*2.1.2 Dynamics of biological red cells model*

The population of red blood cells in a healthy human being oscillates within a certain tolerance interval in normal circumstances. But, sometimes, in presence of a disease such as anemia, this behavior fluctuate dramatically. A discrete model of blood cell populations, Martelli, ([73], p: 35), presented here.

Let *xn*, *xn*þ<sup>1</sup> representing quantities of cells per unit volume (in millions) at time *n* and *n* þ 1, respectively and *pn*, *dn* are, respectively, the number of cells produced and destroyed during the nth generation then

$$\mathbf{x\_{n+1}} = \mathbf{x\_n} + \mathbf{p\_n} - \mathbf{d\_n} \tag{5}$$

Then, assuming that

**Correlation Dimension:**

attractors, **Figure 3**, obtained as:

figure obtained as

**Figure 6.**

**190**

Where,

Extending further the numerical study, correlation dimensions of system (1)

Consider an orbit *O*ð Þ *x***<sup>1</sup>** = f g *x*1, *x*2, *x*3, *x*<sup>4</sup> … … , of a map *f* : *U* ! *U*, where *U*s an open bounded set in *<sup>R</sup>n*. To compute correlation dimension of *O x*ð Þ<sup>1</sup> , for a given

> Xn i6¼j

H xð Þ¼ 0, x <sup>&</sup>lt;<sup>0</sup>

is the unit-step function, (Heaviside function). The summation indicates number of pairs of vectors closer to *r* when1≤i, j≤n and *i* 6¼ *j*.C rð Þ measures the density

�

1, x≥0

log C rð Þ

H r � <sup>∥</sup>**xi** � **xj**<sup>∥</sup> � � (3)

log r (4)

calculated for a chaotic attractor by using Mathematica codes, [73].

1 n nð Þ þ 1

positive real number *r*, we form the correlation integral,

C rð Þ¼ limn!<sup>∞</sup>

*A Collection of Papers on Chaos Theory and Its Applications*

of pair of distinct vectors **xi** and **xj** that are closer to *r*.

*Plot of correlation integral curve for A = 5.3, Q = 2.76 and x0 = 0.5.*

The correlation dimension Dc of Oð Þ **x1** is then defined as

Dc <sup>¼</sup> limr!<sup>0</sup>

To obtain Dc, log C rð Þ is plotted against log *r*, **Figure 6**, and then we find a straight line fitted to this curve. The intercept of this straight line on y-axis provides the value of the correlation dimension DC. Correlation dimensions of time series

a. For first attractor, Q = 2.76, A = 5.3, a plot of the correlation integral curve is shown in **Figure 6**. Then, the linear fit of the correlation data used in this

*y* ¼ 0*:*95661*x* þ 0*:*687605

$$\mathbf{d\_n} = \mathbf{a}\,\mathbf{x\_n}, \mathbf{a} \in [0, 1]$$

$$\mathbf{p\_n} = \mathbf{b(x\_n)}^\mathbf{r}\mathbf{e^{-sx\_n}},$$

where b, r, s all positive parameters. With these our one-dimensional discrete model for blood cells populations comes as

$$\mathbf{x\_{n+1}} = (\mathbf{1} - \mathbf{a})\ \mathbf{x\_n} + \mathbf{b} \ (\mathbf{x\_n})^\mathbf{r}\mathbf{e^{-s}x\_n} \tag{6}$$

The case a ¼ 1 , means that during the time interval under consideration all cells that were alive at time *n* are destroyed. In such a case, above models simply comes as

$$\mathbf{x\_{n+1}} = \mathbf{b(x\_n)}^\mathbf{r} \mathbf{e^{-s\ x\_n}} \tag{7}$$

For *<sup>a</sup>* <sup>¼</sup> <sup>0</sup>*:*8, *<sup>b</sup>* <sup>¼</sup> 10 , *<sup>r</sup>* <sup>¼</sup> 6 and *<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*:*5 , three fixed points *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> <sup>¼</sup> 0, *<sup>x</sup>*<sup>∗</sup> <sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*989813, *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>¼</sup> <sup>3</sup>*:*53665 obtained for system (6) of which only x<sup>∗</sup> <sup>0</sup> ¼ 0 is stable and other two are unstable. Chaotic motion observed for values of parameter *a* ¼ 0*:*8, *b* ¼ 10,*r* ¼ 6, *s* ¼ 2*:*5, as shown in the time series plot, **Figure 8**, with initial condition *x*<sup>0</sup> ¼ 1*:*5.

**Figure 7.** *Plots of correlation dimensions: (a) with Q = 2.76 and varying A, (b) with A = 5.4 and varying Q.*

**Figure 8.** *Chaotic time series plot of map (6) for a = 0.8, b = 10, r = 6, s = 2.5 and x*<sup>0</sup> ¼ 1*:*5*.*

Interesting bifurcations observed for this map: For *<sup>b</sup>* = 1.1 � <sup>10</sup><sup>6</sup> , *r* = 8, two bifurcation diagrams are drawn; (a) in one for *s* =16 and 0 ≤*a*≤1, and (b) in another for *a* ¼ 0*:*8 and 3*:*5≤*s* ≤16*:*0 and shown in **Figure 9**. In former case one finds initially period doubling bifurcation followed by loops before emergence of chaos. In later case, one finds some typical type of bifurcation showing chaos adding, folding and the bistability like phenomena. A magnification of right figure, **Figure 10**, for smaller range, 4*:*5≤*s*≤ 8*:*5, justifying chaos adding behavior.

**Figure 10.**

**Figure 11.**

**Figure 12.**

**193**

*Bifurcation of Blood Cell model when* <sup>3</sup>*:*<sup>0</sup> <sup>≤</sup>s≤7*:*<sup>5</sup> *and* <sup>a</sup> <sup>¼</sup> <sup>0</sup>*:*8, r <sup>¼</sup> <sup>8</sup>*,* <sup>b</sup> *= 1.1* � *106*

*Chaos and Complexity Dynamics of Evolutionary Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.94295*

*above the zero line show the regular and chaotic zones of parameter space.*

*Topological entropy plot for r* <sup>¼</sup> 6, *<sup>s</sup>* <sup>¼</sup> <sup>16</sup> *and b = 1.1* � *106 and* <sup>0</sup>≤*<sup>a</sup>* <sup>≤</sup>1*.*

*LCE Plots for* <sup>¼</sup> <sup>6</sup> *, s* <sup>¼</sup> <sup>16</sup> *and b = 1.1* � *106 , negative and positive values of LCEs, respectively, below and*

*.*

Regular and chaotic motion experienced through bifurcation diagrams, **Figures 9** and **10**, again confirmed by plots of Lyapunov exponents, **Figure 11**. This system, bears enough complexity and, as its measure, plot of topological entropies, **Figure 12**, obtained for values *<sup>r</sup>* <sup>¼</sup> 6, *<sup>s</sup>* <sup>¼</sup> 16 and *<sup>b</sup>* = 1.1 � <sup>10</sup><sup>6</sup> and 0≤*a*≤1. Fluctuations in increase of topological entropies appear, approximately, in the region 0*:*25≤*a*≤0*:*95 indicate existence of complexity.

The correlation dimension of its chaotic attractor for values *a* ¼ 0*:*78, when *<sup>r</sup>* <sup>¼</sup> 6, *<sup>s</sup>* <sup>¼</sup> 16 and *<sup>b</sup>* = 1.1 � 106 is obtained as *Dc* ffi <sup>0</sup>*:*253.
