29. Differentil equtions. The conceptul bsis of llometr Did it occur to ou in Lecture 3 wh Fiboncci would even cre how rpidl rbbit popultion grows? Mbe he wnted to et the rbbits. If so, then he would be less concerned bout totl number of rbbit pirs thn totl weight of rbbit met, which vries continuousl (s opposed to discretel). Let W(t) be weight t time t. Then W is n ordinr function. But wht kind of function is W? In this lecture we investigte. Over ver short time intervl [t, t + h], weight increses b Diff(W, [t, t+h]) W(t+h) W(t), with verge rte of increse DQ(W, [t, t+h]) Diff(W, [t, t+h])/h. Although weight chnges on [t, t+h], it chnges onl little, so it is lws W(t) + O[h]. Therefore verge increse per rbbit on [t, t+h] is DQ(W, [t, t+h])/{w(t) + O[h]}. As h pproches zero, this expression pproches DQ(W,[t,t + h]) lim h 0 W(t) + O[h] W (t) W(t) r(t), (29.1) s, where r hs the sme domin s W nd W. We cll r the reltive growth rte. The preceding rgument pplies virtull without chnge to the growth of n mss of tissue. In prticulr, it pplies to the growth of single orgn or orgnism. Tht is, if W(t) denotes size t time t of n orgn or orgnism, then W (t) r(t)w(t) (29.2) where r is reltive growth rte. Eqution (2) embodies Julin Huxle's first "essentil fct" bout growth, nmel, "tht it is process of self-multipliction of living substnce i.e. tht the rte of growth of n orgnism growing equll in ll its prts is t n moment proportionl to the size of the orgnism" (Huxle, 1932, p. 6). Wht kind of function is r? Tking our cue from Lecture 5, where Fiboncci rbbit pirs reproduce t constnt rte (of 12 times per nnum), we conjecture tht r m be constnt, s r λ (> 0). Then (2) implies W (t) λw(t) or, if W(t), d λ. (29.3) This is ver simple exmple of n ordinr differentil eqution, or ODE. How do we "solve" this ODE? In other words, how do we find if we know d/? Hitherto we hve used the fundmentl theorem. But we cnnot ppl it to (3), becuse the right-hnd side is n unknown function of time. On the other hnd, if W is invertible with inverse U, so tht W(t) is exctl the sme thing s t U(), then from (20.39) we hve d d. (29.4) So we cn rewrite (3) s d 1 λ, (29.5) nd now it follows from the fundmentl theorem tht t C + where C is constnt. To obtin of C, we set ), 0 U() 1 du C + 1 1 λu λ u du, (29.6) the vlue W(t (29.7) 0 1
M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 2 where t0 is initil time, nd so is initil weight. Then (6), with, implies U() C + C + 0. But W(t0) implies U() t0, functions. So (8) implies C t0, from which (6) implies 1 λu du (29.8) becuse U nd W re inverses t t 0 + 1 λ 1 du u t 0 + 1 λ ln(u) (29.9) t 0 + 1 λ t 0 + 1 λ {ln() ln()} ln( / ) on using Tble 22.6 nd Exercise 22.1. So λ{t t0} ln(/), or exp(λ{t t0}) /. Tht is, In other words, with exp(λ{t t 0}) eλ(t t 0 ) e λt e λt 0. (29.10) W(t) Ae λt (29.11) A e λt 0. (29.12) When W(t) A e λt, it is trditionl to s tht W exhibits exponentil growth t rte λ (lthough wht is ctull ment is tht W hs reltive growth rte λ). There is evidence tht orgnisms do grow exponentill during erl development, lthough reltive growth rte ppers to decrese lter on. The crux of the evidence is tht (11) implies ln( ) ln( W(t) ) ln( Ae λt ) ln(a) + ln(e λt ) (29.13) ln(a) + λt. Tht is, if growth is exponentil, then ln() is liner function of t. So we cn test n exponentil growth hpothesis b plotting (t, ln()) dt pirs nd drwing the stright line tht fits them best. The closer the fit, the more confident we re in the hpothesis. TIME t (ds) WEIGHT (grms) ln( ) TIME t (ds) WEIGHT (grms) ln( ) 6 1 0 53 42 3.738 18 4 1.386 60 62 4.127 30 9 2.197 74 71 4.263 39 17 2.833 93 74 4.304 46 26 3.258 Tble 29.1 Bckmn's dt on growth in weight of mize. Source: Exercise 5.10 For exmple, from Thompson (1942, p. 115) we hve Bckmn's dt on weight (grms) of mize t time t (ds); see Tble 1. From Figure 1(), where ln() is plotted ginst t, the dt points fll ver close to stright line for times between 18 nd 60
M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 3 ds. This six-week period is n exponentil growth phse. The dotted line in Figure 1(), determined b the method of Appendix 2A, hs eqution ln() 0.2279 + 0.06574t. (29.14) Compring with (13), λ 0.06574, ln(a) 0.2279 nd so A exp(0.2279) 1.256. Thus W(t) 1.256e 0.06574t (29.15) provides n excellent description of growth in weight of mize on [18, 60]; see Figure 1(b). On the other hnd, (15) provides ver poor description t lter times. It isn't difficult to see wh. B (2), constnt reltive growth rte implies tht W keeps incresing. But the dt in Figure 1 nd Tble 1 impl tht W eventull pproches zero, in which cse, r pproches zero, too. Intuitivel, growth per unit weight declines with weight becuse more cells compete for the sme resources. The simplest possibilit is tht r decreses linerl with W, s r(t) λ 1 W(t) (29.16) where is nother prmeter. We ssume tht W(t0) <, (29.17) where t0 is the initil time; i.e., W hs domin [t0, ). Then r is positive nd decreses towrd zero s W(t) increses towrd. Substituting (16) into (2) nd setting W(t), we obtin new ODE d λ 1 (29.18) in plce of (3). Assuming s before tht W is invertible with inverse U, we hve d λ( ), (29.19) in plce of (5), nd it follows from the fundmentl theorem tht t C + u) where C is constnt. s C t0. Thus U() λu( du, (29.20) Exctl bove, W(t0) implies t U() t 0 + 1 λ Tble 22.6 nd Exercise 22.1 now impl tht t t + 1 0 λ ln u u t + 1 0 λ ln t 0 + 1 λ So λ{t t0} ln({ }/{ }), impling { } { } ln { } du. (29.21) u( u) ln. (29.22) e λ(t t 0 ) (29.23) { } or, fter strightforwrd lgebric mnipultions (Exercise 1),
M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 4 In other words, where Note tht s required. Correspondingl, W(t) e λ(t t 0 ) e λ(t t 0 ) +. (29.24) (29.25) 1 + Ae λt A { / 1}e λt 0. (29.26) W(t 0), (29.27) W (t) λw(t) 1 W(t) λae λt (29.28) (1 + Ae λt ) 2 from (18) nd (25). Growth ccording to (24) is trditionll clled logistic growth. From (17) nd (28), / > 1, impling A > 0. So W(t) is less thn but pproches smptoticll s t becuse e λt 0 s t. We interpret s mximum possible weight. In Figure 2, the functions W nd W re plotted ginst Bckmn's dt with A 389, 75.5 nd λ 0.119 (vlues were obtined b method similr to tht of Appendix 2A). We find tht the logistic model provides much better overll fit, with mximum growth during d 51; see Exercise 2. We conclude this lecture b observing tht Huxle used pir of ODEs to secure conceptul foundtion for llometr. He resoned tht "the growth-rte of n prticulr orgn is proportionl simultneousl () to specific constnt chrcteristic of the orgn in question, (b) to the size of the orgn t n instnt, nd (c) to generl fctor dependent on ge nd environment which is the sme for ll prts of the bod" (Huxle, 1932, p. 6). If is the size of n llometric orgn nd x the size of its bod, nd if k2, k1 re the specific constnts for prt nd bod, respectivel, then becuse both orgn nd bod re exposed to the sme environment, () (c) impl nd d k 1 xg(t) (29.29) k 2 G(t) (29.30) where t denotes time nd G is some unknown function. Dividing (30) b (29), we hve where d 1 β x (29.31) β k 2. (29.32) k 1 Note tht the orgn is positivel or negtivel llometric ccording to whether k2 > k1 or k1 > k2, nd hence ccording to whether β > 1 or β < 1, s in Lecture 22. Assuming tht x is n invertible function of time, nd hence t n invertible function of x, 1 (29.33)
M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 5 b nlog with (4). So (31) cn be rewritten s x. (29.34) If is function of t, however, t function of x, then is lso function of x. So the chin rule reduces (34) to x, (29.35) nother ODE. Compring with (3) we find tht the right-hnd sides of (3) nd (18) involve onl the dependent vrible, wheres the right-hnd side of (35) involves the independent vrible s well; nevertheless, (35) is no more difficult to solve. We proceed s follows. According to the chin rule, if is positive function of x then defines composition with derivtive on using Tble 22.6. So (36) It now follows from the theorem tht d β nd is d β z ln() (29.36) dz dz d d { d d ln() } d d, (29.37) reduces (35) to dz β x. (29.38) x z C + β 1 du C + βln(u) u C + β{ln(x) ln()}, so tht (36) nd properties of the logrithm impl ln() C + β ln x Properties of the exponentil now impl tht x C + ln x ( ) exp(c)exp( ln({x / )} β ) exp C + ln({x / )} β exp(c) x β. β On setting β exp(c) α we hve αx β, greeing with (22.32). 1 β (29.39). (29.40) (29.41) References Huxle, Julin S. (1932). Problems of Reltive Growth. The Dil Press, New York Thompson, D'Arc W (1942). On Growth nd Form. Cmbridge Universit Press.
M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 6 Exercises 29 29.1 Verif (24)-(27). 29.2 Show tht the logistic growth curve W(t) defined b (25) hs n inflection point t t t*, defined b t* ln(a)/λ. Verif tht W(t*) /2. 29.3 Show tht (28) implies W (t) λ 3 W(t) Hence show tht the curve W (t) hs inflection points where t 1 λ ln ± 3 i.e., where t 39 nd t 61 in 1 W(t) 6W(t) 1 W(t) ({ 2 ± 3 }A), Figure 2(b)..
M. Mesterton-Gibbons: Bioclculus, Lecture 29, Pge 7 Answers nd Hints for Selected Exercises 29.2 From (28), we hve So W (t) λw(t) 1 W(t) W (t) d. { W (t)} λ d λ W (t) 1 W(t) λ W (t) 1 W(t) W(t) 1 W(t) + W(t) d 1 W(t) + W(t) 0 1 W (t) λ 2 W(t), impling W 0 where W(t*) 0 or 1 W(t*)/ 0 or 1 2W(t*)/ 0. From (25), however, we hve W(t*) /{ 1 + Ae λt* }, so tht W(t*) 0 is impossible, nd 1 W(t*) so tht 1 W(t*)/ 0 is impossible (except in the limit s t* ). So the inflection point must be determined b 1 2W(t*)/ 0. But λ W (t) 1 W(t) 1 W(t) (t) λ W (t) 1 2W(t) 1 W(t) 2W(t) (t*) 1 1 Ae λt* + 1 Ae λt* 1 + Ae, λt* 1 2 W(t*) 1 2 Ae λt* + 1 Ae λt* 1 1 + Ae. λt* Thus 1 2W(t*)/ 0 implies Ae λt* 1 or e λt* 1/A. Hence λt* ln(1/a) ln(a), nd λt* ln(a) or t* ln(a)/λ. Note tht, becuse 0 < W(t) <, the sign of W (t) is completel determined b 1 2W(t)/, which is positive if t < t* but negtive if t > t*. Thus W hs mximum W (t*). You cn verif tht W (t*) 0 corresponds to minimum (s opposed to mximum) of W b checking the sign of W (t*): from Exercise 3, W (t*) λ 3 W(t*) which is 1 W(t*) negtive. 1 6W(t*) 1 W(t*) λ 3 8,