Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier.com/locate/cnsns

Research paper

A time-based switching scheme for nomadic-colonial

alternation under noisy conditions

Kang Hao Cheong

a , ∗, Zong Xuan Tan

b , Yan Hao Ling

a

a Engineering Cluster, Singapore Institute of Technology, 10 Dover Drive, Singapore, 138683, Singapore b Yale University, New Haven, CT 06520, United States

a r t i c l e i n f o

Article history:

Received 14 November 2017

Revised 25 December 2017

Accepted 26 December 2017

Available online xxx

a b s t r a c t

Recently, a population model that incorporates the Allee effect was developed to explain

how behavior-switching organisms can survive even when each behavior would individ-

ually lead to extinction. The model exhibits nonlinear dynamics in ecologically realistic

settings. The organisms modelled could periodically exploit their habitat as a colony, then

switch to a nomadic lifestyle to allow the environment to regenerate. This strategy was

shown to be viable as long as colonies grew sufficiently quickly when environmental re-

sources were abundant, and colonists switched to a nomadic lifestyle before resource levels

dipped dangerously low. Here, we extend the applicability of this model by incorporating

noise that can arise due to uncertainty, external interference, or seasonal variations. Im-

portantly, we have demonstrated that time-based switching between behaviors can ensure

that survival is robust to noise. In this way, time-based switching scheme can be used to

complement the previous model by emulating the success of nomadic-colonial strategies

under real life conditions more closely.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

Parrondo’s paradox states that playing two losing games in a random or periodic order can result in a winning outcome

[1] . The existence of a winning combination relies on the fact that at least one of the losing Parrondo games exhibits either

capital-dependence (dependence upon the current amount of capital) or history-dependence (dependence upon the past

history of wins or losses) [2,3] . The Parrondo’s games are related to the Brownian ratchets [4–7] and have applications in

physics, biology, engineering and on the financial market. Applications have also been linked to quantum models [8,9] where

the classical coin toss is replaced by the measurement of a qubit, and a generalized form of Parrondo games has been

implemented for the logistic map to obtain stable orbits [10] . The study of switching problems has also been understood

under the framework of Parrondo’s paradox [11,12] . We have also seen how the Parameter Switching algorithm can lead

to a generalization of Parrondo’s paradox [13] . There have been many studies exploring the paradox [14–22] , whether in

the context of complex social networks [23] , bacterial random phase variation [24] , the evolution of less accurate sensors

[25] , or in the analysis of tumor growth [26] . Recently, the periodic alternation of organisms between nomadic and colonial

behaviors has also been analyzed in terms of the paradox, exhibiting nonlinear dynamics in ecologically realistic settings

∗ Corresponding author.

E-mail address: kanghao.cheong@singaporetech.edu.sg (K.H. Cheong).

https://doi.org/10.1016/j.cnsns.2017.12.012

1007-5704/© 2017 Elsevier B.V. All rights reserved.

2 K.H. Cheong et al. / Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8

with both capital (an ecological analogy of which is population size) and history-dependent dynamics (in an ecological

context of growth and decline) [27] .

In the aforementioned study, it was demonstrated that alternating between nomadism and colonialism could ensure

survival even when each behavior would lead to extinction on its own [27] . Survival was ensured as long as colonies grew

sufficiently quickly when resources were abundant, and colonists switched to a nomadic lifestyle before over-exploiting their

environment. It was assumed however that the influence of environmental noise was negligible compared to the amount

of organisms and resources. This assumption may not always hold, especially when considering ecological dynamics and

resilience [28] . Noise can arise due to inherent uncertainties such as seasonality, interference by other species, or variation

in nutrient inputs [29–32] . In the present study, we investigate the effects of noise on the nomadic-colonial model [27] and

develop a novel time-based switching scheme to overcome these effects.

In the following sections, we reformulate the original population model such that it incorporates noise. We then discuss

the limitations of the population model using the original switching rule and motivate the need for a better switching

scheme. Finally, we introduce the time-based switching scheme, and then comment on the robustness of this scheme in

response to changes in the noise characteristics, as well as the implications for modeling real-life systems. The definitions

and notations used in this paper follow closely those given in the original model [27] .

2. Population model

As in the original model [27] , there exist two sub-populations: the nomadic organisms, and the colonial ones. Organisms

that exist in multiple sub-populations can be modeled as follows:

dn i

dt = g i (n i ) +

∑

j

s i j n j −∑

j

s ji n i (1)

where n i is the size of sub-population i, g i is the function describing the growth rate of n i in isolation, and s ij is the rate of

switching to sub-population i from sub-population j .

2.1. Nomadism

Let n 1 be the nomadic population size. In the absence of behavioral switching, the nomadic growth rate is given by

g 1 (n 1 ) = −r 1 n 1 (2)

where r 1 is the nomadic growth constant. Nomadism is modelled as a losing strategy by setting −r 1 < 0 , such that n 1 decays

with time.

2.2. Colonialism

Colonial population dynamics are modelled using a modified logistic equation which incorporates the Allee effect [33] ,

where the carrying capacity is K and the critical capacity or ‘Allee capacity’ is A . Let n 2 be the colonial population size. In

the absence of behavioral switching, the colonial growth rate is given by

g 2 (n 2 ) = r 2 n 2

(n 2

min (A, K) − 1

)(1 − n 2

K

)(3)

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 min ( A, K ) term ensures that when K < A, g 2 is always zero or negative, as would be expected. It

will be shown later that as long as A > 1, pure colonialism results in extinction.

On top of these dynamics, 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 main difference here from the original

model is that an additional term of γ sin ( ωt ) is incorporated to account for the presence of noise. The rate of change of K

with respect to time is thus

dK

dt = α − βn 2 + γ sin (ωt) (4)

where α > 0 is the default growth rate of K , and β > 0 is the per-organism rate of habitat destruction. The short-term carry-

ing capacity K can be thought of as dependent on some essential resource in the environment, which slowly depletes at an

average rate proportional to βn 2 , but with fluctuations proportional to γ sin ( ωt ).

2.3. Time-based behavioral switching

Biological clocks help regulate sleep patterns, feeding behaviour and hormone release in animals [34–39] . They control

the physiological activities of organisms according to a periodic cycle, which might be daily, yearly, or any other length. A

K.H. Cheong et al. / Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8 3

Fig. 1. Population dynamics with the resource-based switching rule under noisy conditions. Initial conditions are n 1 = 0 , n 2 = 2 , K = 5 , shared parame-

ters are r s = 10 0 0 , L 1 = 3 , L 2 = 4 . In (a), dK dt

= 1 − n 2 + sin (20 t) . In (b), dK dt

= 1 − n 2 + 10 sin (20 t) . Survival is possible under noise of high but not lower

amplitudes.

periodic switching scheme can thus be defined in terms of t mod T , the time of a cycle with respect to its period T . For ex-

ample, if T = 1 day, then t mod T measures the time of the day. Mathematically, t mod T satisfies the following properties:

0 ≤ t mod T < T

t − (t mod T ) = kT where k is some integer

With this, we can model organisms which switch periodically between nomadism and colonialism. Specifically, we can

define the phase boundaries 0 < c 1 < c 2 < c 3 , together with the corresponding phases:

0 ≤ t mod T < c 1 ( nomadic-to-colonial switching )

c 1 ≤ t mod T < c 2 ( colonial phase )

c 2 ≤ t mod T < c 3 ( colonial-to-nomadic switching )

c 3 ≤ t mod T < T ( nomadic phase )

Let r s > 0 be a fixed switching constant that captures the speed at which behavioral switching occurs. The switching rates

s 12 and s 21 in Eq. (1) at a given time t can thus be expressed as

s 12 =

{r s if c 2 ≤ t mod T < c 3 0 otherwise

s 21 =

{r s if 0 ≤ t mod T < c 1 0 otherwise

(5)

For illustrative purposes, we use T = 1 , c 1 = 0 . 1 , c 2 = 0 . 2 , c 3 = 0 . 3 for several examples shown in this paper. When T = 1 ,

the expression t mod T reduces to the function frac( t ), which returns the fractional part of the real number t .

3. Results

Numerical simulations implementing the population model and time-based switching scheme, were performed in MAT-

LAB R2017a with the included ode23 ordinary differential equation (ODE) solver. ode23 implements the Runge-Kutta (2,3)

formula pair by Bogacki and Shampine [40] .

3.1. Non-robustness of resource-based switching to noise

Under the resource-based switching rule described by Eq. (5) of our initial study [27] , the survival of the population is

not entirely resistant to noise. The simulation results in Fig. 1 predict that the population goes extinct in the presence of

low amplitude noise ( Fig. 1 (a)), but surprisingly survives when noise amplitudes are increased ( Fig. 1 (b)).

4 K.H. Cheong et al. / Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8

Fig. 2. Survival in the presence of large-amplitude ( γ = 10 ) high-frequency ( ω = 20 ) noise. Initial conditions are n 1 = 0 , n 2 = 2 , K = 5 . For (a), r s = 100 , and

for (b), r s = 40 .

One explanation for this phenomenon is that low-amplitude noise of high frequency triggers switching at inopportune

times under the resource-based rule. For example, a downward spike in the resource levels might delay the switch from no-

madism to colonialism, such that there are insufficient numbers of organisms for a self-sustaining colony after the switch.

This can be observed to occur in Fig. 1 (a) shortly before t = 2 . However, such effects cease to occur when the noise am-

plitude is sufficiently high, as in Fig. 1 (b). This suggests the complicating involvement of resonance and forced oscillations.

While interesting, these effects are beyond the scope of the present study, which focuses instead on developing a switching

rule that is robust to noise of both low and high amplitudes.

3.2. Survival and robustness under time-based switching

Under the time-based switching scheme given by Eq. (5) , simulation results predict that organisms can survive in the

presence of large-amplitude high-frequency noise. Examples are shown in Fig. 2 , for both (a) a high switching rate ( r s = 100 )

and (b) a low switching rate ( r s = 40 ). Further simulations also show that organisms can survive in the presence of high-

frequency noise at reduced amplitudes. Similarly, simulations predict that time-based switchers can survive in the presence

of small-amplitude low-frequency noise ( Fig. 3 ), demonstrating the robustness of a time-based switching scheme to noise of

varied frequencies and amplitudes. Preliminary simulation results have predicted that high-amplitude low-frequency noise

will lead to extinction of the organisms. This is not a serious concern, because organisms can adapt by matching the rate

of resource depletion to the frequency of noise, such that ‘low-frequency’ noise is no longer ‘low’ relative to the rate of

resource depletion.

To further demonstrate the robustness of the time-based scheme, noise of different frequencies and amplitudes can be

superimposed. As an illustration, we consider three noise functions of varying frequency and amplitude, such that we have:

dK

dt = 1 − n 2 + 10 sin (20 t) + 5 sin (5 t) + sin (t)

Under a high switching rate of r s = 100 , the results in Fig. 4 predict that the time-based scheme is robust even to this

superimposed noise. Altogether, the time-based scheme enjoys much higher noise robustness than the original resource-

based scheme.

3.3. Long-term growth through time-based switching

It was shown in our previous work [27] that long-term population growth is possible through strategic resource-based

switching. However, this strategy requires detection of when population levels exceed carrying capacity. Here, we modify

our newly proposed time-based switching scheme, giving a more elegant and biologically feasible method that allows for

reliable long-term growth even under noisy conditions.

Notice that Eq. (5) defines a switching rule where every phase in the cycle has fixed duration. As Fig. 5 depicts, such a

switching rule cannot result in long-term population growth. This is because resource depletion occurs during the colonial

phase c ≤ t mod T < c in an amount proportional to the average value of n . When the colonial phase is fixed in duration,

1 2 2

K.H. Cheong et al. / Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8 5

Fig. 3. Survival in the presence of small-amplitude ( γ = 1 ) low-frequency ( ω = 1 ) noise. Initial conditions are n 1 = 0 , n 2 = 2 , K = 5 . For (a), r s = 100 , and

for (b), r s = 40 .

Fig. 4. Noise of various frequencies is superimposed, giving a carrying capacity growth rate of dK dt

= 1 − n 2 + 10 sin (20 t) + 5 sin (5 t) + sin (t) . For (a), r s = 0

(no switching) and for (b), r s = 100 .

the colonial population n 2 cannot grow unboundedly large, because this would imply an unboundedly large depletion of K

— a contradiction, since n 2 cannot grow large if K does not also grow large.

In order to obtain long-term growth, the colonial phase must instead decrease in duration as t increases. With shorter

colonial phases, less depletion of K occurs as a result of the colonists, allowing K to grow even as n 2 increases. This can be

achieved via the following modification to Eq. (5) :

s 12 =

{r s if C 2

K ≤ t mod T < T

0 otherwise

s 21 =

{r s if 0 ≤ t mod T <

C 1 K

0 otherwise

(6)

6 K.H. Cheong et al. / Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8

Fig. 5. Fixed frequency timers cannot achieve long-term growth. d K/d t = 1 − n 2 + 10 sin (20 t) + 5 sin (5 t) + sin (t) , r s = 100 . The initial conditions have been

changed to n 1 = 0 , n 2 = 10 , K = 20 to show that despite more favourable initial conditions, the population still drop to an equilibrium value.

Fig. 6. Simulation using the modified switching rule in Eq. (6) , such that the duration c 2 − c 1 of the colonial phase decreases with time. Parameters are

T = 1 , C 1 = 0 . 3 , C 2 = 0 . 7 , C 3 = 1 , and r s = 10 0 0 . Noise is captured by dK dt

= 1 − n 2 + 10 sin (20 t) + 5 sin (5 t) + sin (t) . Note that r s is increased compared to

other figures shown because faster switching is required to ensure that larger population levels can switch behaviors sufficiently rapidly.

With this switching scheme, the phase boundaries are altered such that c 1 = C 1 /K, c 2 = C 2 /K, c 3 = T , where C 1 < C 2 are

constants. The phases in each cycle are thus:

0 ≤ t mod T <

C 1 K

( nomadic-to-colonial switching )

C 1 K

≤ t mod T <

C 2 K

( colonial phase )

C 2 ≤ t mod T < T ( colonial-to-nomadic switching / nomadic phase )

K

K.H. Cheong et al. / Commun Nonlinear Sci Numer Simulat 0 0 0 (2018) 1–8 7

As can be seen, the duration of the colonial phase C 2 −C 1

K is now inversely proportional to K . Hence, as long as K increases ini-

tially, the colonial phase duration will decrease with t . This then allows for a greater increase in K (by reducing the amount

of colonist-caused depletion), enabling the possibility of long-term growth. Also important is the fact that the third and

fourth phases in the original time-based scheme have been merged by setting c 3 = 1 , which means that s 21 = r s through-

out the interval C 2 K ≤ t mod T < T . This modification ensures that there are absolutely no colonists that linger after each

colonial phase, who might otherwise exploit the increased levels of K that occur during the nomadic phase. Despite these

modifications, the overall period is maintained at T .

Fig. 6 shows the simulation results for this modified switching rule with T = 1 , C 1 = 0 . 3 , C 2 = 0 . 7 , and r s = 10 0 0 . This

means that the colonial phase lasts while 0 . 3 /K ≤ t mod 1 < 0 . 7 /K, implying that the phase gets shorter as K grows. Fur-

thermore, colonial-to-nomadic switching occurs while 0 . 7 /K ≤ t mod 1 < 1 , which means that this duration gets longer as

K grows. The overall period, T is maintained at 1.0. It can be seen that, in accordance with reasoning above, long-term

growth is achieved up to t = 350 . Our simulation results predict that this holds under a wide range of parameters. Thus,

not only is time-based switching able to ensure population survival, it is also able to achieve long-term population growth

while remaining robust to noise, as depicted in Fig. 6 .

4. Conclusion

Given the ubiquity of noise in the natural world (e.g. seasonal variation), robustness to noise is essential for any strategy

which attempts to ensure survival and growth. By demonstrating the noise robustness of time-based alternation between

colonialism and nomadism, the present study extends the range of realistic biological situations under which nomadic-

colonial alternation allows an organism to thrive. Furthermore, the time-based switching rule provides an efficient mech-

anism by which organisms might undergo behavioral alternation — one that relies upon the well-established existence of

biological clocks. Rather than alternating behaviors by carefully tracking the resources available in the environment (which

introduces sensitive dependence to noise), we have shown that organisms can produce functionally similar behavior by syn-

chronizing their biological clocks to the average rhythm of resource depletion and regeneration, thereby achieving survival

and long-term growth. This is feasible, with or without the consideration of noise. Importantly, we have demonstrated how

a time-based switching scheme can be used to complement the nomadic-colonial population model and in extending its

applicability to more real-life scenarios.

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