Octavio Flores
David Pardo
Randall Dalbey
Troy Kozlowski
Herd Size Distribution
One of the ideas talked about in class was that starting with an equal distribution of cattle in the simulation was unnatural and leads to a strong bias when disasters would occur in the early years versus the later years of the simulation. After running the simulation for 50 years and starting each herd at 70 we got the following results:
Figure 1a: This figure shows the percent of the herds still alive against time for a herd distribution of 70:70. / Figure 1b: This figure shows the percent of herds that died in each decade based on the surviving herd size with a herd distribution of 70:70.The results from our simulation and the simulation from the Osotua paper are approximately the same. The way we quantified the herd bias was to look at the herd deaths at ten years and twenty years. At the ten year mark 64% of the cattle herds remained. When we look at the twenty year mark only 48% of the herds remained. This is a 16% difference of the herd size dying over ten years. It is pretty clear to see that early disasters are having significant impact on the survival rates of the herds.
Since we discussed in class that starting each herd size at 70 was unnatural. Another part of our project was try out several methods that would generate uneven herd sizes while the mean was still approximately seventy. The first idea our group came up with for generating starting herd sizes was to use a gaussian distribution, with a mean of 70 with a floor of 65 and ceiling of 75. We had to create the artificial floor because it would not make sense to start any herds at or below the death threshold of 64. This then forced us to create the ceiling of 75 in order to maintain the mean 70 cattle. The second idea we had was using a uniformly random distribution with numbers between 65 and 75. The third idea for setting initial herd sizes was to use an exponential generator that would set herd sizes with a minimum of 65 since starting a herd at or below the threshold of 64 was not optimal. We set up the exponential distribution so that the average number of cattle was 70. This exponential allowed for a few herds to begin with up to 120 cattle, while the majority of initial herds were close to the threshold of 64. The exponential model makes the most anthropological sense because it allows for outliers to begin with large herds, but keeps most herds close to the threshold.
Figure 2: This shows the initial distributions for one of the herds over 10,000 trials using the various distributions. The graph was generated using integer increments.
Figure 3a: This shows each comparison of initial distribution over 50 years. The different distributions were: 70-70, uniform random, gaussian, and exponential respectively.
Figure 3b: This shows each comparison of initial distribution focusing on the first 10 years.
We had initially thought that varying the method of initial distributions would affect the bias towards early death that the model showed. But from the figure 3b, this is shown to not be the case. They are almost all identical. From figure 3b it is shown that the different methods have little to no effect on the herds as they reach maturity. This illustrates the the initial distribution has little to no effect on the model as long as the mean of the initial herds remains at 70.
Next we decided to create a generational model of the Osotua scheme. We decided to work from the exponential distributions because they make the most sense from an anthropological perspective. So we took the average of the herd size of dyads that were both alive after 50 years. We then used that average to create an new initial exponential distribution. We repeated this process one more time to create a third generation. We chose each generation to last 50 years because that represents an approximate working lifetime of one rancher.
Figure 4: The generational comparison using the exponential distributions. The time in years vs. percent alvie value of cattle.
Table 1: The summary of the mean initial size of cattle and median longevity for each generation.
From these results, we can clearly see from figure 4 that as the generation of the herd increases the herd death rate decreases. The difference from generation 1 to generation 2 is very large. Then from generation 2 to generation 3 there is less pronounced. The rate at which herds die appears to adopt a more linear relationship in the 3rd generation rather than the exponential decay model that the 1st generation clearly displays. This change is likely observed because the average initial herd size has increased significantly, as shown in table 1. Because each herd starts so large, they can retain their own risk for many years without having to bother with an Osotua partner. In addition, it does not make much sense to run the simulation for more generation, because the model stops making realistic sense for the real world situation that the model is based on. Herds that last greater the 150 years would likely outlive multiple generations of the herders that tend to them.
In conclusion, we found that the early bias for herd death rates did not change over the first 50 years no matter which distribution we used. Our initial thinking that using the exponential distribution was the most realistic method and would reduce the bias. However, it clearly showed not to be the case. Essentially what we determined was it doesn’t matter what distribution is used. As long as the mean was 70 the results were going to be similar. However, the when the mean was changed using our generational model, this significantly affected the longevity of the herds. In addition, increasing the average greatly decreased the bias towards early death that the model showed.