Nonlinear Dynhttps://doi.org/10.1007/s11071-018-4436-2

ORIGINAL PAPER

Nomadic-colonial switching with stochastic noise:subsidence-recovery cycles and long-term growth

Jin Ming Koh · Neng-gang Xie · Kang Hao Cheong

Received: 14 February 2018 / Accepted: 16 June 2018© Springer Nature B.V. 2018

Abstract In this paper, we analyze a populationmodel comprising organisms that alternate betweennomadic and colonial behaviours by considering stochas-tic noise. Incorporating stochastic noise into the popu-lation dynamicswill allow awide range of environmen-tal fluctuations to be modelled. The theoretical frame-work is also generalized to include resource deple-tion by both nomadic and colonial sub-populations,and an ecologically realistic population size-dependentswitching scheme is nowproposed.Wedemonstrate therobustness of the present model to stochastic noise, andthe use of novel generalized pure time-based switchingschemes to achieve consecutive subsidence-recoverycycles and long-term proliferation. Our results are ofrelevance to many physical and biological systems.

Keywords Allee effect · Parrondo’s paradox ·Stochastic noise · Population dynamics · Ecologicalapplications · Game Theory · Complexity · Nomadic-colonial

J. M. Koh · K. H. Cheong (B)Engineering Cluster, Singapore Institute of Technology, 10Dover Drive, Singapore 138683, Singaporee-mail: Kanghao.Cheong@SingaporeTech.edu.sg

N. XieDepartment of Mechanical Engineering, Anhui University ofTechnology, Ma’anshan 243002, Anhui, China

1 Introduction

The diversification of available resources is an instinc-tive survival strategy amidst inexorably disadvanta-geous conditions. Indeed, it is known that diversityin behavioural traits and phenotypic expression isconducive for the proliferation of ecological popu-lations [1,2]. Stochastic switching between pheno-types, intrinsically emergent from genetic pathways,can also enhance resilience towards exogenous envi-ronmental fluctuations [3,4]. Alternate migrations isalso observed to enable population persistence, evenwhen constrained to sink habitats only [5,6].

Such mechanisms are reminiscent of the game-theoretic Parrondo’s paradox, in which two losinggames are combined to produce a winning outcome[7–9]. In these paradoxes, stochastic perturbationsfrom one game enable the exploitation of asymme-try in the other, to result in sustained average capi-tal growth [10,11]. To date, the Parrondo frameworkhas been exceedingly valuable in understanding a widerange of physical phenomena and processes, includ-ing bulk drifts in granular and diffusive flow [12,13],chaos control via mixing of two or more chaotic pro-cesses [14–16], and entropy in information thermo-dynamics [17–19]. Quantum variants of Parrondo’sgames, which incorporate entanglement and interfer-ence effects, promise applicability in quantum com-puting and information processing [20–22]. Switchingproblems have also been analyzed under the frame-

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work [23,24], with a class of parameter switching algo-rithms yielding generalizations of Parrondo’s paradox[25]. In biology, the Parrondo effect has been rele-vant in explaining transitions between unicellular andmulticellular life [26], epistatic genetic selection [27],tumour growth dynamics [28], and in modelling eco-evolutionary processes [29,30]. There have also beenmany studies exploring the fundamental mathematicalproperties of the paradox [31–34]. It is hence appar-ent that the Parrondo’s paradox has broad implicationsacross many disciplines.

Recently, the periodic alternation of ecological pop-ulations between nomadic and colonial behaviours hasbeen analyzed in the context of the paradox [35,36].The nonlinear mechanics implemented accounts forboth the Allee effect [37,38] and the potential pres-ence of environmental fluctuations [39,40], the lat-ter via sinusoidal noise functions coupled to the envi-ronmental carrying capacity. It was demonstrated thatalternating between nomadism and colonialism couldresult in population persistence, despite each strategybeing losing individually. These paradoxical survivalscenarios occur, as long as colonies grow sufficientlyquickly when resources are abundant, and switch suffi-ciently fast to nomadic lifestyle before environmentalover-depletion occurs and resource levels become dan-gerously low [35,36].

In the present study, we investigate the effects ofnumerous types of stochastic environmental noise, andgeneralize the theoretical framework to accommodaterealistic environmental depletion by both the colonialand nomadic sub-populations. For instance, weatherpatterns are known to be chaotic [41–43], and inter-species interference is inherently uncertain, depen-dent upon complexmulti-agent dynamics [44,45]. Thismotivates the use of aperiodic stochastic noise in placeof sinusoidal noise in our theoretical framework, so thata wider range of fluctuation conditions can be mod-elled. We demonstrate the robustness of the reformu-lated switching rules to stochastic noise, and presentthe previously unexplored phenomenon of consecutivepopulation subsidence-recovery cycles. The possibilityof long-term proliferation (t ≥ 10,000) via nonlineartime-dependent transition period scaling is also pre-sented. The underlying mechanisms of these intriguingphenomena are elucidated, and are coherent in forminga common qualitative understanding of the populationdynamics involved.

Section 2 proposes a population model incorporat-ing stochastic noise. Section 3 presents the numeri-cal results, starting with system response and robust-ness to stochastic noise in Sect. 3.1, and the effects ofnomadic environmental depletion in Sect. 3.2. Next,we present the observed phenomenon of subsidence-recovery cycles in Sect. 3.3 and the possibility ofachieving long-term growth via nonlinear transforma-tions to the transition periods in Sect. 3.4.

2 Population model

The formalism used in this paper closely follows thosegiven in the original model [35,36]. The coexistence oftwo sub-populations is considered: the nomadic organ-isms, and the colonial ones. These populations are ableto transition from one behaviour to the other, over someperiod of time. Their population sizes can be modelledas

dnidt

= ρi (ni ) +∑

j �=i

χi j n j −∑

j �=i

χ j i ni (1)

where ni is the size of sub-population i , ρi is the func-tion describing the growth rate of ni in isolation, andχi j

is the rate of switching to sub-population i from sub-population j . Coupling between the sub-populations isconsequent of these switching terms, in general time-dependent or resource-dependent; other types of non-linear coupling have been studied in existing literature[46–48].

2.1 Nomadism and colonialism

We let n1 and n2 be the nomadic and colonial popula-tion sizes, respectively. In the absence of behaviouralswitching, the nomadic growth rate can be written as

ρ1(n1) = − ξ1n1 (2)

where ξ1 is the nomadic growth constant. Nomadismcan be made a losing strategy by setting ξ1 > 0, suchthat n1 decays with time. This is consistent with fieldobservations that nomadic phases seldom support pop-ulation growth, as evidenced by a study on slimemouldDictyostelium discoideum [49], dimorphic fungi Can-dida albicans [50], and jellyfish Aurelia aurita [51].

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On the other hand, colonial population dynamics canbe modelled using a modified logistic equation whichincorporates the Allee effect [37,38]. Letting ξ2, K ,and � denote the colonial growth constant, environ-mental carrying capacity, and the critical Allee capac-ity, respectively, growth rate in isolationmay bewrittenas [36]:

ρ2(n2) = ξ2n2

(n2

min(�, K )− 1

) (1 − n2

K

). (3)

Realistically, we must have the growth rate of the colo-nial phase ξ2 > 0. It has also been shown in our previ-ous study [35,36] that � > 1 always results in extinc-tion. In other words, both the nomadic and coloniallifestyles are losing individually.

The carrying capacity K can be expected to declineat a rate dependent upon both the colonial and nomadicpopulation sizes n1 and n2, due to resource depletion.Importantly, we add a stochastic noise function f (t),to account for environment fluctuations. The rate ofchange of K can then be expressed as

dK

dt= μ − ν1n1 − ν2n2 + f (t) (4)

whereμ > 0 is the default growth rate of K , and νi > 0is the per-organism rate of habitat destruction. Withoutloss of generality, all variables may be scaled such thatμ = ν2 = 1.

2.2 Stochastic environmental noise

Setting f (t) = ∑ni=1 γi sin (ωi t + φi )withγi , ωi , φi ∈

R yields the periodic noise functions proposed in theprevious study [36]. As discussed in Sect. 1, we useaperiodic stochastic noise models in the present study,in order to capture a wider range of environmental fluc-tuations.

A natural extension is to replace the sinusoidalsum with an infinite series, with amplitude coefficientsγi being normally distributed with identical variance.Interpreted as a Fourier series, this is analogous to thecanonical Gaussian white noise with associated vari-ance σ 2. Notably, statistics based upon white noise areused in phylogenetics to analyze evolutionary pathways[52,53], and white noise is similarly relevant in the the-oretical understanding of ecological dynamics [54,55].

Theuse ofwhite noise in our populationmodel is, there-fore, biologically motivated.

It is characteristic of white noise to have a flat powerspectral density—that is, the power contained in everyfrequency interval is constant. However, there is exist-ing evidence that fluctuations in ecological systems arebetter modelled using noise with power spectral densi-ties inversely proportional to frequency [56,57]. Sucha type of stochastic noise is termed 1/ f noise, or pinknoise. The relevance of 1/ f noise extends far beyondecology, to human cognition [58], chaos analysis [59],computer networks [60], microelectronics and quan-tum information [61], physical self-organizing systems[62], and even to gravitational-wave astronomy [63]. Inthe present study, we consider 1/ f noise in addition towhite noise, along with a third combined noise model,consisting of superimposed white and 1/ f noise withvariance renormalization of σ 2/2 each.

2.3 Behavioural switching

Biological clocks control an exceedingly wide rangeof physiological activities within animals. It can beexpected that nomadic-colonial transitions are drivenin a similar fashion. As in the original model [36], wehave defined a time-based switching scheme of periodT in terms of t mod T and switching constant rs > 0:

χ12 ={rs if κ2 ≤ t mod T < κ3

0 otherwise

χ21 ={rs if 0 ≤ t mod T < κ1

0 otherwise

(5)

As a way of example, the phase boundaries 0 <

κ1 < κ2 < κ3 were taken to be κ1 = C1/K , κ2 =C2/K , and κ3 = T = 1 previously [36]. Real-worldecological populations indeed respond to changes inenvironmental resources, as the proposed scheme sug-gests. The sensing of environmental capacity K bythese populations represents a possible scenario. How-ever, organisms can also detect their population sizeswith relative ease too. For instance, it is known that antcolonies track sub-population sizes to facilitate the del-egation of roles, via a pheromone mechanism [64,65];a large variety of animals are also known to adjustfeeding and mating behaviours in response to popu-lation size fluctuations [66,67]. We, therefore, propose

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κ1 = C1/(n1 + n2) and κ2 = C2/(n1 + n2) as viablephase boundaries, which can realistically be achievedin real-world ecological populations. For illustrativepurposes, we use C1 = 0.3 and C2 = 0.7 for severalexamples shown in this paper.

3 Results

Numerical simulations implementing the populationmodel and time-based switching scheme were per-formed inMathematica 11with the included nonlineardifferential system solver, which implements a gener-alized Implicit Differential-Algebraic Solver package.Runge–kutta pairs of adaptive order are invoked whenthe differential system is non-stiff; a backward differ-entiation formula scheme is invoked when stiffness isdetected. The generation of white and 1/ f noise wasalso implemented using native functions.

3.1 Stochastic environmental noise

We present in Fig. 1 simulation results for all threestochastic noise models, at low and high amplitudes.Clearly, the new switching scheme is robust towardsenvironmental noise, and it is important to point outthat this remains so even when the noise is of compa-rable magnitude to the initial population sizes and K .Even during periods of major environmental decline,the populations respond sufficiently quickly to avoidexceeding the carrying capacity. Recovery thereafter issimilarly swift.

It is notable that the extinctions observed under low-amplitude periodic fluctuations [36] no longer appearswith stochastic environmental noise. A plausible expla-nation is that the suspected mechanisms of resonanceand forced oscillations can no longer occur effectivelywith stochastic noise, whose power spectrum is spreadthinly over a wide range of frequencies. As a result,the nomadic-colonial switching scheme appears morestable on stochastic noise.

Extinction instead occurs when the switching con-stant rs is insufficiently large, such that the popula-tions are not able to respond to environmental fluctu-ations quickly. This is demonstrated in Fig. 2. Inade-quate rs hinders the rate of transition between nomadicand colonial lifestyles; as a result, the population mightexceed the environmental carrying capacity during

periods of environmental volatility, potentially overmultiple clock cycles. The consequent penalty on colo-nial growth rate might be so large that the populationis unable to recover, hence leading to extinction.

To further demonstrate this effect, we present inFig. 3 the probability of survival Ps in rs-σ parame-ter space. It is clearly observed that as the magnitudeof environmental noise increases, steep increases in rsare necessary to ensure a reasonable chance of sur-vival. In other words, good motility between nomadicand colonial behaviours is important to ensure survival,especially when environmental fluctuations are large,say, due to inter-species competition or volatile climate.

3.2 Nomadic environmental depletion

The coupling of both nomadic and colonial populationsizes to the carrying capacity is also implemented inthe present study [36]. Figure 4 shows the effect ofnomadic environmental depletion, which is unexploredto date. It can be observed that heavy depletion duringthe nomadic phase (ν1 ∼ ν2) results in extinction.

Fundamentally, nomadic-colonial behavioural switch-ing enables paradoxical population growth becausethe over-exploitation of environmental resources isavoided [35,36]. The colonial sub-population growssufficiently fast to offset stagnation during nomadicphases, depleting the environment in the process; thespecies then switches to nomadism before resourcesdip to dangerous levels, sacrificing growth to enablethe environment to regenerate. This is analogous to theagitation-ratcheting mechanism in classical Parrondo’sgames, in which asymmetry in one game is exploitedrepeatedly by activity from the other.

Any breakage in this environmental depletion-recovery cyclewill disrupt sustainedpopulationgrowth,as is indeed seen in Fig. 4. Exceedingly heavy resourceconsumption during the nomadic phase leaves insuf-ficient carrying capacity for breakeven growth duringthe colonial phase; this debt accumulates into an even-tual extinction. For survival, the nomadic lifestyle musthave a depletion coefficient ν1 about an order of mag-nitude smaller than that of colonialism ν2.

3.3 Subsidence-recovery cycles

While the ecological relevance of the population size-dependent switching scheme has been justified in

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Nomadic-colonial switching with stochastic noise

Fig. 1 Numerical simulations of the modified time-basedswitching scheme incorporating population sizes, for low noiseamplitude σ = 0.1 (top row) and high noise amplitude σ = 1

(bottom row). Initial conditions are n1 = 0, n2 = 2, K = 5,with rs = 1000 and ν1 = 0.05. The scheme is robust towardsstochastic noise

Fig. 2 Populationdynamics with switchingconstant (a) rs = 15 and (b)rs = 20. Extinction occurswhen rs is insufficientlylarge. Initial conditions aren1 = 0, n2 = 2, K = 5,with ν1 = 0.05. Identicalcombined stochastic noiseof σ = 0.5 is imposed onboth simulation runs

Sect. 2.3, it is nonetheless likely that some organisms,especially primitive ones,may not have the cognitive orcommunicative traits necessary for tracking populationsize. It is also plausible that some species choose not toutilize population size as an indicator for behaviouralswitching, despite having the ability to do so. In thesecases, a switching mechanism based purely on biolog-ical clocks may be more relevant, without the need forsensory information on exogenous conditions such as

resource levels, carrying capacity, or overall populationsize.

It was shown in our previous work that long-termpopulation growth can be achieved, as long as the colo-nial phase decreases in duration as t increases [36]. Thiscan bemademanifest via the following pure time-basedswitching scheme:

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Fig. 3 Plot of survival probability Ps in rs -σ parameter space,at t = 100. Initial conditions are n1 = 0, n2 = 2, K = 5,with ν1 = 0.05. Numerical simulation results are averaged over10,000 repetitions

χ12 ={rs if κ2 ≤ (1 + t/η) t mod T < κ3

0 otherwise

χ21 ={rs if 0 ≤ (1 + t/η) t mod T < κ1

0 otherwise

(6)

where η > 0 is a timescale constant, and κ1, κ2, and κ3are constants. As a way of illustration, we use η = 5,κ1 = κ2 = 0.15, κ3 = 0.45, and T = 1.

Under such a switching scheme, species survivalis possible with reasonable colonial growth rates(ξ2 < 20). With larges values of ξ2, however, over-exploitation of environmental resources occurs, result-ing in species extinction. Remarkably, with even largervalues of ξ2, consecutive cycles of near-extinction andrecovery are observed, with the nomadic and colonialpopulation sizes oscillating in a sawtooth-like man-

ner before final extinction. Figure 5 demonstrates thisresult. It should be noted that this phenomenon ofsubsidence-recovery cycles occurs bothwith orwithoutenvironmental noise—it is an intrinsic property of theswitching structure, not one induced by external noise.

First, the rapid colonial growth rate results in sus-tained population inflation. At some point in time, anomadic-to-colonial transition introduces such a largecolonial population size that the environmental car-rying capacity is greatly exceeded, thus triggering adecline in population size. As the nomadic-to-colonialtransition is still ongoing, the carrying capacity remainson a declining trend. This sets up a positive feedbackthat forces the collapse of the species essentially withina single clock cycle, the steepness of which is indeedobserved in Fig. 5.

Near the final moments of collapse, the populationbegins the transition to nomadism, essentially allowingthe species to preserve some individuals alive as colo-nialism plunges into extinction. This switch is aidedby the temporal compression of colonial phases as tincreases. By the time the next switch occurs, the envi-ronmental carrying capacity has recovered, thus allow-ing the species to repopulate. This accounts for therecovery phase of the cycle.

Such subsidence-recovery cycles are reminiscent ofthe population dynamics of bacteria under the influenceof antibiotics [68–70]. Upon the introduction of antibi-otics, sub-populations that are not resistant quicklydiminish, leaving only the resistant strains preservedalive. These preserved resistant strains can then repop-ulate when conditions are favourable, typically after

Fig. 4 Comparison of population dynamics with (a) ν1 = 0.05light nomadic environmental depletion and (b) ν1 = 0.30heavy nomadic environmental depletion. Extinction occurs if thenomadic sub-population consumes environmental resources too

heavily. Initial conditions are n1 = 0, n2 = 2, K = 5. Identi-cal combined stochastic noise of σ = 0.2 is imposed on bothsimulation runs

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Fig. 5 On a pure time-based switching scheme, large colonialgrowth rates can result in multiple subsidence-recovery cyclesbefore eventual extinction. Comparisons between (a) ξ2 = 30,(b) ξ2 = 50, and (c) ξ2 = 120 are presented. Initial conditions

are n1 = 0, n2 = 2, K = 5, with rs = 500 and ν1 = 0.01.Identical combined stochastic noise of σ = 0.05 is imposed onall three simulation runs

the drug has been metabolized. The antagonistic pres-sure from the drug mirrors the environmental carry-ing capacity in our context, and the diversification ofbacterial phenotypes can be considered analogous tothe nomadic-colonial switching strategy, with exter-nal environmental noise modelling fluctuations in theimmune response of the host. It is also known, more-over, that certain types of micro-organisms can transi-tion into non-reproductive, incredibly resilient dormantstates termed as endospores [71,72]. This allows sur-vival amidst harsh conditions, in a similar fashion tonomadic-colonial switching. The subsidence-recoverycycles observed in our theoretical framework, thus,have potential applications in microbiology and ecol-ogy, particularly in the quantitative modelling of pop-ulation dynamics.

3.4 Long-term asymptotic growth

In Sect. 3.3, we have observed how a pure time-basedswitching scheme with a linear scaling factor (1+ t/η)

can give rise to consecutive cycles of subsidence andrecovery, ultimately leading to extinction. In general,however, the pure time-based scheme may be general-ized to be dependent on other real powers of t , whichwe denote p below. Such a generalized scheme can bewritten as:

χ12 ={rs if κ2 ≤ (1 + t p/η) t mod T < κ3

0 otherwise

χ21 ={rs if 0 ≤ (1 + t p/η) t mod T < κ1

0 otherwise

(7)

Such a generalized scheme allows for the com-plete avoidance of the subsidence-recovery cycles andcan result in extreme long-term population prolifer-ation for certain p. As before, we choose constantsκ1 = κ2 = 0.15, κ3 = 0.45, and T = 1 for the pur-pose of illustration. We present in Fig. 6 two exam-ples of such a phenomenon, in which we may observepopulation growth towards limiting asymptotes up tot = 10,000without anydecline of population size. Thisis in clear contrast with the results in Fig. 5, in whichthere is clear subsidence-recovery behaviour and even-tual extinctions at t < 150.

The underlying principles behind this bear lim-ited resemblance to the extinction phenomenon causedby insufficient switching constant rs , as discussed inSect. 3.1. We note that lowering the power p causesthe scaling factor (1 + t p/η) to grow more slowlyin t , and as a result χ12 and χ21 will be nonzero ingreater intervals of time. In other words, the periods oftime spent transitioning between colonial and nomadicphases will increase. This is in effect similar to increas-ing rs , in the sense that a greater number of individualscan complete the transition. When p is too large, thepopulation is not nimble enough to transition betweennomadic and colonial behaviours sufficiently, and runthe risk of either over-depleting the environment, orunder-utilizing the procreative potential of availableindividuals, hence explaining the eventual extinctions.Addressing the subsidence-recovery cycles in Sect. 3.3,the observed population collapses would not have hap-pened if the organisms responded sufficiently fast todeclines in carrying capacity.

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Fig. 6 Population dynamics with generalized pure time-basedswitching scheme,with (a) power p = 1/2 and (b) p = 1/4. Thepopulations exhibit extreme long-term sustained proliferation,displayed up to t = 10,000 in these graphs, without any subsi-

dence or population size declines. Initial conditions are n1 = 0,n2 = 2, K = 5, with rs = 500 and ν1 = 0.01. Identical com-bined stochastic noise of σ = 0.05 is imposed on both simulationruns

Fig. 7 (a) Plot of total population size envelope n∗ = n∗1 + n∗

2in η-p parameter space at t = 2000, showing regions of suffi-ciently large η and sufficiently small p where long-term growthis possible. Black regions indicate extinction, and a red line is

drawn to indicate the transition boundary between extinction andsurvival. In (b), a plot of n1 and n2 and their envelopes n∗

1 andn∗2 is presented, to illustrate how the envelopes are computed.

(Color figure online)

In general, the feasible values of p in order forsuch long-term proliferation to be possible is depen-dent on η. From the aforementioned mechanism, wecan expect that increasing η will also allow for theavoidance of subsidence and eventual extinction, forit increases the durations of the colonial and nomadictransitions. Indeed, simulation results on the popula-tion size envelope n∗ = n∗

1 + n∗2 presented in Fig. 7a

are consistent with this reasoning, whereby increasingη or decreasing p sufficiently will allow for long-termsustained proliferation. The computation of envelopesn∗1 and n∗

2 is shown in Fig. 7b.At the same time, we also note that the popula-

tion size envelope n∗ exhibits a local maximum atsome intermediate value of η for every p (Fig. 7a).This represents optimal states for the discussed mech-anism, where the speed of colonial-nomadic transitionis favourable. At greater values of η there is a gradual

decrease in the population size envelope, and at smallervalues of η there is a very steep decline, leading towardspopulation extinction.

4 Conclusion

At the temporal scales relevant to ecological dynam-ics, non-periodic stochastic noise can allow a widerrange of environmental fluctuations to be modelled.In general, these fluctuations arise from a combina-tion of chaotic weather patterns, inter-species compe-tition, and complex interactions within the food web.By demonstrating the robustness of the modified time-based switching strategy to stochastic environmentalnoise, we extend the relevance of nomadic-colonialalternation to many other real-world biological sys-tems. Our results also highlight the importance of good

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behavioural motility, such that responses to environ-mental fluctuations are enacted sufficiently quickly;this is indeed consistent with observations in evolu-tionary biology [73,74].

Throughout the present study, a realistic couplingfor the environmental carrying capacity involving boththe nomadic and colonial population size is considered.(a) Species survival is found to be possible, as long asenvironmental depletion is not excessively heavy dur-ing the nomadic phase. (b) We have also demonstratedthe occurrence of consecutive subsidence-recoverycycles, under a pure time-based switching scheme.(c) Further generalization of the time-based schemereveals the possibility of avoiding these subsidence-recovery cycles, and even achieving long-term asymp-totic growth, via nonlinear time-dependent scaling onthe nomadic-colonial transition rates. Our simulationsindicate that these results hold for a wide range of ini-tial conditions. The findings presented in this paperare relevant in ecological and microbiological pop-ulation modelling; the generalizability of the frame-work also aids greatly in extending the model intoother disciplines, for instance, in themodelling of wild-fire and reforestation, human resource consumption,and generic environmental sustainability, all of whichinvolve multiple stochastic temporal phases. The rel-evance of the framework is similarly extendable tophysical multi-body systems, such as self-organizingautomata and swarm robotics [75,76], which mayexploit switching strategies to achieve collective goals.

Compliance with ethical standards.

Conflict of interest The authors declare that they have no con-flict of interest.

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