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:

$$X\_{n+1} = X\_n \; \operatorname{Exp} \left[ r(\mathbf{1} - X\_n - Y\_n) \right] \tag{9}$$

$$Y\_{n+1} = CX\_n(Y\_n) \tag{10}$$

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 Co = 2.10, associated with aperiodic dynamics. Figure 10 zooms into the region for which "chaos + chaos = order" holds. From this diagram, we see a narrow region of

We then use Eqs. (11) and (12) with Co = 2.26, for which Figure 11 hones in on the relevant Ce parameter values. The interval of Ce values is significantly wider in this case than the previous one, since we find "chaos + chaos = order" for 2.74–2.80. We finish this section by introducing one case in which "periodic + periodic = chaos". We pick the periodic parameter Co = 2.333. In some maps, it is harder to find periodic windows, although they could usually be found sometimes but an extra significant figure is necessary such as in this case. We find chaos in this map

from Ce = 1.71 to 2.00, as shown in Figure 12, periodic values when using

Bifurcation diagram for Eqs. (9) and (10) and Eqs. (11) and (12) with the Co value 2.10.

Bifurcation diagram for Eqs. (9) and (10) and Eqs. (11) and (12) with the Co value 2.26.

Bifurcation diagram for Eqs. (9) and (10) and Eqs. (11) and (12) with the Co value 2.333.

periodicity from C = 2.775 to 2.790.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

Eqs. (11,12).

Figure 10.

Figure 11.

Figure 12.

75

which is in essence modified and extended to 2-D Ricker-like map [22]. The corresponding switched map is defined below:

$$X\_{n+1} = \begin{cases} f\_n(X\_n, Y\_n) = X\_n \operatorname{Exp} \left[ r(\mathbf{1} - X\_n - Y\_n) \quad \text{if } n \text{ even} \right. \\ f\_n(X\_n, Y\_n) = X\_n \operatorname{Exp} \left[ r(\mathbf{1} - X\_n - Y\_n) \quad \text{if } n \text{ odd} \right. \end{cases} \tag{11}$$

$$Y\_{n+1} = \begin{cases} \mathcal{g}\_n(Y\_n) = \mathcal{C}\_\epsilon X\_n Y\_n & \text{if n even} \\ \mathcal{g}\_n(Y\_n) = \mathcal{C}\_\omicron X\_n Y\_n & \text{if n odd} \end{cases} \tag{12}$$

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.

Figure 9. Bifurcation map for Eqs. (9) and (10) showing the interval studied, as well as a close up of the chaotic region. Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

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 Co = 2.10, associated with aperiodic dynamics. Figure 10 zooms into the region for which "chaos + chaos = order" holds. From this diagram, we see a narrow region of periodicity from C = 2.775 to 2.790.

We then use Eqs. (11) and (12) with Co = 2.26, for which Figure 11 hones in on the relevant Ce parameter values. The interval of Ce values is significantly wider in this case than the previous one, since we find "chaos + chaos = order" for 2.74–2.80.

We finish this section by introducing one case in which "periodic + periodic = chaos". We pick the periodic parameter Co = 2.333. In some maps, it is harder to find periodic windows, although they could usually be found sometimes but an extra significant figure is necessary such as in this case. We find chaos in this map from Ce = 1.71 to 2.00, as shown in Figure 12, periodic values when using Eqs. (11,12).

Figure 10. Bifurcation diagram for Eqs. (9) and (10) and Eqs. (11) and (12) with the Co value 2.10.

Figure 11. Bifurcation diagram for Eqs. (9) and (10) and Eqs. (11) and (12) with the Co value 2.26.

Figure 12. Bifurcation diagram for Eqs. (9) and (10) and Eqs. (11) and (12) with the Co value 2.333.

some chaotic behavior for values of Ce in the interval 1½ � :75; 2:00 , a region that is

Bifurcation diagram for Eqs. (5) and (6) and Eqs. (7) and (8) with the Co value 2.44.

Another interesting map includes an exponential term, describing the prey growth, with a simple predator–prey interaction term. The map is determined by

which is in essence modified and extended to 2-D Ricker-like map [22]. The

f <sup>n</sup>ð Þ¼ Xn; Yn Xn Exp ½rð Þ 1 � Xn � Yn if n odd

Ynþ<sup>1</sup> <sup>¼</sup> gnð Þ¼ Yn CeXnYn if n even

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

Bifurcation map for Eqs. (9) and (10) showing the interval studied, as well as a close up of the chaotic region.

gnð Þ¼ Yn CoXnYn if n odd

Xnþ<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup>ð Þ¼ Xn; Yn Xn Exp <sup>½</sup>rð Þ <sup>1</sup> � Xn � Yn if n even

Xnþ<sup>1</sup> ¼ Xn Exp r½ ð Þ 1 � Xn � Yn (9)

Ynþ<sup>1</sup> ¼ C Xnð Þ Yn (10)

(11)

(12)

periodic when using Eqs. (5) and (6).

corresponding switched map is defined below:

4. Modified 2-D Ricker map

the following equations:

Figure 8.

Fractal Analysis

Y = 0.

Figure 9.

74

#### 5. Beddington model

Our next map is the Beddington 2-D map defined by the following equations:

$$X\_{n+1} = X\_n \operatorname{Exp} \left( r(\mathbf{1} - X\_n) - Y\_n \right) \tag{13}$$

$$Y\_{n+1} = \mathbb{C}\, X\_n(1 - \operatorname{Exp}\left(-Y\_n\right))\tag{14}$$

"periodic + periodic = chaos" behavior, for Co = 6.0. Figure 16 shows the area of the map that is normally periodic, and shows characteristic chaotic behavior from Ce = 1.7–3.0, although this particular map shows some periodic windows than the

Bifurcation diagram for Eqs. (13) and (14) and Eqs. (15) and (16) with the Co value 4.0.

Bifurcation diagram for Eqs. (13) and (14) and Eqs. (15) and (16) with the Co value 6.0.

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

Ynþ<sup>1</sup> <sup>¼</sup> CXnYn

<sup>n</sup> � CXnYn

<sup>n</sup> � CeXnYn

Xn <sup>þ</sup> <sup>h</sup> if n odd

Xn <sup>þ</sup> <sup>h</sup> if n even

CoXnYn

CeXnYn

Xn <sup>þ</sup> <sup>h</sup> (17)

(19)

(20)

Xn <sup>þ</sup> <sup>h</sup> (18)

Xn <sup>þ</sup> <sup>h</sup> if n even

Xn <sup>þ</sup> <sup>h</sup> if n odd

Xnþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>r</sup> Xn � r X<sup>2</sup>

<sup>f</sup> <sup>n</sup>ð Þ¼ Xn Xnð Þ� <sup>r</sup> <sup>þ</sup> <sup>1</sup> r Xð Þ<sup>n</sup> <sup>2</sup> � CoXnYn

gnð Þ¼ Yn

gnð Þ¼ Yn

other "periodic + periodic = chaos" maps.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

As before, the switched map is shown below.

Ynþ<sup>1</sup> ¼

<sup>f</sup> <sup>n</sup>ð Þ¼ Xn ð Þ <sup>1</sup> <sup>þ</sup> <sup>r</sup> Xn � rX<sup>2</sup>

8 >><

>>:

6. Modified Lotka-Volterra map

than the previous two maps,

Figure 15.

Figure 16.

Xnþ<sup>1</sup> ¼

77

8 >><

>>:

along with the corresponding alternation equation.

$$X\_{n+1} = \begin{cases} f\_n(X\_n, Y\_n) = X\_n \operatorname{Exp} \left( r(\mathbf{1} - X\_n) - Y\_n \right) & \text{if n even} \\ f\_n(X\_n, Y\_n) = X\_n \operatorname{Exp} \left( r(\mathbf{1} - X\_n) - Y\_n \right) & \text{if n odd} \end{cases} \tag{15}$$

$$Y\_{n+1} = \begin{cases} \mathbf{g}\_n(X\_n, Y\_n) = \mathbf{C}\_\mathbf{e} X\_n (\mathbf{1} - \operatorname{Exp}(-Y\_n)) & \text{if } n \text{ even} \\ \mathbf{g}\_n(X\_n, Y\_n) = \mathbf{C}\_\mathbf{e} X\_n (\mathbf{1} - \operatorname{Exp}(-Y\_n)) & \text{if } n \text{ odd} \end{cases} \tag{16}$$

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, Co = 10, for Eqs. (15) and (16). Figure 14 shows the corresponding bifurcation diagram, and we see a relatively wide range of Ce values for which we have "chaos + chaos = order". We find this behavior for most points of Ce between 4.54 and 4.7.

We then use Eqs. (15) and (16), with Co = 4.0, and here we also see a relatively wide range of parameters in which we find that "chaos + chaos = periodicity". Specifically, we see that alternating with Ce = 10.7–10.88 gives us the desired behavior, shown in Figure 15. Our last figure pertaining to this map, Figure 16, shows the

Figure 13.

Bifurcation diagram for Eqs. (13) and (14) showing the interval studied, as well as a close up of the chaotic region.

Figure 14. Bifurcation diagram for Eqs. (13) and (14) and Eqs. (15) and (16) with the Co value 10.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

5. Beddington model

Fractal Analysis

Figure 13.

Figure 14.

76

region.

Our next map is the Beddington 2-D map defined by the following equations:

Xnþ<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup>ð Þ¼ Xn; Yn Xn Exp rð Þ ð Þ� <sup>1</sup> � Xn Yn if n even

Ynþ<sup>1</sup> <sup>¼</sup> gnð Þ¼ Xn; Yn Ce Xnð Þ <sup>1</sup> � Exp ð Þ �Yn if n even

f <sup>n</sup>ð Þ¼ Xn; Yn Xn Exp rð Þ ð Þ� 1 � Xn Yn if n odd

gnð Þ¼ Xn; Yn Co Xnð Þ 1 � Exp ð Þ �Yn if n odd

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

We start with describing our first chaotic value, Co = 10, for Eqs. (15) and (16). Figure 14 shows the corresponding bifurcation diagram, and we see a relatively wide range of Ce values for which we have "chaos + chaos = order". We find this

We then use Eqs. (15) and (16), with Co = 4.0, and here we also see a relatively wide range of parameters in which we find that "chaos + chaos = periodicity". Specifically, we see that alternating with Ce = 10.7–10.88 gives us the desired behavior, shown in Figure 15. Our last figure pertaining to this map, Figure 16, shows the

Bifurcation diagram for Eqs. (13) and (14) showing the interval studied, as well as a close up of the chaotic

Bifurcation diagram for Eqs. (13) and (14) and Eqs. (15) and (16) with the Co value 10.

along with the corresponding alternation equation.

above that parameter, there are only steady state solutions.

behavior for most points of Ce between 4.54 and 4.7.

Xnþ<sup>1</sup> ¼ Xn Exp rð Þ ð Þ� 1 � Xn Yn (13) Ynþ<sup>1</sup> ¼ C Xnð Þ 1 � Exp ð Þ �Yn (14)

(15)

(16)

Figure 15. Bifurcation diagram for Eqs. (13) and (14) and Eqs. (15) and (16) with the Co value 4.0.

Figure 16. Bifurcation diagram for Eqs. (13) and (14) and Eqs. (15) and (16) with the Co value 6.0.

"periodic + periodic = chaos" behavior, for Co = 6.0. Figure 16 shows the area of the map that is normally periodic, and shows characteristic chaotic behavior from Ce = 1.7–3.0, although this particular map shows some periodic windows than the other "periodic + periodic = chaos" maps.

#### 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,

$$X\_{n+1} = (1+r)X\_n - rX\_n^2 - \frac{CX\_nY\_n}{X\_n+h} \tag{17}$$

$$Y\_{n+1} = \frac{\mathbf{C}X\_n Y\_n}{X\_n + h} \tag{18}$$

As before, the switched map is shown below.

$$X\_{n+1} = \begin{cases} f\_n(X\_n) = (1+r)X\_n - rX\_n^2 - \frac{C\_o X\_n Y\_n}{X\_n + h} & \text{if n even} \\ f\_n(X\_n) = X\_n(r+1) - r(X\_n)^2 - \frac{C\_o X\_n Y\_n}{X\_n + h} & \text{if n odd} \end{cases} \tag{19}$$

$$Y\_{n+1} = \begin{cases} g\_n(Y\_n) = \frac{C\_o X\_n Y\_n}{X\_n + h} & \text{if n odd} \\ g\_n(Y\_n) = \frac{C\_o X\_n Y\_n}{X\_n + h} & \text{if n even} \end{cases} \tag{20}$$

#### Fractal Analysis

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 Co = 3.3, and the corresponding bifurcation diagram is shown in Figure 18. There is a somewhat small region of periodicity from Ce = 3.704 to 3.724.

The second to last figure, Figure 19 shows our final odd switching parameter, Co = 3.1 and the corresponding bifurcation diagram, which shows a similar range of periodic parameter values, specifically, Ce = 3.70–3.72.

Our last figure, Figure 20, shows an example of "periodic + periodic = chaos", where we switch with Co = 3.57 and see chaos for most values between Ce = 2.5 and 3.0.

7. Discussion

Figure 20.

plex trajectories.

tion [13, 14].

79

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, com-

Bifurcation diagram for Eqs. (17) and (18) and Eqs. (19) and (20) with the Co value 3.57.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

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 [23–25], which include migration or immigra-

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 rele-

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

vant or applicable to chemical and mechanical systems [26].

described in previous paragraphs.

Figure 17.

Bifurcation diagram for Eqs. (17) and (18) showing the interval studied, as well as a close up of the chaotic region..

Figure 18. Bifurcation diagram for Eqs. (17) and (18) and Eqs. (19) and (20) with the Co value 3.3.

Figure 19. Bifurcation diagram for Eqs. (17) and (18) and Eqs. (19) and (20) with the Co value 3.1.

Figure 20. Bifurcation diagram for Eqs. (17) and (18) and Eqs. (19) and (20) with the Co value 3.57.
