Abstract
Poaching of elephants in Southern Africa is now dominated by international groups following a model of organized crime. This shift, from poaching conducted by small, local groups; with limited mobility, weapons, and technology, to individuals who organize, finance, equip, and transport wellarmed poaching units to previously scouted locations, has made the protection of elephants in Southern Africa much more difficult and dangerous. This paper develops a model of hightech criminal poaching. A poaching organization makes a decision on the number of “planned poaching expeditions.” If a poaching unit is intercepted the entire organization is destroyed, but is replaced by a new organization in the next year. The operating life of a poaching organization is a stochastic process, which in turn induces a stochastic evolution in the elephant population. Under plausible conditions, the number of planned poaching expeditions is highly sensitive to the probability of interception by antipoaching patrols, but is nonresponsive to reductions in the blackmarket ivory price. Thereby it might be better to focus conservation efforts on increasing the probability of intercepting poaching units rather than trying to control black market ivory prices. A benchmark value of poaching expeditions is identified—above which elephants may slowly decline to extinction.
“The kingpins behind these syndicates have masterminded a platform that will take wildlife poaching to new levels across Africa, posing a virtually unpoliceable risk to rhino and elephant populations … The use of a helicopter, a dart gun and knockdown/immobilizing drugs has transformed the traditional method of poaching on foot to that of a hightech aerial attack that is only noticed once the perpetrators have left the scene…. In some instances an aircraft has been used to guide poachers to where the animals are, while keeping an eye for signs of having been detected.” Vitalis Chadenga, Director General, Parks and Wildlife Management Authority, Zimbabwe.
1. Background and overview
In early 1981, the elephant population in Africa was estimated at 1.2 million animals. There are two species of elephant on the African continent; the forest elephant (Loxodonta cyclotis), predominantly found in the forested areas of Central Africa, and the savanna elephant (Loxodonta africana), found primarily in Eastern and Southern Africa. During the 1980s, an estimated 675,000 elephants were poached and by 1989 the elephant population had declined to just over 600,000 animals (Barbier et al. [1990]). In 1989, the African elephant was listed as an Appendix I (i.e., endangered) species by the Convention on International Trade in Endangered Species (CITES) and a ban was placed on the trade of elephant products. Recent estimates place the total elephant population in African range states at 423,000 in the year 2012 (IUCN [2012]).
There is considerable evidence to suggest that the poaching of elephant tusk and rhino horn in Africa is planned and financed by international syndicates using organized crime as a business model. These syndicates sponsor poaching gangs with aircraft and highpowered weapons, and also arrange for rapid shipment of tusk and horn to markets in Asia (Mullen and Zhang [2012], SasRolfes [2012], Shukman [2013], Wassener [2013], Wasser et al. [2008]). Based on data from seizures of illegal ivory shipments amounting to roughly 24 tons in the year 2006 Wasser et al. [2007] use DNA analysis to estimate that approximately 23,000 savanna elephants were illegally harvested from the southern African range states. In June 2002, authorities in Malawi seized a shipment of 6.5 tons of ivory that was bound for Hong Kong—Wasser et al. [2008] estimate that approximately 6,500 elephants were illegally harvested for that shipment alone. Based on evidence from the ivory seizures, Wasser et al. [2008] suggest that at least two syndicates were associated with poaching at the time in different regions of the African range states.
Armed antipoaching patrols and government corruption have been considered important factors that explain changes in elephant population numbers (Balmford et al. [2003], Frank and Maurseth [2006]). In a provocative article Messer [2010] shows that estimated elephant populations in Kenya and Zimbabwe increased after these countries adopted a shootonsight policy when dealing with poachers during the period 1984 through 2002. Elephant populations declined in other African countries that did not adopt a shootonsight policy, and continued to decline even after 1989 when CITES listed Loxodonta africana in Appendix I. Messer [2010] contends that low wages in developing countries impose limits on the potential economic costs for poachers of fines and imprisonment, which are more traditional forms of antipoaching policy. Kremer and Morcom [2000] show that rational expectations amongst poachers may lead to multiple equilibrium settings in an openaccess harvesting model of elephant populations; some equilibria also include elephant populations being driven to extinction. Kremer and Morcom [2000] show that an effective way for governments to eliminate extinction equilibria may be to commit to tough antipoaching measures, should the population fall below some threshold level. And for governments without a credible antipoaching threat, extinction equilibria may be avoided by selling an accumulated stockpile of ivory. Their analysis assumes that the blackmarket price would fall when a government sells a large amount of stockpiled ivory.
The selling of accumulated ivory stockpiles as an antipoaching policy has been opposed by many African leaders and conservation organizations. On July 18th 1989 then President Daniel Arap Moi of Kenya ignited twelve tons of confiscated ivory as a gesture in support of the Appendix I listing of Loxodonta africana by CITES (Perlez [1989]). In 1997 CITES voted to downlist elephant populations in Botswana, Namibia, and Zimbabwe to Appendix II because their elephant populations were considered sustainable (Burton [1999]). The downlisting allowed these countries to engage in a tightly monitored sale of stockpiled ivory. In 2010 a similar request by Tanzania and Zambia was denied (IUCN [2013]). The majority of CITES members, and most conservation groups opposed the change to Appendix II for Tanzania and Zambia on the grounds that it would “flood the ivory market,” increase the use of ivory, and make the detection of illegally acquired ivory more difficult. The effectiveness of CITES trade bans in the recovery of elephant populations has been questioned by several studies—notably Barbier et al. [1990], Bulte and Van Kooten [1996], Kreuter and Simmons [1995], Sugg and Kreuter [1994], and Burton [1999]. For the CITES ban to be effective it would have to stigmatize the trade and the use of ivory so that its demand and blackmarket prices fall (Burton [1999]). Burton [1999] is skeptical about whether the stigma created by CITES trade bans or even a sale of stockpiled ivory would lower black market prices sufficiently to reduce poaching. According to Burton’s openaccess harvesting models the black market price of ivory would have to fall by ninety percent for elephant populations to increase under openaccess equilibrium. The results in this paper show why a reduction in ivory price (possibly through a sale of stockpiled ivory) might not lead to a decline in elephant poaching.
In this paper, I develop a model of organized criminal poaching. In the next section, I first construct a biological model of elephant population dynamics. I then focus on the economic problem for the leader of a poaching organization. The bioeconomic model is calibrated to reflect elephant populations and poaching in the Southern African range states. I identify the plausible conditions under which the number of planned poaching expeditions will be insensitive to the black market price of ivory, but quite responsive to the probability of interception by antipoaching patrols. I also identify a benchmark value for the number of poaching expeditions, which if consistently exceeded, may lead to the elephant population slowly declining to extinction. In the third section, I conduct simulations of the model to estimate elephant poaching population trajectories. I also consider the implications of poaching by multiple poaching organizations with free entry/exit under open access until profits are driven away. I then conduct a sensitivity analysis of the results by changing the values of the model’s economic parameters—such as the probability of interception by an antipoaching patrol, the cost of poaching expeditions, and the black market price of ivory. The fourth section discusses the key results and provides a conclusion for this paper.
2. A model of organized criminal poaching
Let denote a population of elephants in year t. Elephant population dynamics can be described by the iterative map—Equation (1):
 (1)
is an average annual mortality rate, is a purely compensatory growth function, and is the number of elephants killed by the poaching organization. An agestructured model, or a model with delayed recruitment to an adult population might be more appropriate, however this biomass (lumpedparameter) model has been used in previous studies (see Bulte and van Kooten [1996], Cromsigt et al. [2002], MilnerGulland and LeaderWilliams [1992]). Given that this paper tries to assess the trends in aggregate elephant populations an agestructured model might not provide much additional insight into population trends. Moreover, one can easily derive the analytical steadystate population from lumped parameter models and this can provide a useful benchmark when examining stochastic poaching. I will describe each term of Equation (1) in detail.
I specify a form for the population growth function, . Following MilnerGulland and LeaderWilliams [1992] I adopt a skewed logistic where . Because the average survival rate of the elephant population is given by , I treat as the pregnancy rate of adult females giving birth to approximately one offspring every third year. This implies. The average mortality rate varies across countries and depends on the abundance of water and forage. Under normal years, the average mortality rate (including juveniles) has been estimated at (see Armbruster and Lande [1993]). These values would imply a net intrinsic growth rate of , a value very close to the estimate used by Calef [1988]. The parameter is the environmental carrying capacity for maximum elephant population size. A skew parameter greater than one () will cause the population level supporting peak growth to lie to the right of . is used to model large mammal population dynamics because the relationship between population and density, , is observed to be nonlinear, and this relationship becomes more important as the population approaches its carrying capacity (see Cromsigt et al. [2002] and MilnerGulland and LeaderWilliams [1992]); MilnerGulland and LeaderWilliams [1992] set . K can also influence the steady state elephant population in the absence of poaching.1 The parameter values are summarized in Table 1
The number of elephants killed by poachers, , in any year is determined in part by the number of poaching expeditions planned by the leaders of a poaching organization. In deciding the likely number of poaching expeditions, a leader must balance expected net revenue with the probability that members of the poaching unit might be captured by antipoaching patrols, pleabargain, and provide information leading to the destruction of the organization. I assume that if poachers are intercepted by an antipoaching unit then the organization is decommissioned for the rest of the year. Early detection and dismantling of the poaching organization will reduce the number of elephants killed in that year. Furthermore, I assume that poaching can never be eliminated entirely, and that a new poaching organization reappears at the start of the next year, with a new leader who again optimizes the number of planned poaching expeditions. At the start of year t the organization leader gathers information on the size of the elephant herd as well as the number of antipoaching units. The kill rate of a poaching unit for a single expedition is assumed to be proportional to the elephant population, . I consider as an efficiency parameter for a single poaching expedition. The number of elephants killed on a single poaching expedition would therefore be ; with poaching expeditions, the total number of elephants harvested in year t would be .
Let us now develop the optimization problem for the leader of a poaching organization and show how it leads to a target number of planned poaching expeditions. Let denote the number of antipoaching units that the leader of a poaching organization thinks will be deployed in year t. Let be the subjective, Bernoulli probability held by the leader of the poaching organization, that any single poaching expedition will be intercepted by a government antipoaching unit. I assume that is identical and independent for all poaching expeditions in a given year.2 P denotes the average value of a set of two ivory tusks from a single harvested elephant when sold on the black market. denotes the cost of deploying a single poaching expedition; this would be the cost of gasoline, food, ammunition, and other supplies. The poaching unit is comprised of hunters, carriers, drivers, and perhaps pilots. Assume that these individuals collectively receive a payoff from a poaching expedition, provided that it has not been intercepted by an antipoaching unit. At the beginning of each year the gang leader determines the number of planned poaching expeditions, , a nonnegative integer, as per Equation (2): 3
 (2)
One can think of the poaching expeditions as being sent out sequentially during a given year. If the gang leader sets a target of expeditions in year t, then would be the probability that none of the expeditions will be intercepted by an antipoaching unit, given the assumption of independently conducted expeditions. The actual number of poaching expeditions “successfully” completed in year twill be less than or equal to . The number of elephants killed by poaching units in year t is thereby a random variable given by , where is the “realized” number of successful poaching expeditions in year t.
I parameterize the bioeconomic model to explore the elephant population and poaching dynamics in the Southern African region as a case study. According to Blanc [2007] the combined elephant population in the Southern African range states (i.e., Angola, Botswana, Mozambique, Namibia, South Africa, Zambia, and Zimbabwe as shown in Figure 1) in the year 2006 was approximately 297,718. If this number represented the steadystate elephant population with poaching () then would be the steadystate number of elephants killed by poachers. In reality, the elephant population in Southern Africa will never be in steady state because of stochastic poaching and erratic droughts. However these “counterfactual” steadystate values, , and are useful because they give us a benchmark from which to assess the consequences of organized, stochastic, criminal poaching. Recall that Wasser et al. [2007] estimated that 23,000 savanna elephants were illegally harvested from the Southern African range states in the year 2006—this too serves as a useful reference value while assessing the model’s simulations later in this paper.
The black market price for a set of two ivory tusks (weighing a total of twenty kilograms) may be as high as (Messer [2010]). The lack of regular timeseries estimates of ivory prices makes it difficult to model ivory price as a function of ivory supply; for simplicity it is kept constant for the analyses. The cost of outfitting a single poaching expedition will be set at . The efficiency of a poaching unit, as measured by q, was estimated by MilnerGulland and LeaderWilliams [1992] to be for organized gangs in Zambia in 1985. The value of qhas likely increased in the last decade given reports of poaching organizations using hightech equipment like aircraft, darting guns, and knockdown drugs for their operations (see Mullen and Zhang [2012], Shukman [2013], Wassener [2013], and Wasser et al. [2008]). I assume that this efficiency parameter has increased by a factor of 10 and set . The model’s biological and economic parameters are summarized in Table 1.
The gang leader’s decisionmaking process can be summarized with the help of the schematic in Figure 2. At the beginning of time period tthe gang leader forms an assessment of the elephant population . Given the parameters of blackmarket price (P), probability of interception , cost per expedition (c) and the poaching technology or catchability coefficient (q), the leader maximizes the expected profit expression, Equation (2), by numerically solving for the optimal number of planned poaching expeditions . With a given probability of interception , one can simulate for a resulting number of realized poaching expeditions, denoted by . The elephant population in the next time period, , will evolve as per the iterative map, Equation (1), after a realization of poaching/harvest, , in time period t.
Having laid out the theoretical optimization process, I now turn to numerical simulations using parameter values from the literature. MilnerGulland and LeaderWilliams [1992] estimated an average “detection rate” for poaching expeditions in Zambia for the period 1980 to 1983 to be . It is difficult to determine whether this probability is appropriate for our model of organized poaching circa 2014. Assuming that the investment in the number of antipoaching units in the Southern Africa range states has significantly increased since the early 1980s I set a basecase probability of . However, the use of hightech equipment by poaching organizations might also result in a lower probability of detection by antipoaching patrols. In the next section, we will explore the dynamic consequences of more sophisticated, hightech poaching with lower probabilities of detection.
can be determined using the parameter values in Table 1 and numerically solving Equation (2). If 297,718 and , then . If , . With , . If all nine expeditions were completed without interception by an antipoaching unit, net revenue accruing to “management” of the poaching organization would be $7,158,998 from the killing of 6,859 elephants. With , . If all 19 expeditions were completed without interception, a single poaching organization would kill 14,481 elephants for a profit of $14,719,839.
While Equation (2) would imply that , P, c, q, and would all play a role in determining , the sensitivity of to these parameters is not uniform. In fact, Equation (2) has an interesting property:
Proposition 1. If and but sufficiently small, then only depends on
Proof. Define the expected net revenue from planned poaching expeditions as . Divide both sides by so that . Then, if , the integer value of which maximizes is the integer that maximizes and only depends on .
For the parameter values in Table 1, , which is sufficiently small so that changes in P, c, q, and may not change . For instance in this bioeconomic model, lowering P from $3,000 to $750 does not change the optimal number of planned poaching expeditions (a change in the black market price (P) could be brought about indirectly by the government—wherein they could choose to dump confiscated ivory on the market, which would in turn depress the sale price of ivory (Kremer and Morcom [2000] and Burton [1999])). If and falls from to , poaching still generates a positive profit and the number of planned poaching expeditions remains at . The relative insensitivity of to changes in the black market price of ivory or to the elephant population will have important implications for system dynamics and antipoaching policy.
With poaching the dynamics of the elephant population are determined by
 (3)
Insight into the behavior of the elephant population with stochastic poaching can be gleaned from the deterministic case where , and is a constant. Equation (3) can be solved for a steady state at provided. This steady state is locally stable if and only if . If , 0, and the elephant population will ultimately become extinct as a result of poaching. With stochastic poaching, if is frequently above , the population may slowly decline to extinction as . For , , and I calculate to be 23.43. If is frequently above the elephant population might ultimately go extinct because the harvest rate would consistently exceed the elephant population’s reproductive capacity. I refer to this value of as a reference value or benchmark value with which to compare the simulated number of poaching expeditions later in the paper.
In the next section, we will examine elephant population dynamics in the presence of one and multiple poaching gangs. We will assess the economic decisionmaking of poaching organization leaders when the probability of interception by antipoaching units is set at . We will then examine the consequences of economic decision making on elephant population dynamics when the probability of interception is lowered to because of more evasive, hightech poaching.
3. Simulating elephant poaching and population trajectories
Recall that at the start of each year, the poaching gang leader determines , the optimal number of planned expeditions. The number of elephant killed in year t becomes the realization , where is a random variable. I will first examine elephant poaching and population trajectories with the assumption of a single gang looking to maximize expected net returns using the parameters of Table 1. I will then consider the possibility of additional gangs where each gang independently carries out the same maximization process as determined by Equation (2). I determine the number of poaching gangs that would operate under open access conditions when the elephant population settles into a lowpopulationlevel steady state, or goes to extinction, and it is no longer profitable for more gangs to operate. Each gang leader chooses their own optimal planned number of expeditions , where g denotes a gang. Expeditions are sent out by each gang leader independently and sequentially. The total number of elephants harvest will be the sum: , where G is the number of gangs operating in time period t. This sum is subtracted from the growth of the elephant population as per the modified iterative map, Equation (4):
 (4)
Numerical simulations are carried out for time periods according to Equation (2) and Equation (4). Each of these one thousand time period simulations is iterated one thousand times, and the average number of realized poaching expeditions is calculated as for . Also calculated is the average elephant population over the one thousand iterations for each year t, that is for .4 In Figure 3, I depict the distribution of poaching expeditions and the population trajectories for the case of a single poaching gang facing an interception probability of 10%. The lefthand side panel has a boxplot of the distribution of the average number of realized poaching expeditions. The righthand side panel shows the trajectory of the average elephant population level evolving according to Equation (4) with 1. Over the one thousand iterations of the one thousand year realization time period I find that for the one poaching gang, with and , the average number of realized poaching expeditions is 5.52 with a standard deviation of 0.11. The elephant population, for the interval , is centered at a mean of 376,975 with a standard deviation of 453. None of the one thousand iterations resulted in extinction of the elephant population.
I now consider the addition of a second poaching gang with . The boxplot of poaching expeditions and the elephant population trajectory are shown in Figure 4. I note that the average of the number of realized poaching expeditions doubles to 11 and the average elephant population declines to 357,384. The standard deviation of the elephant population increases to 672. The planned poaching expeditions remain at 9 for each gang, and none of the one thousand iterations resulted in elephant population extinction. In Figures 3and 4, I note that the average number of poaching expeditions remained below the benchmark or reference value of for the entire time horizon of one thousand years. The elephant population evolves over time into a stable distribution for the set of basecase parameters in Table 1.
Let us now examine how the dynamics of elephant population and the poaching organizations’ economic decisionmaking change as a result of more hightech poaching. As mentioned earlier I represent more hightech poaching with a lower probability of detection, given the use of modern equipment by poaching organizations. One can also argue that more hightech poaching would increase the cost of poaching expeditions from the basecase value listed in Table 1. I later examine this through a sensitivity analysis of the results where the economic parameters are varied.
Recollect that a reduction in the interception probability to causes the optimal or planned poaching expeditions to increase to . As more poaching gangs participate in the illegal business of elephant poaching, the frequency with which would increase, and so would the likelihood that the elephant population is driven to extinction. I consider the case of two poaching gangs facing 5% and report the expeditions distribution and population trajectory in Figure 5. From the boxplot one sees that the average number of realized poaching expeditions increase to 23.63, which is higher than the reference value of 23.43. The standard deviation of the average elephant population is split into two parts to reflect the downward population trend: 20,791 for t = 100–500, and 11,002 for t = 501–1000. The frequency with which is such that the elephant population exhibits a slow decline toward extinction. The stochastic nature of realized poaching expeditions, where in some years , can significantly slow the descent toward extinction. In years where the elephant population may increase in the following year, 1. The average harvest of the two poaching organizations combined is estimated to be 8,896 elephants over the one thousand year time horizon. A maximum average harvest of 23,932 occurs at the starting period in the simulations—note that this average exceeds the estimated 23,000 elephants killed as reported by Wasser et al. [2007].
When the number of poaching gangs increases to three, the descent to low population levels is much more rapid. In Figure 6, the boxplot shows that the average number of realized poaching expeditions exceeds 23.43. The elephant population goes into an immediate and steep decline and by period 200 the elephant population has declined to below 14,000, and the three poaching gangs have reduced their number of planned poaching expeditions to between 10 and 12. A reduction in the planned poaching expeditions by all three gangs may stabilize the population at a low steadystate level of 1,497 after 150. The average harvest level with three poaching organizations is estimated to be 897 elephants over the time period 151–1,000. The maximum harvest was estimated to be 26,806 elephants at the beginning time period of these simulations.
One can potentially determine the number of poaching organizations that might operate under open access conditions with our basecase set of parameter values. Despite the small population levels with three gangs in operation, poaching is still profitable with median net revenue calculated as $26,295 per year. With the lure of profits additional organizations may enter the illegal business of elephant poaching in the Southern African range states. This would occur until it is no longer profitable for additional organizations to operate, or the elephant population goes extinct. It was noted in Figure 6 that the minimum population was 1,001 over the time period 151–1,000. There might still exist economic incentive for additional poaching organizations to enter the illicit poaching market. This might happen even though the elephant population settles into a low steadystate value. Poaching gangs might target smaller subpopulations of elephant, and this would potentially lead to extinction.
In Table 2, I report the average statistics from simulation exercises with an increase in the number of poaching gangs in operation to ten, fifteen, and twenty gangs. This is done for the two values of the probability of detection, 5% and 10%. The number of planned poaching trips/expeditions drops down to 3, 2, and 1 when the number of gangs increases to 10, 15, and 20. However in the beginning time periods of the simulation when the population is high the total harvest increases to as much as 166,226 by 20 gangs. As a result of high harvest levels initially, the total harvest by these poaching gangs eventually declines to 14 for 20 gangs with 5%. The total number of realized poaching expeditions declines to low as 17, which implies that for 20 gangs the average is less than 1 poaching expedition per year. I also note the number of years it takes for the elephant population to be driven down to extinction or low steady states in each of the scenarios. With the possibility of elephant populations going extinct or settling into low steadystate levels it may be unviable to support hightech, organized poaching, and there might be an evolution back to smallscale, local, lowtech, gangs operating under open access conditions.
Lastly, in Table 3 I vary the economic parameters in our model to analyze how they might affect the key results of this paper. The economic parameters include the black market price for a pair of elephant tusks, P, cost per poaching expedition, c, and catchability coefficient, q. The sensitivity analysis is conducted for a single poaching gang. The parameters (P, c, & q) are varied by doubling their basecase values, and reducing them by a factor of 4 of their basecase values; this provides uniformity for the sake of proper comparison. The second column of Table 3 reports the statistics for the basecase set of parameters, which are listed in Table 1. In column three of Table 3, the black market price, P, is doubled to $6,000 and in column four it is reduced by a factor of 4 to $750. I note that the number of planned poaching expeditions does not change for either of these cases, and remains at 9. Doubling or reducing the cost per poaching expedition, c, does not affect the planned number of poaching expeditions, which remains at 9. I also examine what happens when the catchability coefficient doubles to 5.12 × 10^{−3}. The median harvest increases to 10,035, and the median population declines to 356,849 compared to their respective basecase values. However there is no change in the planned poaching expeditions. When the catchability coefficient reduces to 0.64 × 10^{−3} the median harvest declines to 1,491; however, there is no change in the planned number of poaching expeditions, 9.
4. Discussion and conclusion
This paper has developed a novel model of organized criminal poaching with pertinent economic and biological parameters calibrated to reflect elephant poaching and population dynamics in the Southern African range states. The economic parameters include the black market ivory price (P), the harvest efficiency parameter (q), the cost per poaching expedition (c), and the probability of interception by antipoaching patrols . The decisionmaking process of a poaching organization’s leadership is a function of both the economic parameters and the number of elephants in each time period. I highlight the salient results of this paper. First, the optimal number of planned poaching expeditions was found to be insensitive to the blackmarket price of ivory, but quite sensitive to the probability of interception by antipoaching patrols. A reduction in ivory price can come about by a government mandate to dump confiscated ivory on the market. As noted in the introduction, reducing black market ivory price to lower the incentive to poach might work in theory; however, its effectiveness has been brought into question. In this paper, we have seen both analytically and numerically that lowering the ivory price might not lead to a reduction in the planned number of poaching expeditions. Moreover, as we saw in the sensitivity analysis, changing the ivory price in a wide range also does not affect elephant poaching. Changing the probability of interception does have a dramatic effect on reducing elephant poaching. If the government were to step up its antipoaching efforts, and thereby increase the probability of intercepting poaching gangs, then the planned number of poaching expeditions would most likely decline. This result could have an important bearing on antipoaching policy, wherein conservation managers might want to direct efforts toward increasing antipoaching patrols in national parks instead of trying to lower ivory prices.
The model in this paper provides insight into various possible trajectories of elephant poaching and population with single and multiple poaching organizations. The second salient result of this paper is that increasing the number of poaching gangs not only lowers the average of the elephant population level but also increases the standard deviation of its stationary distribution (when a stationary distribution exists). With additional poaching gangs in operation this could lead to wider fluctuations in the elephant stock level, making elephants more susceptible to extinction. With multiple poaching gangs the likelihood of stable or stationary populations is lowered over time.
The third salient result of this paper was an identification of a benchmark number of realized poaching expeditions, 23.43, which if consistently exceeded might cause the elephant population to slowly descend toward extinction. We noted that if poaching gangs were to become more hightech, thereby lowering the probability of interception by antipoaching patrols, the number of planned poaching expeditions would increase. Moreover the frequency with which the total number of realized expeditions exceeds the benchmark value may also increase. The fourth salient result is that when the probability of interception is low (i.e., ), the addition of a third poaching gang results in the elephant population permanently dropping below a threshold (≅ 1,500 as shown in Figure 6). This can cause the poaching gangs to lower the number of planned poaching expeditions below the benchmark, , thereby allowing the elephant population to stabilize at a low level.
Under open access conditions there is an incentive for additional poaching gangs to operate as long as positive profits are expected, or the elephant population has not yet become extinct. The planned number of poaching expeditions declines to one trip per gang, per year, when twenty gangs operate. With a large number of gangs in operation it increases the likelihood that poaching would be high in the initial few time periods. This could cause elephant populations to decline rapidly toward extinction. With rapidly declining elephant populations the planned expeditions eventually begin to decline. The expected profits also decline toward zero per poaching gang and this reduces the incentive for further entry of gangs.
The population model uses a lumped parameter approach wherein the dynamics of elephant population (biomass) in the seven Southern African range states is examined. The lumped parameter population model has been widely used in other studies where the net intrinsic growth rate ( birth rate – natural death rate) is applied to the entire population and not just to the mature female population. In population models that are agestructured or with delayed recruitment to an adult population, the intrinsic growth rate r can apply more specifically to the mature female population. Given that this paper assesses the trends in aggregate elephant populations, an agestructured model might not provide much additional insight. Moreover, one can easily derive the analytical steadystate population from such lumped parameter models.
The time scale examined in this paper is up to a thousand years. Shorter time scales (25–50 years) might be more relevant for the purposes of conservation policy or from the perspective of poaching markets. However, as we have seen in the model’s simulations there are interesting dynamics of the elephant population over the one thousand year time scale. This time scale allows us to examine whether the elephant population remains stable, recovers from low levels, or goes extinct. The black market price of ivory, P, is held constant in the model because there are no reliable timeseries estimates of it, thus making it difficult to model it as a function of ivory supply. However, as discussed in the main proposition of this paper the black market price does not affect the decision to poach as long as 0.
The poaching efficiency parameter used in the model is 2.56 × 10^{−3}, which implies that if 300,000 then 768 elephants will be killed on a single expedition. While this number is high there are two points to consider. The first point is that the efficiency of a poaching expedition was estimated by MilnerGulland and LeaderWilliams [1992] as being equal to 2.56 × 10^{−4} for organized gangs in Zambia in 1985. However the value of q has likely increased in the last decade with hightech poaching gangs, and I therefore scale it up by a factor of 10. The second point to consider is that the results of the paper find support in reports on elephant poaching in the southern African range states. Wasser et al. [2007] estimated that poaching gangs in southern Africa killed 23,000 elephants in the year 2006—when the estimated number of elephants was approximately 300,000. So with 768 elephants killed on a single expedition, and assuming there are 19 successful expeditions in one year (as possible when the interception probability is set at π = 5%), the total number of elephants killed would be 14,592. While this number is large, it is still lower than the annual estimate derived by the Wasser et al. study. The closest numerical result I derived was 23,932 elephants killed per year in the Southern African range states—as noted in the discussion of Figure 5 of the paper.
This paper has examined conservation policy in terms of the effectiveness of antipoaching patrols in protecting elephant populations. The model provides a bird’s eye perspective of elephant poaching over a large geographical area with an initial population of 297,718 elephants. Elephants are known to travel vast distances and are therefore a spatially mobile resource. A biomass model would have to be more complex to account for spatial variability in the distribution of a resource; the model in this paper has no spatial dimension. More realistic models should incorporate spatial differences between the countries in Southern Africa, including suitable habitat, the size of elephant populations in each of those countries, the migration of elephant between countries, and most importantly, the countryspecific policies to prevent poaching. Based on the location of elephant metapopulations and the expected number of antipoaching patrols, a poaching gang would need to determine not only the number of planned expeditions but also their location. The resulting model might be viewed as a repeated game between poachers and antipoaching patrols with subgame strategies that would depend on the location of the species and the expected deployment of antipoaching patrols. Hightech poaching will require a hightech, gamestrategic response.
This paper has examined stochasticity in terms of elephant poaching. Other forms of stochasticity can also arise—one example would be environmental shocks in natural ecosystems. Droughts, flash floods, and animal diseases can have drastic effects on elephant herds. Random events such as these might affect not just the population size but also the net intrinsic growth rate of a population. This can easily be incorporated in the model by making the population, X, a random variable with an assumed distribution. This paper has focused more on the economic behavior of a poaching organization and the stochasticity associated with being intercepted by antipoaching patrols.
Finally, the model developed in this paper can be extended and broadly applied to the assessment of antipoaching and/or conservation policy for other endangered animal species, plant and tree species, marine and fish species, as well as the illegal mining of nonrenewable resources. Examples of other endangered species for which this model can be extended to would be rhinos in Southern Africa and South Asia, and tigers in South Asia. With rhinos hunted for their horns and tigers for their skins, both animal species have been subject to drastic increases in poaching over the last decade and more. Illegal logging of timber is another conservation issue in many tropical countries. Extending the model to examine illegal logging would involve using a different growth function that represents growth in tree volume over time. The illegal harvesting of timber can be modeled in a similar manner for a criminal gang of loggers. The conservation of marine species such as sea turtles is also under duress since they are illegally harvested for their meat, shells, and eggs. One can apply a similar framework to examine the issues relating to illegal mining in Africa and South America. In this case one would have to model the depletion of a nonrenewable resource over time as a result of illegal mining. Unsafe mining practices in such areas also impose costs to the surrounding natural systems. In a model this can be incorporated to reflect additional environmental damage resulting from illegal mining. The mode of harvesting of such resources is similar in terms of expeditions that are being made by groups of illegal harvesters. Models of such coupled natural and human systems can be calibrated to reflect the characteristic biological or ecological dynamics of such resources. The essence of the economic model will remain the same with the poaching gangs choosing an optimal number of expeditions subject to the risk of being caught. Applying such a model to a more localized context would produce its own characteristic results and may have distinctive implications for conservation policy. Incorporating extensions of the model—such as including spatial variability with metapopulations, environmental stochasticity, and countryspecific antipoaching policy—would be important for understanding how best to protect scarce resources and/or endangered species.
Acknowledgments
I sincerely thank Jon Conrad for his very valuable guidance in the development of this paper. I also thank Ravi Kanbur and Evan Cooch for their helpful comments in improving previous drafts of this paper. I gratefully acknowledge the Joseph Fisher Dissertation Fellowship from Resources for the Future, and the Andrew Mellon Research Grant from the College of Agriculture & Life Sciences at Cornell University for supporting this research.
Parameter  Notation  Value  References 

Intrinsic growth rate  r =  0.33  Armbruster and Lande [1993] 
Natural mortality rate  m =  0.27  Armbruster and Lande [1993] 
Logistic growth skew parameter  z =  7  Bulte and van Kooten [1999], MilnerGulland and LeaderWilliams [1992] 
Initial elephant population in 2006  X_{0} =  297,718 in the Southern African range states  Blanc [2007] 
Carrying capacity  K  500,000  
Black market ivory price  P  $3,000 per set of two tusks  Messer [2010] 
Harvest efficiency  q  2.56 × 10^{–3}  MilnerGulland and LeaderWilliams [1992] 
Probability of interception by antipoaching patrols  π(a_{t})  0.10; 0.05  MilnerGulland and LeaderWilliams [1992] 
Expedition cost  c =  $2,000 
10 gangs  15 gangs  20 gangs  

π = 5%  π = 10%  π = 5%  π = 10%  π = 5%  π = 10%  
Population (median)  363  680  324  417  318  405 
Total harvest (minimum)  18  33  14  20  14  17 
Total harvest (median)  21  41  19  25  19  24 
Total harvest (maximum)  92,144  43,519  131,396  65,164  166,226  90,468 
Total poaching trips (minimum)  22  21  18  20  18  17 
Total poaching trips (median)  26  26  25  25  24  24 
Total poaching trips (maximum)  131  61  184  90  243  118 
Profit minimum (per gang)  984  5,097  406  1,024  315  847 
n*  3  3  2  2  1  1 
Years to elephant extinction  69  125  34  114  19  47 
Base case parameters a  P = 6,000  P = 750  c = 4,000  c = 500  q= 5.12*10^{−3}  q= 6.4*10^{−4}  



Population (minimum)  375,790  367,239  367,932  366,903  373,638  348,187  386,380 
Population (median)  376,975  377,416  377,286  376,431  376,921  356,849  388,558 
Total harvest (median)  5,307  5,741  5,794  5,841  5,330  10,035  1,491 
Total harvest (maximum)  8,028  8,949  8,996  8,983  8,513  15,770  2,253 
Total poaching trips (median)  5  6  6  6  5  5  6 
Total poaching trips (maximum)  6  9  9  9  8  8  9 
n*  9  9  9  9  9  9  9 
Elephant extinction  No  No  No  No  No  No  No 
ENDNOTES
 1
The nopoaching, steadystate population can be shown to equal . This steadystate population will be locally stable provided elephants for all of Southern Africa, , and , the nopoaching, steadystate elephant population is calculated to be . This value is locally stable since .
 2
It would be possible to allow the probability of interception by an antipoaching unit to increase with each completed (i.e., successful) poaching expedition. One would then need a model of how the conditional probability of success for the next poaching expedition depends on the fact that all previous poaching expeditions were successful.
 3
The nonlinear nature of Equation (2) does not permit the derivation of an analytical expression for . The optimal number of planned expeditions can be numerically solved for a positive integer value. The firstorder condition of the maximization process is implicit in a numerical solution of the optimal number of planned poaching expeditions, , in time period t.
 4
I report the average elephant population as calculated for the years t = 100, …, 999; this reduces the effect of initial conditions on the longterm population trend. The figures however depict the population trends over the entire time horizon of t = 0, …, 999.
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Original article: http://onlinelibrary.wiley.com/doi/10.1111/nrm.12058/full