IMPORTANCIA DEL JUEGO EN LA ESCUELA PRIMARIA

The influence of the Neolithic of the Near East is evident in the Neolithic of the Indus Valley sites. Also, an approximate linear relationship is apparent between the geodesic distance of the Indus Valley sites and their distance from the conventional source of the site of Gesher in the Near East, with an average dispersal speed of about 0.65 km/yr (see Chapter 4). Since there are two separate routes of the eastward Neolithic propagation from the Fertile Crescent, it is of interest to see if the Neolithic reached the Indus Valley along the northern or the southern route. First we analyse the southern route sites combined with the Neolithic Indus sites (sites from Tables 5.5 and 5.6). The upper envelope fits give an average value of Neolithic start time of T_SI∗ = 6.60 kyr BCE with an average speed USI = 3.58 km/yr. The Neolithic start time obtained from the analysis is slightly

Figure 5.7: The two main paths of the Neolithic dispersal are shown in this figure. The red arrow shows the Neolithic dispersal from the southern Zagros. The yellow arrow shows the movement of the Neolithic from the north of Zagros Mountains (north-central Fertile Crescent) to the Indian subcontinent, and the small dashed white arrow indicates dispersal further north-east.

speed obtained with inclusion of IVC sites is more than four times faster than 0.66 km/yr obtained for the southern route in Iran. Similar analysis after combining the northern route sites with the Indus sites (Tables 5.4 and 5.6) gives an average starting time T∗

N I = 6.93

kyr BCE with an average speed USI = 2.92 km/yr. This speed is about 1.5 times faster

than the speed obtained for the northern route without the Indus Valley dates. The R2

values are measures of how good the linear fit is to the data. The closer the R2 value to unity, the better is the fit. Table 5.7 shows the R2 values obtained for each of the fits. For the southern route sites with Indus, the goodness of fit values obtained are about 0.10 and that for the northern route sites combined with Indus sites are about 0.30. Thus the Neolithic of the Indus Valley is more likely an outcome of the spread from the northern route through Iran. This route is shown with the long yellow arrow in Fig. 5.7.

One-Site-One Date One Site – Many Dates

Data Number

of sites

Speed (U) Start Time (T∗₎ R2 Speed (U) Start Time (T∗₎ R2 Subset in a sub- set

(km/yr) (kyr BCE) (km/yr) (kyr BCE)

S 18 0.44 7.66 0.88 0.88 6.82 0.41 SI 43 3.19 6.83 0.13 3.97 6.36 0.10 SInM 42 3.81 6.85 0.09 5.08 6.37 0.06 N 34 1.67 7.20 0.81 2.02 6.94 0.86 NI 59 2.95 7.05 0.32 2.89 6.93 0.27 NInM 58 2.84 6.93 0.30 2.85 6.93 0.30 NAS 36 1.51 7.24 0.78 0.91 7.38 0.56 NA 35 1.50 7.24 0.89 2.04 6.94 0.58

Table 5.7: Spread speeds and Starting times calculated by the percentile method, both the ‘One Site – One Date’ and the ‘One Site – Many Dates’ datasets. The abbreviations are as follows: S: Southern route sites excluding the Indus dates; SI: Southern route sites including Indus dates; SInM: Southern route sites with Indus but without Mehrgarh; N: Northern route sites excluding the Indus dates; NI: Northern route sites including Indus dates; NInM: Northern route sites with Indus but without Mehrgarh; NAS: Northern route sites excluding the Indus dates, with Ayakagytma and Sarazm BOTH included; NA: Northern route sites excluding the Indus dates, with Ayak- agytma included but NOT Sarazm. The values are calculated using a bin width of 350 km for the One Site – One Date datasets and a bin width of 200 km for the One-Side-All-Dates datasets.

5.4

Conclusion

The terrain of West and South Asia is geographically diverse. On the one hand, the region contains the very productive Fertile Crescent, and on the other it has two large deserts, Dasht-e Kavir and Dasht-e Lut. In such an environment it is impossible that the Neolithic spread uniformly. Our analysis shows two distinct routes of the Neolithic propagation across southern Asia. The first via northern Iran, and possibly continuing towards the Indus Valley; and the second through southern Iran, possibly terminating in the Kerman province. According to our results, the Neolithic dispersal via the northern route progressed with an average speed of 1.85 km/yr, and that via southern route had an average speed of 0.66 km/yr. These two routes are shown by the long yellow arrow and the short red arrow respectively in Fig. 5.7. The red arrow in Fig. 5.7 ends short

of the Indus Valley, and makes it necessary to find if this wide spatial gap really exists or is due to lack of archaeological investigation. The white arrow in the figure shows another possible path of Neolithic dispersal north-east of Iran, pointing towards one more interesting region in the present day Tajikistan, Kyrgyzstan, Uzbekistan and south east Kazakhstan. Exploration of this region might be of interest as this might shed more light on the Neolithic propagation north of the Himalayas.

Figure 5.8: The plots of change in speed (left column) and starting times (right column) with change in bin widths (from 100 km to 400 km in steps of 50km) for One Site – One Date datasets for South (S), South With Indus (SI), North (N), and North With Indus(NI), respectively from top to bottom.

Figure 5.9: The plots of change in speed (left column) and starting times (T*, right column) with change in bin widths (from 100 km to 400 km in steps of 50km) for One Site – Many Dates datasets for South (S), South With Indus (SI), North (N), and North With Indus(NI), respectively from top to bottom.

Differential Equation Models

and Preliminary Results

In this chapter we explore two mathematical models that are based on continuous differen- tial equation models. Both models are modifications of the Fisher-Kolmogorov-Petrowskii- Piscounov (FKPP) equation. The first model (see Section 6.1) uses the FKPP equation with constant and variable diffusivities. The variability of the diffusivity is achieved by introducing it as a function of the palaeovegetation data. The second model (see Sec- tion 6.2) is based on the idea of spread of a crop that is of significance to a population and thus represents the spread of that population. The model uses modern altitude and precipitation data. Both the models are currently in the testing stage and only the pre- liminary results obtained are given in this chapter. The models still require work which is described in the ‘Future Work’ for each section.

6.1

FKPP Model and Palaeovegetation

The geographically diverse region from the Fertile Crescent to the Indus Valley contains nine different palaeoecological zones. Not all of these zones are favourable for living.

Motivated by this, we introduce a mathematical model for the spread of the Neolithic, which makes use of the palaeovegetation data. Chapter 3 describes the traditional and non-traditional mathematical methods used in studies of population dynamics. Here, the deterministic FKPP model is used to study this Neolithic spread. As given in Section 3.2.2, the Neolithic spread speed is U = 2√γ ν with the net growth rate γ and the diffusivity ν. The larger the growth rate and diffusivity, the faster is the spread.

Here, we study the spread of the Neolithic across the region in three different ways:

1. The FKPP model is applied to the whole region from the Mediterranean to the Indus Valley. Firstly, the growth rate, carrying capacity and diffusion are all assumed to be constants. Secondly, we introduce a heterogeneous diffusion based on the palaeovegetation data for the region.

2. The FKPP model is applied to the southern Neolithic route with a source in the Zagros region.

3. In the third and last part, the FKPP model is applied to the northern Neolithic route. As with the southern region, we study this domain with and without the Indus Valley sites.