Abstract
Organisms often exhibit behavioral or phenotypic diversity to improve population fitness in the face of environmental variability. When each behavior or phenotype is individually maladaptive, alternating between these losing strategies can counterintuitively result in population persistence–an outcome similar to the Parrondo’s paradox. Instead of the capital or history dependence that characterize traditional Parrondo games, most ecological models which exhibit such paradoxical behavior depend on the presence of exogenous environmental variation. Here we present a population model that exhibits Parrondo’s paradox through capital and historydependent dynamics. Two subpopulations comprise our model: nomads, who live independently without competition or cooperation, and colonists, who engage in competition, cooperation, and longterm habitat destruction. Nomads and colonists may alternate behaviors in response to changes in the colonial habitat. Even when nomadism and colonialism individually lead to extinction, switching between these strategies at the appropriate moments can paradoxically enable both population persistence and longterm growth.
eLife digest
Many organisms, from slime molds to jellyfish, alternate between life as freemoving “nomadic” individuals and communal life in a more stationary colony. So what evolutionary reasons lie behind such stark behavioral diversity in a single species? What benefits are obtained by switching from one behavior to another?
Tan and Cheong have now developed a mathematical model that suggests an intriguing possibility: under conditions that would cause the extinction of both nomadic individuals and colonies, switching between these life strategies can enable populations to survive and grow – a counterintuitive phenomenon called Parrondo’s paradox.
Parrondo’s paradox says that it is possible to follow two losing strategies in a specific order such that success is ultimately achieved. For example, slot machines are designed to ensure that players lose in the long run. What the paradox says is that two slot machines can be configured in such a way that playing either slot machine will lead to financial disaster, but switching between them will leave the player richer in the long run.
Most studies of similar phenomena suggest that switching between two ‘losing’ lifestyle strategies can only improve the chances of survival if the environment keeps changing in unpredictable ways. However, Tan and Cheong’s model shows that this unpredictability is an unnecessary condition – paradoxes also occur when organisms form colonies that predictably destroy their habitat.
The basic mechanism for survival is elegant. The organism periodically exploits its habitat as part of a colony, then switches to a nomadic lifestyle to allow the environment to regenerate. Through mathematical analysis and simulations, Tan and Cheong confirm that this strategy is viable as long as two conditions hold: that colonies grow sufficiently quickly when environmental resources are abundant; and that colonists switch to a nomadic lifestyle before allowing the resource levels to dip dangerously low.
The results produced by Tan and Cheong’s model help to explain how behaviorswitching organisms can survive and thrive, even in harsh conditions. Further work needs to be done to adapt this general model to specific organisms and to investigate the possible evolutionary origins of behaviorswitching lifestyles.
Introduction
Behavioral adaptation and phenotypic diversity are evolutionary metastrategies that can improve a population’s fitness in the presence of environmental variability. When behaviors or phenotypes are sufficiently distinct, a population can be understood as consisting of multiple subpopulations, each following its own strategy. Counterintuitively, even when each subpopulation follows a losing strategy that will cause it to go extinct in the longrun, alternating or reallocating organisms between these losing strategies under certain conditions can result in metapopulation persistence, and hence, an overall strategy that wins (Williams and Hastings, 2011). Some examples include random phase variation (RPV) in bacteria across multiple losing phenotypes (Wolf et al., 2005; Kussell and Leibler, 2005; Acar et al., 2008), as well as the persistence of populations that migrate among sink habitats only (Jansen and Yoshimura, 1998; Roy et al., 2005; Benaïm and Schreiber, 2009).
These counterintuitive phenomena are reminiscent of Parrondo’s paradox, which states that there are losing games of chance which can be combined to produce a winning strategy (Harmer and Abbott, 1999). The existence of a winning combination relies on the fact that at least one of the losing Parrondo games exhibits either capitaldependence (dependence upon the current amount of capital, an ecological analog of which is population size) or historydependence (dependence upon the past history of wins or losses, or in an ecological context, growth and decline) (Parrondo et al., 2000; Harmer and Abbott, 2002). There have been many studies exploring the paradox (Cheong and Soo, 2013; Soo and Cheong, 2013, Soo and Cheong, 2014; Abbott, 2010; Flitney and Abbott, 2003; Harmer et al., 2001), including a multiagent Parrondo’s model based on complex networks (Ye et al., 2016) and also implications to evolutionary biology (Cheong et al., 2016; Reed, 2007; Wolf et al., 2005; Williams and Hastings, 2011). However, many biological studies which have drawn a connection to Parrondo games do not necessarily utilize capitaldependence or historydependence in their models (Williams and Hastings, 2011). Furthermore, models of reversal behavior in ecological settings generally rely upon the presence of exogenous environmental variation (Jansen and Yoshimura, 1998; Roy et al., 2005; Benaïm and Schreiber, 2009; Wolf et al., 2005; Kussell and Leibler, 2005; Acar et al., 2008; Levine and Rees, 2004). Without exogenous variation, the paradoxes do not occur. The broader applicability of Parrondo’s paradox to ecological systems thus remains underexplored.
This lacuna remains despite the abundance of biological examples that exhibit historydependent dynamics. The fitness of alleles may depend on the presence of genetic factors and epigenetic factors in previous generations (Reed, 2007). More generally, the fitness of any one gene can depend on the composition of other genes already present in a population, enabling the evolution of complex adaptations like multicellularity through ratcheting mechanisms (Libby and Ratcliff, 2014). Such mechanisms have recently been shown to help stabilize these complex adaptations (Libby et al., 2016). In ecological contexts, the storage effect can ensure that gains previously made in good years can promote persistence in less favorable times (Warner and Chesson, 1985; Levine and Rees, 2004). Speciesinduced habitat destruction or resource production can also have timedelayed effects on population growth, resulting in nonlinear phenomena like punctuated evolution (Yukalov et al., 2009, Yukalov et al., 2014).
In this paper, we present a biologically feasible population model which exhibits counterintuitive reversal behavior due to the presence of historydependent and capitaldependent dynamics. Unlike most other studies, these dynamics do not rely upon the assumption of exogenous environmental variation. In our model, we consider a population that exists in two behaviorally distinct forms: as nomads, and as colonists. Numerous organisms exhibit analogous behavioral diversity, from slime moulds (amoeba vs. plasmodia) (Baldauf and Doolittle, 1997) and dimorphic fungi (yeast vs. hyphae) (Bastidas and Heitman, 2009) to jellyfish (medusae vs. polyps) (Lucas et al., 2012) and human beings. One model organism which exhibits this sort of behavior, to which our study might apply, is the amoeba Dictyostelium discoideum (Annesley and Fisher, 2009).
Nomads live relatively independently, and thus are unaffected by either competition or cooperation. Under poor environmental conditions, they are subject to steady extinction. Colonists live in close proximity, and are thus subject to both competitive and cooperative effects. They may also deplete the resources of the habitat they reside in over time, resulting in longterm death. However, if these organisms are endowed with sensors that inform them of both population density and the state of the colonial habitat, they can use this information to switch from one behavior to another. Significantly, we find that an appropriate switching strategy paradoxically enables both population persistence and longterm growth – an ecological Parrondo’s paradox.
Population model
Two subpopulations comprise our model: the nomadic organisms, and the colonial ones. In a similar vein to habitatpatch models, organisms that exist in multiple subpopulations can be modelled as follows:
where ${n}_{i}$ is the size of subpopulation $i$, ${g}_{i}$ is the function describing the growth rate of ${n}_{i}$ in isolation, and ${s}_{ij}$ is the rate of switching to subpopulation $i$ from subpopulation $j$. Population sizes are assumed to be large enough that Equation 1 adequately approximates the underlying stochasticity.
Nomadism
Let ${n}_{1}$ be the nomadic population size. In the absence of behavioral switching, the nomadic growth rate is given by
where ${r}_{1}$ is the nomadic growth constant. Nomadism is modelled as a losing strategy by setting ${r}_{1}\text{}\text{}0$, such that ${n}_{1}$ decays with time. In the context of Parrondo’s paradox, nomadism corresponds to the ‘agitating’ strategy, or Game A. Importantly, competition among nomads, as well as between nomads and colonists, is taken to be insignificant, due to the independence of a nomadic lifestyle.
Colonialism
Colonial population dynamics will be modelled by the wellknown logistic equation, with carrying capacity $K$, but with two important modifications.
Firstly, the Allee effect is taken into account. This serves two roles: it captures the cooperative effects that occur among colonial organisms, and it ensures that the growth rate is negative when the population falls below a critical capacity $A$. Let ${n}_{2}$ be the colonial population size. In the absence of behavioral switching, the colonial growth rate is given by
where ${r}_{2}$ is the colonial growth constant. Setting ${r}_{2}>0$, we have a positive growth rate when $A<{n}_{2}<K$, and a negative growth rate otherwise. The $\mathrm{m}\mathrm{i}\mathrm{n}\left(A,K\right)$ term ensures that when $K<A$, ${g}_{2}$ is always zero or negative, as would be expected.
Secondly, the carrying capacity $K$ changes at a rate dependent upon the colonial population size, ${n}_{2}$, accounting for the destruction of environmental resources over the long run.
The rate of change of $K$ with respect to $t$ is given by
where $\alpha >0$ is the default growth rate of $K$, and $\beta >0$ is the perorganism rate of habitat destruction. An alternative interpretation of this equation is that $K$, the shortterm carrying capacity, is dependent on some essential nutrient in the environment, and that this nutrient is slowly depleted over time at a rate proportional to $\beta {n}_{2}$.
Let ${n}^{*}=\frac{\alpha}{\beta}$, the critical population level at which no habitat destruction occurs. $\frac{dK}{dt}$ is zero when ${n}_{2}={n}^{*}$, positive when $n}_{2}<{n}^{\ast$, and negative when $n}_{2}>{n}^{\ast$. $n}^{\ast$ can thus also be interpreted as the longterm carrying capacity. Clearly, if the longterm carrying capacity ${n}^{*}<A$, the only stable point of the system becomes ${n}_{2}=0$. Under this condition, colonialism is a losing strategy as well.
Note that ${g}_{2}$ increases as $K$ increases, and that $K$ increases more quickly as ${n}_{2}$ decreases. In the context of Parrondo’s paradox, colonialism can thus serve as a ‘ratcheting’ strategy, or Game B, because the rate of growth is implicitly dependent upon the colonial population in the past. Another way of understanding the ‘ratcheting’ behavior is through the lens of positive reactivity (Williams and Hastings, 2011; Hastings, 2001, Hastings, 2004). In the shortterm, ${n}_{2}=A$ is a positively reactive equilibrium, because small upwards perturbations of ${n}_{2}$ away from $A$ will result in rapid growth towards $K$ before a slow decrease back down towards $A$.
Behavioral switching
Organisms are able to detect the amount of environmental resources available to them, and by proxy, the carrying capacity of the population. Thus, they can undergo behavioral changes in response to the current carrying capacity.
Here, we model organisms that switch to nomadic behavior from colonial behavior when the carrying capacity is low ($K\text{}\text{}{L}_{1}$), and switch to colonial behavior from nomadic behavior when the carrying capacity is high ($K\text{}\text{}{L}_{2}$), where ${L}_{1}\le {L}_{2}$ are the switching levels. Let ${r}_{s}\text{}\text{}0$ be the switching constant. Using the notation from Equation 1, switching rates can then be expressed as follows:
A variety of mechanisms might trigger this switching behavior in biological systems. For example, since the nomadic organisms are highly mobile, they could frequently reenter their original colonial habitat after leaving it, and thus be able to detect whether resource levels are high enough for recolonization. It should also be noted that the decision to switch need not always be ‘rational’ (i.e. result in a higher growth rate) for each individual. Switching behavior could be genetically programmed, such that ‘involuntary’ individual sacrifice ends up promoting the longterm survival of the species.
Reduced parameters
Without loss of generality, we scale all parameters such that $\alpha =\beta =1$. Equation 4 thus becomes:
Hence, ${n}^{*}=\frac{\alpha}{\beta}=1$. All other population sizes and capacities can then be understood as ratios with respect to this critical population size. Additionally, since $\beta =1$, ${r}_{1}$, ${r}_{2}$ and ${r}_{s}$ can be understood as ratios to the rate of habitat destruction. For example, if ${r}_{2}\gg 1$, this means that colonial growth occurs much faster than habitat destruction. Timescale separation between the population growth dynamics and the habitat change dynamics can thus be achieved by setting ${r}_{1},{r}_{2}\gg 1$. Similarly, the separation between the behavioral switching dynamics and the population growth dynamics can be achieved by setting ${r}_{s}\gg {r}_{1},{r}_{2}$.
Results
Simulation results revealed population dynamics that could be categorized into the following regimes:
Without behavioral switching (${r}_{s}=0$)
Extinction for both subpopulations
Extinction for nomadic organisms, survival for colonial organisms
With behavioral switching (${r}_{s}\text{}\text{}0$)
Extinction for both subpopulations
Survival through periodic behavioral alternation
Longterm growth through strategic alternation
Importantly, there were conditions under which both subpopulations would go extinct in the absence of behavioral switching (regime 1a), but collectively survive if behavioral switching was allowed (regime 2b), thereby exhibiting Parrondo’s paradox. The following sections describe the listed regimes in greater detail, with a focus upon the regimes involved in the paradox. Figures generated via numerical simulation are provided as examples of behavior within each regime.
Extinction in the absence of switching
As described earlier, both nomadic and colonial behaviors can be modelled as losing strategies given the appropriate parameters. Simulations across a range of parameters elucidated the conditions which resulted in extinction for both strategies. Figure 1a shows an example when both strategies are losing, resulting in extinction, while Figure 1b shows an example where only the colonial subpopulation survives.
It is clear from Equation 2 that the growth rate of the nomadic population ${n}_{1}$ is always negative, because of the restriction that ${r}_{1}\text{}\text{}0$. Hence, nomadism is always a losing strategy.
However, the conditions under which colonial behavior is a losing strategy are more complicated. Complex dynamics occur when the critical capacity $A$ is just below 1 that can result in either survival or extinction. Nonetheless, it can be shown that when $A\text{}\text{}1$, extinction occurs (as in Figure 1a), and that survival is only possible when $A$ is significantly less than 1 (as in Figure 1b). That is:
The intuition behind this is straightforward. Suppose that initially, $A\text{}\text{}{n}_{2}\text{}\text{}K$, so that the growth rate is positive. When $A\text{}\text{}1$, the colonial population ${n}_{2}$ increases until it reaches the carrying capacity $K$, following which they converge in tandem until stabilizing at the critical population size, ${n}_{2}=K=1$. However, when $A\text{}\text{}1$, ${n}_{2}=K=1$ is no longer a stable equilibrium, since $d{n}_{2}/dt\text{}\text{}0$ when ${n}_{2}\text{}\text{}A$, resulting in the eventual extinction of the population. For a formal proof, refer to Theorem A.3.
Survival through periodic alternation
We now restrict our analysis to the case where $A\text{}\text{}1$. Under this condition, both nomadism (Game A) and colonialism (Game B) are losing strategies when played individually. Paradoxically, it is possible to combine these two strategies through behavioral switching such that population survival is ensured, thereby producing an overall strategy that wins.
Simulation results over a range of parameters have predicted this paradoxical behavior, and also elucidated the conditions under which it occurs. Figure 2a is a typical example where the population becomes extinct, even though it undergoes behavioral switching, while Figure 2b is a typical example where behavioral switching ensures population survival.
Conceptually, this paradoxical survival is possible because the colonial strategy, or Game B, is historydependent. In particular, the colonial growth rate $d{n}_{2}/dt$ is dependent upon the carrying capacity $K$, which in turn is dependent upon previous levels of ${n}_{2}$. Behavioral switching to a nomadic strategy decreases the colonial population size, allowing the resources in the colonial environment, represented by $K$, to recover. Switching back to a colonial strategy then allows the population to take advantage of the newly generated resources. Because switching occurs periodically, as can be seen in Figure 2b, it should be noted that the organisms need not even detect the amount of resources present in the environment to implement this strategy. A biological clock would be sufficient to trigger switching behavior.
The exact process by which survival is ensured can be understood by analysing the simulation results in detail. In the nomadic phase, the colonial population ${n}_{2}$ is close to zero, the nomadic population ${n}_{1}$ undergoes slow exponential decay, and the carrying capacity $K$ undergoes slow linear growth. $K$ increases until it reaches ${L}_{2}$, which triggers the switch to colonial behavior.
The population thus enters the colonial phase. If the colonial population ${n}_{2}$ exceeds the critical capacity $A$ at this point, then ${n}_{2}$ will grow until it slightly exceeds the carrying capacity $K$. Subsequently, ${n}_{2}$ decreases in the tandem with $K$ until $K$ drops to ${L}_{1}$, triggering the switch back to the nomadic phase. However, if ${n}_{2}\text{}\text{}A$ when the colonial phase begins, the colonial population goes extinct, as can be seen in Figure 2a. Hence, a basic condition for survival is that ${n}_{2}\ge A$ at the start of each colonial phase.
This implies that, by the end of the nomadic phase, ${n}_{1}$ needs to be greater by a certain amount than $A$ as well. Otherwise, there will be insufficient nomads to form a colony which can overcome the Allee effect. Under the reasonable assumption that the rate of behavioral switching is much faster than either colonial or nomadic growth (${r}_{s}\gg {r}_{1},{r}_{2}$), it can be shown more precisely that at the end of the nomadic phase, ${n}_{1}$ needs to be greater than a critical level $B$, which is related to $A$ by the equation:
A full derivation is provided in the Appendix (Theorem A.4). Here, ${W}_{0}\left(x\right)$ is the principal branch of the Lambert W function. Qualitatively speaking, $B$ is a function of $A$ on the interval $(1,\mathrm{\infty})$ that increases in an exponentiallike manner, and that approaches 1 when $A$ does as well. Thus, $B\ge A$, as expected.
The greater the difference between the switching levels, the longer the nomadic phase will last, because it takes more time for $K$ to increase to the requisite value for switching, ${L}_{2}$. And the longer the nomadic phase lasts, the more ${n}_{1}$ will decay. If, at the end of the nomadic phase, the value that ${n}_{1}$ decays to happens to be less than $B$, then the population will fail to survive. It follows that there should be some constraint on the difference between the switching levels ${L}_{1}$ and ${L}_{2}$.
Under the same assumption that ${r}_{s}\gg {r}_{1},{r}_{2}$, such a constraint can be derived:
Survival is ensured given the following additional condition:
where ${t}_{0}$ marks the start of an arbitrary colonial phase, and ${t}^{*}$ marks the time of intersection between ${n}_{2}$ and $K$ during that phase. In other words, ${n}_{2}$ has to grow sufficiently quickly during the colonial phase such that it exceeds both $K$ and ${L}_{1}$ before switching begins. This can be seen occurring in Figure 2b. In accordance with intuition, numerical simulations predict that this occurs when the colonial growth constant is sufficiently large $\left({r}_{2}\gg {r}_{1}\right)$, as can be seen in the Figures. (The Figures also show that ${r}_{1}$ close to 1, but this is not strictly necessary.) Collectively, Equations 10–11 are sufficient conditions for population survival. Mathematical details are provided in the Appendix (Theorems A.5 and A.6).
Note that Equation 10 contains an implicit lower bound on ${L}_{1}$. Since ${L}_{2}\ge {L}_{1}$ by stipulation, we must have $\mathrm{ln}\left[{L}_{1}+{W}_{0}\left({L}_{1}{e}^{{L}_{1}}\right)\right]\text{}\text{}\mathrm{ln}\text{}B$ for survival. The following bound is thus obtained:
On the other hand, under the assumptions made, there is no upper bound for ${L}_{1}$, and hence no absolute upper bound for ${L}_{2}$ either. This suggests that given a sufficiently welldesigned switching rule, $K$ can grow larger over time while ensuring population survival. Such a rule is investigated in the following section.
Longterm growth through strategic alternation
Suppose that, in addition to being able to detect the colonial carrying capacity, nomads and colonists are able to detect or estimate their current population size. This might happen by proxy, by communication, or by builtin estimation of the time required for growth or decay to a certain population level. The following switching rule then becomes possible:
That is, ${L}_{1}$ is set to the carrying capacity $K$ whenever ${n}_{2}$ rises to $K$, resulting immediately in a switch to nomadic behavior, and that ${L}_{2}$ is in turn set to $K$ whenever ${n}_{1}$ falls to $B$, resulting in an immediate switch to colonial behavior.
This switching rule is optimal according to several criteria. Firstly, by switching to nomadic behavior just as ${n}_{2}$ reaches $K$, it ensures that $d{n}_{2}/dt\ge 0$ for the entirety of the colonial phase. As such, it avoids the later portion of the colonial phase where $K$ and ${n}_{2}$ decrease in tandem, and maximizes the ending value ${n}_{2}$. Consequently, it also maximizes the value of ${n}_{1}$ at the start of each nomadic phase.
Furthermore, by switching to colonial behavior right when ${n}_{1}$ decays to $B$, the rule maximizes the duration of the nomadic phase while ensuring survival. This in turn means that the growth of $K$ is maximized, since the longer the nomadic phase, the longer that $K$ is allowed to grow.
In fact, this switching rule is a paradigmatic example of how Parrondo’s paradox can be achieved. It plays Game A, the nomadic strategy, for as long as possible, in order to maximize $K$ and hence the returns from Game B. And then it switches to Game B, the colonial strategy, only for as long as the returns are positive ($d{n}_{2}/dt\text{}\text{}0$), thereby using it as a kind of ratchet.
Suppose that $K$ grows more during each nomadic phase than it falls during each colonial phase. Then the switching rule is not just optimal, but it also enables longterm growth. Simulation results predict that this can indeed occur. Figure 3a shows longterm growth of $K$ from $t=0$ to $t=10$, while Figure 3b shows that with the same initial conditions, this continues until $t=300$ with no signs of abating. Together with $K$, the perphase maximal values of ${n}_{1}$ and ${n}_{2}$ increase as well.
In the cases shown, longterm growth is achieved because $K$ indeed grows more during each nomadic phase than it falls during the subsequent colonial phase. As can be seen from Figure 3a, this is, in turn, because the nomadic phase lasts much longer than the colonial phase, such that the amount of environmental destruction due to colonialism is limited. Simulation results predict that this generally occurs as long as the colonial growth rate is sufficiently large (${r}_{2}\gg {r}_{1}$).
An interesting phenomenon that can be observed from Figure 3b is how the nomadic population size ${n}_{1}$, which peaks at the start of each nomadic phase, eventually exceeds the carrying capacity $K$, and then continues doing so by increasing amounts at each peak. This is, in fact, a natural consequence of the population model. When ${n}_{2}$ grows large, the assumption that switching is much faster than colonial growth starts to break down. This occurs even though ${r}_{s}\gg {r}_{2}$, due to the increasing contribution of the $\left(\frac{{n}_{2}}{A}1\right)$ factor in Equation 3.
The result is that when a large colonial population begins switching to nomadism, a significant number of colonial offspring are simultaneously being produced. These offspring also end up switching to a nomadic strategy, resulting in more nomadic organisms than there were colonial organisms before. A particularly pronounced example of this is shown in Figure 4.
However, this same phenomenon also introduces a limiting behavior to the pattern of longterm growth. As Figure 5 shows, when the same simulation as in Figure 3a and b is continued to $t=1000$, peak levels of ${n}_{1}$, ${n}_{2}$ and $K$ eventually plateau around $t=650$.
This occurs because sufficiently high levels of ${n}_{2}$ cause a qualitative change in the dynamics of behavioral switching. Normally, switching to nomadic behavior starts when $K$ falls below ${L}_{1}$, and ends when $K$ rises above it again. $K$ rises towards the end of the switch, when ${n}_{2}$ levels fall below the critical level of ${n}^{*}=1$. But when ${n}_{2}$ is sufficiently large, the faster production of colonial offspring drags out the duration of switching, as seen in Figure 4. The higher levels of ${n}_{2}$, combined with the longer switching duration, causes an overall drop in $K$ by the end of the switching period. Because the increase in $K$ during the subsequent nomadic phase is unable to overcome this drop, $K$ stops increasing in the longrun.
Nonetheless, it is clear that significant longterm gains can be achieved via the optimal switching rule. Under the conditions of fast colonial growth and even faster switching (${r}_{s}\gg {r}_{2}\gg {r}_{1}\simeq 1$, as in Figures 3a–5), these gains are several orders of magnitude larger than the initial population levels, a huge departure from the longterm extinction that occurs in purely colonial or nomadic populations. Limiting behavior eventually emerges, but this is to be expected in any realistic biological system.
Survival and growth under additional constraints
Our proposed model is convenient for the functional understanding of growth and survival, and can be easily modified for a variety of applications. Additional constraints can be imposed under which survival and longterm growth are still observed. For example, in many biological systems, the dynamics of habitat change might occur on a slower timescale than both colonial and nomadic growth (i.e. ${r}_{1},{r}_{2}\gg 1$). Figure 6 shows the simulation results when this timescale separation exists (${r}_{1}=10$, ${r}_{2}=100$ for (a), ${r}_{1}=100$, ${r}_{2}=1000$ for (b)). It can clearly be seen that survival is still possible under such conditions.
Another practical constraint that can be imposed is limiting the growth of the carrying capacity to some maximal value $K}_{\mathrm{m}\mathrm{a}\mathrm{x}$, capturing the fact that the resources in any one habitat do not grow infinitely large. This can be achieved by modifying Equation 6 as follows:
Figure 6 already takes this constraint into account, showing that survival through periodic alternation is achievable under both bounded carrying capacity and slow habitat change, as long as the maximum carrying capacity is sufficiently high (${K}_{\mathrm{m}\mathrm{a}\mathrm{x}}=20$). As Figure 7 shows, even longterm growth is possible, under both fast habitat change (Figure 7a) and slow habitat change (Figure 7b). In both cases, the carrying capacity $K$ converges towards a maximum value as it approaches $K}_{max$.
Discussion
The results presented in this study demonstrate the theoretical possibility of Parrondo’s paradox in an ecological context. Many evolutionary strategies correspond to the strategies that we have termed here as ‘nomadism’ and ‘colonialism’. In particular, any growth model that is devoid of competitive or collaborative effects is readily captured by Equation 2 (nomadism), while any logistic growth model which includes both the Allee effect and habitat destruction can be described using Equations 3 and 4 (colonialism). Many organisms also exhibit behavioral change or phenotypic switching in response to changing environmental conditions. By incorporating this into our model, we have demonstrated that nomadiccolonial alternation can ensure the survival of a species, even when nomadism or colonialism alone would lead to extinction. Furthermore, it has been demonstrated that an optimal switching rule can lead to longterm population growth.
The switching rules which lead to survival and longterm growth are analogous to the periodic alternation between games that produces a winning expectation in Parrondo’s paradox. If one views the carrying capacity $K$ as the capital of the population, then it is clear that Equation 5 is a capitaldependent switching rule. By setting the appropriate amounts of capital at which switching should occur, survival and growth can be achieved. Survival is achieved by ensuring that Game A, or nomadism, is never played beyond the point where extinction is inevitable, that is, the point where ${n}_{1}$ falls below the critical level $B$. Longterm growth is additionally achieved by ensuring that Game B, or colonialism, is only played in the region where gains are positive, that is, when $A\text{}\text{}{n}_{2}\text{}\text{}K$ such that $d{n}_{2}/dt\text{}\text{}0$. The historydependent dynamics of Game B are thus optimally exploited.
Several limitations of the present study should be noted. Firstly, the study only focuses on cases where nomadism and colonialism are individually losing strategies, despite the abundance of similar strategies that do not lose in the real world. This is because assuming individually losing strategies in fact leads to a stronger result – if losing variants of nomadism and colonialism can be combined into a winning strategy, it follows that nonlosing variants can be combined in a similar way too (see Theorem A.7 in the Appendix).
Secondly, the population model does not encompass all variants of qualitatively similar behavior. For example, many other equations can be used to model the Allee effect (Boukal and Berec, 2002). Nonetheless, our proposed model is general enough that it can be adapted for use with other equations and be expected to produce similar results. Even the presence of the Allee effect is not strictly necessary, since the colonial population might die off at low levels because of stochastic fluctuations, rather than because of the effect. Theorem A.7 in the Appendix also demonstrates that paradoxical behavior can occur even without the Allee effect causing longterm death of the colonial population.
Thirdly, though it is trivially the case that pure nomadism and pure colonialism cannot outcompete a behaviorallyswitching population, a more complex analysis of the evolutionary stability of behavioral switching is beyond the scope of this paper. Finally, spatial dynamics are not accounted for in this study. Exploring such dynamics is a goal for future work.
Materials and methods
Numerical simulations were performed using code written in MATLAB (Source code 1) that relied on the ode23 ordinary differential equation (ODE) solver. ode23 is an implementation of an explicit RungeKutta (2,3) pair of Bogacki and Shampine. Simulations were performed with both behavioral switching turned off (${r}_{s}=0$) and turned on (${r}_{s}\text{}\text{}0$). The accuracy of the simulation was continually checked by repeating all results with more stringent tolerance levels, ensuring that the final simulated parameters did not change significantly (by less than 1%). Both the relative error tolerance and absolute error tolerance were determined to be 10^{9}.
In the case of complex switching rules like Equation 13 that required modifying differential equation parameters at specific time points, the Events option of MATLAB’s ODE solvers was used to detect when these points occurred. After each detection, the parameters were automatically modified as per the switching rule, and the simulation continued with the new parameters.
Broad regimes of model behavior were observed by running simulations across a wide range of parameters and initial conditions. General trends and conditions observed within each regime were formalized analytically, the details of which can be found in the Appendix. In these derivations, reasonable assumptions were made in order to make the model analytically tractable. In particular, it was assumed that the rate of behavioral switching was much faster than the rates of either colonial or nomadic growth (${r}_{s}\gg {r}_{1},{r}_{2}$), and that colonial growth rates were in turn much faster than the rate of habitat destruction (${r}_{2}\gg 1$). Initial conditions corresponding to unstable equilibria (e.g. ${n}_{2}=K=1\text{}\text{}A$) were avoided as unrealistic.
Conclusion
Request a detailed protocolOur comprehensive model captures both capital and historydependent dynamics within a realistic ecological setting, thereby exhibiting Parrondo's paradox without the need for exogenous environmental influences. The possibility of an ecological Parrondo’s paradox has wideranging applications across the fields of ecology and population biology. Not only could it provide evolutionary insight into strategies analogous to nomadism, colonialism, and behavioral diversification, it potentially also explains why environmentally destructive species, such as Homo sapiens, can thrive and grow despite limited environmental resources. By providing a theoretical model under which such paradoxes occur, our approach may enable new insights into the evolution of cooperative colonies, as well as the conditions required for sustainable population growth.
Appendix 1
Switching dynamics
Lemma A.1
Let ${t}_{1}$ and ${t}_{2}$ respectively be the start and end of a period of switching from subpopulation $i$ to subpopulation $j$. Then either of the following must hold:
Proof. Switching to nomadism begins when $K$ falls below ${L}_{1}$ and stops when it rises above ${L}_{1}$ again. Similarly, switching to colonialism begins when $K$ rises above ${L}_{2}$ and stops when it falls below ${L}_{2}$ again. In both cases then:
Alternatively, before $K$ rises or falls back to the appropriate level, ${n}_{i}$ may go extinct, prematurely ending the switch, in which case ${n}_{i}\left({t}_{2}\right)=0$.
Theorem A.1 (Colonialism to nomadism)
Let $x$ be the colonial population at the end of a colonial phase, and $y$ be the colonial population at the start of the subsequent nomadic phase. Assuming that switching is much faster than colonial growth (${r}_{s}{n}_{2}\gg {g}_{2}\left({n}_{2}\right)$), $y$ can be expressed in terms of ${x}_{2}$ as
where ${W}_{0}$ is the principal branch of the Lambert W function. Let $z$ be the nomadic population at the start of the subsequent nomadic phase. If switching is also much faster than nomadic growth (${r}_{s}{n}_{1}\gg {g}_{1}\left({n}_{1}\right)$), and if the nomadic population is 0 before switching, we have
Note that $z$ is an increasing function of $x$, and that $z$ converges to $x$ as $x$ grows larger.
Proof. Since ${r}_{s}{n}_{2}\gg {g}_{2}\left({n}_{2}\right)$, we have $d{n}_{2}/dt={r}_{s}{n}_{2}$. Hence, ${n}_{2}\left(t\right)$ decays exponentially from $x$ to $y$ over a duration of $\frac{1}{{r}_{s}}\mathrm{l}\mathrm{n}\left(x/y\right)$. Substituting ${n}_{2}\left(t\right)$ into Equation A1 and solving, we have
This means that $x$ and $y$ are both solutions to the equation $C=w{e}^{w}$, where $C$ is some constant. The roots of this equation are given by the Lambert W function. Since $x\text{}\text{}y$ and both are real numbers, $x$ must lie on the ${W}_{1}$ branch, while $y$ lies on the principal ${W}_{0}$ branch. Equation A3 follows.
For Equation A4, if ${r}_{s}{n}_{1}\gg {g}_{1}\left({n}_{1}\right)$, all the new nomadic organisms must have switched over from colonialism. It follows that $z=xy$. This completes the proof.
Theorem A.2 (Nomadism to colonialism)
Let $x$ be the nomadic population at the end of a nomadic phase, and $y$ be the nomadic population at the start of the subsequent colonial phase. Assuming that switching is much faster than both nomadic and colonial growth (${r}_{s}{n}_{1}\gg {g}_{1}\left({n}_{1}\right),{r}_{s}{n}_{2}\gg {g}_{2}\left({n}_{2}\right)$ ), and that the colonial population is 0 before switching, $y$ can be expressed in terms of $x$ as
for $x\text{}\text{}1$. If $x\le 1$, then $y=0$, i.e., the nomads go extinct during switching. Let $z$ be the colonial population at the start of the subsequent colonial phase. We have
for $x\text{}\text{}1$, and $z=x$ otherwise. Note that in the first case, $z$ is an increasing function of $x$.
Proof. By our assumptions, we have $d{n}_{1}/dt={r}_{s}{n}_{1}$ and $d{n}_{2}/dt={r}_{s}{n}_{1}$. Hence, ${n}_{1}\left(t\right)$ decays exponentially from $x$ to $y$ over a duration of $\frac{1}{{r}_{s}}\mathrm{l}\mathrm{n}\left(x/y\right)$, and ${n}_{2}\left(t\right)$ increases correspondingly from 0 to $\left(xy\right)$. Substituting ${n}_{2}\left(t\right)$ into Equation A1 and solving, we have
Suppose that $x\text{}\text{}1$, $y$ can be expressed in terms of $x$ using the principal branch of the Lambert W function, and Equation A5 follows. If $x\le 1$, then there is no way to satisfy Equation A1 because ${n}_{2}$ will not exceed 1 during the switching period. It follows that ${n}_{1}$ must go extinct, in which case $y=0$. The results for $z$ follow since $z=xy$. This completes the proof.
Extinction and survival
Theorem A.3 (Colonists go extinct when $A\text{}\text{}1$)
When $A\text{}\text{}1$, pure colonialism leads to extinction in the long run (${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to \mathrm{\infty}}{n}_{2}\left(t\right)=0$), assuming initial conditions are not set to the unstable equilibrium ${n}_{2}=K=1<A$.
Proof. There are three possible equilibria for ${n}_{2}$, at which $d{n}_{2}/dt=0:\text{}{n}_{2}=0$, ${n}_{2}=A$, and ${n}_{2}=K$ (refer to Equation 3). However, when ${n}_{2}=A\text{}\text{}1$ or ${n}_{2}=K\text{}\text{}1$, we have $dK/dt=1{n}_{2}\text{}\text{}0.\text{}K$ decreases until ${n}_{2}\text{}\text{}K$, which implies $d{n}_{2}/dt\text{}\text{}0.\text{}{n}_{2}$ then decreases until ${n}_{2}\text{}\text{}A$, which implies convergence to 0 as $t\to \mathrm{\infty}$.
Similarly, if ${n}_{2}=K\text{}\text{}1$, then $dK/dt\text{}\text{}0$, such that ${n}_{2}\text{}\text{}K\text{}\text{}1\text{}\text{}A$ an infinitesimal moment later, which then leads to extinction. This leaves only ${n}_{2}=0$ and ${n}_{2}=K=1$ as equilibria, the latter of which is unstable to small perturbations of ${n}_{2}$ towards zero. This completes the proof.
Theorem A.4 (Survival imposes a lower bound on ${n}_{1}$)
For survival to occur, ${n}_{1}\left({t}_{N}\right)\text{}\text{}B$ is a necessary condition, where ${t}_{N}$ denotes the end of any nomadic phase, and $B$ is a critical level related to $A$:
Proof. Let ${t}_{C}$ denote the start of the subsequent colonial phase. We know that ${n}_{2}\left({t}_{C}\right)\text{}\text{}A$ is required for the new colony to survive. $B$ is defined using Equation A6 such that ${n}_{2}\left({t}_{C}\right)=A$ if ${n}_{1}\left({t}_{N}\right)=B$. Since ${n}_{2}\left({t}_{C}\right)$ is an increasing function of ${n}_{1}\left({t}_{N}\right)$, it follows that ${n}_{1}\left({t}_{N}\right)\text{}\text{}B$ in order for ${n}_{2}\left({t}_{C}\right)\text{}\text{}A$. This completes the proof.
Theorem A.5 (Survival imposes constraints on ${L}_{1}$ and ${L}_{2}$)
Under the reasonable assumptions that switching is much faster than colonial or nomadic growth (${r}_{s}{n}_{1}\gg {g}_{1}\left({n}_{1}\right),{r}_{s}{n}_{2}\gg {g}_{2}\left({n}_{2}\right)$), and that the difference between ${n}_{2}$ and $K$ is negligibly small once ${n}_{2}$ grows to exceed $K$ during the colonial phase, the following constraint on ${L}_{1}$ and ${L}_{2}$ is a necessary condition for survival through periodic alternation:
This also implies a lower bound on ${L}_{1}$, given by:
Proof. Consider a sequence of consecutive phases which we will term as CP_{1}, NP, CP_{2}, where CP_{1} and CP_{2} are colonial phases and NP is the nomadic phase between them. Since the nomadic phase NP is characterized by exponential decay, it is clear that $n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}\right)={n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right){e}^{{r}_{1}T$, where
is the duration of the nomadic phase. During this period, the colonial population ${n}_{2}$ is close to or equal to 0. Since $dK/dt=1{n}_{2}\le 1$, it follows that
Note that $K\left(\mathrm{N}\mathrm{P}\text{}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}\right)={L}_{1}$ and $K\left(\mathrm{N}\mathrm{P}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)={L}_{2}$ by definition. Combining this with the two equations above, we obtain
For survival to occur, the colonial population at the start of CP_{2} needs to be greater than $A$. By Theorem A.4, this implies that ${n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)\text{}\text{}B$. Substituting into the above, we have
Notice that by Theorem A.1, ${n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}\right)$ is a function of ${n}_{2}\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)$, the colonial population at the end of CP_{1}. If CP_{1} lasts long enough that ${n}_{2}$ exceeds $K$, then by assumption we have ${n}_{2}\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)\simeq K\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)$, otherwise we have ${n}_{2}\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)<K\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)$. Note that $K\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)={L}_{1}$. Thus, $n}_{2}\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)\le {L}_{1$. Then by Equation A4 in Theorem A.1, we have ${n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}\right)\le {L}_{1}+{W}_{0}\left({L}_{1}{e}^{{L}_{1}}\right)$. Substituting this into Equation A11 gives us Equation A8, as desired.
Finally, recall that ${L}_{2}>{L}_{1}$ by definition. It thus needs to be the case that
Solving this gives us the lower bound on ${L}_{1}$. This completes the proof.
Theorem A.6 (Survival is ensured if ${n}_{2}$ converges to $K$ quickly)
Under the reasonable assumption that ${n}_{2}\simeq 0$ during each nomadic phase, then together with Equation A8 as well as the assumptions made in A.5, the following constraint is sufficient to ensure survival:
where ${t}_{0}$ marks the start of an arbitrary colonial phase, and ${t}^{*}$ marks the time of intersection between ${n}_{2}$ and $K$ during that phase. In other words, ${n}_{2}$ has to grow sufficiently quickly during the colonial phase such that it exceeds both $K$ and ${L}_{1}$ before switching begins.
Proof. Let CP_{1} denote the colonial phase in question. Given Equation A12, all switching to the nomadic phase only happens after ${n}_{2}$ exceeds $K$ during CP_{1}. We have already presumed in Theorem A.5 that once ${n}_{2}$ grows to exceed $K$, we have ${n}_{2}K\ll 1$. This implies that $n}_{2}\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)\simeq K\left({\text{CP}}_{1}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)={L}_{1$, which in turn implies that ${n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}\right)\simeq {L}_{1}+{W}_{0}\left({L}_{1}{e}^{{L}_{1}}\right)$ by Theorem A.1.
Given the new assumption that ${n}_{2}\simeq 0$ during each nomadic phase, any value of ${L}_{2}$ that satisfies Equation A10 will be sufficient for survival. Substituting ${n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{e}\mathrm{n}\mathrm{d}\right)>B$ along with our new expression for ${n}_{1}\left(\mathrm{N}\mathrm{P}\text{}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}\right)$ into Equation A10 retains this property, which means that any value of ${L}_{2}$ that satisfies Equation A8 is sufficient for survival (as long as ${L}_{2}>{L}_{1}$). Thus, given our assumptions, Equation A8 and Equation A12 collectively ensure survival. This completes the proof.
Theorem A.7 (Behavioral alternation improves survival levels)
Suppose that pure nomadism or pure colonialism (or both) result in population survival. That is,
where $i=1$ or 2 or both. This can happen if ${r}_{1}=0$ (for nomadism) or $A<1$ (for colonialism). Then there are conditions under which behavioral alternation between the two pure strategies will result in a total population size with higher longterm periodic maxima. That is, given the appropriate conditions, then for any $t\ge 0$, there exists ${t}^{*}\ge t$ such that
for both $i=1$ and $i=2$.
Proof. As shown in the main text, there exist cases where periodic alternation between the levels $K={L}_{1}$ and $K={L}_{2}$ result in survival. Furthermore, there exists cases where ${n}_{2}\left({t}^{*}\right)=K\left({t}^{*}\right)\ge {L}_{1}$ for some ${t}^{*}$ during each colonial phase. For any such case where both nomadism and colonialism are losing strategies, notice that we would still have ${n}_{2}\left({t}^{*}\right)=K\left({t}^{*}\right)\ge {L}_{1}$ even if nomadism or colonialism were modified so that they did not lose.
To be precise, if ${r}_{1}=0$ (nomadism does not lose), then there would be no decay over each nomadic phase, which means that in the subsequent colonial phase, ${n}_{2}$ would start out higher and grow to reach $K$ even more quickly. If $A<1$ (colonialism does not lose), then ${g}_{2}\left({n}_{2}\right)$ would be even larger in magnitude, so ${n}_{2}$ would grow towards $K$ more quickly as well. In both cases, $K$ would decay less than it would have originally by the time ${n}_{2}\left({t}^{*}\right)=K\left({t}^{*}\right)$, so we would still have $n}_{2}\left({t}^{\ast}\right)\text{}\text{}{L}_{1$. It follows that as long as ${L}_{1}\text{}\text{}{n}_{i,\text{pure}}\left(\mathrm{\infty}\right)$, Equation A14 will always be true. Such cases can be easily found for $i=1$ (ensuring ${L}_{1}\text{}\text{}{n}_{1,pure}\left(0\right)$) and for $i=2$ (ensuring ${L}_{1}\text{}\text{}{n}_{2,pure}\left(\mathrm{\infty}\right)=1$). This completes the proof.
References

Asymmetry and disorder: A decade of parrondo'S PARADOXFluctuation and Noise Letters 09:129–156.https://doi.org/10.1142/S0219477510000010

Stochastic switching as a survival strategy in fluctuating environmentsNature Genetics 40:471–475.https://doi.org/10.1038/ng.110

Dictyostelium discoideuma model for many reasonsMolecular and Cellular Biochemistry 329:73–91.https://doi.org/10.1007/s1101000901118

Persistence of structured populations in random environmentsTheoretical Population Biology 76:19–34.https://doi.org/10.1016/j.tpb.2009.03.007

Singlespecies models of the allee effect: extinction boundaries, sex ratios and mate encountersJournal of Theoretical Biology 218:375–394.https://doi.org/10.1006/jtbi.2002.3084

Construction of novel stochastic matrices for analysis of Parrondo’s paradoxPhysica A: Statistical Mechanics and its Applications 392:4727–4738.https://doi.org/10.1016/j.physa.2013.05.048

Quantum models of Parrondo's gamesPhysica A: Statistical Mechanics and its Applications 324:152–156.https://doi.org/10.1016/S03784371(02)01909X

Brownian ratchets and Parrondo’s gamesChaos: An Interdisciplinary Journal of Nonlinear Science 11:705–714.https://doi.org/10.1063/1.1395623

A review of parrondo'S paradoxFluctuation and Noise Letters 2:R71–R107.https://doi.org/10.1142/S0219477502000701

Transients: the key to longterm ecological understanding?Trends in Ecology & Evolution 19:39–45.https://doi.org/10.1016/j.tree.2003.09.007

Effects of temporal variability on rare plant persistence in annual systemsThe American Naturalist 164:350–363.https://doi.org/10.1086/422859

Stabilizing multicellularity through ratchetingPhilosophical Transactions of the Royal Society B: Biological Sciences 371:20150444.https://doi.org/10.1098/rstb.2015.0444

New paradoxical games based on brownian ratchetsPhysical Review Letters 85:5226–5229.https://doi.org/10.1103/PhysRevLett.85.5226

Parrondo’s paradox and complementary Parrondo processesPhysica A: Statistical Mechanics and its Applications 392:17–26.https://doi.org/10.1016/j.physa.2012.08.006

Occurrence of complementary processes in parrondo’s paradoxPhysica A: Statistical Mechanics and its Applications 412:180–185.https://doi.org/10.1016/j.physa.2014.06.010

Coexistence mediated by recruitment fluctuations: A field guide to the storage effectThe American Naturalist 125:769–787.https://doi.org/10.1086/284379

Paradoxical persistence through mixedsystem dynamics: towards a unified perspective of reversal behaviours in evolutionary ecologyProceedings of the Royal Society B: Biological Sciences 278:1281–1290.https://doi.org/10.1098/rspb.2010.2074

Diversity in times of adversity: probabilistic strategies in microbial survival gamesJournal of Theoretical Biology 234:227–253.https://doi.org/10.1016/j.jtbi.2004.11.020

Punctuated evolution due to delayed carrying capacityPhysica D: Nonlinear Phenomena 238:1752–1767.https://doi.org/10.1016/j.physd.2009.05.011

Population dynamics with nonlinear delayed carrying capacityInternational Journal of Bifurcation and Chaos 24:1450021.https://doi.org/10.1142/S0218127414500217
Decision letter

Michael DoebeliReviewing Editor; University of British Columbia, Canada
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Thank you for submitting your article "Nomadiccolonial alternation can enable population growth despite habitat destruction: An ecological Parrondo's paradox" for consideration by eLife. Your article has been favorably evaluated by Ian Baldwin (Senior Editor) and three reviewers, one of whom is a member of our Board of Reviewing Editors. The reviewers have opted to remain anonymous.
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
All reviewers agree that this is an interesting paper that introduces Parrondo's paradox to an ecological setting. In this setting, the paradox consists of the fact that even though each of two subpopulations go extinct when left alone, migration between the subpopulations can lead to persistence. The mechanism enabling persistence is that one of the habitats can regenerate itself while the population spends time declining in the other habitat, so that after some time the regenerated habitat can harbor a growing population again for some time, before that habitat deteriorates again, and so on. I think this phenomenon is worth being brought to the attention of ecologists, and will likely enhance the conceptual toolbox of people concerned with conservation issues.
As you will see the reviewers have raised a number of concerns about the paper. Chief among those is the biological realism and relevance of some of the model assumptions and interpretations. These concerns need to be addressed in a major revision. After we have received the revision, we will make a decision as to the suitability of your paper for eLife based on how each of the following points were addressed:
1) There were concerns about the realism of some of the modelling assumptions. Specifically, the paper assumes that the carrying capacity in the habitat of the colonizers changes on the same rapid ecological time scale as the population density itself changes. This seems unrealistic. Is it possible to formulate an alternative model, e.g. with a time scale separation between population dynamics and the dynamics of the carrying capacity? If not, what is the rationale behind assuming the same time scales for those two dynamics?
2) In addition, the assumptions about the dynamics of the carrying capacity imply that in the absence of colonizers (i.e., if the population density of the colonizers is 0), the carrying capacity increases without bounds. Clearly, this is not realistic, and the question is whether it is possible to obtain the same results with more realistic assumptions, which would e.g. ensure that the carrying capacity saturates at some level even in the absence of colonizers.
3) As referee 1 points out, the assumed switching behavior may also not be very realistic. At the very least, this kind of switching behavior needs to be justified biologically. Are there other, biologically more relevant switching behaviors that would lead to the same results?
4) In general, all other reviewer comments and concerns need to be addressed in a satisfactory manner before a final decision regarding publication can be made.
Reviewer #1:
This is an interesting paper that introduces Parrondo's paradox to an ecological setting. In this setting, the paradox consists of the fact that even though each of two subpopulations go extinct when left alone, migration between the subpopulations can lead to persistence. The mechanism enabling persistence is that one of the habitats can regenerate itself while the population spends time declining in the other habitat, so that after some time the regenerated habitat can harbour a growing population again for some time, before that habitat deteriorates again, and so on. I think this phenomenon is worth being brought to the attention of ecologists, and will e.g. enhance the conceptual toolbox of people concerned with conservation issues.
The paper appears to be well executed technically, but I do have some concerns regarding the biological significance of the work. Chief among those is the fact that the carrying capacity is assumed to change on the same time scale as the population density (i.e., the "t" in eqs (1) and (4) is the same). On intuitive grounds, I would expect that the dynamics of the carrying capacity to be much slower than the population dynamics. Also, in the absence of any population in the habitat in which the carrying capacity is changing, the carrying capacity will grow without bounds according to eq. (4), which is obviously unrealistic. I think one could obtain a biologically more reasonable model by assuming some sort of time scale separation between carrying capacity dynamics and population dynamics (e.g. by assuming that \α and \β in equation (4) are <<1), and by assuming that the carrying capacity saturates as the population size in the habitat goes to 0. The question would then be whether the phenomenon observed by the authors would still be present under such conditions.
I am also not really convinced by the proposed switching behaviour (subsection "Behavioral switching" under "Population model"). The authors purpose that individuals switch from nomadic to colonial behaviour when the carrying capacity for the colonial dynamics is high enough. But how would nomadic individuals be able to assess the carrying capacity for the colonial life style (I can understand how the colonial individuals would assess that carrying capacity)? The more elaborate switching rule given by (13) is even less intuitive: why would colonial individuals switch to monadic life style once they reach carrying capacity? After all, this means that individuals trade a per capita growth rate of 0 (because they treat carrying capacity) with a negative per capita growth rate (as is the case for nomadic life style by assumption). I think it would make more sense to switch once the colonial growth rate is "negative enough", e.g. more negative than the (constant) negative per capita growth rate on the nomadic life style. Again, the question then becomes whether persistence can still be observed under such more realistic assumptions.
Overall, I think the paper is interesting, but needs a much more careful treatment of the biological realism underlying various assumptions, and as a consequence, a much more careful discussion of the biological relevance of the results obtained.
Reviewer #2:
This is a cleverly laid out paper that will introduce a relatively new perspective to ecological and evolutionary biologists. That is, this paper lays out an example of how switching "colonial and nomadic" behavior can lead to persistence despite each singular case not allowing persistence. While I think the general idea of organisms responding to variation in a manner that yields persistence or coexistence is far from new to ecologists and evolutionary ecologists, the perspective put forth is interesting and novel. My guess is that there are numerous existing models that may, in hindsight and with a little work, fit into this "capital" or "history dependent" version of Parrondo's paradox. I do think people studying complex systems may not be overwhelmingly surprised by the notion that an overall average growth rate less than zero in two strategies, or two of whatever (say competitors), can respond to variation (environmentally driven or internally driven) in a manner that yields persistence or coexistence but, again, the paper framed within the Parrondo's paradox seems potentially profitable for understanding and looking for this type of persistence outcome.
I have no major comments and I think this paper is well written, well analyzed (at least for the ideas laid out), and so very close to publishable as is.
Reviewer #3
General Assessment:
The manuscript under review ("Nomadiccolonial alternation can enable population growth despite habitat destruction: An ecological Parrondo's paradox") presents and analyzes a novel model for a population where individuals are one of two behavioral types, which are modelled as subpopulations in an ODE model. Colonial types cooperate, compete and (over the long term) reduce their own carrying capacity. Nomads slowly die off (in the parameter space analyzed). Offspring of each type can "switch" to the other and in the model they do so based on the carrying capacity of the colonial subpopulation.
Through a careful and detailed analysis, the paper shows that the model demonstrates a paradoxical behavior: in situations where pure colonial or nomadic strategies would die out, switching can result in persistence. Such results are well known in ecology and evolution, but most often occur due to environmental variation [1]. Here, the results occur not because of environmental variation but due to the carryingcapacitydependent switching rule modeled. This provides a closer analog to Parrondo games than in many ecological studies, which (where it is biologically realistic) might help unify various studies of such reversal behaviors (as called for in [1]).
The particular situation studied in the model of alternation between nomadic and colonial types is of great evolutionary interest as it mirrors both the evolution of cooperative colonies from modular organisms and even the evolution of multicellularity itself. Indeed, the language of ratcheting has been invoked in describing the latter process [2]. More recently, that same group has examined the stabilization of multicellularity through ratcheting in a genetic model ([3] which should be cited and discussed in any revision).
The model in the manuscript under review, although not explicitly evolutionary, provides an interesting avenue through which these questions might be explored in the future. However, the behavior demonstrated rests on some peculiarities of the model and its assumptions.
Concerns:
Although promising, the model as developed displays some strange and possibly unrealistic behavior, which may be due to atypical assumptions. In particular, modelling the carrying capacity K as a function of the population itself (eqn 3) in a singlespecies model is not common in ecology (e.g., no such model appears in Table 1 of [4]). One reason may be that, as seen here, K will diverge to infinity when populations become small (e.g., if the colonists die off; Figure 1A). The dynamics of K also play a role in the strange behavior explored in Figures 3 and 4, where a vast overabundance of nomads is produced during a longlived period of switching, during which K is reduced. Because the switching dynamics (eqn 5) are based on the dynamics of K, and key to driving the analyzed behavior the generality of these behaviors is tied to the reasonableness and generality of this assumption.
[1] Paul David Williams and Alan Hastings. Paradoxical persistence through mixedsystem dynamics: towards a unified perspective of reversal behaviours in evolutionary ecology. Proceedings of the Royal Society of London B: Biological Sciences, page rspb20102074, 2011.
[2] Eric Libby and William C. Ratcliff. 346(6208):426427, 2014.
[3] Libby E, Conlin PL, Kerr B, Ratcliff WC. 2016 Stabilizing multicellularity through ratcheting. Phil. Trans. R. Soc. B 371: 20150444. http://dx.doi.org/10.1098/rstb.2015.0444
[4] Abbot, Ives. 2012 Single Species Population Models in "Encyclopedia of Theoretical Ecology" Hastings, Gross, Eds. UC Press.
https://doi.org/10.7554/eLife.21673.sa1Author response
[…] As you will see the reviewers have raised a number of concerns about the paper. Chief among those is the biological realism and relevance of some of the model assumptions and interpretations. These concerns need to be addressed in a major revision. After we have received the revision, we will make a decision as to the suitability of your paper for eLife based on how each of the following points were addressed:
1) There were concerns about the realism of some of the modelling assumptions. Specifically, the paper assumes that the carrying capacity in the habitat of the colonizers changes on the same rapid ecological time scale as the population density itself changes. This seems unrealistic. Is it possible to formulate an alternative model, e.g. with a time scale separation between population dynamics and the dynamics of the carrying capacity? If not, what is the rationale behind assuming the same time scales for those two dynamics?
We started with a simpler population model for the functional understanding of growth and survival, and ensured that this model could be easily modified for different applications. We have now added a new subsection “Survival and Growth under Additional Constraints”, implementing all the constraints suggested by the reviewers. In particular, subsection "Reduced parameters" and subsection "Survival and growth under additional constraints", first paragraph address how time scale separation between the population dynamics and the dynamics of the carrying capacity can be achieved by modifying the rate parameters of the existing model. We note that there are many species that exhibit nomadiclike behavior with growth dynamics that could easily occur on the same time frame as habitat change. The spores of fungi and slime molds are one such example. Nonetheless, to demonstrate the generalizability of our model, we performed new simulations using r_{2} >> 1 (r_{2} = 100), and r_{1} >> 1 (r_{1} = 10), such that the rate of both colonial growth and nomadic decay were at least an order of magnitude faster than the rate of habitat destruction. The corresponding results are shown in subsection "Survival and growth under additional constraints" as well. Similar results were also obtained for r_{1} = 100 and r_{2} = 1000. As can be seen from the results, both survival and longterm growth are still possible under the suggested time scale separation. These new results would satisfactorily address the concerns of the reviewers.
2) In addition, the assumptions about the dynamics of the carrying capacity imply that in the absence of colonizers (i.e., if the population density of the colonizers is 0), the carrying capacity increases without bounds. Clearly, this is not realistic, and the question is whether it is possible to obtain the same results with more realistic assumptions, which would e.g. ensure that the carrying capacity saturates at some level even in the absence of colonizers.
Yes, similar results can be obtained even without the carrying capacity increasing without bounds. Using a logisticlike equation (Equation 14), we performed additional simulations that limited the growth of the carrying capacity to some maximal value K_{max}. Please refer to the new subsection "Survival and growth under additional constraints" and Figures 6 and 7. For sufficiently high values of K_{max}, both survival and longterm growth remain possible.
3) As referee 1 points out, the assumed switching behavior may also not be very realistic. At the very least, this kind of switching behavior needs to be justified biologically. Are there other, biologically more relevant switching behaviors that would lead to the same results?
Response: A variety of mechanisms might trigger the assumed switching behavior in biological systems. For example, since the nomadic organisms are highly mobile, they could frequently reenter their original colonial habitat after leaving it, and thus be able to detect whether resource levels are high enough for recolonization. Alternatively, in the case of survival through periodic alternation, a biological clock would be sufficient to implement the periodic switching strategy, eschewing the need to detect resource levels altogether. It should also be noted that the decision to switch need not always be 'rational' (i.e. result in a higher growth rate) for each individual. Switching behavior could instead be genetically programmed (hardwired), such that 'involuntary' individual sacrifice ends up promoting the long term survival of the species. How this sort of 'selfless' behavior might have evolved is beyond the scope of our study. We simply note that sacrificial behavior is both possible and common in nature, as in the case of cellular slime molds which sacrifice themselves when collectively forming a fruiting body. Subsection "Behavioral switching", last paragraph and subsection "Survival through periodic alternation", third paragraph now include our justification.
4) In general, all other reviewer comments and concerns need to be addressed in a satisfactory manner before a final decision regarding publication can be made.
Reviewer #1:
[…] The paper appears to be well executed technically, but I do have some concerns regarding the biological significance of the work. Chief among those is the fact that the carrying capacity is assumed to change on the same time scale as the population density (i.e., the "t" in eqs (1) and (4) is the same). On intuitive grounds, I would expect that the dynamics of the carrying capacity to be much slower than the population dynamics.
As noted above, time scale separation between the population dynamics and the dynamics of the carrying capacity can be achieved by modifying the rate parameters of the existing model. Specifically, this can be achieved by setting r_{1} >> 1 and r_{2} >> 1, such that colonial growth and nomadic decay occur much more rapidly than habitat change. Subsection "Reduced parameters" and subsection "Survival and growth under additional constraints", first paragraph now address this point. As can be seen from the new results in subsection "Survival and growth under additional constraints", both survival and longterm growth are still possible under the suggested time scale separation.
Also, in the absence of any population in the habitat in which the carrying capacity is changing, the carrying capacity will grow without bounds according to eq. (4), which is obviously unrealistic. I think one could obtain a biologically more reasonable model by assuming some sort of time scale separation between carrying capacity dynamics and population dynamics (e.g. by assuming that \α and \β in equation (4) are <<1), and by assuming that the carrying capacity saturates as the population size in the habitat goes to 0. The question would then be whether the phenomenon observed by the authors would still be present under such conditions.
We thank reviewer #1 for the suggestions, which we have since taken into account in subsection "Survival and growth under additional constraints". Time scale separation has already been addressed above. Saturation of the carrying capacity was implemented by modifying Equation 6 (the reduced version of Equation 4) to incorporate logisticlike behavior, as stated in Equation 14. This limits the growth of the carrying capacity to a maximal value K_{max}, capturing the fact that the resources do not grow infinitely large.
Figure 6 takes this constraint into account, showing that survival through periodic alternation is achievable under both bounded carrying capacity and slow habitat change, as long as the maximum carrying capacity is sufficiently high (K_{max} = 20). As Figure 7 shows, longterm growth is also possible under these constraints, with the carrying capacity K converging towards a maximum value as it approaches K_{max}. Please refer to subsection "Survival and growth under additional constraints" for a full description of these results.
I am also not really convinced by the proposed switching behaviour (subsection "Behavioral switching"). The authors purpose that individuals switch from nomadic to colonial behaviour when the carrying capacity for the colonial dynamics is high enough. But how would nomadic individuals be able to assess the carrying capacity for the colonial life style (I can understand how the colonial individuals would assess that carrying capacity)?
As noted above, a variety of mechanisms might trigger the assumed switching behavior. For example, since the nomadic organisms are highly mobile, they could frequently reenter their original colonial habitat after leaving it, and thus be able to detect whether resource levels are high enough for recolonization. This is now explained in subsection "Behavioral switching" , last paragraph.
Alternatively, in the case of survival through periodic alternation, a biological clock would be sufficient to implement the periodic switching strategy, eschewing the need to detect resource levels altogether. This is now discussed in subsection "Survival through periodic alternation", third paragraph. Both colonial and nomadic organisms might simply be ‘hardcoded’ to switch behaviors after a certain amount of time, generating periodic behavior that synchronizes with the regeneration of the colonial environment. This could well explain the existence of multiform lifecycles like that of jellyfish, which alternate between colonial polyps and nomadic medusa.
The more elaborate switching rule given by (13) is even less intuitive: why would colonial individuals switch to monadic life style once they reach carrying capacity? After all, this means that individuals trade a per capita growth rate of 0 (because they treat carrying capacity) with a negative per capita growth rate (as is the case for nomadic life style by assumption). I think it would make more sense to switch once the colonial growth rate is "negative enough", e.g. more negative than the (constant) negative per capita growth rate on the nomadic life style. Again, the question then becomes whether persistence can still be observed under such more realistic assumptions.
We thank reviewer #1 for bringing up this concern. As noted earlier, this concern is only valid if we consider the colonial organisms to be selfish 'rational agents' which never choose to take a loss. This is now addressed in subsection "Behavioral switching", last paragraph. It is common knowledge that seemingly irrational 'selfless' behavior can be genetically programmed (hardcoded), so it is not unrealistic that colonial organisms might trade a temporary decrease in growth rate for a longterm gain. For example, cellular slime molds sacrifice themselves during the formation of a colonial fruiting body, even though it is not guaranteed that the spores released by the fruiting body will immediately encounter more favorable conditions. As such, we assess that it is entirely possible that colonial organisms might trade a growth rate of zero for a potentially negative one.
How such 'selfless' behavior could have evolved is beyond the scope of our current study, but is a potential area for future work. One possibility is that nomadic growth rates are not always negative, which could drive evolution to locally optimize towards switching to nomadic behavior when the colonial carrying capacity is reached. Then in harsh times, even when nomadic growth rates are negative, the species would still be able to survive and grow by employing the same switching strategy. As noted in "Discussion", fourth paragraph, and in Appendix Theorem A.7, our proposed switching strategy is beneficial for the species regardless of whether nomadism is modelled as a losing game or not.
Overall, I think the paper is interesting, but needs a much more careful treatment of the biological realism underlying various assumptions, and as a consequence, a much more careful discussion of the biological relevance of the results obtained.
We started with a simpler population model in order to better elucidate the basic mechanisms by which Parrondo’s paradox can occur in species with both nomadic and colonial behaviors. A minimal amount of constraints were imposed to demonstrate the paradox, and so the theoretical results obtained are widely generalizable. As we now show in subsection "Survival and growth under additional constraints", when imposing additional constraints to more accurately model features of certain biological systems, the paradox of survival and longterm growth can still be observed. We believe that we have now satisfactorily responded to the comments made by our reviewers.
Reviewer #3:
General Assessment:
The manuscript under review ("Nomadiccolonial alternation can enable population growth despite habitat destruction: An ecological Parrondo's paradox") presents and analyzes a novel model for a population where individuals are one of two behavioral types, which are modelled as subpopulations in an ODE model. Colonial types cooperate, compete and (over the long term) reduce their own carrying capacity. Nomads slowly die off (in the parameter space analyzed). Offspring of each type can "switch" to the other and in the model they do so based on the carrying capacity of the colonial subpopulation.
Through a careful and detailed analysis, the paper shows that the model demonstrates a paradoxical behavior: in situations where pure colonial or nomadic strategies would die out, switching can result in persistence. Such results are well known in ecology and evolution, but most often occur due to environmental variation [1]. Here, the results occur not because of environmental variation but due to the carryingcapacitydependent switching rule modeled. This provides a closer analog to Parrondo games than in many ecological studies, which (where it is biologically realistic) might help unify various studies of such reversal behaviors (as called for in [1]).
Thank you reviewer #3 for the close reading. Indeed, many biological studies have drawn a connection to this paradox but often do not incorporate the capitaldependence and historydependence which are characteristic of Parrondo games. The paradox often emerges due to the presence of exogenous environmental variation. The main contributions of our model are in that it captures capital and historydependent dynamics within a realistic ecological setting, thereby exhibiting the paradox without the need for exogenous environmental influences.
The particular situation studied in the model of alternation between nomadic and colonial types is of great evolutionary interest as it mirrors both the evolution of cooperative colonies from modular organisms and even the evolution of multicellularity itself. Indeed, the language of ratcheting has been invoked in describing the latter process [2]. More recently, that same group has examined the stabilization of multicellularity through ratcheting in a genetic model ([3] which should be cited and discussed in any revision).
[Libby et al., 2016] ‘Stabilizing multicellularity through ratcheting’ is a highly relevant study and we have included it as a reference to motivate our work. The work is discussed in the Introduction, third paragraph.
The model in the manuscript under review, although not explicitly evolutionary, provides an interesting avenue through which these questions might be explored in the future. However, the behavior demonstrated rests on some peculiarities of the model and its assumptions.
Concerns:
Although promising, the model as developed displays some strange and possibly unrealistic behavior, which may be due to atypical assumptions. In particular, modelling the carrying capacity K as a function of the population itself (eqn 3) in a singlespecies model is not common in ecology (e.g., no such model appears in Table 1 of [4]). One reason may be that, as seen here, K will diverge to infinity when populations become small (e.g., if the colonists die off; Figure 1A).
Thank you reviewer #3 for raising these concerns. Reasoning from first principles suggests that modelling the carrying capacity K as a function of the population is entirely within the realm of possibility, since organisms affect the environment that they live in. Another study that backs up this approach is Yukalov et al., 2009, which demonstrates how this sort of feedback can result in punctuated evolution and growth. Models where K depends on the population size can thus be highly productive for understanding complex ecological phenomena.
Nonetheless, we acknowledge that it is unrealistic for K to diverge towards infinity, assuming a finite habitat size. To address this concern, we have added a new subsection “Survival and Growth under Additional Constraints”, imposing all the constraints suggested by the reviewers. As noted in our response to reviewer #1, saturation of the carrying capacity was implemented by modifying Equation 6 (the reduced version of Equation 4) to incorporate logisticlike behavior. The new dynamics of K are stated in Equation 14. With this equation, the growth of K is limited to a maximal value K_{max}. K might thus be thought of as the shortterm carrying capacity, and K_{max} as the longterm carrying capacity.
It is important to note that when K_{max} >> K, Equation 14 reduces to Equation 6. This means that all of our previously presented results still hold (and are completely valid), given this condition. Even when K_{max} is set to be considerably lower (K_{max} = 20), Figures 6 and 7show that survival and longterm growth are still possible. We believe that these results should adequately address the reviewers’ concerns about biological realism.
Vyacheslav I. Yukalov, E. P. Yukalova, and Didier Sornette. Punctuated evolution due to delayed carrying capacity. Physica D: Nonlinear Phenomena, 238(17):1752–1767, 2009
The dynamics of K also play a role in the strange behavior explored in Figures 3 and 4, where a vast overabundance of nomads is produced during a longlived period of switching, during which K is reduced. Because the switching dynamics (eqn 5) are based on the dynamics of K, and key to driving the analyzed behavior the generality of these behaviors is tied to the reasonableness and generality of this assumption.
After limiting the growth of K to a maximal value K_{max}, the nomadic growth spikes observed in Figures 3 and 4 can still be observed in Figure 7A. We acknowledge that the presence of growth spikes may be an artifact of the specific model used, and may be unique to biological systems that adhere closely to such a model. However, this does not detract from our broader and more important conclusion that both survival and longterm growth are possible. As our new subsection "Survival and growth under additional constraints" now shows, these remain possible even under slightly different modelling assumptions. In sum, our findings still hold under a full simulation incorporating all of our earlier assumptions and conditions suggested by our reviewers.
https://doi.org/10.7554/eLife.21673.sa2Article and author information
Author details
Funding
No external funding was received for this work.
Reviewing Editor
 Michael Doebeli, University of British Columbia, Canada
Publication history
 Received: September 20, 2016
 Accepted: January 11, 2017
 Accepted Manuscript published: January 13, 2017 (version 1)
 Version of Record published: February 21, 2017 (version 2)
 Version of Record updated: February 22, 2017 (version 3)
 Version of Record updated: March 1, 2017 (version 4)
Copyright
© 2017, Tan and Cheong
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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