Physics Letters A 375 (2011) 3124–3129
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Physics Letters A www.elsevier.com/locate/pla
Seasonality as a Parrondian game Enrique Peacock-López 1 Department of Chemistry, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
a r t i c l e
i n f o
Article history: Received 13 April 2010 Received in revised form 24 June 2011 Accepted 28 June 2011 Available online 5 July 2011 Communicated by C.R. Doering
a b s t r a c t Switching strategies can be related to the so-called Parrondian games, where the alternation of two losing games yields a winning game. We consider two dynamics that by themselves yield undesirable behaviors, but when alternated, yield a desirable oscillatory behavior. In the analysis of the alternate-logistic map, we prove that alternating parameter values yielding extinction with parameter values associated with chaotic dynamics results in periodic trajectories. Ultimately, we consider a four season logistic model with either migration or immigration. © 2011 Elsevier B.V. All rights reserved.
1. Introduction For the past ten years, switching strategies in dynamical systems has been reconsidered [1–7]. In the context of the so-called Parrondo’s Paradox, it has been proven that two losing games can be combined in a deterministic or random order to yield a winning game [8–10]. Originally, the Parrondo’s Paradox, which is a discrete version of continuous ratchet systems, was motivated by molecular motors, and Brownian and flashing ratchets [11,12]. For a recent review, see Abbot’s paper [13]. The fundamental idea of switching parameter values is not new in chemical systems. For instance, we have used an on–off switching to study polymerization of hydrocarbons with the purpose of modifying the polymer’s length distribution [14,15]. We have also considered the hypothalamus square-wave regulation of the pituitary gland [16]. In these cases, the alternate dynamics allow modifications of the final output, whereas the all-on or all-off conditions yield uninteresting dynamics or products. Other researchers have studied chemical systems under thermal cycling as Parrondian systems [17,18]. Moreover, microbial survival [19] and sexually antagonistic selection [20] have been analyzed as Parrondian games. We want to emphasize that there are two distinct types of Parrondian games. One type has been studied by Tang et al. [21], where the authors considered chaotic switching strategies. The second type considers random and periodic switching strategies, and it has been discussed by Almeida et al. [3]. We use the latter type to model seasonality. On the one hand, the idea of “lose + lose = win” has been studied in maps when looking for conditions or situations where “chaos + chaos = order” [3]. On
the other hand, we have considered the logistic map due to its relevance in modeling ecological systems, and we have studied situations where “undesirable + undesirable = desirable” dynamical behaviors, in the context of populations of non-overlapping generations like in the case of insects [22]. Therefore alternate dynamics in the logistic map is a natural approach to seasonality. In our analysis, we use a straightforward numerical construction of the bifurcation diagrams to find intervals in parameter space that meet our expectations [23]. This method can be extended to the logistic map but under different considerations, as in the case of controlling chaos. In our analysis, we consider periodic reduction pulses, where for the “ P th” iteration one multiplies the output by a constant “R”. Notice that, as a controlling technique, we use large pulse perturbations rather than small perturbative pulses. In particular we are interested in the periodic large pulse technique considered by Matías et al. [24] and others [25,26]. In the case of small perturbations, one could use the well-known Ott, Grebogi and Yorke (OGY) [27], chaos controlling method, where the chaotic attractor is driven onto a periodic attractor by repeated small perturbations of a relevant parameter. The literature related to chaos controlling strategies is vast, and we refer our readers to Boccaletti’s review [28] for a more detailed discussion and further references. In Section 2, we study the logistic map under switching strategies, and we examine different periodicities and find parameter values with relevant dynamics. In Section 3, we implement an extension of periodic parameter switching and consider a four-season logistic model that includes harvesting or immigration. Finally in Section 4, we discuss our results. 2. Switching in the logistic map
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In this section, we start by considering the well-known logistic map defined as:
E. Peacock-López / Physics Letters A 375 (2011) 3124–3129
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Fig. 1. Bifurcation diagram for the logistic map, where we plot 200 values of X n after dropping 500 iterations. We also blow-up the chaotic region of the diagram.
Fig. 2. Bifurcation diagram for P = 2, G 1 ( X , C , R ) and G 2 ( X n , C , R ), where we plot 200 values of X n after dropping 500 iterations. We consider C = C o = 4, which is related to chaotic trajectories of the logistic map, and we make R our bifurcation parameter.
Fig. 3. Lyapunov exponents for P = 2, G 1 ( X , C , R ) and G 2 ( X n , C , R ), where we plot 200 values of X n after dropping 500 iterations. We consider C = C o = 4, which is related to chaotic trajectories of the logistic map, and we make R our bifurcation parameter.
X n+1 = C X n (1 − X n ) = F ( X n , C )
(1)
where X n is less than unity and C lies in the interval [0, 4]. Using the logistic map, we can now define the alternate map as:
X n +1 =
G k ( Xn , C ) F ( Xn , C )
if mod[n, P ] = 0 otherwise
(2)
where mod[n, P ] stands for the modulus P function of n, and G k for the harvesting function, with two possible definitions:
G k ( Xn , C , R ) =
R F ( Xn , C ) R Xn
if k = 1 if k = 2
(3)
In Eq. (3), F ( X n , C ) represents the population growth function during the nth cycle or iteration. In the first case, k = 1, we reduced or increment the output of the P th iteration by a factor R, while in the second case, k = 2, the P th iteration consist in a reduction of the ( P − 1) population by a factor R. We discuss the former case in more detailed, but our results are easily extendable to the latter case. Notice that R = 1 represents the case of no action, which
would grow chaotically, and that harvesting, R < 1, or immigration, R > 1, could be useful strategies to stabilize the population. In Fig. 1, we construct the bifurcation diagram for the logistic map by taking 500 transient iterations and plotting the subsequent 200 values. We want to emphasize that for C values less than unity, the map drives the population X n to extinction. A second parameter region of interest relates the C values that are associated with chaotic trajectories, which occur for C values greater than 3.569945672 . . . . Therefore, from Fig. 1, we can easily select C values for our switching strategy and, in general, R becomes the bifurcation parameter. Recently we considered the case P = 2 and G 1 (xn , C ) = R F ( X n , C ) with a slightly different but equivalent definition [23]:
X n +1 =
f C e ( X n ) = C e X n (1 − X n ) f C o ( X n ) = C o X n (1 − X n )
if n even if n odd
(4)
If we compare Eqs. (3), (4), we notice that C o = C , C e = RC o , or C e is the bifurcation parameter. Notice that the compositions f C e ◦ f C o or f C o ◦ f C e are not equivalent because they imply two iterations
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Fig. 4. Bifurcation diagram for G 1 ( X , C , R ) and G 2 ( X n , C , R ), P = 3. We consider C = 4.
Fig. 5. Bifurcation diagram for G 1 ( X , C , R ), and P = 3. We consider C = 3. In one case R = 0.6550 is related to a steady state in the logistic map, and R = 0.9645 to chaos, resulting in a “chaos + chaos = periodic” case.
Fig. 6. Bifurcation diagram for G 1 ( X , C , R ) and P = 4. We consider C = 4, and the region of “chaos + chaos = periodic”.
at a time and they do not recover Fig. 1 when C e = C o = C , while Eq. (4) reproduces Fig. 1 when C o = C e = C . Using Eq. (3), we can recover our previous results [23] by setting C = C o and varying R from zero to unity, as in Fig. 2. In Fig. 2, we also consider the case G 2 ( X n , C , R ) = R X n , where for the P th iterations, the population is reduced by a factor R. If we compare the two periodic strategies, we observe that direct reduction of the population during the P th iteration has a more stabilizing effect than the case where we reduce the logistic map’s P th iteration output. Finally, we recall that RC < 1 is associated with extinction, while RC > 3.570 is associated with chaotic trajectories. For a more detailed discussion of Fig. 2, we refer our readers to our previous work [23]. It is also clear from the bifurcation diagram in Fig. 2 that for values R < 1/4, which are related to extinction, the switching strategy induces stable oscillations. The alternation of undesirable conditions leading to extinction with optimal conditions leading to chaotic oscillations gives rise to desirable population dynamics.
Therefore, we get a case of “undesirable + undesirable = desirable”. We also notice that for R = 0.90875, associated to chaotic oscillation in the logistic equation, the switching strategy yields stable oscillation, which is an example of “chaos + chaos = order”. For completeness, we include in Fig. 3 the corresponding Liapunov exponents associate with Fig. 2. Next, we consider the case P = 3, and we depict the bifurcation diagrams as a function of R in Fig. 4. As before, in Fig. 4 we consider both G functions, and we emphasize that R X n is more stabilizing than R F (C , X n ). In Fig. 5, we show that, by blowing up the diagram, it is easy to find parameter values for stable periodic trajectories. In one case, we consider the narrow range around R = 0.65, and, in the second case R = 0.96. In the latter case, we find that R = 0.9645 yields periodic oscillations but, RC = 3.858 yields a chaotic trajectory in the absence of switching. Therefore, R = 0.9645 is an example of “chaos + chaos = periodic”, which can be considered as a Parrondian game.
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Fig. 7. Bifurcation diagram for G 1 ( X , C , R ), G 2 ( X n , C , R ), and P = 6. We consider C = 4, and R = 1/3.
Fig. 8. Bifurcation diagram for G 1 ( X , C , R ), G 2 ( X n , C , R ), and P = 6. We consider C = 4 for R 1/20.
Fig. 9. Bifurcation diagram for G 1 ( X , C , R ), C = 4, and P = 6. We consider a “chaos + chaos = periodic” case.
For the case P = 4, in Fig. 6, we depict the case G 1 ( X , C , R ) for C = 4. It is clear from the figure that there are windows of stable oscillations for R < 1/4, which yield cases of “undesirable + undesirable = desirable”. In Fig. 6, we also include a blow up of the region around 0.915, showing a window of periodicity associated to “chaos + chaos = periodic”. As one would expect, with larger values of P the periodic windows become quite narrow. In Fig. 7 we depict the bifurcation diagram for P = 6, C = 4, and both G k functions, but only consider R < 1/3. As before, in the case G 2 , we notice wider periodic windows than in the case of G 1 . To emphasize the differences between these two choices, we blow up the region R < 1/20 in Fig. 8, where we can easily select parameter values associated with periodic oscillations. Therefore, by considerably reducing the output of every sixth iteration, we can stabilize and control an otherwise chaotic system. Finally, we consider larger values of R and look for periodic windows. In Fig. 9 we consider values around R = 0.9290 and find
a narrow interval of values. Consequently, we can find examples where the parameter of the sixth iteration would also yield chaotic trajectories for the logistic map, but when alternated every sixth time, it yields periodic trajectories. In such a case, we have a Parrondian game of “chaos + chaos = periodic”. For completeness, we depict, in Fig. 10, two periodic trajectories that exemplify the “undesirable + undesirable = desirable” in an ecological equivalent to “extinction + chaos = periodic”. The second trajectory in Fig. 10 relates to “chaos + chaos = periodic”. Both cases can be cast as controlling chaos strategies because every sixth iteration we modify the output. We can also these two trajectories as examples of the Parrondian games where two loosing strategies can be alternated to give a winning strategy. From our analysis, we can describe Eq. (3) as Parrondian game or as a proportional pulse chaos controlling strategy. Although, as a chaos controlling technique, we would only be interested in “chaos + chaos = periodic” cases, where as a Parrondian game we can extend our focus to “undesirable + undesirable = desirable” cases, which is more ecological relevant. 3. Four season switching model As an application and extension of our analysis, we consider a four season logistic model with harvesting or migration. In Section 2, harvesting or migration is represented by a reduction or increment of the population every fourth iteration. In contrast with our P = 4 case, we define a population cycle of 4 units of time, where each season is represented by a different value of the parameter C in the logistic equation,
⎧ R F ( Xn , C F ) ⎪ ⎨ F ( Xn , C W ) X n +1 = ⎪ ⎩ F ( Xn , C Sp ) F ( X n , C Su )
if if if if
mod[n, P ] = 0 mod[n, P ] = 1 mod[n, P ] = 2 mod[n, P ] = 3
(5)
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Fig. 10. Trajectories for G 1 ( X , C , R ), P = 6, and C = 4, which are related to “undesirable + undesirable = desirable” and “chaos + chaos = periodic” cases, R = 0.0150, and R = 0.9289, respectably.
Fig. 11. A four season model with harvesting (R < 1) and immigration (R > 1) for C F = 3, C W = 1/2, C Sp = 3.85, C Su = 4.
Fig. 12. Lyapunov exponents for the four season model with harvesting (R < 1) and immigration (R > 1) for C F = 3, C W = 1/2, C Sp = 3.85, C Su = 4.
In this model, we set harvesting or immigration, during the season, to be associated with a steady state, 1 < C F < 3, and parameter values such that for R = 1 the system displays chaotic trajectories. In Eq. (5), we consider harvesting or migration, represented by R, during the fall, where the population does not change, and the parameter C F ranges between unity and three. We follow with a winter season associated with extinction, so C W is less than unity. For the spring and summer, we consider values of C Sp , and C Su greater than 3.570, which are associated with abundance and chaotic trajectories. As an example, we consider the following parameter values: C F = 3, C W = 1/2, C SP = 3.85, and C Su = 4. The bifurcation diagram for the harvesting case is depicted in Fig. 11 as R < 1, and for immigration as R > 1. From Figs. 11, 12, we can easy pick two parameter values that yield periodic solutions. We highlight that in the case, R = 1, we get chaotic trajectories from Eq. (5). For R = 0.22, which allows for 78% harvesting, and R = 1.22, which allows for a 22% increase in the population, however, Eq. (5) yields a periodic oscillation as depicted in Fig. 13.
From the oscillations in Fig. 13, we notice that the minima is always associated to the winter, the maxima is achieved at the end of summer for R = 0.22, followed by a decrease due to the harvesting. In contrast, for R = 1.22 the maxima is achieved during fall due to migration and followed with a decrease due to winter. 4. Discussion In the context of ecological applications, seasonality can be modeled as switching between environmental conditions, which may drive the population to desirable behaviors. Given its ecological relevance, we considered the logistic map as a population model and include parameter switching. As in the case of proportional pulses technique to control chaos [29], we consider a parameter change for the P th iteration. Although we allow two possible changes at the P th iteration, we only detailed the consequence of reducing the output of the P th iteration as R F ( X P 1 , C ). We noticed that, in this case, the result of parameter switching is
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Fig. 13. Trajectories for four-season model (C F = 3, C W = 1/2, C Sp = 3.85, C Su = 4) with harvesting (R = 0.22) and migration (R = 1.22).
less stabilizing than in the case where the P th iteration is substituted by a population reduction, R X P −1 . By constructing the bifurcation diagrams for P = 3, 4, 6, we easily find parameter values, related to extinction, that could yield periodic oscillations, which are considered desirable behaviors. By enlarging the bifurcation diagrams, we are also able to find parameter values that relate to Parrondian games such as the case of “chaos + chaos = periodic”. In the case of the Parrondo’s Paradox, alternation of two losing games results in a winning game, and in the case of discrete maps, the combination of two chaotic dynamics may result in order. Consequently, we can cast the periodic pulse technique as a Parrondian game or the Parrondian game as a chaos control technique. From the ecological point of view, the switching of dynamics is a desirable strategy to model seasonality [30]. We considered a four season model, where we allow harvesting or immigration only in one of the four seasons or every fourth iteration, as in the case of the periodic pulse technique. We selected parameters that yielded chaotic trajectories when harvesting or migration is not allowed, R = 1, but yield periodic trajectories for harvesting, R = 0.22, or immigration R = 1.22 alike. Therefore, based on the logistic model, we can stabilize a chaotic population by either allowing harvesting or immigration. On the one hand, we can consider switching maps as a chaos controlling technique. On the other hand, we have observed that the same technique can be recast as a Parrondian strategy. The main difference could be described as follows: in Parrondian games, we only alternate parameter values without changing the underlying map. In contrast as in some chaos controlling strategies, we alternate the underlying map with a different map for the P th iteration. While in Parrondian games we explore the “chaos + chaos = periodic”, as in the chaos controlling case, we also explore the “extinction + chaos = periodic”, and the “periodic + periodic = chaos”. In summary, we observe that alternate dynamic strategies, such as Parrondian games and periodic pulse chaos controlling strategy, are similar. In other words, we want to recognize that Parrondian games may be described as a chaos control strategy or the periodic pulse technique for controlling chaos as a Parrondian game.
Acknowledgements The author would like to thank Professor Allen Rodgers, and the Chemistry Department of the University of Cape Town for their hospitality during my sabbatical leave. The author would also like to thank Ms. Alex Y. Peacock-Villada for helpful comments, and Williams College and the National Science Foundation (CHE0911380) for their financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
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