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Ecological Modelling 250 (2013) 134–142 Contents lists available at SciVerse ScienceDirect Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel Impacts of maximum sustainable yield policy to prey–predator systems T.K. Kar ∗ , Bapan Ghosh Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India a r t i c l e i n f o Article history: Received 14 September 2012 Received in revised form 9 November 2012 Accepted 12 November 2012 Available online 13 December 2012 Keywords: Prey–predator Intraspecific competition Ratio-dependent Maximum sustainable total yield Persistence a b s t r a c t This article investigates the effects of reaching the maximum sustainable yield (MSY) in prey–predator systems where the prey population follows logistic law of growth. Two different models are proposed: (i) first model involves linear prey–predator interaction and intraspecific competition among predator populations, and (ii) the second one is a ratio-dependent prey–predator system. In the first model, our results suggest that the introduction of intraspecific competition among predator population has important consequences for the fishing to reach MSY from prey species and maximum sustainable total yield (MSTY) for combined harvesting of both prey and predator species. On the otherhand, in the second model, our results suggest that though the harvesting of prey species at MSY level shall be guaranteed the coexistence of both the species, but the combined harvesting of both the species at MSTY level may cause extinction of the predator species. However, for both the models, predator harvesting at MSY level may be a sustainable fishing policy. Therefore, based on our results we can conclude that MSY (or MSTY) policy in prey–predator systems in nature are not likely to fit requirements of Conservation of Biological Diversity (CBD, 1992) in all cases. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The exploitation of biological resources and harvesting of population are commonly practiced in fisheries, forestry, and wildlife management. In particular, the management of a fishery is a decision with multiple objectives. Some of the desirable objectives are the provision of the good bio-mass yield, the conservation of fish population, the provision of good economic returns and the provision of recreation. The formulation of good harvesting policies which take into account these objectives is a complex and difficult task. Maximum sustainable yield (MSY) is a simple way to manage resources taking into consideration that over-exploited resources lead to a loss in productivity. But, we have seen that the traditional MSY approach to fishery management relies on single species model of population dynamics and ignores species interactions with other member of the communities. Single species assessments and management controls may produce misleading prediction and pathological changes in the ecosystems. Even simple food-chain models predict stronger compensatory responses for any species when interactions with the rest of the system are accounted for, owing to the negative impact of harvesting on its predators, and the positive impacts on its food organisms. ∗ Corresponding author. Tel.: +91 3326684561; fax: +91 33 26682916. E-mail addresses: [email protected] (T.K. Kar), keshab [email protected] (B. Ghosh). 0304-3800/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2012.11.015 We have come through a long way from targeting MSY using equilibrium population level. Schaefer (1954) was first to introduce MSY policy for a single species fishery having logistic law of growth and subject to proportional harvesting. Clark (1990) also discussed the importance of the concept of MSY policy for fishery management. Recently, Kar and Matsuda (2007) investigated the MSY policy for single species fishery with strong Allee effect. Legovic (2008) mentioned the effects of harvesting of MSY policy in a single species fishery. Walters et al. (2005) show that the widespread application of single species model MSY policy would in general cause severe deterioration in ecosystem structure, in particular the loss of top predator species. All the above results suggest that MSY exists for a single isolated population living in nature but there are significant dangers of adopting single-species management approaches that may be myopic. Therefore, can we really get away with considering exploitation of one species at a time, and ignoring interactions among species? This is the subject of our study in this paper. Individual species do not live in isolation, which means that the population dynamics of different species are inevitably linked. Therefore, it is not at all obvious whether we should expect higher or lower yields from an entire ecosystem than would be predicted from application of single species harvest control policy. Recently, Matsuda and Abrams (2006) examined various food web models, such as two-prey one predator system, three species with three trophic levels, six species systems with multiple trophic levels and asserted that independent harvest of species within the web can cause extinction, but that this is less likely to happen. In their work T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 (2010b), Legovic and Gecek investigated a community of independent and logistically growing populations under a common harvesting effort which leads to the maximum sustainable total yield (MSTY). Their results suggest that in the case of two independent populations with approximately equal carrying capacities, both the populations persist at its MSTY level. Their results also suggest that the MSTY with a common harvesting effort implies suboptimal fishing of some population, overfishing of others and extinction of the rest of the populations. Legovic et al. (2010a) show that approaching MSY in ecosystem means that most likely fish species will be driven to extinction in every fishery that includes exploitation of at least one trophic level which is directly or indirectly used as food for a higher trophic level. In addition, Legovic and Gecek (2012) studied the impact of MSY policy in mutualistic communities and interestingly noticed that harvesting of all species to the MSTY level will induce extinction of the species with lower biotic potentials and carrying capacities. In this paper, we have studied the impacts of MSY policy on two different prey–predator systems under different harvesting scenarios. Section 2 provides the general understanding about MSY policy and revision of the outcomes due to Legovic et al. (2010a). In Section 3, we introduce an intraspecific competition term to the predator equation and study the impacts of MSY policy when either prey or predator species is harvested or MSTY policy when the combined harvesting effort of both prey and predator species are taken into account. In Section 4, we study the impacts of MSY (and MSTY) policy in a ratio dependent prey–predator system. Section 5 provides some concluding remarks of our investigated models. 2. Effect of MSY policy in a simple prey–predator system We first consider a single species population model followed by proportional harvesting as follows: x dx = rx 1 − dt k − ex, (1) where x is the population size at any time t, r is the intrinsic growth rate, k is the environmental carrying capacity of the population and e is the harvesting effort. A basic assumption of most of the models is to determine a catch level rate for which a stock can be sustained indefinitely based upon the average productivity of the stock. Therefore, largest average catch that can be employed on a sustainable basis from a stock under existing environmental conditions has become the main goal to manage exploited populations. The maximum sustainable yield (MSY) is based on equilibrium biomass of the species and yield. For a single species model as given in Eq. (1), MSY = rk/4 and it is occurred at effort level e = r/2. In fact, even for this single species fishery, fishing at the MSY level does not ensure constant catches in the future, because of the substantial variability in reproductive success and recruitment. In good years, fishers may prosper with MSY based catches, but in years when the environment is less favorable and recruitment and productivity decline, the stock will diminish and MSY may quickly lead overfishing. Most management measures are directed at individual stock of a single species and these do not take into account species interactions, such as predator-prey relationship. To study the consequences of MSY policy in multispecies system, Legovic et al. (2010a) considered the following prey–predator system: x dx = rx 1 − dt k − axy (2) dy = axy − my, dt where x and y are respectively the prey and predator biomass at any time t. r is a constant intrinsic growth rate (biotic potential) 135 and k is the environmental carrying capacity of prey population, a is the predation rate, m is the natural mortality rate of the predator. The prey–predator model (2) is simple in the sense that predator consumes prey population according to Holling type I functional response and the intraspecific competition of the predator is not taken into account. Introducing proportional harvesting to either prey or predator or both the populations with combined fishing effort we have the following results proposed by Legovic et al. (2010a) as: In any prey–predator system, fishing to reach MSY of the prey population only will cause extinction of the predator population. In any prey–predator system, fishing to reach MSY of the predator population only, is unlikely to causes extinction of either prey or predator species. In any prey–predator system subject to equal fishing effort on both prey and predator populations, the ultimate MSTY will be the MSTY of a single species population composed of prey only, which means that the predator population has gone to extinction. 3. Impacts of MSY policy under intraspecific competition Legovic et al. (2010a) extends our knowledge on effects of MSY policy in prey–predator models, however, they have considered predation as the only interaction in their prey–predator systems. But there are various other factors that may be taken into account when modeling of such a system. One such key and somewhat novel feature is the intraspecific competition in the predator growth dynamics (Kuang et al., 2003; Ruan et al., 2007). This intraspecific competition is assumed to induce additional instantaneous deaths to the predator population and the increased death rate is proportional to the square of the predator density. To completely eliminate density-independent mortality is as biologically unrealistic as eliminating density-dependent mortality (Caswell and Neubert, 1998). Recently, Hixon and Jones (2005) also found this density-dependent mortality in demersal marine fishes, which is often caused by the interplay of predation and competition. As this intraspecific competition enhances the mortality rate of the species, it is expected that the MSY policy will be significantly differ from the MSY policy in a traditional prey–predator system, and hence, in this section, we proceed to study the MSY policy in a prey–predator system having intraspecific competition in the predator growth dynamics when either prey or predator or both the prey and predator populations are subject to harvest. Therefore, our modified model becomes dx x = rx 1 − dt k − axy (3) dy = axy − my − y2 , dt where is the coefficient of intraspecific competition. Coexistence equilibrium of the system (3) is P(x*, y*), where x∗ = k and y∗ = am + r a2 k + r r(ak − m) a2 k + r , provided ak > m. Since the environment cannot support a population size above its carrying capacity, therefore x* < k would be ecologically meaningful prey abundance. We now examine whether MSY or MSTY policy can be implemented in our prey–predator system for a possible sustainable fishing activity. Various scenarios associated with MSY and MSTY are observed successively. 136 T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 3.1. Prey harvesting 5 4.5 Suppose the target species is the prey only. Then after introducing the proportional harvesting of prey species the system (3) becomes 3.5 − axy − ex (4) dy = axy − my − y2 , dt where e is the harvesting effort. For simplicity we consider the catchability coefficient as unity. The coexistence equilibrium for a fixed effort is P1 (x1∗ (e), y1∗ (e)), where x1∗ (e) =k and y1∗ (e) = am + r − e a2 k + r ak(r − e) − mr a2 k + r r − e r 2 1 0 y*1 0 e MY 0.5 1 e . >0 re(ak − m − e) . a2 k + r If the predator is driven to extinction, then the yield is obtained as Y (e) = yield curve 2.5 0.5 if the effort applied to the fishery is less than the biotic potential of prey species. The yield function at equilibrium is Y (e) = ex1∗ = MY 3 1.5 Here it is observed that both prey and predator populations at equilibrium depend linearly on employed effort. Also x1 *(e) is positive if e < r + am/ and y1 *(e) is positive if e < r − mr/ak. Hence the coexistence equilibrium exists if the effort e < r − mr/ak. Increasing of effort diminishes both prey and predator biomass, and ultimately predator biomass reduces to zero if e = r − mr/ak, even though the prey population persists. There exists another equilibrium R(x̂1 (e), 0), where x̂1 (e) = k biomass, yield 1 4 m (am + r). a Since Y is a quadratic function of the effort e, there is a possibility to have a single maximum. Therefore MSY may be available from prey exploitation. We explicitly calculate the effort, eMSY for which MSY is reached. The fishing policy that maximizes Y must be satisfied dY/de = 0 and d2 Y/de2 < 0 at e = eMSY ,and MSY is obtained at eMSY = (r + am)/2. Now if eMSY > r − mr/ak, then the predator species goes to extinction to reach the MSY level. It is difficult to explain analytically whether MSY policy can be applied or not. Here we give some examples to illustrate various possibilities. Example 1. Select the ecological parameters as r = 2, k = 5, a = 0.25, m = 0.9 and = 0.4 in appropriate units, then the predator goes to extinction at e = 0.55 as shown in Fig. 1. The yield curve is increasing for any effort lying in (0, 1), but if effort is greater than 0.55, then the coexisting equilibrium disappears and the predator free equilibrium becomes stable. Therefore, e = 0.55 is a threshold value and causes the change of stability behavior between the equilibrium points P1 and R. The corresponding maximum of the yield curve occurs when e = 1 and this effort is nothing but equal to r/2. The maximum sustainable yield is equal to 2.5 and theoretically it is rk/4. We may state that fishing of prey species to reach MSY of the 1.5 2 Fig. 1. Yield curve and equilibrium values of x1∗ and y1∗ as function of the effort e. MY occurs at e = 1. prey–predator system (4) is not a safe stock conservation process and MSY does not exists in such multispecies system. For better understanding of our analysis we introduce a term ‘maximum yield’ (MY) which means the maximum harvested biomass from ecosystem with the extinction of at least one species. By and large, the MY of the prey–predator system (4) is equal to MSY of the single species model (1). Example 2. Let us now fix the parameters as r = 1, k = 5, a = 0.5, m = 0.5 and = 0.4, then both prey and predator population coexist and equilibrium densities of both the species decrease linearly for e ∈ (0, 0.8). The yield curve is increasing and possess its maximum as MY = 0.8 at e = 0.8 (see Fig. 2). If the effort is slightly increased then the coexistence equilibrium disappears and only predator free stable equilibrium exists. Within the effort interval (0.8, 1), the prey population decreases with increasing rate unlike observed in the Example 1. The yield curve also decreases with respect to effort. Therefore MSY is not accessible form prey–predator system. Since, in a single species model, MSY occurs at e = r/2, hence fishing of prey to reach MSY level of the prey–predator system (4) is not equivalent to reach MSY associated with the system (1). If it is, then MSY would 2 1.8 * 1 x 1.6 1.4 biomass, yield dx x = rx 1 − dt k x* 1.2 MY 1 0.8 0.6 yield curve 0.4 e MY 0.2 y 0 0 0.2 0.4 * 1 0.6 0.8 1 e Fig. 2. Yield curve and variation of equilibrium values of prey and predator populations. Maximum yield occurs when e = 0.8 causing the extinction of predator species. T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 137 4.5 x* 4 1 * x2 3.5 yield curve biomass, yield biomass, yield 3 2.5 MSY 2 1.5 1 MSY 0 0.5 1 e 1.5 e Example 3. Now we consider r = 2, k = 100, a = 0.5, m = 0.5 and = 0.4, then the condition (r + am)/2 < r − mr/ak is satisfied. YMSY = 2.67 occurs for eMSY = 1.33 with stable coexistence equilibrium at (2, 1.28) (see Fig. 3). The term ‘sustainable fishing’ guarantees the persistence of all species at the expanse of effort to achieve MSY. From the above Examples 1–3, we notice that fishery agency should take advice before applying the MSY policy while harvesting prey population only. It is also observed that the success of fishing to reach MSY level depends on the mortality rate of predator fish, biotic potential of prey, prey carrying capacity and other ecological parameters. 3.2. Predator harvesting − axy (5) dy = axy − my − y2 − ey. dt The coexisting equilibrium is =k and = am + r + ae a2 k + r r(ak − m − e) a2 k + r 0 Fig. 4. Yield curve and equilibrium values of x2∗ and y2∗ as function of effort. MSY can be obtained. This interior equilibrium exists when e ∈ (0, (ak − m)). From x2∗ (e) and y2∗ (e) it is observed that prey population at equilibrium increases with effort while predator population at equilibrium decreases and goes to extinction when e = (ak − m). However, the prey biomass does not crosses its carrying capacity as in principle x2∗ (e) → k when predator goes to extinction. The yield at equilibrium is given by Y= re(ak − m − e) . a2 k + r Now dY/de = 0 and d2 Y/de2 = − r/(a2 k + r) < 0 at e = (ak − m)/2) < (ak − m). Hence the yield curve has a single maximum in presence of both the species as shown in Fig. 4 and we have eMSY = (ak − m)/2. The following theorem exhibits the impact of fishing to reach the MSY from predator. Theorem 1. In system (5), fishing to reach MSY from the predator population is a sustainable fishing policy. The effects of applying the combined harvesting effort are investigated in the following section. 3.3. Both prey and predator harvesting If both the species are subjected to proportional harvesting with equal effort, then the system (3) becomes Let us consider the target species is the predator species only. Then introducing a proportional harvesting, system (3) becomes MSY e Now we shall attempt to analyze the effect of fishing on prey population to reach MSY when (r + am)/2 < r − mr/ak holds. In this case, yield curve attends its maximum value at eMSY = (r + am)/2 and this effort is less than the effort which causes the extinction of the predator population. Therefore, MSY policy can be employed to manage both the prey and predator populations. A numerical example is provided to support the result using a different set of parameters. x dx = rx 1 − dt k 0 2 occur at e = 0.5. In fact, MY from both species of the system (4) is smaller than the MSY from the prey species of the system (1). y2∗ (e) * 2 y Fig. 3. Equilibrium values of prey and predator biomass exist when fishing reaches to maximum sustainable yield. x2∗ (e) yield curve e 0.5 0 MSY y* 1 P2 (x2∗ (e), y2∗ (e)), where dx x = rx 1 − dt k . − axy − ex (6) dy = axy − my − y2 − ey, dt The coexisting equilibrium of the system (6) is denoted by P3 (x3∗ (e), y3∗ (e)), where x3∗ (e) = k and y3∗ (e) = (am + r) − ( − a)e a2 k + r r(ak − m) − (ak + r)e a2 k + r . 138 T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 3 After some manipulations, the criteria for the coexistence of both species is obtained as e<r ak − m ak + r 2.5 . < r. This condition produces that predator is ruined from ecosystem before the effort reaches the biotic potential of the prey population. In this context, we address the interesting phenomenon of Volterra’s first principle for fishing on both the species. It was found that the proportion of predator fish increases and the proportion of prey fish decreases during 1st world war, while the fishing is reduced. The principle can be stated (Pireddu and Zanolin, 2008; Legovic, 2008) as ‘If one tries to destroy uniformly and proportionally to their numbers the individuals of the two species, then the average of the number of individuals of the eaten species increases and the one of the eating species decreases’. One can easily verify this principle in Lotka–Volterra predator–prey model. The total yield at equilibrium becomes Y (e) = e k (am + r) − ( − a)e a2 k + r + r(ak − m) − (ak + r)e a2 k + r when eMSTY = k + r a2 k +r 2(k + r) y* 1 3 eMY 0.5 0 0 0.5 1 e 1.5 2 Fig. 5. Yield curve and variation of prey and predator population at equilibrium. Predator goes to extinction before the existence of MSTY. 4. Effect of MSY policy in a ratio-dependent prey–predator system In the previous sections we have analyzed the impacts of fishing to reach MSY (or MSTY) level in a prey–predator system having linear interaction. In this section, we consider a ratio-dependent prey–predator system as follows: dx x = rx 1 − k dt <0 k(am + r) + r(ak − m) 1.5 . Our goal is to maximize the total yield as well as the conservation of both the species. We observe that dY/de = 0 and d2 Y =− de2 biomass, yield ak + r 2 − axy cy + x (7) dy baxy − my, = cy + x dt . Analytically it is very difficult to test whether eMSTY > eend or eMSTY < eend . If eMSTY > eend , prey species survives but predator species goes to extinction; and maximum sustainable total yield (MSTY) cannot be obtained. Otherwise, MSTY is achieved through the exploitation of both prey and predator populations. These two different scenarios are supported by the following examples. Example 4. Let us take r = 2, k = 5, a = 0.5, m = 0.5 and = 0.4. These parameters satisfy eMSTY > eend . Now from Fig. 5, we observe that prey biomass increases and predator biomass decreases when effort increases from zero and ultimately predator goes to extinction when effort is 0.88. Therefore, fishing to reach MSTY eliminates predator population with prey density as x3∗ = 2.777. If effort is increased further, then fishing is established only from the single species population model (1). In this case, MY is obtained at e = 1 which is the MSY obtained from single species only. Therefore application of MSTY rule in such prey–predator system may not be beneficial tool from ecological aspect. It is to be noted that Volterra’s principle has been established here. Example 5. Reset the parameters as r = 2, k = 100, a = 0.5, m = 0.5 and = 0.9. Then we have eMSTY < eend . In this case, attempt to reach MSTY is a good management tool to conserve both prey and predator populations, through the biomass of both the prey and predator population have a decreasing nature as effort increases (see Fig. 6). This observation is not found in Example 4. The equilibrium density is P3 (5.187, 0.493) when fishing reaches at MSTY and eMSTY = 1.65. It is also to be noted that eend < r. This example contradicts the celebrated Volterra’s principle. where x and y are respectively the prey and predator biomass at any time t. r is the constant intrinsic growth rate (biotic potential) and k is the environmental carrying capacity of prey population, a is the predation rate. b(< 1) refers as conversion rate of prey biomass into predator biomass, c is known as half saturation constant and m is the natural mortality rate of the predator. 10 9 MSTY 8 7 biomass, yield ak − m yield curve MY Suppose eend = r * x3 x* 3 6 yield curve 5 * y3 4 3 2 eMSTY 1 0 0 0.2 0.4 0.6 0.8 1 e 1.2 1.4 1.6 1.8 Fig. 6. Equilibrium values of both the prey and predator population decrease with respect to combined harvesting effort. Maximum sustainable total yield can be obtained before predator goes to extinction. T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 k y2 biomass, yield biomass, yield x1 139 y1 MSY x2 yield curve MSY e MSY yield curve 0 eMSY 0 0 e 0 e Fig. 7. Prey and predator biomass at equilibrium and yield curve for varying effort. Fig. 8. Represents the yield-effort curve and the variation of equilibrium values of both prey and predator populations. 4.1. Prey harvesting If prey is harvested through the rule h = ex, then the system (7) takes the following form: x dx = rx 1 − dt k − axy − ex cy + x (8) dy baxy = − my, cy + x dt 4.2. Predator harvesting If the target species is predator then we have The coexisting equilibrium of the system (8) is F1 (x1 (e), y1 (e)), with k (bc(r − e) − (ab − m)) bcr x1 (e) = Theorem 2. In ratio dependent prey–predator system (8), fishing to reach MSY of the prey population only guarantees the coexistence of both the species. x dx = rx 1 − dt k − axy cy + x and We have the coexisting equilibrium F2 (x2 (e), y2 (e)), where k(ab − m) y1 (e) = (bc(r − e) − (ab − m)) . bc 2 rm x2 (e) = It is observed that both as x1 (e) and y1 (e) decrease with effort and ultimately tend to zero as Hence prey and predator population exist under prey harvesting if (ab − m) . bc e<r+ The exploited yield at equilibrium is Y (e) = ke (bc(r − e) − (m − ab)) . bcr Obviously dY/de = 0 and e= 1 2 r+ ab − m bc d2 Y/de2 < r+ ab − m bc . This demonstrates that MSY exists and we have eMSY = 1 2 r+ ab − m bc . Therefore attempt to reach MSY level is always possible and sufficient to conserve prey–predator system (see Fig. 7). This phenomenon is not experienced for the system (4) in any circumstances. Hence we can state this finding through the following theorem. k(ab − m − e) (bcr − (ab − m) + e) . bc 2 r(m + e) In the absence of harvesting, we have (bcr − (ab − m)) > 0 from the existence condition of equilibrium. Therefore, F2 (x2 (e), y2 (e)) exists if e < (ab − m). Predator biomass ate equilibrium is a nonlinear function of effort and it is not straight forward to comment on its variation with respect to effort. The yield function Y (e) = = − k/r < 0 at k (bcr − (ab − m) + e) bcr and y2 (e) = ab − m . bc e→r+ (9) baxy dy = − my − ey, cy + x dt ek(ab − m − e) (bcr − (ab − m) + e) bc 2 r(m + e) is again highly nonlinear function of effort. Obviously Y(0) = 0 and Y(e) has only positive zero at e = ê = (ab − m). Also Y(0 +) >0. Hence Y(e) has a maximum at some value of e = ē ∈ (0, ê). Therefore, both prey and predator population can coexist while fishing increases at MSY level. Fig. 8 exhibits a possible graphical representation of the equilibrium biomass of prey and predator and the yield curve. The prey biomass increases linearly and predator biomass decreases but not linearly. Hence we can state the following theorem. Theorem 3. In ratio dependent prey–predator system (9), fishing to reach MSY of the predator population would guarantee sustainability of both the species. 140 T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 4.3. Prey–predator harvesting 200 180 If both prey and predator populations are harvested simultaneously with combined harvesting effort e, then we have the following system (10) dy baxy = − my − ey. cy + x dt The coexisting equilibrium of the system (10) is F3 (x3 (e), y3 (e)), where x3 (e) = k (bcr − (ab − m) − (bc − 1)e) bcr bc − 1 . Predator population at equilibrium exists if e < (ab − m). Hence both the species coexist if e < min bcr − (ab − m) bc − 1 min bcr − (ab − m) bc − 1 , (ab − m) . It can be conclude that , (ab − m) = (ab − m) (A) if r > (ab − m) and min bcr − (ab − m) bc − 1 , (ab − m) = bcr − (ab − m) bc − 1 (B) if r < (ab − m). Hence we have the following theorem for the coexistence of both the species. Theorem 4. Both the prey and predator populations coexist if either of (H1) or (H2) or (H3) holds, where (H1): bc < 1 and e < (ab − m), (H2): bc > 1, r > (ab − m) and e < (ab − m) and (H3): bc > 1, r < (ab − m) and e < bcr−(ab−m) . bc−1 The yield at equilibrium is given by Y (e) = ek [ab − m + cm − (1 − c)e][bcr − (ab − m) bc 2 r(m + e) + (1 − bc)e]. 80 60 0 k(ab − m − e) (bcr − (ab − m) − (bc − 1)e) . bc 2 r(m + e) bcr − (ab − m) x3 100 yield curve eMY 20 Two different cases may arise as (i) bc < 1 and (ii) bc > 1. Case I. bc < 1. If bc < 1, then the prey biomass at equilibrium increases with respect to the effort. Again predator population at equilibrium exists if e < (ab − m). Therefore, as (1 − bc) is a proper fraction, both prey and predator populations at equilibrium coexist if e < (ab − m). Case II. bc > 1. In this case, prey biomass at equilibrium decreases with effort but remains positive if e< 3 120 40 and y3 (e) = y 140 axy − − ex cy + x biomass, yield x dx = rx 1 − dt k 160 0 0.5 1 e 1.5 2 Fig. 9. Increasing effort causes the extinction of the predator population. MSTY is no longer achieved. The impacts of fishing to reach MSTY with equal harvesting effort on both the populations can be investigated through the following examples. Example 6. If we select the parameters as r = 2, k = 100, a = 0.9, b = 0.8, m = 0.1 and c = 1, then bc < 1 holds and both the species coexist for e < 0.62. From Fig. 9 we observe that prey biomass increases and predator biomass decreases as fishing effort increases; and predator biomass becomes zero at e = 0.62. Hence fishing is not reached to MSTY level in presence of positive abundance of both the populations. If we increase the fishing effort further, then MY is achieved when effort is half of the biotic potential of the prey, but predator goes to extinction. Therefore attempt of fishing to reach MSTY level gives the extinction of predator population. MY exists at e = 1 and is equal to MSY of the system (1). Example 7. Let r = 1 and other parameters remain same as in Example 6. Then bc < 1 holds and both prey and predator population coexist if e < 0.62. We observe that prey biomass increases and predator biomass decreases as fishing effort increases; and predator is eliminated from the system if e = 0.62. The yield curve increases for 0 < e < 0.62, but if effort is increased further, then the character of the yield curve is changed dramatically compare to the above example and actually it decreases as shown in Fig. 10. MY exists at e = 0.62 and hence MY of the system (10) is not equal to MSY of the system (1). From Examples 6 and 7, we observe that biotic potential has a vital role to introduce MSTY policy in a ratio-dependent prey–predator system while dealing with equal harvesting effort to both the species. Fishing to reach MSTY is equal to MSY of prey harvesting if biotic potential is sufficiently large (see Fig. 9), but neither MSTY nor MSY exists if biotic potential is significantly small (see Fig. 10). Examples 6 and 7 are arguing the Volterra’s Principle. Example 8. Let us consider the biological parameters as r = 1, k = 100, a = 0.9, b = 0.8, m = 0.1 and c = 2, then all the conditions of (H2) in Theorem 4 are satisfied. Again both MSTY for the combined harvesting of both the species and MSY from prey harvesting do not exist as already found in Example 7, but the fundamental difference is that both prey and predator populations diminish (see Fig. 11) as effort increases in 0 < e < 0.62. Example 9. Take the parameters as r = 0.6, k = 100, a = 0.9, b = 0.8, m = 0.1 and c = 1.5, then all conditions of (H3) of Theorem 4 are satisfied. In this case both prey and predator populations T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 simultaneously are driven to extinction when effort e = 0.5. Also biomass of both the populations decreases as effort increases. The yield curve is concave downward with respect to effort and possesses a maximum at e = 0.12 protecting both the populations from extinction (see Fig. 12). Hence MSTY can be pursued in such situation and Volterra’s Principle no longer works for Examples 8 and 9. 140 120 100 biomass, yield 141 80 x3 y3 60 5. Conclusions MY 40 20 yield curve 0 0 0.2 0.4 0.6 0.8 1 e Fig. 10. Maximum yield occurs when predator is removed from ecosystem resulting the nonexistence of MSTY. 150 y 100 biomass, yield 3 x3 MY 50 yield curve 0 0 0.2 0.4 0.6 0.8 1 e Fig. 11. The yield curve and the variation of equilibrium values of both prey and predator populations. 15 y3 biomass, yield 10 x 3 MSTY 5 yield curve 0 0 0.1 0.2 0.3 0.4 0.5 e Fig. 12. Represents the yield curve and equilibrium values of prey and predator stock. Yield curve reaches at MSTY level. The objective of this study is to analyze the impacts of fishing with the aims of both yield maximization and species conservation. It is well described by Legovic et al. (2010a) that maximum yield may drive to species extinction from a traditional prey–predator system. In the models considered there are more cases when the predator will not go to extinction at MSY or MSTY. We also compared our outcomes with the results due to Legovic et al. (2010a) and other references there in. Some potential consequences of MSY or MSTY policy has been drown for both the models. Introducing density-dependent mortality term into traditional prey–predator system makes the system behavior differently in view of species coexistence while yield is maximized. Harvesting of prey species according to catch-pre unit-effort (CPUE) hypothesis may collapse the predator species from the system to achieve the maximum yield from target species. This result is very much consistent with the previous study of Legovic et al. (2010a). But the yield from the prey population in absence of the predator species may (Example 1) or may not (Example 2) be equal to MSY as can be obtained from the single species population model. In some occasions, harvesting of prey species at MSY level ensures the species coexistence, which is not observed in traditional prey–predator system (2). The equilibrium prey biomass at MSY effort level has decreased to exactly half of its biomass before the exploitation. Attempt to reach MSTY from both the prey and predator populations with combined harvesting effort ensures the extinction of predator species when density-dependent mortality term is absent in the predator (Legovic et al., 2010a). But we observe that densitydependent mortality term does matter when we activate MSTY policy to harvest both the species. May be MSTY will not drive predator to extinction in a prey–predator system in nature (see Examples 4 and 5). This result differs from the results as described by Legovic et al. (2010a). We also consider a ratio-dependent prey–predator model to provide some vital impacts of harvesting to maximize biomass yield. It is observed that both prey and predator populations may be driven to extinction together when the prey harvesting effort crosses some threshold value (see Fig. 7). This is quite unlike to the case of prey–predator systems like (2) and (3). Both smaller and higher harvesting effort produces lower yield from the prey species and in between these efforts we can set up the effort which maximize the harvested biomass. Therefore, harvesting of prey species to reach MSY level must prevent extinction of the predator species and it removes the uncertainty of the species coexistence for prey harvesting as stated in our prey–predator system (3). Two folded impacts (extinction and persistence of predator) of harvesting both the species from the system to obtain MSTY are also discussed. There is a possibility to conserve predator species from extinction to reach MSTY level. It is also observed that the existence of MSTY indicates that both the species die out with a common threshold effort which is unlike the case in traditional prey–predator system (see Fig. 12). Harvesting of both the species, with increasing effort, reduces the predator biomass and increases the prey biomass at equilibrium, which is termed as Volterra’s principle, can be found in Lotka-Volterra model (Legovic, 2008). Our systems may agree with 142 T.K. Kar, B. Ghosh / Ecological Modelling 250 (2013) 134–142 the Volterra’s principle when predator goes to extinction as yield approaches to MSTY (see Figs. 5, 9 and 10), but it may contradict the Volterra’s principle when MSTY exists in both the models (see Figs. 6 and 12). This study also gives us three potential open questions in our future research direction which we would like to include in our next work. (i) Whether the above prey–predator system can be applied to a food chain or food web model consisting of more than two species? (ii) Are Legovic et al.’s (2010a) results for one-prey one-predator system are well established tool to implement into the food chain or food web model as they already pointed out? (iii) Is harvesting of both the prey and predator species under combined harvesting effort always sufficient to prevent extinction of the prey species? Acknowledgements The research work of T. K. Kar is supported by the University Grants Commission (UGC), India (F. No. 40-239/2011(SR), dated 29th June, 2011) and the research work of Bapan Ghosh is financed by the Council of Scientific and Industrial Research (CSIR), India (File No. 08/003(0077)/2011-EMR-I, dated 23rd March, 2011). The authors are really grateful to the anonymous referees for their careful reading, constructive comments, and helpful suggestions. References CBD, 1992. Convention on Biological Diversity. UNEP, Nairobi. http://www.cbd.int/ (accessed 11.11.10). Caswell, H., Neubert, M.G., 1998. Chaos and closure terms in plankton food chain models. Journal of Plankton Research 20 (9), 1837–1845. Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd ed. 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