Abstract
An interesting problem in nonlinear dynamics is the stabilization of chaotic trajectories, assuming that such chaotic behavior is undesirable. The method described in this chapter is based on the Parrondo’s paradox, where two losing games can be alternated, yielding a winning game. The idea of alternating parameter values has been used in chemical systems, but for these systems, the undesirable behavior is not chaotic. In contrast, ecological relevant map in one and two dimensions, most of the time, can sustain chaotic trajectories, which we consider as undesirable behaviors. Therefore, we analyze several of such ecological relevant maps by constructing bifurcation diagrams and finding intervals in parameter space that satisfy the conditions to yield a desirable behavior by alternating two undesirable behaviors. The relevance of the work relies on the apparent generality of method that establishes a dynamic pattern of behavior that allows us to state a simple conjecture for two-dimensional maps. Our results are applicable to models of seasonality for 2-D ecological maps, and it can also be used as a stabilization method to control chaotic dynamics.
Keywords
- chaos control
- Parrondo’s paradox
- switched dynamic systems
- ecological maps
- seasonality
1. Introduction
In population dynamics, discrete dynamic systems have been used to model the dynamics of ecological systems. One of the first maps used in ecology that suggested to study the new,
In the case of 1-D discrete dynamics, for the last 18 years, alternate dynamics strategies have been the center of attention due to the so-called Parrondo paradox [8, 9, 10], where two losing games can be combined to yield a winning game. Furthermore, the idea that “lose + lose = win” has been extended to “chaos+chaos = periodic” in one-dimensional maps [11]. Just recently and for the first time, we were able to find the Parrondo dynamics in two 2-D maps [12]. In the contest of seasonality, we consider the alternation of undesirable dynamical behaviors yield a desirable behavior [13, 14]. So in the context of population dynamics we have considered cases where “undesirable + undesirable = desirable” dynamical behaviors occur as a result of a simple alternation of parameters [15, 16, 17, 18, 19, 20, 21].
In our present discussion, we extend our seasonality modeling strategy to several two-dimensional ecologically relevant maps and find that the “undesirable + undesirable = desirable”, the “chaos + chaos = periodic”, as well as, the “periodic + periodic = chaos” behaviors are not unique to 1-D maps. In Section 2, we consider a delayed logistic map, and in Section 3, we analyze a Lotka-Volterra map. In Section 4, we study a modified 2-D Ricker map, and in Section 5, we analyze the Beddington map. In Section 6, we discuss a modified Lotka-Volterra map, which includes a logistic prey growth. We conclude in Section 7 with a discussion and a summary of our results.
2. Delayed logistic equation
In our analysis of two-dimensional maps, we begin with the extended logistic map that incorporates a delay in population growth, defined by the following relation:
where
From Figure 1, we define our parameter value regions associated with complex or non-complex dynamics. The map on the left of the figure shows the whole bifurcation map; while the right magnifies the complex region. On the right hand figure, which is the magnified map, we can clearly see some periodic windows, but we pick parameter values associated with complex dynamics.
Next, we switch, or alternate, the parameter values between even and odd iterations through the following relation:
The equation above describes our switching strategy in which we pick one parameter for every odd iteration, which we name
For our analysis, we pick a
For our first example of “chaos + chaos = periodic”, we consider the parameter value,
Another combination of parameters yielding “chaos+chaos = periodic” uses
We complete our analysis of delayed logistic map with one case in which “periodic + periodic = chaos”. As mentioned beforehand, we pick our value associated with periodic trajectories from the area associated with chaotic trajectories, and focus on
3. Lotka-Volterra model
We begin our next section by discussing a discretized form of the Lotka-Volterra model. The Lotka-Volterra map describes predator prey interactions, assuming that the prey has a relatively high initial population, and that the predator’s growth rate is directly proportional to the prey’s growth rate.
The model follows a relation defined by the map below
In Figure 5, showing Eqs. (5) and (6), we look at the unswitched map, defined by showing the ranges of periodic and aperiodic behavior. As in the previous section, we use the unswitched bifurcation map as a comparison to the switched map when using certain parameters. For this section, we focus on the interval
As before, we pick a
We conclude the present section with an example of “periodicity + periodicity = chaos”, using Eqs. (7) and (8) and
4. Modified 2-D Ricker map
Another interesting map includes an exponential term, describing the prey growth, with a simple predator–prey interaction term. The map is determined by the following equations:
which is in essence modified and extended to 2-D Ricker-like map [22]. The corresponding switched map is defined below:
Figure 9, showing Eqs. (9) and (10), considers the range of C values we focus on, from C = 0 to 2.8. We want to remark however, that this map also shows some interesting behavior beyond the interval of study, but we choose this interval to get a close up of the intervals of periodicity, since this interval is where we find our relevant behavior. For the X function we study, at higher values, the function stays at unity for values of C = 28 and higher, while the Y function stays at extinction, or Y = 0.
As in previous cases, we start with finding parameter values satisfying the “chaos + chaos = order” relation. To begin, we use Eqs. (11) and (12) with
We then use Eqs. (11) and (12) with
We finish this section by introducing one case in which “periodic + periodic = chaos”. We pick the periodic parameter
5. Beddington model
Our next map is the Beddington 2-D map defined by the following equations:
along with the corresponding alternation equation.
Figure 13, showing Eqs. (13) and (14), shows the parameter range we use to analyze the map. We pick points between 0 and 14, and show the corresponding bifurcation diagrams within that range. We pick 14 as our maximum value because above that parameter, there are only steady state solutions.
We start with describing our first chaotic value,
We then use Eqs. (15) and (16), with
6. Modified Lotka-Volterra map
Our last 2-D map considers a logistic growth, and an interaction term, and only a predation term for the predator. The dynamics of this map is considerably different than the previous two maps,
As before, the switched map is shown below.
Aside from the r parameter, this map also has the h parameter, which we set equal to unity. Unlike the previous two maps we study that have relevant behaviors past C = 10, the max value of the unswitched map is C = 3.85, but chaos is only present above C = 3.0, as shown in Figure 17.
Our first chaotic point is
The second to last figure, Figure 19 shows our final odd switching parameter,
Our last figure, Figure 20, shows an example of “periodic + periodic = chaos”, where we switch with
7. Discussion
In previous sections, we have analyzed five relevant ecological 2-D maps, setting a pattern of dynamic behavior similar to the well studied “chaos + chaos = periodic” in switched 1-D maps. Therefore, with the results discussed in this chapter, we can extend the 1-D maps conjecture to 2-D maps. The conjecture asserts that given a map with chaotic dynamics, we can find two parameters associated to chaotic trajectories that, when alternated yield a periodic trajectory. In general, we can consider these kinds of maps as nonautonomous maps because one of the parameters is a function of the iterations. In most case, we pick a parameter value for the even iterations and a different parameter for the odd iterations. But the connection with the Parrondo’s paradox is associated with the kind of alternating parameters, which in the conjecture are parameter associated with chaotic, or, in general, complex trajectories.
The case of “chaos + chaos = periodic” was presented for the first time by Almeida et al. [16] for simple 1-D maps, and just recently for 2-D maps by Mendoza et al. [12]. The implication of the so-called Parrondo’s dynamics has been used to model seasonality, but with the observation that, under the Parrondo dynamics, the case of “periodic + periodic = chaos” is also possible [15]. As generalization we have consider cases of “undesirable + undesirable = desirable” dynamics behaviors to analyze simple models of seasonality [23, 24, 25], which include migration or immigration [13, 14].
In the present analysis, we emphasize the use of bifurcation diagrams to find intervals of values in parameter space that could satisfy the “undesirable + undesirable = desirable” or “periodic + periodic = chaos” dynamics. Although we are interested in modeling ecological systems and in particular the effect of seasonality, one could use our results to look at the switched maps as a way to control chaotic dynamics. In particular an extension to continuous dynamic systems may be relevant or applicable to chemical and mechanical systems [26].
In summary, our approach of building bifurcation diagrams readily yield intervals of parameter values that can show the so-called Parrondian dynamics for 1-D and 2-D maps. We have concentrated on ecological relevant maps, but the approach applies to any kind of maps. In particular, we can easily find parameters that show desirable dynamics in switched maps, controlling complex or undesirable dynamics, with the by product that we can also avoid the alternation of desirable dynamics that could yield undesirable dynamics in switched maps. Finally, we believed that we have stablished a pattern of dynamic behavior that supports the conjecture described in previous paragraphs.
8. Conclusions
In previous sections, we have stablished a pattern of dynamic behavior for 2-D maps, which have been used to model ecological systems. The dynamic pattern allows to state that for any 2-D maps that shows chaotic dynamics for a set of parameters, we can always find two of such parameters that, when alternate, yield a periodic trajectory. This conjecture is an extension of the so-called Parrondo’s paradox, in the sense that two undesirable dynamics can be alternate to yield a desirable dynamics. In other words, we can always find a region in parameter space, where we can select a pair of such parameters. Therefore, we the developed methodology can be use, in general, as a chaos control approach, and, in particular, we can use it to model, in the case of ecological maps, seasonality. Although we interested in ecological relevant 2-D maps, we believed that our conjecture can be extended to other type of 1-D and 2-D maps. Finally, we consider that the major application of the methodology is in controlling chaotic dynamics.
Acknowledgments
The authors will like acknowledge the financial support of the National Science Foundation (CHE-0911380), and the Bronfman Science Center. One of us (EPL) would like to thank Professor Allen Rodgers, and the Chemistry Department of Cape Town University for their hospitality during my sabbatical leave.
References
- 1.
May RM. Simple mathematical models with very complicated dynamics. Nature. 1976; 261 :459-467 - 2.
May RM. Stability and Complexity in Model Ecosystems. Princeton: Princeton University Press; 1974 - 3.
Kot M. Elements of Mathematical Ecology. Cambridge: Cambridge University press; 2001 - 4.
Turchin P. Complex Population Dynamics. Princeton: Princeton University Press; 2003 - 5.
Mangel M. The Theortical Biologist’s Toolbox. Cambridge: Cambridge University press; 2006 - 6.
Allen LJS. An Introduction to Mathematical Biology. Upper Saddle River, NJ: Pearson Prentice Hall; 2007 - 7.
May RM, McLean AR. Theoretical Ecology: Principles and Applications. Oxford: Oxford University Press; 2007 - 8.
Harmer GP, Abbott D. Losing strategies can win by Parrondo’s paradox. Nature. 1999; 402 :864 - 9.
Harmer GP, Abbott D. Parrondo’s paradox. Statistical Science. 1999; 14 :14 - 10.
Harmer GP, Abbott D, Taylor PG. The paradox of Parrondo’s games. Proceedings of the Royal Society. 2000; 456 :247-259 - 11.
Cánovas JS, Linero A, Peralta-Salas D. Dynamic Parrondo’s paradox. Physica D. 2006; 218 :177-184 - 12.
Mendoza SA, Matt EW, Guimaraes-Blandon DR, Peacock-López E. Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems. Chaos, Solitons & Fractals. 2018; 106 :86-93 - 13.
Peacock-López E. Seasonality as a Parondian game. Physics Letters A. 2011; 375 :3124-3129 - 14.
Silva E, Peacock-López E. Seasonality and the logistic map. Chaos, Solitons & Fractals. 2017; 95 :152-156 - 15.
Percus OE, Percus JK. Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox. Mathematical Intelligencer. 2002; 24 :68-72 - 16.
Almeida J, Peralta-Salas D, Romera M. Can two chaotic systems give rise to order? Physica D. 2005; 200 :124-132 - 17.
Behrends E. Stochastic dynamics and Parrondo’s paradox. Physica D. 2008; 237 :198-206 - 18.
Boyarsky A, Góra P, Aslam MS. Randomly chosen chaotic maps can give rise to nearly ordered behavior. Physica D. 2005; 210 :284-294 - 19.
Amengual P, Meurs P, Cleuren B, Toral R. Reversal of chance in paradoxical games. Physica A. 2006; 371 :641-648 - 20.
Levinohn EA, Mendoza SA, Peacock-López E. Switching induced complex dynamics in an extended logistic map. Chaos, Solitons & Fractals. 2012; 45 :426-432 - 21.
Maier MPS, Peacock-López E. Switching induced oscillations in the logistic map. Physics Letters A. 2010; 374 :1028-1032 - 22.
Ricker WE. Stock and recruitment. Journal of the Fisheries Research Board of Canada. 1954; 11 :559-623 - 23.
Blasius B, Kurths J, Stone L, editors. Complex Population Dynamics. Singapore: World Scientific; 2007 - 24.
Allen LJS. An Introduction to Mathematical Biology. Upper Saddle River: Prentice Hall; 2007 - 25.
Kot M, Schaffer WM. The effects of seasonality on discrete models of population growth. Theoretical Population Biology. 1984; 26 :340-360 - 26.
Kohar V, Ji P, Choudhary A, Sinha S, Kurths J. Synchronization in time-varying networks. Physical Review E. 2014; 90 :022812