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Case2000Harvesting - 232 Chapter 10 not always easy to...

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Unformatted text preview: 232 Chapter 10 not always easy to find the optimum level of sheep on the pasture. If the optimum num- ber of sheep, T2, in Figure 10.15(b) is exceeded by even a single sheep, then wool pro- duction falls drastically, to a much lower stable level. If that single sheep is then removed, the system does not return to its previous level, but instead rides the thin red curve in Fig- ure 10.15(c). Only modest increases in wool yield are produced as more sheep are removed from the pasture until the number of sheep drops to T1. When the number of sheep is below T1, the wool production takes a giant leap upward. However, wool pro- duction still does not reach the level of production that was achieved at exactly T2 sheep. HARVESTING FISH An Example of the Peruvian Anchovy Peruvian anchovy live in the cool, upwelling, nutrient-rich waters along the coast of Peru and northern Chile. Reproductive maturity is reached at about 1 year of age and the typical anchovy lives about 3 years. Anchovy occur in large schools and are caught near the surface with nets. This was the largest fishery in the world until it collapsed in 1972 in association with an El Nifio that disrupted these cold currents and diminished productivity. Fishing was suspended to allow the stocks to recover, but there was no immediate return of the anchovy. Populations of seabirds that feed on the anchovy also remained low. Since anchovy have a rapid generation time, we can assume that there has been adequate time for the population to return to its former level since the fishing suspension. But, as shown in Figure 10.16, it hasn’t. Why hasn’t it? In the sheep—grazing example, we had complete control over the number of sheep that were grazed on the pasture. When we harvest trees from a forest or fish or whales from the sea, natural forces beyond the control of the forester or the fisher or the Whaler influence the size of the resource available for harvesting. However, the theory that we have already developed will help us see possible outcomes of different resource harvest- ing strategies. Often the goal is to achieve the maximum sustainable yield (MSY). The word sustainable is an important qualifier since the maximum yield in any single year is to simply harvest all the resource that exists at that time. However, this is not sustainable since next year there may be no resource. A bit later, we show how to estimate the MSY, but for now we accept that the estimated MSY for Peruvian anchovy was about 9 to 10 million tons/year, based on average conditions for several years before the El Nifio episode in 1972. Note that the harvest exceeded MSY in 1970 and in 1966—67. Two common ways of regulating natural resources like fisheries are explained in Box 10.1. H U.) H ._: 3 Anchovy catch (million of tons) \l 1 1955 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 Year Figure 10.16 . Total catch for the Peruvian anchovy fishery. After the collapse in 1972, fishing was greatly reduced but the population had only ‘1 slightly recovered by 1989. After Krebs 1994. Exploited Resources Box 10.1 Methods of Regulating a Fishery or Other Dynamic Natural Resources As with the sheep, we begin by setting up a growth and loss expression for fish: Net growth of a fish population in : a fishing season WU“) - L K logistic total losses recruitment minus from (a) curve of the fish fishing (per season) (per season). Now we focus on the loss part of this equation, which is the part that governments can regulate. We may distinguish two methods for regulating the losses due to fishing. 1. Fixed quota. Total losses to fishing, L, are regulated by setting a limit, Q, on the number of fish harvested per season, regardless of the number of fish in the popula- tion, as illustrated in the following diagram. Logistic growth of fish / Regulators set the total fishing loss per season to Q, regardless of population size, N. Here are three possible levels of Q. 2. Fixed effort quotas. We can expand the loss part of Eq. (a) as follows: L 2 (fishing effort)(C)(N). (b) Here the losses, L, are a product of three factors: the amount of fishing effort (which increases with the amount of time spent fishing, the number of fishing boats, the area the boats cover per day, etc.), a parameter of “fish catchability,” C, and the size of the fish population, N. Fish catchability measures the efficiency of each unit of fishing effort. Fish catchability therefore has units of fish caught per time period per effort expended per fish in the population. We assume that C is a constant. Then, Eq. (b) says that all else being equal, the more fish in the population, the more effort expended trying to catch those fish, and the easier the fish are to catch, then the more fish will be caught in a given time period. Equation (b) also says that the harvested losses repre- sent a type 1 functional response by the consumers—the people doing the fishing. The numbers of fishers and the “foraging” effort of each fisher are combined into a single term: the “fishing effort.” Thus the analog of Figure 10.14 in this case, where the number of fish caught increases linearly with the fish population size (as by Eq. b), looks something like the following. The slopes of these lines = (C )(effort), and effort is regulated. Here are three different effort levels. Regulation is effected by limiting the “effort” term in Eq. (b). This may be done by restricting the number of fishing permits, the length of the fishing season, or restricting regions where fishing is allowed. Variations in fishing effort from year to year will lead to different catches, which will trace out the recruitment curve (assuming that it remains constant from year to year). For example, the following dia- gram shows five different efforts in five different years applied to a fish population and the resulting catches for each. Catch 2 dN Catch 3 7t— Catch 1 Catch 4 The following diagram shows those same fish catches plot— ted against the effort values instead of against fish popula- tion size, N. 233 234 Chapter 10 Predicted maximum }/ catch = MSY Catch 2 Fish caught Catch 3 season Catch 1 , Preéictedbptimum, ‘ f . effort: abovethis tiaeflah : catchdeclines- ‘_ _ « This plot also yields a hump—shaped curve, like the logistic curve. This is a key result since this figure shows that, even without knowing the absolute fish population size, N, the optimum effort and the MSY can still be estimated simply by the relative position of the yield—effort curve. The opti- mum effort and the MSY are shown with arrows in the pre- ceding figure. If, additionally, the catchability constant, C, is known, then fish population size, N, and the corresponding r and K may also be estimated from the yield—effort curve (we work through some examples later in this chapter). The catch versus Peruvian anchovy data and a fitted curve are shown in Figure 10.17. Note that during this period the anchovy did not appear to be severely overex- ploited. Obviously several complications may affect all this. Fishing effort is normally modified from year to year—but with climatic changes, the recruitment curve, and the r and K for the fish population are also changing yearly. Addition- ally, the fish population may not grow according to a logis— tic equation. A full model might require knowledge of the age structure of the fish population. Density-dependence Total catch (million tons/year) t—‘NW-kaOVOO 51015 20 25 30 35 4O 45 50 55 60 Total fishing efl'ort (million trips) Figure 10.17 The relationship between total fishing effort and total catch for the Peruvian anchovy fishery. The effects of humans and sea birds are combined in these data. The parabola is based on a continuous logistic equation fitted to these data. Arrows indicate the estimate for MSY and appropriate fishing effort. After Krebs (1994) from Boerema and Gulland (1973). might be different for different ages, and the functional form of the density-dependence could contain several higher- order terms. Recall the more complicated single-species models that we explored in Chapter 5. Finally, the catcha- bility, C, may not be a constant but instead a function of effort and/or fish population size, N. In this case, we would not have type 1 functional response curves but perhaps type 2 or type 3 curves. As we have already shown, the latter can yield break points and alternative stable states. Finally, the fish population is also being exploited by several natural predators, and we usually have little knowledge of their abundance and functional responses. The task for government resource regulators is to prevent a tragedy of the com- mons by keeping the fishing industry from “killing the goose that laid the golden egg.” It is in the best interest of each individual to prevent the overexploitation of the com- mon fishery that they collectively depend on, yet at any particular time it is in the short- term economic best interest of each fisher to maximize his or her catch. Thus we have the classic problem of what is best in the short term for the individual, if followed by all individuals, leads to the collapse of the fishery and thus a bad situation for all who depend on it for their livelihood. As we have indicated, one possible way to prevent the collapse of the fishery is for governments to impose a fixed-quota harvest (Box 10.1). This imposes a limit on short- term profits so as to maximize longer term profits and the continued viability of the nat- ural resource. For example, regulatory agencies could put a limit on the number of fish that can be caught per season. The fishing season would close after the limit is reached. Exploited Resources 235 Figure 10.18 Growth of a logistically growing resource under a fixed- harvesting quota. In the central red region, thelogistie , ‘growthfof thelresource exceeds the ,imposed’harvesung mortality, Q, so the , net growth of the population is positive. d_N . dt Fixed harvesting quota, Q 0 0 x \ ,1 / Population size, N \ \ , \ / However, in these regions of N, the net growth rate is negative. (3) The em sto create; three equilibrium 3 points, twostable and one unstable. d_N dt Fixed harvesting quota, Q ’ Unstable // Stable equilibrium / equilibrium / 0 x X P 1 t' ' N Stable / 0 N*2 Nakl opu a non Size, equilibrium }_—___> at N = 0 I I For any population in this range of N, and with the fixed quota, Q, depicted, However, if population size falls the population size, N, will eventually 1nto thls low range of N, then the stabilize at N*1. population size will collapse further to 0 unless harvesting is reduced. (b) ble, and therefore the yield at N 1* is sustainable. However, the one to the left is unsta- ble, and therefore the yield associated with N2* would be unsustainable at the same quota level. If the quota were increased, the horizontal line would march up the y axis, giving higher allowed yields. The MSY occurs when the quota is raised to the point that it just equals the peak of the logistic hump, as depicted in Figure 10.19. For logistic recruitment, the population size will be at K/2 when the MSY is reached. Therefore the MSY will equal MSY=d—N at £=rK/2(K—£) dt 2 K 2 (101) -15 4 . It shouldn’t take long to see a major problem with regulation for a fixed—quota level Qmax. If Qmax is exceeded in a single year, perhaps because it was misidentified by gov- ernment regulators or because environmental conditions change so that the recruitment parabola becomes lower, then resource levels will be depressed the following year to a level below K/2. Now, assuming the same quota level, the fish population will 236 Chapter 10 Figure 10.19 A much higher fixed-quota harvest. This harvest will produce the MSY. Figure 10.20 Harvesting at the maximum sustainable level for four different initial population sizes: 90, 99, 120, and 200. For this example, r = 1, K = 200, and Qmax is 50. Since K / 2 = 100, any population slipping below N = 100 eventually crashes unless Q is reduced. Semistable equilibrium point Fixed / harvesting quota, Q, is (ii—1;] now at Qmax. 0 0 N*1= K/2 = MSY Population size, N 1‘» I——I \ . _ . f For any population in th1s range . _ . . of N, then with the quota, Qmax, If the populatlon Size falls Into this depicted, the population size, N, low range of N, then with this quota. will eventually stabilize at N>|<lv Qmax, the population size, N, will collapse further to 0. 200 N0 = 120 and 200 0 50 100 150 200 Time approach the stable equilibrium of zero, sliding the fish population into extinction. Therefore harvesting will need to be suspended or greatly reduced until the resource level can return to a level greater than R* so that the quota Qma1x will again produce a stable equilibrium. An example of a logistic population being harvested at the MSY is shown in Figure 10.20 for K = 200. The initial fish population size is shown at four alternative levels, and for each the population is followed over time. In this example, the two populations that dipped below N = 100 (= K/ 2) fell to extinction. The MSY can be approached only from population sizes above K/Z. If reg- ulators take quick action and reduce the harvest rate as soon as they notice that the fish population is being overexploited, it will rebound, as Figure 10.21 illustrates. Why didn’t the Peruvian anchovy population rebound like this after fishing was suspended? The anchovy had not gone extinct but were at very low levels. One of our assumptions may be wrong, or there may be more going on then is accounted for by this simple logistic model of fish growth. We will see how this question might be answered a little later in the chapter. In real systems we typically have little knowledge of the actual level of the resource population, let alone the exact shape of the recruitment curve, and yearly variations in climate influence the shape of this curve anyway. Thus it is easy to overestimate Qmax, setting off a population decline and thus producing less than optimal long-term yields (May et al. 1979). A fixed—effort harvesting scheme avoids some of the stability pitfalls of the fixed—quota method, as demonstrated in Boxes 10.2 and 10.3). A fixed—effort Figure 10.21 200 This is the same example as in Figure 10.20 for an initial population size of 90. The fish population is on its way to a crash. Now, however, at time 15, the harvest rate is reduced from 50 to 15. With this adjustment the fish N 100 population recovers. Box 10.2 In Figure 10.22, we develop a curve similar to that in Figure 10.15 for a type 1 functional response instead of a type 3 response. This situation corresponds to the fixed—effort regula- tion of a fishery discussed in Box 10.1. We label the resource population size R and the consumer population as predators R For example, governments could regulate the number of fish— ermen, P, as a means of controlling fishing effort. Solution: Now we plot the relationship between the levels of R at equilibria (0n the x axis) and the number of predators, P, in Exploited Resources 237 Fish Population Harvest Rate 50 100 1 50 200 Time Figure 10.23. The dots on the x axis in Figure 10.22 are car- ried over here for each level of predator, and projected (by dashed lines) to the appropriate number of predators. Then a line has been drawn to connect these dots. Figure 10.23 is the analog of Figure 10.12 for a type 1 functional response. The final step is to produce the analog of Figure 10.15. The open squares on the y axis in Figure 10.22 show the equilibrium level of dR/dt associated with the indicated predator numbers on each total consumption curve. These values are plotted on the y axis in Figure 10.24 against the corresponding predator numbers on the x axis. Logistic prey recruitment The straight \ lines are the total consumption rates for different numbers of predators, P. Equilibrium levels of prey for these different numbers of predators Resources, R Figure 10.22 238 Chapter 10 , . . V . . (Hf Line of cqurhbna E02“ prey: ~ [A :1 (3 {if Predators, P DJ 0 H— CU! LII Prey population, R Figure 10.23 Box 10.3 Derive an algebraic expression for the graphical relationship between dR/dt and predator numbers that was developed in Box 10.2. Solution: Given a type 1 functional response of predators, with a pop— ulation size 'P and a resource population R that grows logis- tically in the absence of harvesting, we may write £2 rR(l—£)—aPR dt K gar—’EH The first term The second term is the (C) is the logistic consumption rate of R by P equation for R. predators, each with encounter rate a: a type 1 functional response. Predator productivity (proportional to dR/dt when R reaches an equilibrium) 5 1015 30 60 Predators, P Figure 10.24 We let R* be the equilibrium level of R and solve for R* by setting Eq. (c) to 0. rR*£1—R )=aPR*. K Rearranging, we get R* =K(l—£j. I” At R*, consumer productivity 2 aPR* (from Eq. (c)), and thus the turnover of resources at this equilibrium is dR * dt =aPK(1—£). (d) r Equation ((1) describes the parabola plotted in Figure 10.24. method is based on the assumption that the harvesters have a type 1 functional response. Unlike the situation for a type 3 functional response, a type 1 functional response pro- duces no unstable equilibrium points and thus no hysteresis. However, overexploitation of the resources (i.e., diminishing returns with increasing effort) is still possible, as we have already shown in Box 10.1. More about Fixed-Effort Harvesting Because of the desirable stability features of the type 1 functional response (Box 10.3), the regulation of total resource harvesting can be accomplished most successfully by a Figure 10.25 Fixed-effort harvesting of a resource, R. The medium harvesting effort produces the MSY and an equilibrium resource level of K / 2. Figure 10.26 The only equilibrium R for the very high effort is zero resource. Exploited Resources 239 Medium effort Recruitment High effort curve dR — MSY dt Low effort 0 R Very high effort Medium effort dR — MSY dt fixed-effort regulation of harvesting. Recall from Box 10.1 that the total yield (or catch) of resource to all harvesters is the product of three different terms: capture effort expended Total yield 2 [by all harvesters [catchability] [resource level]. (10.2) Regulating the first term sets the total yield, and the resource level adjusts accordingly. An example of the overall harvest in terms of dR/dt based on three different levels of capture effort per resource (assuming that catchability remains constant) is shown in Figure 10.25. Here, regulatory agencies do not fix the total quota per season, but instead try to regulate the amount of capture effort expended. To do this successfully, they must be able to obtain a census of the number of fish, R, and adjust effort accordingly so that the MSY can be reached. In Figure 10.25, the medium effort corresponds to the MSY. Note, however, from the arguments developed in Box 10.1, that the equilibria fish numbers corresponding to all three efforts are stable. Moreover, unlike the situation with fixed-quota harvest— ing, if the medium effort is being maintained through regulation, but the resource level temporarily falls below R*med, the recruitment rate still exceeds the harvest rate and resource levels will increase back to R*med when conditions return to normal. Resources will not become extinct unless very high capture efforts are maintained, as illustrated in Figure 10.26. With fixed-effort harvesting, the total yield varies with the resource population size according to Eq. (10.2) If the population size were to drop below R*, then the yield would drop below MSY. The appropriate response by regulators would be to reduce the allowable effort to let resource levels increase back to R* or beyond. However, co" .. tition among the harvesters would act in the opposite direction. In the face of ' ishing catch, harvesters will apply political pressure to increase allowable efl’ofl 240 Chapter 10 , yields can go up again, restoring their profit margins. At the same time, with a dimin- ' shed supply of resources...
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