## Abstract

Counterintuitive dynamics of various biological phenomena occur when composite system dynamics differ qualitatively from that of their component systems. Such composite systems typically arise when modelling situations with time-varying biotic or abiotic conditions, and examples range from metapopulation dynamics to population genetic models. These biological, and related physical, phenomena can often be modelled as simple financial games, wherein capital is gained and lost through gambling. Such games have been developed and used as heuristic devices to elucidate the processes at work in generating seemingly paradoxical outcomes across a spectrum of disciplines, albeit in a field-specific, ad hoc fashion. Here, we propose that studying these simple games can provide a much deeper understanding of the fundamental principles governing paradoxical behaviours in models from a diversity of topics in evolution and ecology in which fluctuating environmental effects, whether deterministic or stochastic, are an essential aspect of the phenomenon of interest. Of particular note, we find that, for a broad class of models, the ecological concept of equilibrium reactivity provides an intuitive necessary condition that must be satisfied in order for environmental variability to promote population persistence. We contend that further investigations along these lines promise to unify aspects of the study of a range of topics, bringing questions from genetics, species persistence and coexistence and the evolution of bet-hedging strategies, under a common theoretical purview.

## 1. Introduction

Environmental heterogeneity has long been recognized as a fundamentally important aspect of ecology and evolutionary biology [1,2]. The systematic study of the influences that this variability can have in shaping natural systems has led to the development of numerous research avenues, including bet-hedging [3], the evolution of dispersal [4], source–sink theory [5] and species invasion and coexistence [6,7]. While all distinct, these areas share a number of common features: first, each deals with a situation in which organisms are faced with time-varying environmental conditions; and second, in each case environmental heterogeneity can drive ‘paradoxical’ behaviours. For example, coupling together two demographic sink environments through dispersal [8], or expressing multiple phenotypes, each individually insufficient for long-term population viability [3], can lead, counterintuitively, to species persistence.

In many cases, an organism's interactions with a suite of environments can be interpreted as the playing of multiple gambling games, with population numbers playing the role of the games' capital. Counterintuitive effects can then often be interpreted as an overall expected gain in capital despite expected losses when restricted to either of the component environments. This obvious metaphorical correspondence between the biological concept of fitness and the financial concept of capital growth has long influenced modelling in evolutionary ecology [9]. For example, most works on the topical issues of persistence and dynamics of a species experiencing random or periodic environmental conditions, while not explicitly using this language, can be easily interpreted in a game-playing framework (e.g. [10,11]).

These ‘games’ can all be represented as dynamical systems that track changes in capital (or, equivalently, changes in population numbers) over the course of repeated application of subsets of a collection of updating rules. The systems can be linear or nonlinear; modelled in either discrete or continuous time; and the switching between updating rules can be either deterministic or stochastic (table 1). Of particular, interest is when two losing games of chance are combined in such a way as to produce a paradoxical or counterintuitive winning expectation. Such capital gains and losses models have been used to, among other things, clarify the mechanisms at work in producing the seemingly paradoxical dynamical behaviours demonstrated in diverse settings of interest in the biological (e.g. [8]), financial [12–14] and physical sciences [15–17].

While the utility of these various paradoxical games for studying numerous issues in evolutionary ecology has been suggested [15,18,19] and, to some extent, investigated, the framework they provide for investigating ecological and evolutionary questions pertaining to persistence remains under-appreciated and largely unexplored. Here, our aim is twofold: first, to provide a synthetic overview of these paradoxical games and to illustrate their merits as mechanistic descriptors of evolutionary and ecological systems; and second, to demonstrate how these simple formalisms can provide a framework that can help to identify some deeper connections between previously unrelated fields of study in evolutionary ecology. By this route, we hope to provide conceptually novel insights into the role of environmental variability in promoting species persistence as well as maintaining diversity in both ecological and evolutionary settings.

## 2. Growth through volatility: stock markets, sources and sinks

Perhaps, the prototypical example of counterintuitive dynamical behaviour in an ecological setting is provided by the classical Levins' model of metapopulation dynamics [20], d*p*/d*t* = *cp*(1 − *p*)− *ep*, where *p* is the fraction of population patches occupied at time *t*, *c* is the rate at which colonists are produced per occupied patch and *e* is the extinction rate of occupied patches. Since the patch extinction rate is non-zero, any population bound exclusively to any given patch will eventually succumb to extinction, and all patches are thus population sinks in the long run. However, connections between patches that allow for the re-colonization of extinguished ones can result in metapopulation persistence.

While Levins' model ignores the details of within-patch dynamics and has constant *per capita* rates of colonization and extinction, matrix population models have been used to explicitly demonstrate the importance of stochasticity in generating paradoxical positive long-term growth of a species in time-varying environments [8,21–24]. For example, Jansen & Yoshimura [8] present a spatially implicit, discrete-time model of offspring allocation to two environments, both of which are long-term demographic sinks, so that exclusive use of either environment eventually leads to population extinction. Conditions in the first of these environments are constant but insufficient to allow population persistence, while in the second environment conditions fluctuate between favourable (source) and unfavourable (sink). The authors show that, despite the long-term unsuitability of either of the component environments, the entire metapopulation can persist by allocating a fixed proportion of offspring to each environment every generation. Further elaborations have led to a large body of research on growth and persistence in metapopulations composed entirely of sink habitats [8,21–24].

These ecological models have important antecedents in the engineering [17] and finance literatures [25,26], where mechanisms of persistence of this type were first studied under the guise of gambling or capital investment games. Here capital, volatile stocks and proportions of capital invested in each stock replace numbers of surviving offspring, fluctuating survival rates and proportions of offspring allocated to each environment, respectively.

All such models can be formulated as simple, discrete-time, stochastic matrix models of the type
2.1where *n*_{t} = (*n*_{1}_{t}, *n*_{2}_{t}, … ,*n*_{ω}_{t}) is a vector of population abundances (stock capitals) at time *t*, **D** is a time-independent dispersal matrix and **M**_{t} is a time-dependent diagonal matrix of growth rates.

Of primary interest is the question of whether dispersal between component populations can allow for long-term persistence when each component population is a long-term demographic sink. Since growth rates are time-dependent, the relevant measure of persistence is the stochastic growth rate [27], given by
2.2where , and *N*_{0} is the total initial population size (total capital investment).

For the case of two populations, in the absence of fitness differences between migrants and residents and without any cost to dispersal, equation (2.1) becomes 2.3

When *γ* = 0, this describes two populations independently evolving in time. If population growth rates each take on one of two values, *m*_{j}(*t*) = *m*_{j}_{1} with probability *p*_{j} and *m*_{j}_{2} with probability for *j* = 1, 2, then the stochastic growth rate is given by
2.4Thus, if both rates on right-hand side of equation (2.4) are less than zero then both environments are demographic sinks, and extinction is the long-term expectation in the absence of dispersal [12–14].

Alternatively, when *γ* = 1, proportions *ϕ* and *ϕ*′ (=1 − *ϕ*) from either environment are reallocated to populations 1 and 2, respectively, at each time step. Ecologically, this corresponds to a situation in which offspring from all sites enter a common pool prior to dispersing, in fixed proportions, to all existing populations [28]. This also has been studied as an investment scheme in the financial literature, and is referred to as a ‘fixed proportions rebalancing strategy’ [12,13]. In this case, the long-term growth rate is given by
2.5

Since *a*_{r} = 〈log *m*_{1}〉_{1} or 〈log *m*_{2}〉_{2} when *ϕ* = 1 or 0, respectively, and since *a*_{r} is a concave function of *ϕ* (d^{2}*a*_{r}/d*ϕ*^{2} < 0) a rebalancing strategy outperforms a buy-and-hold-strategy (i.e. *γ* = 1) when *a*_{r} has a maximum for *ϕ* ≠ 0, 1. This occurs when the slope of *a*_{r} is increasing from the left and decreasing from the right. That is
2.6where *H*_{1}(*m*_{1}) = 1/(*p*_{1}/*m*_{11} + *p*^{′}_{1}/*m*_{12}) *H*_{2}(*m*_{2}) = 1/(*p*_{2}/*m*_{21} + *p*^{′}_{2}/*m*_{22}) are the harmonic means of *m*_{1} and *m*_{2}, respectively. Thus, paradoxical behaviour results when 〈*m*_{1}〉_{1} > *H*_{2}(*m*_{2}) and 〈*m*_{2}〉_{2} > *H*_{1}(*m*_{1}). Moreover, redistribution of offspring among sink environments (respectively, reinvestment in losing stocks) can sometimes lead to persistence (respectively, capital growth) [8,12–14] (figure 1).

### (a) Evolutionary game playing: bet-hedging via phenotype switching

Thus far, we have largely considered situations in which fluctuating environmental conditions impose ecological games on populations. However, such fluctuations can also promote the evolution of game-playing strategies. Just as successful asset management can require a strategy of diversified stock investment when faced with market uncertainty, fluctuating selection owing to temporally varying environments can promote the evolution of risk-spreading strategies through phenotypic diversification [29].

In evolutionary ecology this is given the label ‘diversified bet-hedging’, and an intriguing potential example of its manifestation, interpreted in the present context, is provided by recent work on stochastic phenotype-switching in micro-organisms [30–33]. This process, effected through a suite of (cellular) mechanisms [34], essentially describes the random switching, over time, of a given individual organism's trait, which is chosen from some permissible phenotype set. For example, the soil bacterium *Bacillus subtilis* exists in either of two phenotypic states in nature: a competent form, engaged in DNA-scavenging activity or a quiescent phase [35,36]. Switches between these two states are governed by the stochastically determined levels of a single intracellular protein, such that a given bacterium enters the active phase when this protein concentration exceeds some threshold. Other important medical examples of this type of phenomenon include the induction of the persistence phenotype in *Escherichia coli*, allowing for prolonged survival in the face of antibiotic challenge [37], toxin production in *Mycobacterium pulmonis* [38] and the expression of cell-surface pili in various pathogenic micro-organisms [39].

A collection of recent theoretical works has established the conditions under which stochastic phenotype-switching is selectively favoured. Such flip-flopping between phenotypic states is expected to evolve in situations where the maintenance of sensory organs is prohibitively costly and environmental fluctuations are not too rapid [30–32]. When these conditions are met, strains that exhibit any pure strategy (the competent and dormant behaviours) are doomed to extinction, while those displaying a stochastically determined mixture of these losing strategies (the phenotype-switching behaviour) can endure [31].

With just two alternative phenotypes and two environmental conditions, this situation can be representatively modelled (in continuous time) as 2.7

Here *n*_{j} is the number of individuals exhibiting phenotype *j* at time *t*, *m*_{j} gives the fitness of type *j* individuals at time *t* and *k*_{j} is the phenotypic switching rate into type *j* from the alternate phenotype, *j* = 1, 2 (e.g. [32]). For sufficiently rapid environmental change, the *m*_{j} can be replaced by its time-averaged values [40], , both of which are negative if the system goes to extinction in the absence of phenotype switching. However, within a range of environmental switching rates, non-zero rates of switching between the two phenotypes (*k*_{j} ≠ 0) can promote persistence [31].

## 3. Parrondo games

In 1996, the Spanish physicist Juan M. R. Parrondo presented another class of gambling games originally devised as a heuristic description of the counterintuitive behaviours exhibited by Brownian ratchets and molecular motors [41–43]. These are physical systems in which particles subject to random Brownian perturbations move under the influence of a periodic saw-toothed potential (a periodically varying spatial function of the particle's potential energy) that is switched on and off in either a regular, or random, fashion. If flow is biased in a particular direction by tilting the potential, particles tend to drift downhill under the action of either the saw-tooth (on) or flat (off) settings. However, by ‘flashing’ on and off between the two settings, particles can be ratcheted ‘uphill’ against the gradient resulting in a counterintuitive net flow [42].

Parrondo games are renderings of these ratchet systems into mathematical abstractions, interpretable as games of chance in which capital is gained or lost depending on the outcomes of flipping biased coins ([15,44]; figure 2). The game corresponding to the flat potential, game A, is a simple coin toss, biased towards losing. The saw-tooth potential is represented by a more complex two-component, overall losing, game, B, composed of sub-games with losing, B_{1}, and winning, B_{2}, biases, and a rule determining which of these to play on any given iteration. This rule can, as in Parrondo's original game, depend on some property of the total present winnings, such as the remainder of the capital upon division by some integer, *M* (termed capital dependent games) [15,44]; or on other features, like the history of wins and losses (history dependant games) [45]. The situation can be represented in matrix form as
3.1where *σ*_{t} is a Bernoulli random variable that determines whether game A or B is played, *α*_{t}, *β*_{1}_{t} and *β*_{2}_{t} are Bernoulli random variables whose parameters give the probability of winning game A, B_{1} or B_{2}, respectively, *γ* is a variable that determines history or capital dependence such that game B_{1} (respectively, B_{2}) is played if *γ* = 1 (respectively, *γ* = 0), and *X*′ = 1 − *X* for all variables *X*.

Over the course of the last decade, these eponymously named Parrondo games have elicited a good deal of excitement across a diverse breadth of disciplines ([18,46,47]; and references therein). The relevance of Parrondo effects for questions in the biological realm has been the source of informal musings since their original description [15,19,44]. It has been suggested that parrondian phenomena might also have a bearing on such staples of evolutionary ecology as the evolution of sex and costly signalling, as well as the fundamental question of the origins of life from non-living precursors [18]. Even the process of evolution by natural selection itself has been offered as an example of a Brownian ratchet [18].

Despite this ambitious casting of Parrondo effects as a potential evolutionary ecology panacea, most of these particular suggestions remain purely speculative at present. However, some recent verbal models and analytical work suggest that the ratchet-like effect of Parrondo might provide a natural framework for certain types of population genetics investigations. Atkinson & Peijnenburg [48] present a toy model (without explicit genetics) wherein two separate populations suffer from a mitochondrial genetic disorder with an inheritance pattern that depends on both mother and grandmother genotypes. In one, males carry an allele that can epistatically disrupt the expression of the disorder. At equilibrium, approximately 56 per cent of females in the population with the epistatic allele, and all females in the second population, are expected to suffer from the disorder. However, the population resulting from a randomly mating mix of both populations experiences a much diminished genetic load, with only 33 per cent of females affected.

A more genetically explicit model by Reed [49] studies the dynamics at an autosomal locus under the influence of epistasis and sexually antagonistic selection, and shows how a Parrondo-type effect can contribute to the fixation of an allele with a smaller sex-averaged fitness than an alternative allele. Both these models make use of a recent reformulation of Parrondo games, in which winning and losing probabilities are determined by the history of wins and losses (history dependence) rather than their original dependence on accrued capital (capital dependence). Such an extension has long been thought to be a vital step in applying Parrondo games to a wider class of biological situations [45,46], wherein history dependence arises more naturally and with a more obvious interpretation than does capital dependence.

As originally constructed, each iteration of the Parrondo game (see equation (3.1)) results in a fixed return of either +1 or −1 to the total capital. However, extending the definition of ‘Parrondo game’ to include proportional returns, while retaining capital or history dependence, leaves the essential aspects of the game intact, while allowing for a more natural application to ecological and/or evolutionary phenomena. This also has the advantage of making the games amenable to a standard linear system representation that characterizes all the games we consider here.

For example, although not explicitly recognized as such, work by Levine & Rees [50] provides a nice illustration of the relevance of Parrondo games to the modelling of community structure. Their model considers the factors influencing the persistence of a native forb species in competition with an exotic grass in a Californian grasslands system. The model invokes the ‘storage effect’ mechanism [51], through which competing species can coexist in a temporally variable environment as long as each species benefits from different, ephemeral, environmental conditions and can ‘store’ the effects of good years in particular life stages. In Levine & Rees [50], forbs achieve storage through the use of a seed bank. Competition between the forb and exotic grass is represented as
3.2aand
3.2bwhere F_{t} and G_{t} are the number of forb and grass seeds in their respective populations at the beginning of the growing season each year, *λ*_{F}_{,t} (respectively, *λ*_{G}_{,t}) is the *per capita* number of forb (respectively, grass) seeds produced per individual, *d* is the death rate of ungerminated seeds in the soil (*d*′ = 1 − *d*), *σ*_{t} is the fraction of seeds that germinate in year *t* (), and *α*_{FG} (respectively, *α*_{GF}) is the competition coefficient quantifying the effect of grass on forb (respectively, forb on grass).

To consider the conditions under which the forb's seed bank allows for its persistence even when it is an inferior competitor (*α*_{FG} >1), the authors looked at the condition under which the forb can increase when initially rare (F_{t} ≈ 0), so that the dynamical system (3.2) becomes
3.3

Dividing total forb seeds at time *t* into those produced, F_{1,t}, and those from the seed bank, F_{2,t}, this can be expressed in the same fashion as (2.3):
3.4

The correspondence with a Parrondo game is made obvious by assuming that the environment switches stochastically between favourable (*σ* _{t} = 1) and unfavourable years (*σ*_{t} = 0), so that *λ*_{F}_{,t} = *λ*_{G}_{,t} = *λ*_{+} or *λ*_{−}, respectively. When rare, the forb thus experiences three possible growth rates: *d*′ = (1 − *d*) in unfavourable years, corresponding to game A; 1/*α*_{FG} in favourable years following a favourable year, corresponding to game B_{1}; and *λ*_{+}/*α*_{FG}*λ*_{−} in favourable years following an unfavourable year, corresponding to game B_{2}. Note that the conditional growth rate in favourable years following a favourable year introduces a dependence on the history of the environmental conditions, and thus provides the necessary ingredients for a Parrondo's paradox to arise.

## 4. Reversal behaviour, Simpson's Paradox and equilibrium reactivity

From a synthetic perspective, the question of to what degrees do the mechanisms presented in the preceding sections represent the same underlying phenomenon naturally arises. In at least one important sense, a deeper conceptual unity between all various manifestations of paradoxical reversal phenomena can be demonstrated. Astumian [16] provides an illuminating discussion of this point, arguing that all counterintuitive reversals of fortune resulting from the combination of two losing (or winning) games of chance can be understood as being owing to the influence of hidden correlations. In this sense, reversals of fortune in mixed dynamical games can thus be viewed as realizations of the much-studied Simpson's Paradox [52], a well-known source of reversal behaviour in statistical relationships derived from numerical data comprised of several distinct groups and with recognized implications for studies in evolutionary ecology [53–55].

Simpson's paradox, also known as Yule's paradox or the reversal paradox [56], arises when a relationship between two variables changes when a third, often ‘hidden’ variable, is introduced into their relationship. For example, suppose there are two games, A and B, played *x*_{1} and *y*_{1} times, with the winning proportions given by *f*_{1} and *g*_{1}, respectively. Upon repeating the experiment, Games A and B are played *x*_{2} and *y*_{2} times, with a winning proportions *f*_{2} and *g*_{2}. Overall, the winning proportion of game A is given by *f*_{1}*ρ* + *f*_{2}(1 − *ρ*), where *ρ* = *x*_{1}/(*x*_{1} + *x*_{2}), while the overall winning proportion of game B is *g*_{1}*ϕ* + *g*_{2}(1 − *ϕ*), with *ϕ* = *y*_{1}/(*y*_{1} + *y*_{2}).

In this setting, Simpson's paradox manifests when, for example, the inequalities *f*_{1} < *g*_{1} and *f*_{2} < *g*_{2} both hold, indicating that game B is advantageous, but *ρ* and *ϕ* are such that the inequality *f*_{1}*ρ* + *f*_{2}(1 − *ρ*) > *g*_{1}*ϕ* + *g*_{2}(1 − *ϕ*) also holds, so that the overall winning proportions seem to indicate that game A is advantageous (figure 3). Reversal behaviour occurs owing to the games' results being combined rather than acknowledging the two distinct outcomes of the games as a variable.

In the case of Parrondo games, it is clear that history (or capital) dependence introduces the correlations that are, at least partly, responsible for generating counterintuitive reversal behaviours. However, these factors are absent in the other mixed matrix games we consider here, but paradoxical behaviour remains possible. What are the hidden correlations in these cases? An obvious, though critical, necessary condition for positive growth to be achieved in cases of mixed dynamics is that the inequality **A**_{t}_{+1} **A**_{t} ≠ **A**_{t} **A**_{t}_{+1} must typically hold, since otherwise matrices could be rearranged to produce a long enough run of one dynamic to guarantee extinction. More generally, component dynamics cannot all commute if positive growth is attainable (asymptotically) via dynamical switching, a constraint that holds for nonlinear systems as well. In either case this switching determines the temporal order of environments, which clearly cannot be highly positively autocorrelated: environments must switch sufficiently often in order to interrupt the approach to equilibrium under either dynamic (although some degree of positive autocorrelation can enhance the probability of metapopulation persistence; see [21]).

If a positive growth rate is unachievable at all times, then persistence is impossible. Intuitively then, at least some of the component systems must allow for transient periods of growth. More precisely, for a population with *m* distinct classes, ** n** = (

*n*

_{1},

*n*

_{2}, … ,

*n*

_{m}), subject to a sequence of stochastically determined environments,

**A**

_{j},

*j*= 1, 2, … ,

*t*, the long-term average growth rate,

*λ*

_{s}, [27] is given by 4.1awhere is the 1-norm for the vector

**n**= (

*n*

_{1},

*n*

_{2}, … ,

*n*

_{m}). By the properties of the 1-norm, it follows that 4.1bwhere , the maximum absolute column sum of the matrix

**A**, is the matrix 1-norm [57]. Letting and , where

**A**

^{T}is the matrix transpose of

**A**, be the vector and matrix 2-norms, respectively, and

*λ*

_{max}(

**A**) is the dominant eigenvalue of the matrix

**A**, the properties of the matrix 1- and 2-norms [57] imply that 4.1c 4.1d 4.1e

Therefore, if *λ*_{s} > 1, then *λ*_{max} () > 1 for at least one *j*. This condition is known as positive reactivity [58], and any stable equilibrium point possessing positive reactivity is termed reactive. Small perturbations away from such equilibria can result in a transient initial period characterized by motion away from the asymptotically stable state before eventually returning [59,60]. Thus, a unifying, necessary condition for paradoxical persistence in these matrix games is that at least some of the environmental conditions must have * n* =

**0**as a reactive equilibrium. For example, for the model presented in equation (2.3) with

*γ*= 1, equation (4.1

*e*) evaluates to 4.2

Taking the logarithm of this quantity and rearranging gives
4.3where the stochastic growth rate, *a*_{r}, is as given in equation (2.5). Since the four logarithm terms in equation (4.3) are all non-negative, the whole expression is greater than *a*_{r}, and persistence (*a*_{r} > 0) implies reactivity.

### (a) Nonlinear models

The language of paradoxical games has also been invoked in nonlinear settings of ecological relevance as well. For example, an epidemiological study of the influence of seasonality on outbreaks of Hantavirus in deer mice [61] shows that environmental shifts, if rapid enough to disrupt the system's relaxation towards the disease-free equilibrium, can result in disease outbreaks that could not be supported in either of the component environments alone. Previous empirical studies implicate seasonal environmental disturbances in Hantavirus outbreaks in the southern US in 1993 and 1998 [62], and provide some support for this mechanism as an explanation for unexpected disease persistence in natural systems.

Reversal behaviour in this study is understood to be a function of relaxation time versus switching time: long periods under one dynamic allow the system to relax towards the disease-free equilibrium, while sufficiently rapid switching continually buoys the system away from it. Despite this similarity with the discussion of the previous section, an analysis of the system in Buceta *et al*. [61] near the disease-free equilibrium shows that linear considerations are insufficient to explain disease persistence, indicating that nonlinear effects play a prominent role in determining reversal behaviour (P. D. Williams & A. M. Hastings 2011, unpublished data). However, in the numerical example performed by Buceta *et al*. [61], the disease-free equilibrium is always reactive for one of the two sets of dynamics (P. D. Williams & A. M. Hastings 2011, unpublished data), suggesting a possible role for reactivity here as well. More fully discerning the connections that exist between the generation of paradoxical behaviour in linear and nonlinear systems is an important task for future investigations.

## 5. Discussion

The works we survey here present a variety of mechanisms that are thematically unified by their common finding of mixed system dynamics that differ from those displayed by their component systems. In many cases, simple financial-type models can capture the essential dynamical features of these situations, clarifying the reasons for apparently paradoxical behaviours. In particular, despite suggestions of their potentially broad utility [15,19,44], Parrondo games have received little explicit attention in the ecology or evolutionary biology literatures. However, given the applications they have found thus far [48,49], history-dependent Parrondo games might be a novel and integrative framework for modelling a host of phenomena involving nonlinear genetic interactions. Even the community structure model given in equations (3.2*a*,*b*), through its utilization of the diversity-promoting storage effect [11,51], might inspire application to the study of some aspects of genotypic diversity maintenance [63]. Moreover, the explicit recognition of a Parrondo game in the modelling of processes shaping community ecology (equations 3.2*a*,*b*) might lead to further insights in this area as well. The initial promise seen in Parrondo games for modelling ecological and evolutionary questions might yet be realized.

While all the phenomena discussed herein are interesting in their own right, the question of whether or not these various instances of counterintuitive behaviour share some fundamental properties that can allow them to be understood and analysed as various manifestations of the same underlying principle naturally arises. Clarification of these properties, and a more thorough understanding of these underlying principles, is an important synthetic step in the study of mixed system dynamics. Moreover, since mixed dynamics arise naturally in many ecological and evolutionary settings in which environments are changing, either periodically or stochastically, such a synthesis represents an important conceptual advance in the study of phenomena for which time-varying dynamics are essential features.

Here we argue that, for many ecological or evolutionary systems, the concept of equilibrium reactivity provides a simple, intuitive and general necessary condition for determining whether mixed dynamics can generate counterintuitive reversals. While the general pertinence of this condition to nonlinear systems is, at present, unclear, other, more mechanistically constructive perspectives might also be possible. For instance, although some of the population models we consider are linear, growth rates derived from these models are not, and some authors [12] have stated that it is the nonlinear nature of the geometric mean (or its stochastic counterpart) that allows for certain reversal behaviours to manifest.

Similar nonlinear effects, arising in the calculation of winning and losing probabilities of mixed games, have been suggested as factors underlying the classical version of Parrondo's paradox [46]. These nonlinear effects might be interpretable with respect to the physical concept of stochastic resonance [19], in which nonlinearities in a dynamical system interact with noise to enhance a weak signal. This connection has been explicitly explored in the Parrondo game setting [64], and suggested in the financial literature [12]. Further work along these lines could provide a more precise link between these various ideas, as well as the concept of reactivity, allowing for a deeper understanding of the pervasiveness of noise-induced phenomena [65] and a more synthetic exploration of its implications in evolutionary and ecological settings. Indeed, the potential for noise to have ‘positive’ effects is a concept of enduring interest across a broad spectrum of disciplines [19] and paradoxical games can transparently demonstrate precisely how noise generates positive outcomes, like species' persistence [66].

### (a) Extending the paradigm?

The idea of studying formalized, capital-based games to understand general properties of counterintuitive reversal behaviour in ecological/evolutionary models can potentially be extended to include other qualitative phenomena. The first explicit attempt to extend the ‘like + like = unlike’ observation from a capital-based, quantitative paradox to include other qualitative properties of a dynamical system appears to be attributable to Allison & Abbott [64], who show how a stable system can be constructed by switching between two unstable ones. However, earlier, conceptually related issues appear throughout various literatures, including a numerical investigation demonstrating that periodic forcing of the bifurcation parameter in the logistic map can, depending on initial conditions, result in either regular dynamics when component dynamics reside in the chaotic realm or vice versa [67].

Other, more recent, works have shown that regular behaviour (i.e. systems characterized by a negative Lyapunov exponent) can result from the composition of chaotic maps [68–70]. As noted by Almeida *et al*. [69], a necessary condition for generating this type of reversal behaviour in the composite dynamical system is sufficiently rapid switching between the component systems in order to interrupt the approach to the stationary regime under either map, which in both cases is a chaotic attractor.

Similar ideas have also been explored, in spatially extended systems, with respect to pattern formation. In a number of publications, Buceta and co-workers [71–73] discuss how pattern formation in fluid mechanical systems can result from just such an alternation between two sets of dynamics, each of which generates spatial homogeneity when acting alone. Indeed, two distinct mechanisms have been proposed. In the first case [72], the mechanism is similar to that discussed by Almeida *et al*. [69], and can be understood in terms of two characteristic time parameters: switching time, *t*_{s}, which is the average time the system spends experiencing either dynamic; and relaxation time, *t*_{r}, which is the minimum time to reach equilibrium under either dynamic. When *t*_{s} ≫ *t*_{r,} the system will go to equilibrium under either set of dynamics before the switch is made between the component systems and will thus alternate between homogeneous states. Otherwise the system will not evolve long enough under either single dynamic to achieve a homogeneous equilibrium, allowing stationary (*t*_{s} ≪ *t*_{r}) or oscillatory (*t*_{s} ≅ *t*_{r}) patterns to develop.

In the second case [71], component systems share some common homogeneous equilibrium, implying that switching time versus relaxation time arguments play no role in pattern formation. Nevertheless, competing time-scale effects can again be invoked to understand the mechanism by which patterns can emerge. Here, the relevant comparison is between switching time, *t*_{s}, and the time it takes for an initial Turing instability to develop in the system, *t*_{I}. As above, pattern formation can occur if *t*_{s} ≫ *t*_{I}; that is, if the switching time between component dynamics is sufficiently fast.

In all these works, sufficiently rapid shifts between the component global dynamics is the requisite ingredient for counterintuitive behaviours to be possible, mirroring the similar observations made regarding the necessity of negative autocorrelation between environmental conditions for reversal behaviours in capital-based models. The possibility of further, more constructive similarities are hinted at by a closer inspection of the details of the model by Buceta & Lindenberg [71], where pattern formation is generated by the global alternation of two sets of reaction–diffusion dynamics. When the switching rate between component systems is rapid enough, the resultant dynamics can be described by a single dynamical system having the form
5.1in which model parameters are suitably defined weighted averages of the parameters of the component systems [71]. Here, ** n**(

**x**,

*t*) = (

*n*

_{1}(

**x**,

*t*),

*n*

_{2}(

**x**,

*t*), … ,

*n*

_{m}(

**x**,

*t*)) is a vector of species abundances at time

*t*and spatial location

**x**= (

*x*

_{1},

*x*

_{2}, … ,

*x*

_{n}),

*f*(

**) is the reaction term that summarizes species birth and death processes,**

*n***D**is the matrix whose diagonal entries are the species' diffusion coefficients and

*Δ*is the Laplace operator, defined as . Such systems can exhibit pattern formation via Turing instabilities, in which an asymptotically stable equilibrium, , of the system of ordinary differential equation given by 5.2becomes unstable in the presence of dispersal [74], despite the absence of pattern development in the component systems [71].

Interestingly, the existence of a Turing instability at can tell us something about the behaviour of perturbations to the same equilibrium of the non-spatial model (5.2). In particular, as demonstrated by Neubert *et al*. [75], if is an unstable homogeneous equilibrium of the spatial model in equation (5.1), then it must also be a reactive equilibrium of the non-spatial model in equation (5.2), so that small perturbations away from this stable equilibrium must initially grow before eventually shrinking to zero [58]. This is quite a remarkable connection, as the former concept pertains to the long-term behaviour of perturbations to a spatially homogeneous equilibrium, the latter to the short-term, transient behaviour of perturbations to a stable equilibrium of a non-spatial model. Moreover, the necessity of reactivity in this setting also provides some hint of the deeper mechanistic relationship between counterintuitive behaviours in linear, capital-based models and nonlinear models of qualitative phenomena.

## 6. Conclusions

Work towards a more synthetic understanding of reversal behaviour in ecological and evolutionary phenomena is challenging: first, many of the examples highlighted in this survey appeal to the language introduced in the games of Parrondo [15,44] to classify the mechanisms promoting paradoxical or counterintuitive reversal behaviours in mixed dynamical systems. However, it is history or capital-dependent switching rates that define Parrondo games in their seminal form, and most of these examples involve underlying models lacking either of these elements. Thus, while the parrondian terminology does offer the advantage of providing a concise lexicon for discussing reversal behaviour phenomena, it fails to address the possibility of a more fundamental commonality among the various examples.

Second, discussions of these games occur across disparate sets of literature, often with little reference to one another. To use these games as a way to get at fundamental issues in evolutionary ecology, some consolidation of this pluralistic approach to their study is thus required. One major benefit of this perspective is that it has the potential to stimulate much interest and work into the direct application of specific models of paradoxical behaviour from the physics and finance literatures to long-standing questions of interest in various fields of evolutionary ecology, including species persistence and coexistence, the evolution of bet-hedging strategies and population genetics. While such applications are beginning to be studied, published works appear across diverse sets of literature that rarely reference one another. We hope this review will serve as a timely presentation of these results under the same purview, facilitating the dissemination of these ideas to an ecological/evolutionary biology audience. Perhaps even more importantly, this approach will aid in fostering a truly interdisciplinary effort in the study of stochastic dynamics, and will provide immediate benefits in understanding phenomena in evolutionary ecology in terms of the hidden correlations that are at the heart of paradoxical games.

## Acknowledgements

We thank two anonymous reviewers for helpful comments. We received support from the National Science Foundation (grant EF 0742674 to A.H) and from the Natural Sciences and Engineering Research Council of Canada (Postdoctoral fellowship to P.D.W.).

- Received September 27, 2010.
- Accepted December 23, 2010.

- This Journal is © 2011 The Royal Society