The maximum sustainable yield of Allee dynamic system

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1 Ecological Moelling 154 (2002) The maximum sustainable yiel of Allee ynamic system Zhen-Shan Lin a, *, Bai-Lian Li b a Department of Geography, Nanjing Normal Uni ersity, Nanjing , People s Republic of China b Department of Biology, The Uni ersity of New Mexico, Albuquerque, NM , USA Receive 11 April 2001; receive in revise form 19 October 2001; accepte 31 October 2001 Abstract In this stuy an attempt is mae to investigate comprehensively the maximum sustainable yiel (MSY) of Allee population ynamic system. The results show that: (1) because the net prouction rate of Allee system is a function of the shape parameter, all Allee systems which possess ifferent B (B=A/K, here K is the carrying capacity, an A the parameter of Allee effect) are aapte themselves to the environments by controlling their prouction rate. Allee effect oes not change the interaction between the system an environment, but it will reuce the MSY; (2) the MSY of Allee system is approximately one-nineteenth of the height of the allometry curve of boy size for all A/K However when the net prouction rate R takes the values from 3 to 15, the MSY of Allee system is also approximately 1/19 of the height of the allometry curve of boy size for all A/K Elsevier Science B.V. All rights reserve. Keywors: Population ynamics; Allee-effect; MSY 1. Introuction Surplus prouction moels for biomass yiel have a long, useful history in fishery management (Ricker, 1983). By weing this management technique to certain results known from the stuy of boy size scaling or allometry (Caler, 1984; Damuth, 1987; Charnov, 1991, 1992, 1993a,b), Charnov (1993a) consiere that MSY (in weight) is approximately 1/6 ofc, where C is efine by K=CW 0.75 (here K is the carrying capacity an W the ault boy weight), i.e. the maximum * Corresponing author. aress: zslin@ .njnu.eu.cn (Z.-S. Lin). sustainable yiel (MSY) is inepenent of boy size for mammals. Suppose that population growth is governe by the equation: N t =r mn K N (1) K where N is the population size (aults per unit area), is the shape parameter an r m is the maximum intrinsic rate of increase. Let u be the relative population size (N/K), the equation can then be rewritten as: (NW) =Wr t m (1 u) uk=f(u, r m, ) (2) where (NW)/t is the surplus prouction available for harvesting from a population hel at size N /02/$ - see front matter 2002 Elsevier Science B.V. All rights reserve. PII: S (01)

2 2 Z.-S. Lin, B.-L. Li / Ecological Moelling 154 (2002) 1 7 Letting f u =0 one obtains the steay state: u m =1/(1+), or =(1 u m )/u m (3) an MSY=Wr m (1 u m ) u m K=f(u m, r m, ) (4) Suppose R m is the net reprouction rate of the steay state, from Fowler s approximation (Fowler, 1989), u m = y, y=log e (log e R m ) (5) Consiering the maximum intrinsic rate of increase r m an the carrying capacity K, the scaling laws (Charnov, 1993a) can be generally shown as following: r m =A 3 W 0.25, K=CW 0.75 (6) where both C an A 3 are the parameters, an A 3 = ln R m, A 1 =M 1 /W 0.25, A 1 +A 2 A 2 = /W 0.25, A 1 +A (7) here M 1 is the mean ault lifespan an is the age of first reprouction. From Eq. (6) we have r m KW=CA 3 (8) So, MSY=C(1 u m ) u m ln R m /(A 1 +A 2 ) (9) The term (1 u m ) u m ln R m is almost a constant for R m in the range 3 15, which inclues all the vertebrates in Fowler s (1989) ata set. The term equals 0.37 at R m =3 an rises to 0.58 at R m =15 (Charnov, 1993a,b). This means that for mammals (R m =3), MSY C 0.4 (10) A 1 +A 2 Estimating A 1 +A 2 for non-primate (Charnov, 1993a,b; units are kg, year) makes A 1 +A 2 =2.4 or MSY=0.17C ( C/6). Thus, Charnov (1993a) foun that MSY is approximately 1/6 of the height of allometry curve of the prouction ensity boy size (MSY C/6), which means that the MSY of non-allee population system is also inepenent of the shape parameter, or the MSY is inepenent of the internal interaction. Two things account for the results: Charnov use Eq. (3) (which is not a function of boy size by Fowler s rule) to etermine the shape parameter, an Eq. (5) to control u m. However, the interactions of ifferent species (mammals an others) with the environment are ifferent in a certain local region, the shape parameter will change with ifferent moels, i.e. the MSY is a function of the shape parameter. So, Charnov s results (MSY C/6) are not suitable for all mammals. On the other han, it is known that an Allee effect can occur in many population systems (Lewis an Kareiva, 1993; Amarasekare, 1998). Wang et al. (1999) set up a competitive ynamics of population subject to an Allee effect an Stephens an Sutherlan (1999) showe some consequences of Allee effect on behavior, ecology an conservation. We usually want the maximum sustainable yiel of Allee system with the minimum effort. Is MSY of Allee system also inepenent of the boy size for mammals an others? How the MSY epens on the shape parameter will be shown in Section 2, an some extreme values of Allee population ynamic system will be iscusse in the Sections 3 an Some extreme values of population growth moel at ifferent shape parameters Letting Y m (u)=(1 u m ) u m (11) Y a *=(1 u m ) u m ln R m (12) where Y m (u) shows the effect of environment, an Y a * is the interaction between environment an the mammals population. Some extreme values of the system at ifferent shape parameters are shown in Table 1. From Table 1, we fin that the MSY increases with the increase of shape parameter. Suppose

3 there are ifferent species (mammals an others) in a certain local region, because of their ifferent interactions with the environment, the shape Table 1 Some extreme values of population growth moel at ifferent shape parameters Z.-S. Lin, B.-L. Li / Ecological Moelling 154 (2002) u m Y m (u) Y* a MSY R m C C/ C C/ C C/ C C/ C C/ C C/3 Fig. 2. The curves ya1, ya2 an ya3 show, respectively, the ya, or MSY m of Allee system insensitive to the maximum net reprouctive rate R m at B= 0.001, 0.01, an 0.1. Table 2 The MSY of Allee population system when takes an extreme value (Eq. (16)) B R m ya MSY A C C/ C C/ C C/ C C/ C C/ C C/23 Table 3 The MSY of Allee population system when net reprouction rate R m =3 B R m ya MSY A C C/ C C/ C C/ C C/ C C/ C C/23 Fig. 1. (a) The relation between the relative population size u m an the shape parameter at the steay state. (b) The relation between the shape parameter an the net prouction rate R m at the steay state. parameter will change with the ifferent moels. So, the MSY of ifferent species are ifferent, unless they have the same shape character. However, when a species of mammals is at an equilibrium state, is a constant for the species, then its MSY is a constant. An the further calculation shows that Charnov s results MSY C/6 is hel

4 4 Z.-S. Lin, B.-L. Li / Ecological Moelling 154 (2002) 1 7 only when =0.625 an the population size N m =u m K=0.615K. Fig. 1 shows the relationships u m (a) an R m (b). When the nonlinear interaction of the system is enhance, the value of increases an u m ecreases (Fig. 1(a)), then the population size N m = u m K is reuce. In orer to maintain the ecological equilibrium, the species must increase its net reprouction rate R m (Fig. 1(b)) so that the weaker increases its net reprouction rate. Because the net reprouction rates of many mammals are less than 145 an greater than 1.6, we consier that power numbers or the shape parameters of the system usually are in the range which tallies with the fact (Charnov, 1993b, p. 103). = (1 u Am)(2u Am B) u Am (u Am B) (16) Eq. (14) will possess the largest (NW)/t, then the maximum sustainable yiel: MSY A = C (1 u A Am ) u Am (u Am B)ln R m 1 +A 2 (17) Here Eqs. (5) (8) are use to euce Eq. (17). In orer to compare with Charnov s (1993a) results about non-allee system, we first use Eq. (5) to etermine u Am, then Eq. (16) to etermine just as Charnov s (Charnov, 1993a). Letting y(u)=(1 u Am ) u Am (u Am B) (18) 3. The maximum sustainable yiel of Allee system when takes an extreme value Suppose that population growth with an Allee effect is governe by the equation N m t =Nr N N B 1, B=A/K (13) K K i.e. (NW) =Wr m K(1 u) u(u B) (14) t When u satisfies the following conition: 2 u Am (2+) u Am (2+B+B)+B=0 (15) or Table 4 MSY of Allee population system when R m =15 B R m ya MSY A C C/ C C/ C C/ C C/ C C/ C C/25 ya=(1 u Am ) u Am (u Am B)ln R m =y(u)ln R m (19) Fig. 2 shows the relation between ya an R m at ifferent B. Comparing with Charnov s Figure 1 (1993a), some ifferences between Population Moel an Allee Population Moel are foun: 1. the MSY of Allee population system is also inepenent of the boy size for mammals (an others), an it is less than that of population moel; 2. the MSY of non-allee system monotonically increases with the net prouction rate, an there is a maximum value in the MSY of Allee system. When the net prouction rate is greater than the extreme point, the MSY of Allee system will monotonically ecrease with the net prouction rate. Larger B will inuce faster ecrease in MSY. Tables 2 4 show some MSY values of Allee population ynamic system in ifferent conitions.from Tables 2 4, (1) the MSY of Allee population system is only 1/3 of that of non-allee system (C/6, Charnov, 1993a); (2) when takes an extreme value, i.e. = (1 u Am)(2u Am B), u Am (u Am B) the MSY of Allee system is approximately 1/19 of

5 Z.-S. Lin, B.-L. Li / Ecological Moelling 154 (2002) Fig. 3. The relation between effect of environment y(u) an the relative population size u of Allee system; where y(u) is note as y i (u), an i=1, 2, 3, 4, 5 inicates, respectively, = 0.5, 1, 1.5, 2, an 3. (a) B=0.1; (b) B=0.01. the height of the allometry curve of population ensity boy size for all B 0.01, an when net reprouction rate R m takes the values from 3 to 15, the MSY of Allee system is also approximately 1/19 of the height of the allometry curve of population ensity boy size for all B

6 6 Z.-S. Lin, B.-L. Li / Ecological Moelling 154 (2002) Some extreme values of Allee system at ifferent shape parameters In Section 3 the maximum sustainable yiel of Allee system has been iscusse when takes an extreme value. Where is etermine by u Am (Eq. (16)) which is etermine by R m (Eq. (5)). That means the inepenent variable of the system is R. However, the MSY is etermine by u which is associate with the largest (NW)/t. So u is exactly the inepenent variable of the system. Fig. 3 shows the variations of y(u) with u in ifferent when B=0.1 (a) an B=0.01 (b). Accoring to Fig. 3, Eqs. (5), (15) an (16), some extreme values of Allee system when B= Table 5 Some extreme values of Allee system at ifferent shape parameter when B=0.1 u m R m y(u) ya MSY A C C/ C C/ C C/ C C/ C C/28 Table 6 Some extreme values of Allee system at ifferent shape parameter when B=0.01 u m R m y(u) ya MSY A C C/ C C/ C C/ C C/ C C/21 Fig. 4. The x axis is R m,any axis is. The curves show the relation, respectively, between the shape parameter an net prouction rate R m (at the steay state) of Allee system when B=0.01, 0.001, an , 0.01, an are shown in Tables 5 7, respectively. Form Tables 5 7 it is foun that: when is the inepenent variable about the system, both u m an R m are inepenent of B, but the MSY is relate to B. For any exact, the MSY of Allee system will increase with the ecrease of B. The fact that the net prouction rate is a function of shows that all Allee systems possessing ifferent B aapt themselves to the environment by controlling their prouction rate. Allee effect oes not change the interaction between the system an environment, but it will reuce the MSY. Fig. 4 shows the relation between an R of Allee system when B takes ifferent values. From Fig. 4, three curves of uner ifferent B values (0.0001, 0.001, 0.01) are almost coincient, which means that Allee effect oes not change the interaction between the system an environment. Table 7 Some extreme values of Allee system at ifferent shape parameter when B=0.001 u m R m y(u) ya MSY A C C/ C C/ C C/ C C/ C C/21 5. Conclusions This paper has iscusse the MSY of Allee population ynamic system. The most significant finings are as follows: 1. By calculating an simulating, it is foun that the MSY of non-allee population growth increases with the increase of shape parameter, an Charnov s results about MSY C/6 is hel only when =0.625.

7 Z.-S. Lin, B.-L. Li / Ecological Moelling 154 (2002) When the nonlinear interaction of the system is enhance, the value of the shape parameter increases an the relative population size u m at the steay state ecreases, then the population size N m =u m K is reuce. In orer to maintain the ecological equilibrium, the species must increase its net reprouction rate R m.sothe weaker always has a big net reprouction rate. 3. Because the net reprouction rates of many mammals are between 145 an 1.6, the power numbers or the shape parameters of the system are in the range 0.3 2, which tallies with the fact. 4. Because the net prouction rate of Allee system is a function of, all Allee systems which possess ifferent B are aapte themselves to the environment by controlling their prouction rate. Allee effect oes not change the interaction between the system an environment, but it will reuce the MSY. 5. The MSY of Allee system is approximately one-nineteenth of the height of allometry curve of the population ensity boy size for all B However, when R takes the values from 3 to 15, the MSY of Allee system is also approximately 1/19 of the height of allometry curve of the population ensity boy size for all B So we consier that as the Allee effect relates to populations at low ensities it is ifficult to see how it can be relevant to the MSY. Acknowlegements This research has been supporte in part by Ministry of Science an Technology of China uner grant NKPDBS G part 1, an US National Founation uner grant DEB References Amarasekare, P., Allee effects in metapopulation ynamics. Am. Nat. 152, Caler, W.A., Size, Function an Life History. Harvar University Press, Cambrige, MA. Charnov, E.L., Evolution of life history variation among female mammals. Proc. Natl. Aca. Sci. USA 88, Charnov, E.L., Allometric aspects of population ynamics: a symmetry approach. Evol. Ecol. 6, Charnov, E.L., 1993a. Is maximum sustainable yiel inepenent of boy size for mammals (an other)? Evol. Ecol. 7, Charnov, E.L., 1993b. Life History Invariants: Some Exploration of Symmetry in Evolutionary Ecology. Oxfor University Press, New York, p Damuth, J., Interspecific allometry of population ensity of mammals an other animals: the inepenence of boy mass an population energy use. Biol. J. Linn. Soc. 31, Fowler, C.W., Population ynamics as relate to rate of increase per generation. Evol. Ecol. 2, Lewis, M.A., Kareiva, P., Allee ynamics an the sprea of invaing organism. Theor. Popul. Biol. 43, Ricker, W.E., Computation an interpretation of biological statistics of fish population. Bull. Fish. Res. Boar Can. 191, Stephens, P.A., Sutherlan, W.J., Consequences of the Allee effect for behaviour, ecology an conservation. TREE 14, Wang, G., Liang, X., Wang, F., The competitive ynamics of populations subject to an Allee effect. Ecol. Moel. 124,