Nomadic-Colonial Life Strategies Enable Paradoxical Survival And

1 Nomadic-colonial life strategies enable paradoxical

2 survival and growth despite habitat destruction

1 2,* 3 Zhi Xuan Tan and Kang Hao Cheong

1 4 Yale University, New Haven, CT 06520, United States 2 5 Engineering Cluster, Singapore Institute of Technology, 10 Dover Drive, S138683, Singapore * 6 Corresponding Author: [email protected]

7 Abstract

8 Organisms often exhibit behavioral or phenotypic diversity to improve population ﬁtness in the 9 face of environmental variability. When each behavior or phenotype is individually maladaptive, 10 alternating between these losing strategies can counter-intuitively result in population persistence–an 11 outcome similar to Parrondo’s paradox. Instead of the capital or history dependence that characterize 12 traditional Parrondo games, most ecological models which exhibit such paradoxical behavior depend 13 on the presence of exogenous environmental variation. Here we present a population model that 14 exhibits Parrondo’s paradox through capital and history-dependent dynamics. Two sub-populations 15 comprise our model: nomads, who live independently without competition or cooperation, and 16 colonists, who engage in competition, cooperation, and long-term habitat destruction. Nomads 17 and colonists may alternate behaviors in response to changes in the colonial habitat. Even when 18 nomadism and colonialism individually lead to extinction, switching between these strategies at the 19 appropriate moments can paradoxically enable both population persistence and long-term growth.

20 1 Introduction

21 Behavioral adaptation and phenotypic diversity are evolutionary meta-strategies that can improve a

22 population’s ﬁtness in the presence of environmental variability. When behaviors or phenotypes are

23 suﬃciently distinct, a population can be understood as consisting of multiple sub-populations, each

24 following its own strategy. Counter-intuitively, even when each sub-population follows a losing strategy

25 that will cause it to go extinct in the long-run, alternating or reallocating organisms between these

26 losing strategies under certain conditions can result in meta-population persistence, and hence, an overall 1 27 strategy that wins. Some examples include random phase variation (RPV) in bacteria across multiple 2–4 5–7 28 losing phenotypes, as well as the persistence of populations that migrate among sink habitats only.

29 These counter-intuitive phenomenon are reminiscent of Parrondo’s paradox, which states that there 8 30 are losing games of chance which can be combined to produce a winning strategy. The existence of a

31 winning combination relies on the fact that at least one of the losing Parrondo games exhibits either

32 capital-dependence (dependence upon the current amount of capital, an ecological analog of which is

33 population size) or history-dependence (dependence upon the past history of wins or losses, or in an 9, 10 11–16 34 ecological context, growth and decline). There have been many studies exploring the paradox, 17 35 including a multi-agent Parrondo’s model based on complex networks and also implications to evolu- 1, 2, 18, 19 36 tionary biology. However, many biological studies which have drawn a connection to Parrondo 1 37 games do not necessarily utilize capital-dependence or history-dependence in their models. Further-

38 more, models of reversal behavior in ecological settings generally rely upon the presence of exogenous 2–7, 20 39 environmental variation. Without exogenous variation, the paradoxes do not occur. The broader

40 applicability of Parrondo’s paradox to ecological systems thus remains under-explored.

41 This lacuna remains despite the abundance of biological examples that exhibit history-dependent

42 dynamics. The ﬁtness of alleles may depend on the presence of genetic factors and epigenetic factors in 19 43 previous generations. More generally, the ﬁtness of any one gene can depend on the composition of other

44 genes already present in a population, enabling the evolution of complex adaptations like multicellularity 21 45 through ratcheting mechanisms. Such mechanisms have recently been shown to help stabilize these 22 46 complex adaptations. In ecological contexts, the storage eﬀect can ensure that gains previously made 20, 23 47 in good years can promote persistence in less favorable times. Species-induced habitat destruction

48 or resource production can also have time-delayed eﬀects on population growth, resulting in non-linear 24, 25 49 phenomena like punctuated evolution.

1 50 In this paper, we present a biologically feasible population model which exhibits counter-intuitive

51 reversal behavior due to the presence of history-dependent and capital-dependent dynamics. Unlike most

52 other studies, these dynamics do not rely upon the assumption of exogenous environmental variation. In

53 our model, we consider a population that exists in two behaviorally distinct forms: as nomads, and as

54 colonists. Numerous organisms exhibit analogous behavioral diversity, from slime moulds (amoeba vs. 26 27 28 55 plasmodia) and dimorphic fungi (yeast vs. hyphae) to jellyﬁsh (medusae vs. polyps) and human

56 beings. One model organism which exhibits this sort of behavior, to which our study might apply, is the 29 57 amoeba Dictyostelium discoideum.

58 Nomads live relatively independently, and thus unaﬀected by either competition or co-operation.

59 Under poor environmental conditions, they are subject to steady extinction. Colonists live in close

60 proximity, and are thus subject to both competitive and co-operative eﬀects. They may also deplete

61 the resources of the habitat they reside in over time, resulting in long-term death. However, if these

62 organisms are endowed with sensors that inform them of both population density and the state of the

63 colonial habitat, they can use this information to switch from one behavior to another. Signiﬁcantly,

64 we ﬁnd that an appropriate switching strategy paradoxically enables both population persistence and

65 long-term growth – an ecological Parrondo’s paradox.

66 2 Population model

67 Two sub-populations comprise our model: the nomadic organisms, and the colonial ones. In similar vein

68 to habitat-patch models, organisms that exist in multiple sub-populations can be modelled as follows:

dni X X = g (n ) + s n − s n (1) dt i i ij j ji i j j

69 where ni is the size of sub-population i, gi is the function describing the growth rate of ni in isolation, 70 and sij is the rate of switching to sub-population i from sub-population j. Population sizes are assumed 71 to be large enough that Equation1 adequately approximates the underlying stochasticity.

72 2.1 Nomadism

73 Let n1 be the nomadic population size. In the absence of behavioral switching, the nomadic growth rate 74 is given by g1(n1) = −r1n1 (2)

75 where r1 is the nomadic growth constant. Nomadism is modelled as a losing strategy by setting −r1 < 0, 76 such that n1 decays with time. In the context of Parrondo’s paradox, nomadism corresponds to the 77 ‘agitating’ strategy, or Game A. Importantly, competition among nomads, as well as between nomads

78 and colonists, is taken to be insigniﬁcant, due to the independence of a nomadic lifestyle.

79 2.2 Colonialism

80 Colonial population dynamics will be modelled by the well-known logistic equation, with carrying ca-

81 pacity K, but with two important modiﬁcations.

82 Firstly, the Allee eﬀect is taken into account. This serves two roles: it captures the cooperative

83 eﬀects that occur among colonial organisms, and it ensures that the growth rate is negative when the 84 population falls below a critical capacity A. Let n2 be the colonial population size. In the absence of 85 behavioral switching, the colonial growth rate is given by n n g (n ) = r n 2 − 1 1 − 2 (3) 2 2 2 2 min(A, K) K

86 where r2 is the colonial growth constant. Setting r2 > 0, we have a positive growth rate when A < n2 < 87 K, and a negative growth rate otherwise. The min(A, K) term ensures that when K < A, g2 is always 88 zero or negative, as would be expected. 89 Secondly, the carrying capacity K changes at a rate dependent upon the colonial population size, n2, 90 accounting for the destruction of environmental resources over the long run. The rate of change of is

91 given by dK = α − βn (4) dt 2

2 92 where α > 0 is the default growth rate of K, and β > 0 is the per-organism rate of habitat destruction.

93 An alternative interpretation of this equation is that K, the short-term carrying capacity, is dependent

94 on some essential nutrient in the environment, and that this nutrient is slowly depleted over time at a 95 rate proportional to βn2. ∗ α dK 96 Let n = β , the critical population level at which no habitat destruction occurs. dt is zero when ∗ ∗ ∗ ∗ 97 n2 = n , positive when n2 < n , and negative when n2 > n . n can thus also be interpretted as the ∗ 98 long-term carrying capacity. Clearly, if the long-term carrying capacity n < A, the only stable point of 99 the system becomes n2 = 0. Under this condition, colonialism is a losing strategy as well. 100 Note that g2 increases as K increases, and that K increases more quickly as n2 decreases. In the 101 context of Parrondo’s paradox, colonialism can thus serve as a ‘ratcheting’ strategy, or Game B, because

102 the rate of growth is implicitly dependent upon the colonial population in the past. Another way of 1, 30, 31 103 understanding the ‘ratcheting’ behavior is through the lens of positive reactivity. In the short- 104 term, n2 = A is a positively reactive equilibrium, because small upwards perturbations of n2 away from 105 A will result in rapid growth towards K before a slow decrease back down towards A.

106 2.3 Behavioral switching

107 Organisms are able to detect the amount of environmental resources available to them, and by proxy,

108 the carrying capacity of the population. Thus, they can undergo behavioral changes in response to the

109 current carrying capacity.

110 Here, we model organisms that switch to nomadic behavior from colonial behavior when the carrying 111 capacity is low (K < L1), and switch to colonial behavior from nomadic behavior when the carrying 112 capacity is high (K > L2), where L1 ≤ L2 are the switching levels. Let rs > 0 be the switching constant. 113 Using the notation from Equation1, switching rates can then be expressed as follows: ( ( rs if K < L1 rs if K > L2 s12 = s21 = (5) 0 otherwise 0 otherwise

114 A variety of mechanisms might trigger this switching behavior in biological systems. For example,

115 since the nomadic organisms are highly mobile, they could frequently re-enter their original colonial

116 habitat after leaving it, and thus be able to detect whether resource levels are high enough for recolo-

117 nization. It should also be noted that the decision to switch need not always be ‘rational’ (i.e. result

118 in a higher growth rate) for each individual. Switching behavior could be genetically programmed, such

119 that ‘involuntary’ individual sacriﬁce ends up promoting the long-term survival of the species.

120 2.4 Reduced parameters

121 Without loss of generality, we scale all parameters such that α = β = 1. Equation4 thus becomes: dK = 1 − n (6) dt 2 ∗ α 122 Hence, n = β = 1. All other population sizes and capacities can then be understood as ratios with 123 respect to this critical population size. Additionally, since β = 1, r1, r2 and rs can be understood as 124 ratios to the rate of habitat destruction. For example, if r2 1, this means that colonial growth occurs 125 much faster than habitat destruction. Time-scale separation between the population growth dynamics 126 and the habitat change dynamics can thus be achieved by setting r1, r2 1. Similarly, separation 127 between the behavioral switching dynamics and the population growth dynamics can be achieved by 128 setting rs r1, r2.

129 3 Methods

130 Numerical simulations were performed using code written in MATLAB (Source code 1) that relied on

131 the ode23 ordinary diﬀerential equation (ODE) solver. ode23 is an implementation of an explicit Runge-

132 Kutta (2,3) pair of Bogacki and Shampine. Simulations were performed with both behavioral switching 133 turned oﬀ (rs = 0) and turned on (rs > 0). The accuracy of the simulation was continually checked by 134 repeating all results with more stringent tolerance levels, ensuring that the ﬁnal simulated parameters

135 did not change signiﬁcantly (by less than 1%). Both the relative error tolerance and absolute error −9 136 tolerance were determined to be 10 .

3 137 In the case of complex switching rules like Equation 13 that required modifying diﬀerential equation

138 parameters at speciﬁc time points, the Events option of MATLAB’s ODE solvers was used to detect

139 when these points occurred. After each detection, the parameters were automatically modiﬁed as per

140 the switching rule, and the simulation continued with the new parameters.

141 Broad regimes of model behavior were observed by running simulations across a wide range of param-

142 eters and initial conditions. General trends and conditions observed within each regime were formalized

143 analytically, the details of which can be found in the Appendix. In these derivations, reasonable as-

144 sumptions were made in order to make the model analytically tractable. In particular, it was assumed

145 that the rate of behavioral switching was much faster than the rates of either colonial or nomadic growth 146 (rs r1, r2), and that colonial growth rates were in turn much faster than the rate of habitat destruction 147 (r2 1). Initial conditions corresponding to unstable equilibria (e.g. n2 = K = 1 < A) were avoided as 148 unrealistic.

149 4 Results

150 Simulation results revealed population dynamics that could be categorized into the following regimes:

151 1. Without behavioral switching (rs = 0)

152 (a) Extinction for both sub-populations

153 (b) Extinction for nomadic organisms, survival for colonial organisms

154 2. With behavioral switching (rs > 0)

155 (a) Extinction for both sub-populations

156 (b) Survival through periodic behavioral alternation

157 (c) Long-term growth through strategic alternation

158 Importantly, there were conditions under which both sub-populations would go extinct in the absence

159 of behavioral switching (regime 1a), but collectively survive if behavioral switching was allowed (regime

160 2b), thereby exhibiting Parrondo’s paradox. The following sections describe the listed regimes in greater

161 detail, with a focus upon the regimes involved in the paradox. Figures generated via numerical simulation

162 are provided as examples of behavior within each regime.

163 4.1 Extinction in the absence of switching

164 As described earlier, both nomadic and colonial behaviors can be modelled as losing strategies given

165 the appropriate parameters. Simulations across a range of parameters elucidated the conditions which

166 resulted in extinction for both strategies. Figure1a shows an example when both strategies are losing,

167 resulting in extinction, while Figure1b shows an an example where only the colonial sub-population

168 survives. 169 It is clear from Equation2 that the growth rate of the nomadic population n1 is always negative, 170 because of the restriction that r1 > 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 > 1, extinction occurs (as in Figure1a, and that survival is only possible when A is signiﬁcantly less than 1 (as in Figure1b). That is:

A > 1 (colonial extinction guaranteed) (7) A < 1 (colonial survival possible) (8)

171 The intuition behind this is straightforward. Suppose that initially, A < n2 < K, so that the growth 172 rate is positive. When A < 1, the colonial population n2 increases until it reaches the carrying capacity 173 K, following which they converge in tandem until stabilizing at the critical population size, n2 = K = 1. 174 However, when A > 1, n2 = K = 1 is no longer a stable equilibrium, since dn2/dt < 0 when n2 < A, 175 resulting in the eventual extinction of the population. For a formal proof, refer to Theorem A.3.

4 Eventual extinction for both strategies Survival for colonial strategy No switching, A = 1.001 No switching, A = 0.500 5 5 n n 4.5 1 4.5 1 n n 4 2 4 2 K K 3.5 3.5 3 3 2.5 2.5 2 2

Population size 1.5 Population size 1.5 1 1 0.5 0.5 0 0 0 5 10 15 0 5 10 15 Time Time (a) (b)

Figure 1: In the absence of switching, (a) eventual extinction for both strategies vs. (b) survival for the colonial strategy. Initial conditions for both are n1 = 2, n2 = 2, K = 5. Shared parameters are rs = 0, r1 = 1, r2 = 10. For (a), A = 1.001. For (b), A = 0.5.

176 4.2 Survival through periodic alternation

177 We now restrict our analysis to the case where A > 1. Under this condition, both nomadism (Game A)

178 and colonialism (Game B) are losing strategies when played individually. Paradoxically, it is possible

179 to combine these two strategies through behavioral switching such that population survival is ensured,

180 thereby producing an overall strategy that wins.

181 Simulation results over a range of parameters have predicted this paradoxical behavior, and also

182 elucidated the conditions under which it occurs. Figure2a is a typical example where the population

183 becomes extinct, even though it undergoes behavioral switching, while Figure2b is a typical example

184 where behavioral switching ensures population survival.

Eventual extinction despite alternation Survival through perodic alternation Switch to nomadic if K < 3.00; Switch to colonial if K > 4.50 Switch to nomadic if K < 3.00; Switch to colonial if K > 4.00 5 5 n n 4.5 1 4.5 1 n n 4 2 4 2 K K 3.5 3.5 3 3 2.5 2.5 2 2 Population size Population size 1.5 1.5 1 1 0.5 0.5 0 0 0 2 4 6 8 0 2 4 6 8 Time Time (a) (b)

Figure 2: Behavioral switching which results in (a) extinction vs. (b) survival. Initial conditions for both are n1 = 2, n2 = 2, K = 5. Shared parameters are rs = 1000, r1 = 1, r2 = 10. For (a), L1 = 3, L2 = 4.5. For (b), L1 = 3, L2 = 4.

185 Conceptually, this paradoxical survival is possible because the colonial strategy, or Game B, is history- 186 dependent. In particular, the colonial growth rate dn2/dt is dependent upon the carrying capacity K, 187 which in turn is dependent upon previous levels of n2. Behavioral switching to a nomadic strategy 188 decreases the colonial population size, allowing the resources in the colonial environment, represented by

189 K, to recover. Switching back to a colonial strategy then allows the population to take advantage of the

190 newly generated resources. Because switching occurs periodically, as can be seen in Figure2b, it should

191 be noted that the organisms need not even detect the amount of resources present in the environment

192 to implement this strategy. A biological clock would be suﬃcient to trigger switching behavior.

193 The exact process by which survival is ensured can be understood by analysing the simulation results 194 in detail. In the nomadic phase, the colonial population n2 is close to zero, the nomadic population

5 195 n1 undergoes slow exponential decay, and the carrying capacity K undergoes slow linear growth. K 196 increases until it reaches L2, which triggers the switch to colonial behavior. 197 The population thus enters the colonial phase. If the colonial population n2 exceeds the critical 198 capacity A at this point, then n2 will grow until it slightly exceeds the carrying capacity K. Subsequently, 199 n2 decreases in the tandem with K until K drops to L1, triggering the switch back to the nomadic phase. 200 However, if n2 < A when the colonial phase begins, the colonial population goes extinct, as can be seen 201 in Figure2a. Hence, a basic condition for survival is that n2 ≥ A at the start of each colonial phase. 202 This implies that, by the end of the nomadic phase, n1 needs to be greater by a certain amount than 203 A as well. Otherwise, there will be insuﬃcient nomads to form a colony which can overcome the Allee

204 eﬀect. Under the reasonable assumption that the rate of behavioral switching is much faster than either 205 colonial or nomadic growth (rs r1, r2), it can be shown more precisely that at the end of the nomadic 206 phase, n1 needs to be greater than a critical level B, which is related to A by the equation: B B A = B − (1 − B)W exp (9) 0 1 − B 1 − B

207 A full derivation is provided in the Appendix (Theorem A.4). Here, W0(x) is the principal branch of the 208 Lambert W function. Qualitatively speaking, B is a function of A on the interval (1, ∞) that increases

209 in an exponential-like manner, and that approaches 1 when A does as well. Thus, B ≥ A, as expected.

210 The greater the diﬀerence between the switching levels, the longer the nomadic phase will last, because 211 it takes more time for K to increase to the requisite value for switching, L2. And the longer the nomadic 212 phase lasts, the more n1 will decay. If, at the end of the nomadic phase, the value that n1 decays to 213 happens to be less than B, then the population will fail to survive. It follows that there should be some 214 constraint on the diﬀerence between the switching levels L1 and L2. 215 Under the same assumption that rs r1, r2, such a constraint can be derived:

−L1 1 L1 + W0(−L1e ) L2 < L1 + ln (10) r1 B

216 Survival is ensured given the following additional condition:

∗ ∗ ∗ There exists t ≥ t0 : n2(t ) = K(t ) ≥ L1 (11)

∗ 217 where t0 marks the start of an arbitrary colonial phase, and t marks the time of intersection between n2 218 and K during that phase. In other words, n2 has to grow sufficiently quickly during the colonial phase 219 such that it exceeds both K and L1 before switching begins. This can be seen occurring in Figure2b. 220 In accordance with intuition, numerical simulations predict that this occurs when the colonial growth 221 constant is sufficiently large (r2 r1), as can be seen in the Figures. (The Figures also show that r1 222 close to 1, but this is not strictly necessary.) Collectively, Equations 10 to 11 are sufficient conditions

223 for population survival. Mathematical details are provided in the Appendix (Theorems A.5 and A.6). 224 Note that Equation 10 contains an implicit lower bound on L1. Since L2 ≥ L1 by stipulation, we −L1 225 must have ln[L1 + W0(−L1e )] > ln B for survival. The following bound is thus obtained:

BeB L > (12) 1 eB − 1

226 On the other hand, under the assumptions made, there is no upper bound for L1, and hence no absolute 227 upper bound for L2 either. This suggests that given a suﬃciently well-designed switching rule, K can 228 grow larger over time while ensuring population survival. Such a rule is investigated in the following

229 section.

230 4.3 Long-term 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 built-in estimation of the time required for growth or decay to a certain population level. The following switching rule then becomes possible:

When n2 = K, dn2/dt > 0, set L1 = K,L2 = ∞ (13) When n1 = B, dn1/dt < 0, set L2 = K

6 Long-term growth Long-term growth Strategic switching, up to t=10 Strategic switching, up to t=300 12 350 n n 1 1 n 300 n 10 2 2 K K 250 8 200 6 150 4 Population size Population size 100

2 50

0 0 0 2 4 6 8 10 0 50 100 150 200 250 300 Time Time (a) (b)

Figure 3: Long-term growth through strategic alternation. Initial conditions are n1 = 0, n2 = 2, K = 5. Parameters are A = 1.001, L1 = 3, L2 = 4, rs = 1000, r1 = 1, r2 = 10.

Mechanism behind nomadic growth spikes Long-term growth with limiting behavior Strategic switching Strategic switching, up to t=1000 8000 n n 7000 1 7000 1 n n 2 2 6000 K 6000 K

5000 5000

4000 4000

3000 3000 Population size Population size 2000 2000

1000 1000

0 0 844.6 844.62 844.64 844.66 844.68 844.7 844.72 844.74 0 200 400 600 800 1000 Time Time

Figure 4: Zoomed-in portion of Figure5 that Figure 5: Long-term growth through strategic shows the mechanism behind the spikes in n1. alternation, up to t = 10. Initial conditions are When n2 is large, switching takes longer, causing a n1 = 0, n2 = 2, K = 5. Parameters are A = 1.001, drop in K, and a large increase in n1. rs = 1000, r1 = 1, r2 = 10.

231 That is, L1 is set to the carrying capacity K whenever n2 rises to K, resulting immediately in a switch 232 to nomadic behavior, and that L2 is in turn set to K whenever n1 falls to B, resulting in an immediate 233 switch to colonial behavior.

234 This switching rule is optimal according to several criteria. Firstly, by switching to nomadic behavior 235 just as n2 reaches K, it ensures that dn2/dt ≥ 0 for the entirety of the colonial phase. As such, it avoids 236 the later portion of the colonial phase where K and n2 decrease in tandem, and maximizes the ending 237 value n2. Consequently, it also maximizes the value of n1 at the start of each nomadic phase. 238 Furthermore, by switching to colonial behavior right when n1 decays to B, the rule maximizes the 239 duration of the nomadic phase while ensuring survival. This in turn means that the growth of K is

240 maximized, since the longer the nomadic phase, the longer that K is allowed to grow.

241 In fact, this switching rule is a paradigmatic example of how Parrondo’s paradox can be achieved.

242 It plays Game A, the nomadic strategy, for as long as possible, in order to maximize K and hence the

243 returns from Game B. And then it switches to Game B, the colonial strategy, only for as long as the 244 returns are positive (dn2/dt > 0), thereby using it as a kind of ratchet. 245 Suppose that K grows more during each nomadic phase than it falls during each colonial phase. Then

246 the switching rule is not just optimal, but it also enables long-term growth. Simulation results predict

247 that this can indeed occur. Figure3a shows long-term growth of K from t = 0 to t = 10, while Figure

248 3b shows that with the same initial conditions, this continues until t = 300 with no signs of abating. 249 Together with K, the per-phase maximal values of n1 and n2 increase as well. 250 In the cases shown, long-term growth is achieved because K indeed grows more during each nomadic

251 phase than it falls during the subsequent colonial phase. As can been from Figure3a, this is, in turn,

252 because the nomadic phase lasts much longer than the colonial phase, such that the amount of environ-

253 mental destruction due to colonialism is limited. Simulation results predict that this generally occurs as

7 254 long as the colonial growth rate is suﬃciently large (r2 r1). 255 An interesting phenomenon that can be observed from Figure3b is how the nomadic population size 256 n1, which peaks at the start of each nomadic phase, eventually exceeds the carrying capacity K, and 257 then continues doing so by increasing amounts at each peak. This is, in fact, a natural consequence of 258 the population model. When n2 grows large, the assumption that switching is much faster than colonial 259 growth starts to break down. This occurs even though rs r2, due to the increasing contribution of the n2 260 ( A − 1) factor in Equation3. 261 The result is that when a large colonial population begins switching to nomadism, a signiﬁcant

262 number of colonial oﬀspring are simultaneously being produced. These oﬀspring also end up switching

263 to a nomadic strategy, resulting in more nomadic organisms than there were colonial organisms before.

264 A particularly pronounced example of this is shown in Figure4.

265 However, this same phenomenon also introduces limiting behavior to the pattern of long-term growth.

266 As Figure5 shows, when the same simulation as in Figures3a and3b is continued to t = 1000, peak 267 levels of n1, n2 and K eventually plateau around t = 650. 268 This occurs because sufficiently high levels of n2 cause a qualitative change in the dynamics of 269 behavioral switching. Normally, switching to nomadic behavior starts when K falls below L1, and ends 270 when K rises above it again. K rises towards the end of the switch, when n2 levels fall below the critical ∗ 271 level of n = 1. But when n2 is sufficiently large, the faster production of colonial offspring drags out the 272 duration of switching, as seen in Figure4. The higher levels of n2, combined with the longer switching 273 duration, causes an overall drop in K by the end of the switching period. Because the increase in K

274 during the subsequent nomadic phase is unable to overcome this drop, K stops increasing in the long-run.

275 Nonetheless, it is clear that signiﬁcant long-term gains can be achieved via the optimal switching 276 rule. Under the conditions of fast colonial growth and even faster switching (rs r2 r1 ' 1, as in 277 Figures3a to5), these gains are several orders of magnitude larger than the initial population levels,

278 a huge departure from the long-term extinction that occurs in purely colonial or nomadic populations.

279 Limiting behavior eventually emerges, but this is to be expected in any realistic biological system.

280 4.4 Survival and growth under additional constraints

281 Our proposed model is convenient for the functional understanding of growth and survival, and can be

282 easily modiﬁed for a variety of applications. Additional constraints can be imposed under which survival

283 and long-term growth are still observed. For example, in many biological systems, the dynamics of habitat 284 change might occur on a slower timescale than both colonial and nomadic growth (i.e. r1, r2 1). Figure 285 6 shows the simulation results when this timescale separation exists (r1 = 10, r2 = 100 for (a), r1 = 100, 286 r2 = 1000 for (b)). It can clearly be seen that survival is still possible under such conditions.

Survival under slow habitat change (r1 = 10, r2 = 100) Survival under slow habitat change (r1 = 100, r2 = 1000) Switch to nomadic if K < 3.000; Switch to colonial if K > 3.083 Switch to nomadic if K < 3.000; Switch to colonial if K > 3.008 3.5 3.5 n n 1 1 3 n 3 n 2 2 K K 2.5 2.5

2 2

1.5 1.5

Population size 1 Population size 1

0.5 0.5

0 0 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 Time Time (a) (b)

Figure 6: Survival through periodic alternation under slow habitat change. Parameters for (a) are rs = 10000, r1 = 10, r2 = 100,L1 = 3, L2 = 3.083. For (b), rs = 100000, r1 = 100, r2 = 1000,L1 = 3, L2 = 3.008. Shared parameters are Kmax = 20, A = 1.001.

287 Another practical constraint that can be imposed is limiting the growth of the carrying capacity to 288 some maximal value Kmax, capturing the fact that the resources in any one habitat do not grow inﬁnitely 289 large. This can be achieved by modifying Equation6 as follows:

8 Long-term growth Long-term growth Strategic switching, K = 20, r = 1, r = 10 Strategic switching, K = 7.5, r = 10, r = 100 max 1 2 max 1 2 25 8 n n 1 7 1 n n 20 2 2 K 6 K

15 5 4

10 3 Population size Population size 2 5 1

0 0 0 50 100 150 200 0 10 20 30 40 50 Time Time (a) (b)

Figure 7: Long-term growth through strategic alternation with a bounded carrying capacity. Initial conditions are n1 = 0, n2 = 2, K = 5, shared parameters are A = 1.001. For (a), Kmax = 20, rs = 1000, r1 = 1, r2 = 10 (fast habitat change). For (b), Kmax = 7.5, rs = 10000, r1 = 10, r2 = 100 (slow habitat change).

dK K = (1 − n2) 1 − (14) dt Kmax

290 Figure6 already takes this constraint into account, showing that survival through periodic alternation

291 is achievable under both bounded carrying capacity and slow habitat change, as long as the maximum 292 carrying capacity is suﬃciently high (Kmax = 20). As Figure7 shows, even long-term growth is possible, 293 under both fast habitat change (Figure7a) and slow habitat change (Figure7b). In both cases, the 294 carrying capacity K converges towards a maximum value as it approaches Kmax.

295 5 Discussion

296 The results presented in this study demonstrate the theoretical possibility of Parrondo’s paradox in

297 an ecological context. Many evolutionary strategies correspond to the strategies that we have termed

298 here as ‘nomadism’ and ‘colonialism’. In particular, any growth model that is devoid of competitive

299 or collaborative eﬀects is readily captured by Equation2 (nomadism), while any logistic growth model

300 which includes both the Allee eﬀect and habitat destruction can be described using Equations3 and4

301 (colonialism). Many organisms also exhibit behavioral change or phenotypic switching in response to

302 changing environmental conditions. By incorporating this into our model, we have demonstrated that

303 nomadic-colonial alternation can ensure the survival of a species, even when nomadism or colonialism

304 alone would lead to extinction. Furthermore, it has been demonstrated that an optimal switching rule

305 can lead to long-term population growth.

306 The switching rules which lead to survival and long-term growth are analogous to the periodic al-

307 ternation between games that produces a winning expectation in Parrondo’s paradox. If one views

308 the carrying capacity K as the capital of the population, then it is clear that Equation5 is a capital-

309 dependent switching rule. By setting the appropriate amounts of capital at which switching should occur,

310 survival and growth can be achieved. Survival is achieved by ensuring that Game A, or nomadism, is 311 never played beyond the point where extinction is inevitable, that is, the point where n1 falls below the 312 critical level B. Long-term growth is additionally achieved by ensuring that Game B, or colonialism, is 313 only played in the region where gains are positive, that is, when A < n2 < K such that dn2/dt > 0. The 314 history-dependent dynamics of Game B are thus optimally exploited.

315 Several limitations of the present study should be noted. Firstly, the study only focuses on cases

316 where nomadism and colonialism are individually losing strategies, despite the abundance of similar

317 strategies that do not lose in the real world. This is because assuming individually losing strategies in

318 fact leads to a stronger result – if losing variants of nomadism and colonialism can be combined into a

319 winning strategy, it follows that non-losing variants can be combined in a similar way too (see Theorem

320 A.7 in the Appendix).

321 Secondly, the population model does not encompass all variants of qualitatively similar behavior. For 32 322 example, many other equations can be used to model the Allee eﬀect. Nonetheless, our proposed model

9 323 is general enough that it can be adapted for use with other equations and be expected to produce similar

324 results. Even the presence of the Allee eﬀect is not strictly necessary, since the colonial population might

325 die off at low levels because of stochastic fluctuations, rather than because of the effect. Theorem A.7

326 in the Appendix also demonstrates that paradoxical behavior can occur even without the Allee eﬀect

327 causing long-term death of the colonial population.

328 Thirdly, though it is trivially the case that pure nomadism and pure colonialism cannot out-compete

329 a behaviorally-switching population, a more complex analysis of the evolutionary stability of behavioral

330 switching is beyond the scope of this paper. Finally, spatial dynamics are not accounted for in this study.

331 Exploring such dynamics is a goal for future work.

332 6 Conclusion

333 Our comprehensive model captures both capital and history-dependent dynamics within a realistic eco-

334 logical setting, thereby exhibiting Parrondo’s paradox without the need for exogenous environmental

335 inﬂuences. The possibility of an ecological Parrondo’s paradox has wide-ranging applications across the

336 ﬁelds of ecology and population biology. Not only could it provide evolutionary insight into strate-

337 gies analogous to nomadism, colonialism, and behavioral diversiﬁcation, it potentially also explains why

338 environmentally destructive species, such as Homo sapiens, can thrive and grow despite limited envi-

339 ronmental resources. By providing a theoretical model under which such paradoxes occur, our approach

340 may enable new insights into the evolution of cooperative colonies, as well as the conditions required for

341 sustainable population growth.

342 7 Additional information

343 Competing interests: The authors declare no competing ﬁnancial interests.

344 Supporting information: Supporting information containing analytical derivations can be found in

345 the Appendix.

346 Source code: MATLAB code used for the numerical simulation is attached as additional ﬁle. This

347 contains all the functions used in the analyses presented in the paper. There are accompanying

348 comments that document the parameters used in the model.

349 References

350 [1] Paul David Williams and Alan Hastings. Paradoxical persistence through mixed-system dynamics:

351 towards a uniﬁed perspective of reversal behaviours in evolutionary ecology. Proceedings of the Royal

352 Society of London B: Biological Sciences, page rspb20102074, 2011.

353 [2] Denise M. Wolf, Vijay V. Vazirani, and Adam P. Arkin. Diversity in times of adversity: probabilistic

354 strategies in microbial survival games. Journal of theoretical biology, 234(2):227–253, 2005.

355 [3] Edo Kussell and Stanislas Leibler. Phenotypic diversity, population growth, and information in

356 ﬂuctuating environments. Science, 309(5743):2075–2078, 2005.

357 [4] Murat Acar, Jerome T. Mettetal, and Alexander van Oudenaarden. Stochastic switching as a

358 survival strategy in ﬂuctuating environments. Nature genetics, 40(4):471–475, 2008.

359 [5] Vincent AA Jansen and Jin Yoshimura. Populations can persist in an environment consisting of

360 sink habitats only. Proceedings of the National Academy of Sciences, 95(7):3696–3698, 1998.

361 [6] Manojit Roy, Robert D. Holt, and Michael Barﬁeld. Temporal autocorrelation can enhance the

362 persistence and abundance of metapopulations comprised of coupled sinks. The American Naturalist,

363 166(2):246–261, 2005.

364 [7] Michel Bena¨ımand Sebastian J. Schreiber. Persistence of structured populations in random envi-

365 ronments. Theoretical Population Biology, 76(1):19–34, 2009.

366 [8] Gregory P. Harmer and Derek Abbott. Game theory: Losing strategies can win by parrondo’s

367 paradox. Nature, 402(6764):864–864, 1999.

10 368 [9] Juan MR Parrondo, Gregory P. Harmer, and Derek Abbott. New paradoxical games based on

369 brownian ratchets. Physical Review Letters, 85(24):5226, 2000.

370 [10] Gregory P. Harmer and Derek Abbott. A review of parrondo’s paradox. Fluctuation and Noise

371 Letters, 2(2):R71–R107, 2002.

372 [11] Kang Hao Cheong and Wayne Wah Ming Soo. Construction of novel stochastic matrices for analysis

373 of parrondo’s paradox. Physica A: Statistical Mechanics and its Applications, 392(20):4727–4738,

374 2013.

375 [12] Wayne Wah Ming Soo and Kang Hao Cheong. Parrondo’s paradox and complementary parrondo

376 processes. Physica A: Statistical Mechanics and its Applications, 392(1):17–26, 2013.

377 [13] Wayne Wah Ming Soo and Kang Hao Cheong. Occurrence of complementary processes in parrondo’s

378 paradox. Physica A: Statistical Mechanics and its Applications, 412:180–185, 2014.

379 [14] Derek Abbott. Asymmetry and disorder: A decade of parrondo’s paradox. Fluctuation and Noise

380 Letters, 9(1):129–156, 2010.

381 [15] A. P. Flitney and D. Abbott. Quantum models of parrondo’s games. Physica A: Statistical Mechanics

382 and its Applications, 324(1):152–156, 2003.

383 [16] Gregory P. Harmer, Derek Abbott, Peter G. Taylor, and Juan M. R. Parrondo. Brownian ratchets

384 and parrondo’s games. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(3):705–714,

385 2001.

386 [17] Ye Ye, Kang Hao Cheong, Yu-wan Cen, and Neng-gang Xie. Eﬀects of behavioral patterns and

387 network topology structures on parrondo’s paradox. Scientiﬁc Reports (Nature), 6:srep37028, 2016.

388 [18] Kang Hao Cheong, Zhi Xuan Tan, Neng-gang Xie, and Michael Jones. A paradoxical evolutionary 389 mechanism in stochastically switching environments. Scientiﬁc Reports (Nature), 6:srep34889, 2016.

390 [19] Floyd A. Reed. Two-locus epistasis with sexually antagonistic selection: a genetic parrondo’s para-

391 dox. Genetics, 176(3):1923–1929, 2007.

392 [20] Jonathan M Levine and Mark Rees. Eﬀects of temporal variability on rare plant persistence in

393 annual systems. The American Naturalist, 164(3):350–363, 2004.

394 [21] Eric Libby and William C. Ratcliﬀ. Ratcheting the evolution of multicellularity. Science,

395 346(6208):426–427, 2014.

396 [22] Eric Libby, Peter Conlin, Ben Kerr, and William C. Ratcliﬀ. Stabilizing multicellularity

397 through ratcheting. Philosophical Transactions of the Royal Society B: Biological Sciences,

398 346:rstb.2015.0444, 2016.

399 [23] Robert R Warner and Peter L Chesson. Coexistence mediated by recruitment ﬂuctuations: a ﬁeld

400 guide to the storage eﬀect. The American Naturalist, pages 769–787, 1985.

401 [24] Vyacheslav I. Yukalov, E. P. Yukalova, and Didier Sornette. Punctuated evolution due to delayed

402 carrying capacity. Physica D: Nonlinear Phenomena, 238(17):1752–1767, 2009.

403 [25] Vyacheslav I. Yukalov, E. P. Yukalova, and Didier Sornette. Population dynamics with nonlinear

404 delayed carrying capacity. International Journal of Bifurcation and Chaos, 24(2):1450021, 2014.

405 [26] Sandra L. Baldauf and W. Ford Doolittle. Origin and evolution of the slime molds (Mycetozoa).

406 Proceedings of the National Academy of Sciences, 94(22):12007–12012, October 1997.

407 [27] Robert J. Bastidas and Joseph Heitman. Trimorphic stepping stones pave the way to fungal viru-

408 lence. Proceedings of the National Academy of Sciences, 106(2):351–352, January 2009.

409 [28] Cathy H. Lucas, William M. Graham, and Chad Widmer. Jellyﬁsh Life Histories: Role of Polyps

410 in Forming and Maintaining Scyphomedusa Populations. Advances in marine biology, 63:133, 2012.

411 [29] Sarah J. Annesley and Paul R. Fisher. Dictyostelium discoideum—a model for many reasons.

412 Molecular and Cellular Biochemistry, 329(1):73–91, 2009.

11 413 [30] Alan Hastings. Transient dynamics and persistence of ecological systems. Ecology Letters, 4(3):215–

414 220, 2001.

415 [31] Alan Hastings. Transients: the key to long-term ecological understanding? Trends in Ecology &

416 Evolution, 19(1):39–45, 2004.

417 [32] David S. Boukal and Ludˇek Berec. Single-species models of the allee eﬀect: extinction boundaries,

418 sex ratios and mate encounters. Journal of Theoretical Biology, 218(3):375–394, 2002.

419 A Appendix

420 A.1 Switching dynamics

Colonialism to Nomadism Nomadism to Colonialism Switch to nomadic if K < 3.00; Switch to colonial if K > 4.00 Switch to nomadic if K < 4.00; Switch to colonial if K > 4.80 4 2 n (t ) = x n n 1 1 1 3.5 1 1.8 n n n (t ) = x 2 2 2 1 K 1.6 3 n (t ) = z n (t ) = z 1.4 2 2 1 2 2.5 1.2 2 1

1.5 0.8

Population size Population size 0.6 1 0.4 n (t ) = y 0.5 1 2 n (t ) = y 0.2 t t 2 2 t t 0 1 2 0 1 2 2.5 2.505 2.51 2.515 2.52 2.525 1.15 1.152 1.154 1.156 1.158 1.16 Time Time

Appendix ﬁgure 1: An example of switching Appendix ﬁgure 2: An example of switching from colonialism to nomadism. t1 marks the start from nomadism to colonialism. t1 marks the start of the switch, t2 marks the end. Other parameters of the switch, t2 marks the end. Other parameters are A = 1.001, rs = 1000, r1 = 1, r2 = 10. are A = 1.001, rs = 1000, r1 = 1, r2 = 10.

421 Lemma A.1. Let t1 and t2 respectively be the start and end of a period of switching from sub-population 422 i to sub-population j. Then either of the following must hold:

Z t2 1 − n2(t)dt = 0 (A1) t1 423 ni(t2) = 0 (A2)

Proof. Switching to nomadism begins when K falls below L1 and stops when it rises above L1 again. Similarly, switching to colonialism begins when K rises above L2 and stops when it falls below L2 again. In both cases then: Z t2 dK Z t2 ∆K = dt = 1 − n2(t)dt = 0 t1 dt t1

424 Alternatively, before K rises or falls back to the appropriate level, ni may go extinct, prematurely ending 425 the switch, in which case ni(t2) = 0.

426 Theorem A.1 (Colonialism to nomadism). Let x be the colonial population at the end of a colonial

427 phase, and y be the colonial population at the start of the subsequent nomadic phase. Assuming that 428 switching is much faster than colonial growth (rsn2 g2(n2)), y can be expressed in terms of x2 as

−x y = −W0(−xe ) (A3)

429 where W0 is the principal branch of the Lambert W function. Let z be the nomadic population at the start 430 of the subsequent nomadic phase. If switching is also much faster than nomadic growth (rsn1 g1(n1)), 431 and if the nomadic population is 0 before switching, we have

−x z = x + W0(−xe ) (A4)

432 Note that z is an increasing function of x, and that z converges to x as x grows larger.

12 Proof. Since rsn2 g2(n2), we have dn2/dt = −rsn2. Hence, n2(t) decays exponentially from x to y 1 over a duration of ln(x/y). Substituting n2(t) into Equation A1 and solving, we have rs x − ln x = y − ln y

w 433 This means that −x and −y are both solutions to the equation C = we , where C is some constant.

434 The roots of this equation are given by the Lambert W function. Since x > y and both are real numbers, 435 −x must lie on the W−1 branch, while −y lies on the principal W0 branch. Equation A3 follows. 436 For Equation A4, if rsn1 g1(n1), all the new nomadic organisms must have switched over from 437 colonialism. It follows that z = x − y. This completes the proof.

438 Theorem A.2 (Nomadism to colonialism). Let x be the nomadic population at the end of a nomadic

439 phase, and y be the nomadic population at the start of the subsequent colonial phase. Assuming that 440 switching is much faster than both nomadic and colonial growth (rsn1 g1(n1), rsn2 g2(n2) ), and 441 that the colonial population is 0 before switching, y can be expressed in terms of x as x x y = (1 − x)W e 1−x (A5) 0 1 − x

442 for x > 1. If x ≤ 1, then y = 0, i.e., the nomads go extinct during switching. Let z be the colonial

443 population at the start of the subsequent colonial phase. We have x x z = x − (1 − x)W e 1−x (A6) 0 1 − x

444 for x > 1, and z = x otherwise. Note that in the ﬁrst case, z is an increasing function of x.

Proof. By our assumptions, we have dn1/dt = −rsn1 and dn2/dt = rsn1. Hence, n1(t) decays exponen- 1 tially from x to y over a duration of ln(x/y), and n2(t) increases correspondingly from 0 to (x − y). rs Substituting n2(t) into Equation A1 and solving, we have

(1 − x)(ln x − ln y) + (x − y) = 0

445 Suppose that x > 1, y can be expressed in terms of x using the principal branch of the Lambert W 446 function, and Equation A5 follows. If x ≤ 1, then there is no way to satisfy Equation A1 because n2 will 447 not exceed 1 during the switching period. It follows that n1 must go extinct, in which case y = 0. The 448 results for z follow since z = x − y. This completes the proof.

449 A.2 Extinction and survival

450 Theorem A.3 (Colonists go extinct when A > 1). When A > 1, pure colonialism leads to extinction 451 in the long run (limt→∞ n2(t) = 0), assuming initial conditions are not set to the unstable equilibrium 452 n2 = K = 1 < A.

453 Proof. There are three possible equilibria for n2, at which dn2/dt = 0: n2 = 0, n2 = A, and n2 = K 454 (refer to Equation3). However, when n2 = A > 1 or n2 = K > 1, we have dK/dt = 1 − n2 < 0. 455 K decreases until n2 < K, which implies dn2/dt < 0. n2 then decreases until n2 < A, which implies 456 convergence to 0 as t → ∞. 457 Similarly, if n2 = K < 1, then dK/dt > 0, such that n2 < K < 1 < A an inﬁnitesimal moment later, 458 which then leads to extinction. This leaves only n2 = 0 and n2 = K = 1 as equilibria, the latter of which 459 is unstable to small perturbations of n2 towards zero. This completes the proof.

460 Theorem A.4 (Survival imposes a lower bound on n1). For survival to occur, n1(tN ) > B is a necessary 461 condition, where tN denotes the end of any nomadic phase, and B is a critical level related to A: B B A = B − (1 − B)W exp (A7) 0 1 − B 1 − B

462 Proof. Let tC denote the start of the subsequent colonial phase. We know that n2(tC ) > A is required 463 for the new colony to survive. B is deﬁned using Equation A6 such that n2(tC ) = A if n1(tN ) = B. 464 Since n2(tC ) is an increasing function of n1(tN ), it follows that n1(tN ) > B in order for n2(tC ) > A. 465 This completes the proof.

13 466 Theorem A.5 (Survival imposes constraints on L1 and L2). Under the reasonable assumptions that 467 switching is much faster than colonial or nomadic growth (rsn1 g1(n1), rsn2 g2(n2)), and that the 468 diﬀerence between n2 and K is negligibly small once n2 grows to exceed K during the colonial phase, 469 the following constraint on L1 and L2 is a necessary condition for survival through periodic alternation:

−L1 1 L1 + W0(−L1e ) L2 ≤ L1 + ln (A8) r1 B

470 This also implies a lower bound on L1, given by: BeB L ≥ (A9) 1 eB − 1

Proof. Consider a sequence of consecutive phases which we will term as CP1, NP, CP2, where CP1 and CP2 are colonial phases and NP is the nomadic phase between them. Since the nomadic phase NP is r1T characterized by exponential decay, it is clear that n1(NP-start) = n1(NP-end)e , where 1 n (NP-start) T := NP-end − NP-start = ln 1 r1 n1(NP-end)

is the duration of the nomadic phase. During this period, the colonial population n2 is close to or equal to 0. Since dK/dt = 1 − n2 ≤ 1, it follows that K(NP-end) − K(NP-start) ≤ 1 · (NP-end − NP-start) = T

471 Note that K(NP-start) = L1 and K(NP-end) = L2 by deﬁnition. Combining this with the two 472 equations above, we obtain 1 n1(NP-start) L2 ≤ L1 + ln (A10) r1 n1(NP-end)

473 For survival to occur, the colonial population at the start of CP2 needs to be greater than A. By 474 Theorem A.4, this implies that n1(NP-end) > B. Substituting into the above, we have

1 n1(NP-start) L2 < L1 + ln (A11) r1 B

475 Notice that by Theorem A.1, n1(NP-start) is a function of n2(CP1-end), the colonial population at the 476 end of CP1. If CP1 lasts long enough that n2 exceeds K, then by assumption we have n2(CP1-end) ' 477 K(CP1-end), otherwise we have n2(CP1-end) < K(CP1-end). Note that K(CP1-end) = L1. Thus, −L1 478 n2(CP1-end) ≤ L1. Then by Equation A4 in Theorem A.1, we have n1(NP-start) ≤ L1 + W0(−L1e ). 479 Substituting this into Equation A11 gives us Equation A8, as desired. Finally, recall that L2 > L1 by deﬁnition. It thus needs to be the case that

L + W (−L e−L1 ) ln 1 0 1 > 0 B

480 Solving this gives us the lower bound on L1. This completes the proof.

481 Theorem A.6 (Survival is ensured if n2 converges to K quickly). Under the reasonable assumption 482 that n2 ' 0 during each nomadic phase, then together with Equation A8 as well as the assumptions 483 made in A.5, the following constraint is suﬃcient to ensure survival:

∗ ∗ ∗ There exists t ≥ t0 : n2(t ) = K(t ) ≥ L1 (A12) ∗ 484 where t0 marks the start of an arbitrary colonial phase, and t marks the time of intersection between n2 485 and K during that phase. In other words, n2 has to grow suﬃciently quickly during the colonial phase 486 such that it exceeds both K and L1 before switching begins.

487 Proof. Let CP1 denote the colonial phase in question. Given Equation A12, all switching to the nomadic 488 phase only happens after n2 exceeds K during CP1. We have already presumed in Theorem A.5 that 489 once n2 grows to exceed K, we have n2 − K 1. This implies that n2(CP1-end) ' K(CP1-end) = L1, −L1 490 which in turn implies that n1(NP-start) ' L1 + W0(−L1e ) by Theorem A.1. 491 Given the new assumption that n2 ' 0 during each nomadic phase, any value of L2 that satisfies 492 Equation A10 will be sufficient for survival. Substituting n1(NP-end) > B along with our new expression 493 for n1(NP-start) into Equation A10 retains this property, which means that any value of L2 that satisfies 494 Equation A8 is sufficient for survival (as long as L2 > L1). Thus, given our assumptions, Equation A8 495 and Equation A12 collectively ensure survival. This completes the proof.

14 496 Theorem A.7 (Behavioral alternation improves survival levels). Suppose that pure nomadism or pure

497 colonialism (or both) result in population survival. That is,

ni,pure(∞) := lim ni,pure(t) > 0 (A13) t→∞

498 where i = 1 or 2 or both. This can happen if r1 = 0 (for nomadism) or A < 1 (for colonialism). Then 499 there are conditions under which behavioral alternation between the two pure strategies will result in a

500 total population size with higher long-term periodic maxima. That is, given the appropriate conditions, ∗ 501 then for any t ≥ 0, there exists t ≥ t such that

∗ ∗ n1(t ) + n2(t ) > ni,pure(∞) (A14)

502 for both i = 1 and i = 2.

503 Proof. As shown in the main text, there exist cases where periodic alternation between the levels K = L1 ∗ ∗ ∗ 504 and K = L2 result in survival. Furthermore, there exists cases where n2(t ) = K(t ) ≥ L1 for some t 505 during each colonial phase. For any such case where both nomadism and colonialism are losing strategies, ∗ ∗ 506 notice that we would still have n2(t ) = K(t ) ≥ L1 even if nomadism or colonialism were modiﬁed so 507 that they did not lose. 508 To be precise, if r1 = 0 (nomadism does not lose), then there would be no decay over each nomadic 509 phase, which means that in the subsequent colonial phase, n2 would start out higher and grow to reach K 510 even more quickly. If A < 1 (colonialism does not lose), then g2(n2) would be even larger in magnitude, 511 so n2 would grow towards K more quickly as well. In both cases, K would decay less than it would have ∗ ∗ ∗ 512 originally by the time n2(t ) = K(t ), so we would still have n2(t ) > L1. It follows that as long as 513 L1 > ni,pure(∞), Equation A14 will always be true. Such cases can be easily found for i = 1 (ensuring 514 L1 > n1,pure(0)) and for i = 2 (ensuring L1 > n2,pure(∞) = 1). This completes the proof.