# Rhino Poaching Stats 101

South Africa usually releases reports on the management of its rhinos biannually. Conservationists eagerly anticipate these reports for two reasons. First, they contain many vital statistics, including, often, the number of rhinos poached during the reporting period. Second, the majority of African rhinos, some 80%, reside within South Africa's borders; thus, as South Africa goes, so goes the species. Since Pembient wants to supplant the trade in rhino horns, we, too, take an interest in these reports.

A recent report covers the period from January 1st to December 31st, 2018, and it contains good news: the number of rhinos poached has decreased from 1,028 in 2017 to 769 in 2018. That works out to a 25% year-over-year decline in poaching and represents the continuation of an ongoing trend. An epidemiologist, though, would worry that poaching has decreased because there are fewer rhinos around to poach. She would like to see incidence rates, which take into account population size and time at risk. That is, there is a big difference between, say, 200 animals being poached out of a population of 1,000 throughout half a year versus the same amount being poached out of 10,000 throughout a full year. In order to compute incidence rates, more data are needed than available in a single report; hence, Table 1.

Table 1. Collected Rhino Poaching Data for South Africa
$$\begin{array}{cccc} \hline \text{Year} & \begin{array}{c} \text{Days} \\ \text{Observed} \end{array} & \begin{array}{c} \text{Number} \\ \text{Poached} \end{array} & \begin{array}{c} \text{Population} \\ \text{Size} \end{array} \\ \hline 2007 & 365 & 13^{a} & 17761^{e} \\ 2008 & 366 & 83^{a} & - \\ 2009 & 365 & 122^{a} & - \\ 2010 & 365 & 333^{a} & 20711^{e} \\ 2011 & 365 & 448^{a} & - \\ 2012 & 366 & 668^{a} & 21001^{e} \\ 2013 & 365 & 1004^{a} & - \\ 2014 & 365 & 1215^{a} & - \\ 2015 & 365 & 1175^{a} & 20306^{a} \\ 2016 & 366 & 1054^{b} & - \\ 2017 & 365 & 1028^{c} & - \\ 2018 & 365 & 769^{d} & - \\ \hline \end{array}$$

From looking at Table 1, it is obvious why incidence rates are seldom computed. Namely, there are a lot of missing values for population size. The reason for the missing data is that rhino censuses are costly, and therefore done sporadically. Even when figures are obtainable, they are, at best, estimates or compiled from multiple surveys. Fortunately, Kruger National Park (KNP), where much of South Africa’s rhinos dwell, has begun to regularly publish population estimates for the species it harbors (i.e., black rhino and white rhino). Table 2 shows these estimates.

Table 2. Collected Rhino Population Estimates for KNP
$$\begin{array}{ccc} \hline \text{Year} & \text{Black Rhino} & \text{White Rhino} \\ \begin{array}{c} \\ \hline 2011^{a} \\ 2012^{a} \\ 2013^{a} \\ 2014^{a} \\ 2015^{a} \\ 2016^{b} \\ 2017^{c} \end{array} & \begin{array}{cc} \hline \text{Lower Limit} & \text{Upper Limit} \\ \hline 400 & 620 \\ 328 & 597 \\ 343 & 487 \\ 249 & 371 \\ 313 & 453 \\ 349 & 465 \\ 427 & 586 \end{array} & \begin{array}{cc} \hline \text{Lower Limit} & \text{Upper Limit} \\ \hline 8767 & 12682 \\ 8500 & 12900 \\ 8394 & 9564 \\ 8001 & 9290 \\ 8365 & 9337 \\ 6649 & 7830 \\ 4759 & 5532 \end{array} \\\hline \end{array}$$

Each pair of upper and lower limits in Table 2 forms a confidence interval that should bound the true population size of the given species at the given time. Confidence intervals are constructive, but for this exercise, point estimates are needed. They can be computed using the formula below:

$$\textit{Point Estimate} = \frac{\textit{Lower Limit} + \textit{Upper Limit}}{2}$$

The results are shown, per species, in Table 3, along with population estimates for the entire South African National Parks (SANP) system, including KNP. Additionally, Table 3 shows how the whole population changes in percentages from year to year. As an aside, it turns out that the confidence intervals in Table 2 are empirical, not analytical. Nevertheless, limited calculations indicate that the mean absolute errors of the point estimates for black and white species are less than 3 and 75 rhinos, respectively. These relatively small errors support the use of the aforementioned formula.

Table 3. Breakdown of Rhino Population Estimates for SANP
$$\begin{array}{cccccr} \hline \text{Year} & \text{KNP Black} & \text{KNP White} & \text{Other} & \text{Total} & \text{Change (YoY)} \\ \hline 2011 & 510 & 10725 & 226^{a} & 11461 & \\ 2012 & 463 & 10700 & 293^{a} & 11456 & -0.04\% \\ 2013 & 415 & 8979 & 354^{a} & 9748 & -14.91\% \\ 2014 & 310 & 8646 & 398^{a} & 9354 & -4.04\% \\ 2015 & 383 & 8851 & 489^{a} & 9723 & 3.94\% \\ 2016 & 407 & 7240 & 501^{b} & 8148 & -16.20\% \\ 2017 & 507 & 5146 & 512^{c} & 6165 & -24.34\% \\ \hline \end{array}$$

Assuming the percentage changes in Table 3 are suggestive of the overall trends in South Africa's rhino population, they can be used to reconstruct said trends. More specifically, given the number of rhinos observed in 2012 and 2015 (see Table 1), a cubic equation based on the percentage changes over the intervening years can be written:

$$21001(1 + \alpha(-0.1491))(1 + \alpha(-0.0404))(1 + \alpha(0.0394)) = 20306$$

Here, $$\alpha$$ is a smoothing parameter. Solving for it gives $$\alpha = 0.220029$$. The diminution of the percentage changes induced by $$\alpha$$ is reasonable, especially since roughly half of South Africa's rhinos belong to SANP and the remainder are in private or provincial hands. In other words, SANP should dominate the overall trends, and those trends should generally be slightly downwards due, in part, to a drought affecting KNP's white rhinos.

Imputing values for the population estimates of Table 1 using the percentage changes of Table 3 smoothed by $$\alpha$$ produces Table 4.

Table 4. Partially Imputed Rhino Population Estimates for South Africa
$$\begin{array}{cc} \hline \text{Year} & \begin{array}{c} \text{Population} \\ \text{Size} \end{array} \\ \hline 2007 & 17761 \\ 2008 & - \\ 2009 & - \\ 2010 & 20711 \\ 2011 & - \\ 2012 & 21001 \\ 2013 & 20312 \\ 2014 & 20131 \\ 2015 & 20306 \\ 2016 & 19582 \\ 2017 & 18533 \\ 2018 & - \\ \hline \end{array}$$

Two techniques must be used to handle the remaining missing values in Table 4. One, called interpolation, estimates values between existing data points. The other, called extrapolation, estimates values beyond existing data points. The Wolfram programming language's time series processing functions can do both. The necessary code is shown in Listing 1.

Listing 1. Wolfram Language Code for Interpolation and Extrapolation
popsize = TimeSeries[
{{2007, 17761}, {2010, 20711}, {2012, 21001}, {2013, 20312},
{2014, 20131}, {2015, 20306}, {2016, 19582}, {2017, 18533}}]
model = TimeSeriesModelFit[
TimeSeriesResample[popsize, 1,
ResamplingMethod -> {"Interpolation",
InterpolationOrder -> 2}]]
fig1 = Show[
ListLinePlot[
{model["TemporalData"], TimeSeriesForecast[model, {8}]},
Frame -> {{True, False}, {True, False}},
FrameLabel -> {"Year", "Population Size"}],
ListPlot[popsize, PlotStyle -> {PointSize[Medium], Black}]]


Running the code in Listing 1 generates Fig. 1, wherein all the population estimates are visualized.

Fig. 1. Rhino Population Estimates for South Africa over Time

The blue line traces through interpolated values, and the gold line traces through extrapolated values. Black dots denote data from Table 4.

Although Fig. 1 looks scary, it is important to understand that the extrapolation is based only on previous population estimates. It does not take into account biological growth rates, poaching effort, catchability, and the like. Consequently, the further it extends into the future, the less validity it has. The focus, then, should be on the three interpolated population estimates and the first extrapolated one. Taken together with the population estimates in Table 4, they provide enough information to determine an incidence rate for each year in Table 1.

For this article, incidence rate or, technically, average daily crude poaching rate (CPR) per 20,000 rhinos, is defined as follows:

$$\textit{Average Daily CPR} = \frac{\textit{Number Poached}}{\textit{Days Observed} \times \textit{Population Size}} \times 20000$$

It can be thought of as a way of fixing the rhino population at 20,000 rhinos, allowing a meaningful comparison of the average number of rhinos poached per day between years. Table 5 consolidates all the observations and estimates previously discussed and presents the likely incidence rates of poaching over the last dozen years.

Table 5. Revised Rhino Poaching Data for South Africa
$$\begin{array}{ccccc} \hline \text{Year} & \begin{array}{c} \text{Days} \\ \text{Observed} \end{array} & \begin{array}{c} \text{Number} \\ \text{Poached} \end{array} & \begin{array}{c} \text{Population} \\ \text{Size} \end{array} & \begin{array}{c} \text{Average Daily CPR} \\ \text{(per 20000)} \end{array} \\ \hline 2007 & 365 & 13 & 17761 & 0.04 \\ 2008 & 366 & 83 & 19080 & 0.24 \\ 2009 & 365 & 122 & 20063 & 0.33 \\ 2010 & 365 & 333 & 20711 & 0.88 \\ 2011 & 365 & 448 & 21134 & 1.16 \\ 2012 & 366 & 668 & 21001 & 1.74 \\ 2013 & 365 & 1004 & 20312 & 2.71 \\ 2014 & 365 & 1215 & 20131 & 3.31 \\ 2015 & 365 & 1175 & 20306 & 3.17 \\ 2016 & 366 & 1054 & 19582 & 2.94 \\ 2017 & 365 & 1028 & 18533 & 3.04 \\ 2018 & 365 & 769 & 17221 & 2.45 \\ \hline \end{array}$$

So, has the tide turned on rhino poaching? Plotting the average daily CPR (per 20,000) by year, as in Fig. 2, reveals that poaching has fallen in real terms. It is now at a level not seen since 2012-2013. Moreover, the simple rate of poaching in 2018 is close to two, meaning the oft-quoted conservationist slogan of "three rhinos poached per day" no longer holds.

However, let's not fool ourselves. South Africa has an intentional homicide rate per 100,000 of 34 or so. In comparison, its rhino poaching rate per 100,000 is about 4,465. That means being a rhino in South Africa is approximately 131 times more dangerous than being a human there! Clearly, more needs to be done to tackle the problem.

Fig. 2. Average Daily Crude Poaching Rate (per 20,000 Rhinos) for South Africa over Time

Another takeaway is that it is becoming harder to argue that supply-side initiatives, whether in the form of synthetics or non-lethal harvesting, exacerbate rhino poaching. According to Fig. 2, the era of "big" conservation, starting from 2009, when South Africa banned domestic sales of rhino horn, and culminating in a frenzy of awareness-raising campaigns in 2014, coincides with runaway poaching. In contrast, a leveling off overlaps with Pembient's announcement of its intentions to biofabricate horn in 2015 and runs through to the overturning of South Africa's domestic sales ban in 2017 and the granting of 15 permits for the sale of 1,342 rhino horns in 2018.

The precautionary principle states that evidence of harm should prompt policy action, even if cause-and-effect relationships cannot be established. The "big" conservation organizations, such as Humane Society International, increasingly lack the evidence needed to lobby for prohibitions on supply-side approaches to rhino conservation.