Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance, food, or space). If is he variable for ime, and () he size of he populaion a ime, hen he logisic differenial equaion is d = k (L ) d where he consans k and L depend on he paricular populaion and he deails of is environmen. In fac, he consan L has a clear physical inerpreaion, as we ll see below: i is he carrying capaciy of he environmen, he larges populaion ha he environmen can suppor. One way o ry and undersand he logisic equaion qualiaively is o skech is slope field. To do his, a each poin (, ) we mark a lile line, whose slope is obained by plugging in he coordinaes and ino he equaion above. In he case of he logisic equaion, we have a bi of luck: he equaion k (L ) doesn depend on he variable, so he slopes on he slope field will be he same as we move horizonally. I s worhwhile o look a he equaion k (L ) a bi more carefully. s a funcion of, i is a parabola, wih zeros a = 0 and = L. The relaionship beween he values (he heighs) of he parabola and he slopes of he slope field is shown in he diagram below.... is slope here Heigh here... In order o analyze he soluions, i is easies o simply urn he parabola on is side, and draw i nex o he graph of he slope field so ha he -axes line up. 1 This discussion is aken enirely from rnol d s book Ordinary Differenial Equaions, p 24 27. 1
Here is he corresponding skech of he slope field, and some of he soluion curves. L SLOE FIELD SOME SOLUTION CURVES From he skeches, we see ha if we sar wih a populaion smaller han L, i will climb o approach L. If we sar wih a populaion larger han L, will decrease, again approaching L. This jusifies he claim made earlier ha L represens he carrying capaciy of he environmen. In fac, he observaion ha all saring possibiliies end o L says even more: i says ha = L is a sable equilibrium value. Roughly, sable means ha if we perurb he soluion a lile (pushing upwards or downwards a bi) i will end back o where i sared from. In conras, an unsable equilibrium is one where perurbing he soluion may cause i o move away from he equilibrium value. The idea is perhaps bes explained informally by he wo picures below. STBLE UNSTBLE hysically we never expec o see an unsable soluion. Everyhing in he world is subjec o bumps and flucuaions; he survival of an unsable soluion is unrealisic. 2. Two Quoa policies. Le s suppose ha we are dealing wih a populaion of fish, perhaps confined o a lake, or o a paricular breeding environmen, and ha his populaion of fish saisfies he logisic differenial equaion. We wan o allow harvesing of he fish (i.e., fishing), and need o impose some kind of quoa o ensure ha he fish populaion coninues o survive. We d like o analyze he effecs of wo possible quoa policies. 2
The policies are: OLICY I (FIXED QUOT): We fix a number c, and each year we allow c fish o be caugh. OLICY II (ERCENTGE QUOT): We fix a number f (for fracion of fish) and each year allow f fish o be caugh. We need o analyze he effec of hese policies on he dynamics of he fish populaion and decide beween hem. We also need o decide on he appropriae value of c or f for he policy we choose. Our firs concern is for he susainabiliy of he policy we wan o ensure he long erm survival of he fish. Our second concern (always making sure ha our policy is susainable) is o have he larges yearly yield of fish. m In he he analysis of each of hese wo quoas, he maximum value m = kl 2 /4 (shown in he picure above) achieved by he parabola k (L ) will play a par, so i s useful o give i is own name. 3. Fishing olicy I: Fixed Quoa. Lef alone, he populaion of fish obeys he logisic differenial equaion. If we allow he removal of c fish per year hen he new differenial equaion saisfied by he populaion is: d d = k (L ) c. We can analyze his in he same way ha we did he logisic equaion. The effec of he c erm is o shif he parabola down. convenien graphical way o hink abou his is o draw he line of heigh c on he parabola, and hen hink of his line as forming he new -axis. Here is a skech for olicy I where he yearly quoa c is less han m. c < m B c QUOT BELOW MXIMUM VLUE (STBLE) 3
There are wo possible equilibrium values, = B and =. The value a = B is sable small flucuaions of near he equilibrium head back o B. The equilibrium a = is unsable a small change eiher sends he fish populaion shooing up o B (which is good) or downwards o exincion (which is cerainly bad). If we chose olicy I wih a value of c like his, we would have a susainable policy. The level of fish (as long as hey sared near = B) would remain roughly consan, he reproducion of he fish exacly compensaing for he fish removed each year. In his policy our yearly yield is exacly c (ha s he meaning of c). Wha happens if we ry and increase c? s long as c < m, we ge a picure like he one above, he only difference being he exac locaion of and B and he amoun of space beween hem. Le s look a wha happens if we increase c all he way o m. c = m c QUOT T MXIMUM VLUE (UNSTBLE) Here here is only one equilibrium value, a =. This equilibrium value is however unsable; if ever drops below, hen he resul is exincion of he populaion. racically, even hough i gives he larges yearly yield of fish, c = m is inadmissible as a policy. We expec sligh flucuaions in he fish populaion, and (given he insabiliy of he equilibrium) hese would soon lead o exincion. This is no a susainable soluion. We should also no allow values of c oo close o m, here he values of and B would be close ogeher, and a small flucuaion of he populaion could send i below, again resuling in exincion. Summarizing, i seems ha for olicy I, he maximal heoreical yearly limi is m fish per year, bu pracically we re going o have o insis ha c < m. How far below m is a combinaion of our guess a how large random flucuaions in he fish populaion migh be, and our olerance for risk. 4. Fishing olicy II: ercenage Quoa. Insead of allowing he removal of a fixed amoun c of fish per year, we re going o ry fixing a number f, and allowing he removal of f fish per year. (For insance, f = 0.20 would mean ha we allow 20% of he fish o be removed each year.) 4
Under his policy, he differenial equaion describing he populaion of fish is d d = k (L ) f. The effec of he f erm is again o shif he parabola, alhough he amoun of shifing now depends on he locaion of. Le s jus do he same hing we did before: draw he line y = f over he graph of he parabola, and realize ha we re looking a he difference in he wo equaions. One sriking difference beween his policy and he previous one is ha he equilibrium soluion is sable, no maer which fracion f we pick. EQUILIBRIUM SOLUTION LWYS STBLE The equilibrium value is he coordinae of he inersecion of he line and he parabola. Since he equilibrium is sable, his is a susainable policy. So, once we fix our value of f, he fish populaion will sele down o, and each year we ll be geing f fish. Wha we need o do is figure ou how o pick f so ha f will be he larges. One sligh rouble is ha he value of depends on f. Forunaely here s a way o see he produc f on he graph of he parabola and he line, and ha will make i easy o maximize f. y y = f f Since we re looking for f, and he line is of he form y = f, his is he same as looking for he heigh of he line when =. Bu = is exacly he poin where he line inersecs he parabola (ha s how we found in he firs place), so he number of fish we expec o ge each year is jus he heigh of he inersecion poin. In order o maximize our yearly yield of fish, we jus have o make he inersecion poin he highes possible. nd we clearly do ha by puing he line hrough he verex of he parabola: 5
m LINE WITH MXIMUM YERLY YIELD When we do his, he yearly yield from fishing is exacly m, he heigh of he verex of he parabola. Summarizing, for olicy II, he maximal heoreical yearly limi is m fish per year, and we can do i in a safe, susainable way. Clearly olicy II (wih f = kl/2, which pus he line hrough he verex) is he bes choice: We no only ge more fish han wih olicy I, bu he resuling equilibrium is sable. 5. Concluding remarks. The fac ha olicy II has beer sabiliy properies han olicy I is no surprising: Since we re aking a fracion f of he fish, anyime he populaion drops we are effecively lowering our quoa as well, giving he fish populaion ime o recover. Under olicy I, we keep removing he same number of fish, poenially devasaing he populaion. I s somewha wonderful ha he dynamics of hese differen models of he fish populaion can be undersood by our geomeric argumens wihou ever solving he differenial equaions. Many of he modern insighs ino he naure of differenial equaions come from emphasizing he geomeric properies over explici soluions; he geomeric mehod is a powerful one. This handou can (soon) be found a hp://www.mas.queensu.ca/ mikeroh/calculus/calculus.hml E-mail address: mikeroh@mas.queensu.ca 6