An Evolutionary Game Theoretic Model of Rhino Horn Devaluation
AAn Evolutionary Game Theoretic Model of Rhino Horn Devaluation
Nikoleta E. Glynatsi, Vincent Knight, Tamsin E. Lee
Keywords : Evolutionary dynamics; Evolutionary stability; Game theory;Poachers’ interactions; Rhinoceros; Wildlife
Declaration of interest : None. 1 a r X i v : . [ q - b i o . P E ] O c t bstract Rhino populations are at a critical level due to the demand for rhino horn and the subsequent poaching.Wildlife managers attempt to secure rhinos with approaches to devalue the horn, the most common ofwhich is dehorning. Game theory has been used to examine the interaction of poachers and wildlifemanagers where a manager can either ‘dehorn’ their rhinos or leave the horn attached and poachers maybehave ‘selectively’ or ‘indiscriminately’. The approach described in this paper builds on this previouswork and investigates the interactions between the poachers. We build an evolutionary game theoreticmodel and determine which strategy is preferred by a poacher in various different populations of poachers.The purpose of this work is to discover whether conditions which encourage the poachers to behaveselectively exist, that is, they only kill those rhinos with full horns.The analytical results show that full devaluation of all rhinos will likely lead to indiscriminate poaching.In turn it shows that devaluing of rhinos can only be effective when implemented along with a strongdisincentive framework. This paper aims to contribute to the necessary research required for informeddiscussion about the lively debate on legalising rhino horn trade.
Rhino populations now persist largely in protected areas or on private land, and require intensive protec-tion [12] because the demand for rhino horn continues to pose a serious threat [2]. The illegal trade in rhinohorn supports aggressive poaching syndicates and a black market [28, 34]. This lucrative market enticespeople to invest their time and energy to gain a ‘windfall’ in the form of a rhino horn, through the poachingof rhinos.Standard economic theory predicts extinction through poaching alone is unlikely due to escalating costsas the number of remaining species approaches zero [6]. However, the rarity of rhino horn makes it a luxurygood, or financial investment for the wealthy [13], and thus the increased cost and risk to poach does notincrease as rapidly as the increased gain - the anthropogenic Allee effect [4, 6]. However, the anthropogenicAllee effect was recently revisited [18] to highlight that the relationship is even more complex and pessimistic.The value of rhino horn can inflate, even with a large population size, due to an increase in the cost (i.e.risk) to poach. Therefore measures to protect rhino horn may actually be increasing the gain to poachers.It is not clear whether this relationship has contributed to the escalation in rhino poaching over recentyears. Nonetheless, it is clear that the future existence of rhinos is endangered because of poaching [3, 30].This rationale leads to debate about legalising rhino horn trade, which in turn may reduce demand. In [3]the authors suggest meeting the demand for rhino horn through a legal market by farming the rhino horn2rom live rhinos. In fact recently the actual quantity of horn that could be farmed was estimated by [31].However [7] argues that because the demand for horn is so high, legalising trade may lead to practices thatmaximise profit, but are not suitable for sustainable rhino populations, and thus rhinos may be ‘traded onextinction’. Preventing poaching covers in-country and global issues, and thus legalising rhino horn trade isa controversial and active conversation, which is not limited to rhinos - [17] considered ivory and stated thatby enforcing a domestic ivory trade ban we can reduce the market’s demand.As it stands, for wildlife managers law enforcement is often one of the main methods to deter poachers.Rhino conservation has seen increased militarisation with ‘boots on the ground’ and ‘eyes in the sky’ [10].An alternative method is to devalue the horn itself, one of the main methods being the removal so that onlya stub is left. The potential impact of various policies are nicely summarised in [8], where de-horning isnoted to be promising for ‘in-country intervention’. The first attempt at large-scale rhino dehorning as ananti-poaching measure was in Damaraland, Namibia, in 1989 [25]. Other methods of devaluing the horn thathave been suggested include the insertion of poisons, dyes or GPS trackers [14, 30]. However, like dehorning,they cannot remove all the potential gain from an intact horn (poison and dyes fade or GPS trackers canbe removed and have been found to affect only a small proportion of the horn). In [25, 26] they found theoptimum proportion to dehorn using mean horn length as a measure of the proportion of rhinos dehorned.They showed, with realistic parameter values, that the optimal strategy is to dehorn as many rhinos aspossible. A manager does not need to choose between law enforcement or devaluing, but perhaps adopta combination of the two; especially given that devaluing rhinos comes at a cost to the manager, and theprocess comes with a risk to the rhinos.A recent paper modelled the interaction between a rhino manager and poachers using game theory [21].The authors consider a working year of a single rhino manager. A manager is assumed to have standard yearlyresources which can be allocated on devaluing a proportion of their rhinos or spent on security. It is assumedthat all rhinos initially have intact horns. Poachers may either only kill rhinos with full horns, ‘selectivepoachers’, or kill all rhinos they encounter, ‘indiscriminate poachers’. This strategy may be preferred toavoid tracking a devalued rhino again, and/or to gain the value from the partial horn. If all rhinos are left bythe rhino manager with their intact horns, it does not pay poachers to be selective so they will chose to beindiscriminate since being selective incurs an additional cost to discern the status of the rhino. Conversely,if all poachers are selective, it pays rhino managers to invest in devaluing their rhinos. This dynamic isrepresented in Fig. 1. Assuming poachers and managers will always behave so as to maximise their payoff,there are two equilibriums: either all rhinos are devalued and all poachers are selective; or all horns are intact3nd all poachers are indiscriminate. Essentially, either the managers win, the top left quadrant of Fig. 1, orthe poachers win, the bottom right quadrant of Fig. 1. The paper [21] concludes that poachers will alwayschoose to behave indiscriminately, and thus the game settles to the top left quadrant, i.e., the poachers win.
M P
Horn devaluedManager strategies P o ac h e r s t r a t e g i es S e l ec t i ve I nd i sc r i m i n a t e Horn intact
Figure 1: The game between rhino manager and rhino poachers. The system settles to one of two equilibri-ums, either devaluing is effective or not.At the extremes, we could consider the game as one of opportunistic exploitation [4]. That is, considerintact rhinos and devalued rhinos as two species, where one is more valuable than the other. Opportunisticexploitation advances upon the theory of anthropogenic Allee effect to consider two species which are exploitedtogether. Specifically, when a highly valued species becomes rarer, a secondary, less valuable species is thentargeted. As with opportunistic exploitation on a larger scale, rhino managers need to account for themultispecies system.In this manuscript, we explore the population dynamic effects associated with the interactions describedby [21]. More specifically, the interaction between poachers. In a population full of indiscriminate poachersis there a benefit to a single poacher becoming selective or vice versa? This notion is explored here usingevolutionary game theory [29]. The game is not that of two players anymore (manager and poacher) butnow the players are an infinite population of poachers. This allows for the interaction between poachers overmultiple plays of the game to be explored with the rhino manager being the one that creates the conditionsof the population.Note that poachers are, in practice finite, and each has individual factors that will affect a poacher’sbehaviour. An infinite population model corresponds to either an asymptotic generalisation or overall de-scriptive behaviour. 4n evolutionary game theory, we assume infinite populations and in our model this is represented by( x , x ) with x being the proportion of the population using a strategy of the first type and x of thesecond. We assume there are utility functions u and u that map the population to a fitness for eachstrategy, given by, u ( x , x ) and u ( x , x ) . In evolutionary game theory these utilities are used to dictate the evolution of the population over time,according to the following replicator equations, dx dt = x ( u ( x , x ) − φ ) ,dx dt = x ( u ( x , x ) − φ ) , (1)where φ is the average fitness of the whole population [27]. In some settings these utilities are referred to asfitness and/or are mapped to a further measure of fitness. This is not the case considered here (it is assumedall evolutionary dynamics are considered by the utility measures).Here, the overall population is assumed to remain stable thus, x + x = 1 and dx dt + dx dt = 0 ⇒ x ( u ( x , x ) − φ ) + x ( u ( x , x ) − φ ) = 0 . (2)Recalling that x + x = 1 the average fitness can be written as, φ = x u ( x , x ) + x u ( x , x ) . (3)By substituting (3) and x = 1 − x in (1), dx dt = x (1 − x )( u ( x , x ) − u ( x , x )) . (4)5he equilibria of the differential equation (4) are given by, x = 0, x = 1, and 0 < x < u ( x , x ) = u ( x , x ). These equilibria correspond to stability of the population: the differential equation (4) no longerchanges.The notion of evolutionary stability can be checked only for these stable strategies. For a stable strategyto be an Evolutionary Stable Strategy (ESS) it must remain the best response even in a mutated population( x , x ) (cid:15) . A mutated population is the post entry population where a small proportion (cid:15) > u , u that correspond to a population of wild rhino poachersand we explore the stability of the equilibria identified in [21]. The results contained in this paper are provenanalytically, and more specifically it is shown that: • In the presence of sufficient risk: a population of selective poachers is stable, meaning dehorning is aviable option. • Full devaluation of all rhinos will lead to indiscriminate poachers.
As discussed briefly in Section 1, a rhino poacher can adopt two strategies, to either behave selectively orindiscriminately. To calculate the utility for each strategy, the gain and cost that poachers are exposed tomust be taken into account. The poacher incurs a loss from seeking a rhino, and the risk involved. The gaindepends upon the value of horn, the proportion of horn remaining after the manager has devalued the rhinohorn and the number of rhinos (devalued and not).Let us first consider the gain to the poacher, where θ is the amount of horn taken. We assume rhino hornvalue is determined by weight only, a reasonable assumption as rhino horn is sold in a grounded form [1].Clearly if the horn is intact, the amount of horn gained is θ = 1 for both the selective and the indiscriminatepoacher. If the rhino horn has been devalued, and the poacher is selective, the amount of horn gained is θ = 0 as the poacher does not kill. However, if the poacher is behaving indiscriminately, the proportion ofvalue gained from the horn is θ = θ r (for some 0 < θ r < θ ( r, x ) = x (1 − r ) + (1 − x )(1 − r + rθ r ) (5)6here r is the proportion of rhinos that have been devalued, and x is the proportion of selective poachersand 1 − x is the proportion of indiscriminate poachers. Note that since θ r , r, x ∈ [0 , θ ( r, x ) >
0, thatis, some horn will be taken. Standard supply and demand arguments imply that the value of rhino horndecreases as the quantity of horn available increases [23]. Thus at any given time the expected gain is Hθ ( r, x ) − α , (6)where H is a scaling factor associated with the value of a full horn, α ≥ r increases, so that the supply of horn decreases, the value ishigher and vice versa. We have chosen a simple function to model the demand (and thus gain) of rhino hornvalue, relative to the proportion of rhinos devalued. However, demand for illegal wildlife generally involvesmore factors than simply supply. Additional factors include, but are not limited to, social stigma, tourismrevenues, government corruption, and rich countries being willing to pay to ensure species existence [5, 32]. r H ( r , x ) = 1.0= 0.75= 0.5= 0.25= 0.0 Figure 2: Hθ ( r, x ) − α for values H = 10 , θ r = 0 . x = 0 . . An individual interacts with the population which is uniquely determined by x , the proportion of selectivepoachers. Therefore, the gain for a poacher in the population x is either θ ( r, Hθ ( r, x ) − α selective poacher θ ( r, Hθ ( r, x ) − α indiscriminate poacher (7)7epending on the chosen strategy of the individual.Secondly we consider the costs incurred by the poacher. It is assumed that a given poacher will spendsufficient time in the park to obtain the equivalent of at least a single rhinoceros’s horn. For selectivepoachers this implies searching the park for a fully valued horn and for indiscriminate poachers this implieseither finding a fully valued horn or finding N r total rhinoceroses where N r = (cid:100) θ r (cid:101) .Figure 3 shows a random walk that any given poacher will follow in the park. Both types of poacher willexit the park as soon as they encounter a fully valued rhino, which at every encounter is assumed to happenwith probability 1 − r . However, the indiscriminate poachers may also exit the park if they encounter N r devalued rhinos in a row. Each step on the random walk is assumed to last 1 time unit: during which a rhinois found. To capture the fact that indiscriminate poachers will spend a different amount of time to selectivepoachers with each rhino the parameter τ is introduced which corresponds to the amount of time it takes tofind and kill a rhino (thus τ ≥ r − r r − r − r r − r r − r − r Exitbothbothbothboth both sel. N r ind.Figure 3: Illustrative random walk showing the points at which an indiscriminate or a selective poacher willleave the park.Using this, the expected time spent in the park T , T by poachers of both types can be obtained:For selective poachers: 8 = (1 − r ) τ + r (1 − r )(1 + τ ) + r (1 − r )(2 + τ ) + . . . = (1 − r ) ∞ (cid:88) i =0 r i ( i + τ )= (1 − r ) (cid:32) r ∞ (cid:88) i =0 ir ( i +1) + τ ∞ (cid:88) i =0 r i (cid:33) = (1 − r ) (cid:18) r (1 − r ) + τ − r (cid:19) using standard formula for geometric series = r + τ (1 − r )1 − r (8)For indiscriminate poachers: T = (1 − r ) τ + r (1 − r )2 τ + r (1 − r )3 τ + · · · + r N r − (1 − r )( N r − τ + r N r − N r τ = (1 − r ) τ N r − (cid:88) i =1 ir i − + r N r − N r τ = (1 − r ) τ (cid:32) r ( r − (cid:0) N r rr N r − N r r N r − rr N r + r (cid:1)(cid:33) + r N r − N r τ = τ (1 − r N r )(1 − r ) (9)Figure 4 shows T and T for varying values of r and τ highlighting that τ has a greater effect on T than T . Also, as r increases the overall time spent in the park by both poachers increases and the value of τ atwhich T and T are equal increases.Additionally, the poachers are also exposed to a risk. The risk to the poacher is directly related to theproportion of rhinos not devalued, 1 − r , since the rhino manager can spend more on security if the cost ofdevaluing is low. In real life this is not always the case. The cost of security can be extremely high thus itcannot be guaranteed that much security will be added from the saved money. However, our model assumesthat there is a proportional and negative relationship between the measures.(1 − r ) β , (10)where β ≥ .6 0.8 1.0 1.2 1.40.751.001.251.501.752.002.252.50 T i m e i n p a r k r = 0.4 T i m e i n p a r k r = 0.6 T i m e i n p a r k r = 0.8 Indiscriminate time Selective time
Figure 4: Expected time spent in the park for selective T and indiscriminate poachers T for θ r = . F T i (1 − r ) β for i ∈ { , } (11)where F is a constants that determines the precise relationship. Fig. 5 verifies the decreasing relationshipbetween r and the cost. Notice that the cost to indiscriminate poachers remains fairly consistent, irrespectiveof the proportion of devalued rhinos, until this proportion gets high. Whereas the cost to selective poachersis more sensitive to the proportion of rhinos devalued, especially when the time to kill a rhino is large.One final consideration given to the utility model is the incorporation of a disincentive to indiscriminatepoachers. Numerous interpretations can be incorporated with this: • more severe punishment for indiscriminate killing of rhinos; • educational interventions that highlight the negative aspects of indiscriminate killing; • the possibility of a better alternative being offered to selective poachers.This will be captured by a constant Γ. 10 .0 0.2 0.4 0.6 0.8 1.0 r Selective cost, = 0.95 r Selective cost, = 1.05 r Indisriminate cost, = 0.95 = 1.0 = 1.25 = 1.5 = 1.75 = 2.0
Figure 5: Costs associated to both poachers for F = 5 and varying values of r and τ .11ombining (7) and (11) gives the utility functions for selective poachers, u ( x ), and indiscriminate poachers, u ( x ), u ( x ) = θ ( r, Hθ ( r, x ) − α − ( r + τ (1 − r )) F (1 − r ) β − , (12) u ( x ) = θ ( r, Hθ ( r, x ) − α − τ (1 − r N r ) F (1 − r ) β − − Γ (13)Given a specific individual, let s denote the probability of them behaving selectively. Thus the generalutility function for an individual poacher in the population with a proportion of 0 ≤ x ≤ u ( s, x ) = su ( x ) + (1 − s ) u ( x ) . (14)Substituting (12) and (13) into (14) and using (5) gives, u ( s, x ) = H ( θ r r (1 − s ) − r + 1) θ ( r, x ) − α − F (cid:0) sr + sτ (1 − r ) + (1 − s ) τ (1 − r N r ) (cid:1) (1 − r ) β − − (1 − s )Γ(15)Figure 6 shows the evolution of the system over time for a variety of initial populations and param-eters. This is done using numerical integration implemented in [11]. All the source code used for thiswork has been written in a sustainable manner: it is open source ( https://github.com/Nikoleta-v3/Evolutionary-game-theoretic-Model-of-Rhino-poaching/ ) and tested which ensures the validity of theresults. The source code has also been properly archived and can be found at [16].A summary of all the parameters and their meanings is given by Table 1.In Figure 6 the left column shows parameters sets for which a selective population is stable ( x = 1.The figures on the right correspond to a decrease in τ which decreases the risk associated with actingindiscriminately: in these cases a population of selective poachers is unstable. In one case ( H = 192) a mixedpopulation is stable: some poachers will continue to act selectively.In Section 3, these observations will be confirmed theoretically.12arameter Interpretation θ r the proportion of value gained from a devalued horn r the proportion of rhinos that have been devalued H a scaling factor associated with the value of a full horn α the relationship between the quantity and value of a horn F the cost of retrieving a horn β the relationship between the proportion of devalued rhinos and security τ the time it takes to find and kill a rhinoΓ a disincentive only applied to indiscriminate poachersTable 1: A summary of the parameters used. time x r = 0.54 = 1.5 time x r = 0.54 = 1.1 time x r = 0.6 = 1.8 time x r = 0.6 = 1.5 time x r = 0.2 = 2 time x r = 0.2 = 1.5 x (0) = 1 10 x (0) = 1 10 x (0) = 0.5 x (0) = 0.1 x (0) = 0.01 Figure 6: The change of the population over time with different starting populations. For F = 5 , H =50 , α = 2 , β = . , τ = 1 . , θ r = 0 . , Γ = 0. 13
Evolutionary Stability
By definition, for a strategy to be an ESS it must first be a best response to an environment where the entirepopulation is playing the same strategy. In our model there are three possible stable distributions based onthe equilibria of equation (4): • all poachers are selective; • all poachers are indiscriminate; • mixed population of selective and indiscriminate poachers.An ESS corresponds to asymptotic behaviour near the equilibria of (4), this correspond to the concept ofLyapunov stability [22].For simplicity, denote the right hand side of (4) as f . In this setting, when x is near to some equilibria x ∗ so that f ( x ∗ ) = 0 then the evolutionary game can be linearized (using standard Taylor Series expansion) as: d ( x ∗ + (cid:15) ) dt = J ( x ∗ ) (cid:15) (16)where: J ( a ) = dfdx (cid:12)(cid:12)(cid:12)(cid:12) x = a (17)This gives a standard approach for evaluating equilibria of the underlying game. For a given equilibria x ∗ , J ( x ∗ ) < x ∗ is an ESS.Using equations (12) and (13): J ( a ) = 1( r −
1) ( − arθ r + rθ r − r + 1) α +1 ( J − J ) + Γ(1 − a ) (18)where: 14 = F ( − r + 1) β ( − arθ r + rθ r − r + 1) α +1 (cid:16) art − ar − ar (cid:100) θr (cid:101) t − rτ + r + r (cid:100) θr (cid:101) τ (cid:17) J = Haαr θ r ( − a + 1) ( r −
1) +
Hrθ r (2 a −
1) ( r −
1) ( arθ r − rθ r + r − Theorem 1.
Using the utility model described in Section 2, a population of selective poachers is stable if andonly if: τ > − r (cid:100) θr (cid:101)− F + Hθ r (1 − r ) − α − β − Γ r (1 − r ) − β F (19) Proof.
Direct substitution gives: J (1) = 1( − r + 1) α +1 ( r − (cid:16) F ( − r + 1) β ( − r + 1) α +1 (cid:16) rτ − r − r (cid:100) θr (cid:101) τ (cid:17) − Hrθ r ( r − (cid:17) − Γ= (cid:16) F (1 − r ) β − (cid:16) r − τ ( r − r (cid:100) θr (cid:101) ) (cid:17) + Hrθ r (1 − r ) − α (cid:17) − ΓThe required condition is J (1) < F (1 − r ) β − r + Hrθ r (1 − r ) − α − Γ < F (1 − r ) β − τ ( r − r (cid:100) θr (cid:101) ) rr − r (cid:100) θr (cid:101) F + Hθ r (1 − r ) − β − α − Γ r (1 − r ) − β F < τ which gives the required result.Note that the limit of the right hand side of equation (19) tends to infinity as r → − . This means thatdevaluing all rhinos is not a valid approach.Furthermore, we see that the equilibria with poachers acting selectively, predicted in [21] can in fact beobtained in specific settings.Note that similar theoretic results have been obtained about the evolutionary stability of indiscriminatepoachers but these have been omitted for the sake of clarity.In this section we have analytically studied the stability of all the possible equilibria. We have proventhat all potential equilibria are possible. All of these theoretic results have been verified empirically, and15he data for this has been archived at [15]. Figure 7 shows a number of scenarios where F = 5, β = 0 . θ r = 0 .
05 and, unless varied as stated on the x-axis: • Scenario 1: H = 50 r = 0 . α = 2, τ = 2, Γ = 0 • Scenario 2: H = 50 r = 0 . α = 2 . τ = 1 .
8, Γ = 0 • Scenario 3: H = 25 r = 0 . α = 2, τ = 2, Γ = 0 • Scenario 4: H = 25 r = 0 . α = 2 . τ = 1 .
8, Γ = 0 • Scenario 5: H = 25 r = 0 . α = 2, τ = 2, Γ = 4 • Scenario 6: H = 25 r = 0 . α = 2 . τ = 1 .
8, Γ = 4The first plot in Figure 7 shows that when the value of a full horn is low, and there is no disincentivefactor (Scenarios 3 and 4), the poacher strategy can be influenced by the time taken to kill a rhino suchthat a long time can push the poacher to behave selectively (as there is increased risk associated with actingindiscriminately). The second plot, shows that the disincentive factor has most influence when the valueof a full horn is high (Scenarios 1 and 2). Otherwise, poachers will generally be indiscriminate if r is large(Scenarios 5 and 6) or selective otherwise (Scenarios 1, 2, 3 and 4). Most importantly, the third plot confirmsTheorem 1. A high value of r forces the population to become indiscriminate even with a high disincentive.Moreover, for all scenarios a value of r does exist for which a selective population will subsist.This confirms that devaluing alone is not a solution and in fact can potentially have averse consequences:combinations of devaluing and education (creating a disincentive) is needed. In this work the dynamics of a selective population were explored. It was shown that given sufficient riskassociated with killing a rhino, it would be possible for a selective population of poachers to subsist.We have developed a game theoretic model which examines the specific question for rhino managers: howto deter poachers by devaluing horns? One of the main conclusions of the work presented here is that ifthere is sufficient risk associated with indiscriminate behaviour then a population of selective poachers canbe stable. The model also incorporates wider factors in a general manner such as a disincentive factor. Thedisincentive factor may be an increase in the monetary fine for poachers. In fact [9], who identify the mostimportant contributors to the number of rhinos illegally killed in South Africa (between 1900 and 2013),16 .50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.500.00.20.40.60.81.0 E v o l u t i o n a r y s t a b l e x E v o l u t i o n a r y s t a b l e x r E v o l u t i o n a r y s t a b l e x Scenario: 1Scenario: 2Scenario: 3Scenario: 4Scenario: 5Scenario: 6
Figure 7: Evolutionary stable populations for varying values of τ, r,
Γ for 6 difference scenarios.17ound that increasing the monetary fine has a more significant effect than increasing the years in prison.However, the disincentive factor may also include wider influences, such as engaging the rural communitiesthat neighbour wildlife [3], or decreasing the cost of living with wildlife, and supporting a livelihood that isnot related to poaching. Zooming out further, it could include global issues such as an increase in ecotourism,which would provide a sustainable income for the community.Another opportunity for wider factors, such as global issues, to be included in the model is via the supplyand demand function. For example, [9] show that one of the three most important contributors to the numberof rhinos illegally killed was the GDP in Far East Asia, where the demand for rhino horn is at its greatest.This finding supports [20] call for improved law enforcement and demand reduction in the Far East.Note that the proportion of devalued rhinos r is continuous over [0 ,
1] in the model. However, standardpractice of a given park manager in almost all cases is to either devalue all the animals in a defined enclosedarea, or none at all. This is thought to be because partial devaluing tends to disturb rhino social structures.Our results indicate that devaluing all rhinos will only decrease rhino poaching if potential poachers have aviable alternative (even in the case of a large disincentive).The debate about the effectiveness of devaluation for preventing poachers and, is extensive and ongoing.This model answers one aspect of the topic, but larger questions remain. There are many drivers to accountfor, many of which are included in a systems dynamics model presented in [7] which captures the five mostimportant factors: rhino abundance, rhino demand, a price model, an income model and a supply model.Using the optimal dehorning model of [25], the model [7] finds that poachers behaving indiscriminately willalways prevail, which indicates that the risk associated with indiscriminate behaviour might not have beencaptured fully.Following discussions with environmental specialists it is clear that devaluing is empirically thought to beone of the best responses to poaching. This indicates that whilst of theoretic and potential macroeconomicinterest, the modelling approach investigated in this work has potential for further work. For example, adetailed study of two neighbouring parks with differing policies could be studied using a game theoreticmodel, this would require an understanding of the travel times which can be very large and have a nonnegligible effect. Another interesting study would be to introduce a third strategy available to poachers: thiswould represent the possibility of not poaching (perhaps finding another source of income) and/or leavingthe current environment to poach elsewhere. Finally, the specific rhino population could also be modelledusing similar techniques and incorporated in the supply and demand model.18 uthors’ contributions
All authors conceived the ideas and designed the methodology. NG and VK developed the source code neededfor the numerical experiments and generating the data. All authors contributed critically to the drafts andgave final approval for publication.
Acknowledgements
This work was performed using the computational facilities of the Advanced Research Computing @ Cardiff(ARCCA) Division, Cardiff University.A variety of software libraries have been used in this work: • The Scipy library for various algorithms [11]. • The Matplotlib library for visualisation [19]. • The SymPy library for symbolic mathematics [24]. • The Numpy library for data manipulation [33].We would like to express our appreciation to David Roberts and Michael ’t Sas-Rolfes for their valuableand constructive suggestions for this research. We would also like to thank Dr. Jonathan Gillard for hiscareful reading and double checking of the algebra in the paper, and Anna Huzar for Figure 1.Finally, we would like to thank the referees whose comments on the first version of this manuscript greatelyimproved the undertaken research.
Data Accessibility
The data generated for this work have been archived and are available online [15].
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