Crucially, however, the mean extinction time in small systems is increased more than fourfold in the presence of such demographic variability [ 19 , 83 ]. The optimization of efficiency distributions is reminiscent of co-evolutionary arms race scenarios: Yoshida et al studied the consequences of rapid evolution on the predator—prey dynamics of a rotifer-algae system, using experiments and simulations via coupled non-linear differential equations [ , ].

### mathematics and statistics online

Figure Distribution of predation efficencies at steady state, predators and prey , in a predator—prey system with demographic variability and evolutionary dynamics for a finite correlation between the parent and offspring efficiencies, and b uniformly distributed efficiencies. Unraveling what underpins the coexistence of species is of fundamental importance to understand and model the biodiversity that characterizes ecosystems [ ].

In this context, the cyclic dominance between competing species has been proposed as a possible mechanism to explain the persistent species coexistence often observed in Nature, see, e. In the last two decades, these observations have motivated a large body of work aiming at studying the dynamics of populations exhibiting cyclic dominance. The simplest and, arguably, most intuitive form of cyclic dominance consists of three species in cyclic competition, as in the paradigmatic rock-paper-scissors game RPS —in which rock crushes scissors, scissors cut paper, and paper wraps rock.

Not surprisingly therefore, models exhibiting RPS interactions have been proposed as paradigmatic models for the cyclic competition between three species and have been the subject of a vast literature that we are reviewing in this section. As examples of populations governed by RPS-like dynamics, we can mention some communities of E. In the absence of spatial degrees of freedom and mutations, the presence of demographic fluctuations in finite populations leads to the loss of biodiversity with the extinction of two species in a finite time, see, e.

## Fast Variables in Stochastic Population Dynamics

However, in Nature, organisms typically interact with a finite number of individuals in their neighborhood and are able to migrate. It is by now well established both theoretically and experimentally that space and mobility greatly influence how species evolve and how ecosystems self-organize, see e.

The in vitro experiments with Escherichia coli of [ 39 , , , ] have attracted particular attention because they highlighted the importance of spatial degrees of freedom and local interactions. The authors of [ ] showed that, when arranged on a Petri dish, three strains of bacteria in cyclic competition coexist for a long time while two of the species go extinct when the interactions take place in well-shaken flasks. Furthermore, in the in vivo experiments of [ ], species coexistence is maintained when bacteria are allowed to migrate, which demonstrates the evolutionary role of migration.

These findings have motivated a series of studies aiming at investigating the relevance of fluctuations, space and movement on the properties of systems exhibiting cyclic dominance. A popular class of three-species models exhibiting cyclic dominance are those with zero-sum RPS interactions, where each predator replaces its prey in turn [ 45 , 46 , 58 , 59 , , — ] and variants of the model introduced by May and Leonard [ 38 ], characterized by cyclic 'dominance removal' in which each predator 'removes' its prey in turn see below [ , — ].

Particular interest has been drawn to questions concerning the survival statistics survival probability, extinction time and in characterizing the spatio-temporal arrangements of the species. Here, we first introduce the main models of population dynamics between three species in cyclic competition, and then review their main properties in well-mixed and spatially-structured settings. In the context of population dynamics, systems exhibiting cyclic dominance are often introduced at an individual-based level as lattice models often in two dimensions.

Such an approach is the starting point for further analysis and coarse-grained descriptions. Here, for the sake of concreteness we introduce a class of models exhibiting RPS-like interactions between three species by considering a periodic square lattice consisting of nodes L is the linear size of the lattice in which individuals of three species, S i , are in cyclic competition 5. Each node of the lattice is labeled by a vector and, depending on the details of the model formulation, each node is either i a boolean random variable; ii a patch with a certain carrying capacity, iii an island that can accommodate an unlimited number of individuals.

More specifically, these distinct but related formulations correspond to the following settings. In most models, cyclic dominance between the species S i is implemented through one or both of the following binary reactions among nearest-neighbors:. The reactions 18 account for dominance—removal with rate while the scheme 19 accounts for dominance—replacement with rate. While many works have focused exclusively either on the reactions 18 , see e.

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In addition to cyclic dominance, we also consider the processes of reproduction with rate and mutation with rate according to the schemes:. In principle, cyclic dominance, reproduction and mutation occur with different rates for each species, but here as in the vast majority of other works, and unless stated otherwise, we simply assume , i. Furthermore, to account for the fact that individuals can move, the models are endowed with spatial degrees of freedom by allowing individuals to migrate from one node to a neighboring site according to pair-exchange and hopping processes:.

It is worth noting that corresponds to the simplest form of movement in which an individual or a void in is swapped with any other individual or in [ 67 — 70 , — , , , ]. When the pair-exchange and hopping processes are divorced yielding non-linear diffusion effects [ 67 — 70 , ]. This mimics the fact that organisms rarely move purely diffusively, but rather sense and respond to their environment [ ]. By divorcing 22 and 23 , we can discriminate between the movement in crowded regions, where mobility is dominated by pair-exchange, and mobility in diluted regions.

The individual-based models 18 — 23 are defined by the corresponding Markov processes and the dynamics is governed by the underlying master equation [ 57 , 60 , , ]. However, solving the master equation is a formidable task, and one needs to rely on a combination of analytical approximations and simulation techniques to make progress. In the absence of spatial structure, the population is 'homogeneous' or 'well-mixed' and the analysis is greatly simplified by the fact that in this case all individuals are nearest-neighbors and therefore interact with each other.

When the population size is infinitely large, the dynamics is aptly described by its deterministic rate equations, whose predictions are dramatically altered by demographic fluctuations when the population size is finite. In the limit of very large and spatially unstructured populations, the species densities can be treated as continuous variables and any random fluctuations and correlations can be neglected. In such a mean-field setting, the main cyclic dominance scenarios are covered by the rate equations of the generic model 18 — 21 :.

It is worth noting that the processes 22 and 23 are obviously absent from the non-spatial rate equation The deterministic description in terms of 24 is widely used because of its simplicity. Here, it allows us to establish a connection with evolutionary game theory [ 3 , 33 , 37 , , ], see below. It is also worth noting that equations 24 are characterized by a steady state at which all species coexist with the same density and, when there are no mutations , equations 24 admit also three absorbing fixed points corresponding to a population consisting of only one species.

The rate equations 24 encompass three main types of oscillatory dynamics around the coexistence state :. The evolution in well-mixed populations of finite size is usually formulated in terms of birth-and-death processes, see, e. The pairwise interactions between a finite number of discrete individuals lead to fluctuations that alter the mean-field predictions. In particular, the Markov chains associated with the CLVM and MLM admit absorbing states and these are unavoidably reached causing the extinction of all but one species, with the surviving species that takes over and 'fixates' the entire population.

Hence, the above deterministic scenarios a and b are dramatically modified by demographic fluctuations since the dynamics in a finite population always leads to the survival of one species and the extinction of the two others [ 71 , — ]. Questions of great importance, that have been studied in detail, concern the survival or fixation probability.

The former refers to the probability that, starting from a certain initial population composition, a given species survives after an infinitely long time. A related question of great interest is the unconditional mean extinction time which is the average time that is necessary for two of the species to go extinct while the third survives and fixates the population. The question of survival probability is particularly interesting when the species have different reaction rates: For the CLVM with asymmetric rates, when N is large but finite, it has been shown that the species with the smallest dominance rate the 'weakest species' is the most likely to fixate the population by helping the predator of its own predator [ , ], a phenomenon referred to as the 'law of the weakest'.

A similar phenomenon has also been found in other three-species models exhibiting cyclic dominance, see, e. However, it has also been shown that no law of the weakest holds when the number of species in cyclic competition is more than three [ , , ], see also section 4 , and that this law is generally not followed when the rates are subject to external fluctuations [ ]. The existence of quantities conserved by the CLVM rate equations has been exploited to compute the mean extinction time, found to scale linearly with the system size [ 71 , , ], and the extinction probability giving the probability that, starting at , two of the three species go extinct after a time t [ 58 , 59 , 71 , ], see figure Some other aspects of species extinction in well-mixed three-species models exhibiting cyclic dominance have been considered for instance in [ , ], and the quasi-cycles arising in these systems have been studied in [ 71 , ].

In particular, the quasi-cyclic behavior around the coexistence fixed point of the three-species cyclic model with mutations of [ ] was investigated by computing the power-spectrum and the mean escape time from the coexistence fixed point. In related models, the authors of [ ] studied the entropy production in the nonequilibrium steady state while in [ ] it is shown that demographic noise slows down the quasi cycles of dominance.

In [ ], it is shown that for a class of multi-species zero-sum systems the mean-time for the extinction of one species scales linearly with the population size N when the mean-field dynamics predicts the coexistence of all species and logarithmically with N otherwise , see also section 4. Extinction probability starting at in the CLVM with as a function of the rescaled time : symbols are stochastic simulation results for different system sizes : triangles; : boxes; : circles and solid blue dark gray and red light gray lines are analytical upper and lower bounds of , while the black line is the average of these upper and lower bounds.

Most ecosystems are spatially extended and populated by individuals that move and interact locally. It is therefore natural to consider spatially-extended models of populations in cyclic competition. When spatial degrees of freedom are taken into account, the interactions between individuals are limited to their neighborhood and this restriction has far-reaching consequences that have been observed experimentally.

In fact, since prey may avoid to encounter their predators, species can coexist over long periods.

### The Annals of Applied Probability

Furthermore, long-term species coexistence is often accompanied by the formation of spatio-temporal patterns such as propagating fronts or spiral waves, see, e. Spatially-extended models in which immobile agents of three species occupy the sites of a lattice and interact according to the schemes 18 — 20 have received significant interest, see, e. For instance, the authors of [ 45 , 46 ] considered the CLVM with immobile individuals i.

Spatial degrees of freedom also allow us to consider elementary processes associated with species' movement such as 22 and This is particularly important in biology where migration has been found to have a profound impact on the maintenance of biodiversity, see, e. In this context, important issues, both from theoretical and biological perspectives, are:. Some of the main findings are reviewed below. A spatially-extended version of population dynamics with RPS-like interactions is generally formulated at an individual-based level on a regular lattice, most often in two dimensions, which is the natural biologically-relevant choice for the interaction network.

The model's dynamics is thus formally described by the underlying master equation that appears to be intractable on the face of it.

## Fast Variables in Stochastic Population Dynamics | George William Albert Constable | Springer

Yet, when fluctuations can be neglected and the linear size L of the lattice is large, the spatial population dynamics of RPS-like systems on a unit square domain can often be well described in the continuum limit by the following set of partial differential equations for the local densities :.

Here, the density of species S i is a continuous variable, with and position vector. The dimensional Laplacian is. As in most studies we focus on the two-dimensional case, , but see also section 3. The diffusion coefficients in 26 and the migration rates of 22 and 23 are simply related by. In the first line on the right-hand-side of 26 , we recognize the diffusion terms while the second and third lines respectively correspond to the processes of cyclic dominance 18 , 19 and mutation It is worth noting that non-linear diffusive terms arise when the pair-exchange and hopping processes 22 and 23 are divorced and [ 67 — 70 ], whereas regular linear diffusion occurs when any neighboring pairs are exchanged with rate [ — , , ].

It is also noteworthy that within the metapopulation formulation of [ 67 — 70 , , ], equations 26 arise at lowest order in an expansion in the inverse of the carrying capacity. The intriguing role of migration, is well exemplified by a series of in vitro and in vivo experiments: the authors of [ 39 , , , ] showed that when arranged on a Petri dish, three strains of bacteria in cyclic competition coexist for a long time while two species go extinct when the interactions take place in well-shaken flasks.

On the other hand, in [ ] it was shown that mobility allows the bacterial colonies in the intestines of co-caged mice to migrate between mice which help maintain the coexistence of bacterial species. Theoretical aspects related to the above questions have been addressed in a series of works [ — ] on the two-dimensional MLM with at most one individual per lattice site i. In fact, by considering the above spatially-extended model defined with only dominance-removal, no mutation and exchange between any pairs of neighbors, it was shown that below a critical mobility threshold D c all species coexist in a long-lived quasi-stationary state and form spiraling patterns, whereas biodiversity is lost above the mobility threshold with only one surviving species [ — ].

This phenomenon was analyzed by combining lattice simulations, with a description in terms of stochastic partial differential equations and a complex Ginzburg—Landau equation [ ] derived from equation 26 by approximating heteroclinic orbits with limit cycles. By exploiting the properties of the complex Ginzburg—Landau equation, it has been shown that the extinction probability in the MLM is at when.

In this case all species coexist and form spiral waves, whereas only one species survives when and , see figure A similar analysis was then extended to the case of cyclic dominance-removal and dominance-replacement with linear mobility and no mutations, i. The diffusion coefficient D in the two-dimensional MLM with rates and.

D c is the critical value, see text. Initially, individuals are randomly distributed. Increasing D from left to right , the spiraling patterns grow, and outgrow the system size when the diffusion coefficient exceeds D c : biodiversity is lost above D c , see text.

The experimental findings of [ , ] and theoretical results of [ — ] suggest that mobility can both promote and jeopardize biodiversity in systems with RPS interactions. Interestingly, recent experiments and agent-based simulations of the range expansion of the E.

The oscillatory dynamics characterizing the metastable quasi-stationary state of the two-dimensional MLM has also been studied by computing the species density correlation functions and the Fourier transform of the densities [ , — ]. In [ ], it was shown that for the two-dimensional MLM the above results are robust against quenched disorder in either the reaction rates or mobility rates.

Furthermore, the mean extinction time T ex as the mean time for the first species to go extinct in the two-dimensional MLM with linear diffusion was found to grow exponentially with the lattice size, i. Further properties of the spiral waves characterizing the species coexistence in the MLM have been investigated.