ and …
Competition is the main driver of population dynamics, which shapes the genetic composition of populations and the assembly of ecological communities. Neutral models assume that all the individuals are equivalent and that the dynamics is governed by demographic (shot) noise, with a steady state species abundance distribution (SAD) that reflects a mutation-extinction equilibrium. Recently, many empirical and theoretical studies emphasized the importance of environmental variations that affect coherently the relative fitness of entire populations. Here we consider two generic time-averaged neutr…
Rubella is a completely immunizing and mild infection in children. Understanding its behavior is of considerable public health importance because of Congenital Rubella Syndrome, which results from infection with rubella during early pregnancy and may entail a variety of birth defects. The recurrent dynamics of rubella are relatively poorly resolved, and appear to show considerable diversity globally. Here, we investigate the behavior of a stochastic seasonally forced susceptible-infected-recovered model to characterize the determinants of these dynamics and illustrate patterns by comparison wi…
Sedentism was a decisive moment in the history of humankind. In a review article Kay and Kaplan quantified land use for early human settlements and found that sedentism and the emergence of farming go hand in hand. For these settlements two primary land use categories, farming and living, can be identified, whereas for hunter gatherer societies no distinct differences can be made. It is natural to search for this in the behavior of two different groups, settlers and farmers. The development of two distinct zones and the two groups lead us to the hypothesis that the emergence of settlements is …
Immigration can rescue local populations from extinction, helping to stabilise a metapopulation. Local population dynamics is important for determining the strength of this rescue effect, but the mechanistic link between local demographic parameters and the rescue effect at the metapopulation level has received very little attention by modellers. We develop an analytical framework that allows us to describe the emergence of the rescue effect from interacting local stochastic dynamics. We show this framework to be applicable to a wide range of spatial scales, providing a powerful and convenient…
The theory of life history evolution provides a powerful framework to understand the evolutionary dynamics of pathogens in both epidemic and endemic situations. This framework, however, relies on the assumption that pathogen populations are very large and that one can neglect the effects of demographic stochasticity. Here we expand the theory of life history evolution to account for the effects of finite population size on the evolution of pathogen virulence. We show that demographic stochasticity introduces additional evolutionary forces that can qualitatively affect the dynamics and the evol…
In large but finite populations, weak demographic stochasticity due to random birth and death events can lead to population extinction. The process is analogous to the escaping problem of trapped particles under random forces. Methods widely used in studying such physical systems, for instance, Wentzel-Kramers-Brillouin (WKB) and Fokker-Planck methods, can be applied to solve similar biological problems. In this article, we comparatively analyse applications of WKB and Fokker-Planck methods to some typical stochastic population dynamical models, including the logistic growth, endemic SIR, pred…
Over the past century, nonlinear difference and differential equations have been used to understand conditions for species coexistence. However, these models fail to account for random fluctuations due to demographic and environmental stochasticity which are experienced by all populations. I review some recent mathematical results about persistence and coexistence for models accounting for each of these forms of stochasticity. Demographic stochasticity stems from populations and communities consisting of a finite number of interacting individuals, and often are represented by Markovian models …
When a collection of phenotypically diverse organisms compete with each other for limited resources, with competition being strongest amongst the most similar, the population can evolve into tightly localised clusters. This process can be thought of as a simple model of the emergence of species. Past studies have neglected the effects of demographic noise and studied the population on a macroscopic scale, where species formation is found to depend upon the shape of the curve describing the decline of competition strength with phenotypic distance. In the following, we will show how including th…
Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics - such as those determining population extinction, fixation or switching between different states - are presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to thi…
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.
Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviours 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 favouring 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 …
Identifying the critical domain size necessary for a population to persist is an important question in ecology. Both demographic and environmental stochasticity impact a population's ability to persist. Here we explore ways of including this variability. We study populations which have traditionally been modelled using a deterministic integrodifference equation (IDE) framework, with distinct dispersal and sedentary stages. Individual based models (IBMs) are the most intuitive stochastic analogues to IDEs but yield few analytic insights. We explore two alternate approaches; one is a scaling up …
Demographic noise has profound effects on evolutionary and population dynamics, as well as on chemical reaction systems and models of epidemiology. Such noise is intrinsic and due to the discreteness of the dynamics in finite populations. We here show that similar noise-sustained trajectories arise in game dynamical learning, where the stochasticity has a different origin: agents sample a finite number of moves of their opponents in-between adaptation events. The limit of infinite batches results in deterministic modified replicator equations, whereas finite sampling leads to a stochastic dyna…
Existing theories for the evolution of aging and death treat senescence as a side-effect of strong selection for fertility. These theories are well-developed mathematically, but fit poorly with emerging experimental data. The data suggest that aging is an adaptation, selected for its own sake. But aging contributes only negatively to fitness of the individual. What kind of selection model would permit aging to emerge as a population-level adaptation? I explore the thesis that population dynamics is inherently chaotic, and that aging is selected for its role in smoothing demographic fluctuation…
The "Kill the Winner" hypothesis is an attempt to address the problem of diversity in biology. It argues that host-specific predators control the population of each prey, preventing a winner from emerging and thus maintaining the coexistence of all species in the system. We develop a stochastic model for the "Kill the Winner" paradigm and show that the stable coexistence state of the deterministic "Kill the Winner" model is destroyed by demographic stochasticity, through a cascade of extinction events. We formulate an individual-level stochastic model in which predator-prey coevolution promote…
Ecological and evolutionary dynamics have been historically regarded as unfolding at broadly separated timescales. However, these two types of processes are nowadays well documented to much more tightly than traditionally assumed, especially in communities of microorganisms. With this motivation in mind, here we scrutinize recent experimental results showing evidence of rapid evolution of tolerance by lag in bacterial populations that are periodically exposed to antibiotic stress in laboratory conditions. In particular, the distribution of single-cell lag times evolves its average value to app…
Mutualistic interactions are vital constituents of ecological and socio-economic systems. Empirical studies have found that the patterns of reciprocal relations among the participants often shows the salient features of being simultaneously nested and modular. Whether and how these two structural properties of mutualistic networks can emerge out of a common mechanism however remains unclear. We propose a unified dynamic model based on the adaptation of niche relations that gives rise to both structural features. We apply Hutchinson's concept of niche interaction to networked cooperative specie…
Understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology. The first theoretical approach to the problem dates back to the 50's with the KiSS (after Kierstead, Slobodkin and Skellam) model, namely a continuous reaction-diffusion equation for a population growing on a patch of finite size $L$ surrounded by a deadly environment with infinite mortality -- i.e. an oasis in a desert. The main outcome of the model is that only patches above a critical size allow for population persistence. Here, we introduce an individual-based anal…
The concept of fitness is central to evolution, but it quantifies only the expected number of offspring an individual will produce. The actual number of offspring is also subject to noise, arising from environmental or demographic stochasticity. In nature, individuals who are more fecund tend to have greater variance in their offspring number -- sometimes far greater than the Poisson variance assumed in classical models of population genetics. Here, we develop a model for the evolution of two types reproducing in a population of non-constant size. The frequency-dependent fitness of each type i…
Evolutionary branching points are a paradigmatic feature of adaptive dynamics, because they are potential starting points for adaptive diversification. The antithesis to evolutionary branching points are Continuously stable strategies (CSS's), which are convergent stable and evolutionarily stable equilibrium points of the adaptive dynamics and hence are thought to represent endpoints of adaptive processes. However, this assessment is based on situations in which the invasion fitness function determining the adaptive dynamics have non-zero second derivatives at a CSS. Here we show that the scop…
The complexities involved in modelling the transmission dynamics of COVID-19 has been a roadblock in achieving predictability in the spread and containment of the disease. In addition to understanding the modes of transmission, the effectiveness of the mitigation methods also needs to be built into any effective model for making such predictions. We show that such complexities can be circumvented by appealing to scaling principles which lead to the emergence of universality in the transmission dynamics of the disease. The ensuing data collapse renders the transmission dynamics largely independ…
Population dynamics reflects an underlying birth-death process, where the rates associated with different events may depend on external environmental conditions and on the population density. A whole family of simple and popular deterministic models (like logistic growth) support a transcritical bifurcation point between an extinction phase and an active phase. Here we provide a comprehensive analysis of the phases of that system, taking into account both the endogenous demographic noise (random birth and death events) and the effect of environmental stochasticity that causes variations in bir…
