Delay Di erential Equations and Oscillations

Transcription

1 Lecture 2 Delay D erental Equatons and Oscllatons Readng: Most of the materal n today s lecture comes from Chapter 1, pages of James D. Murray (22), Mathematcal Bology I: An Introducton, 3rd edton. If you are at the unversty, ether physcally or va the VPN, you can read an onlne verson at Logstc growth revsted One of the shortcomngs of Malthusan growth law mentoned n yesterday s lecture was that t took no account of ether gestaton (the tme t takes for a chld to grow n ts mother s womb) or the tme t takes for a newborn to grow to sexual maturty. There are, of course, many organsms for whom these are not serous objectons yeast cells reproducng n a lump of rsng bread dough have nether gestaton nor sexual maturaton but today I ll ntroduce a mathematcal tool, the delay d erental equaton (DDE), capable of descrbng growth laws n whch the brth rate at tme t depends on the state of the populaton at earler tmes. The smplest DDE s a d erental equaton of the form = f(n(t), N(t T )) (2.1) where T>ssomefxeddelay. Forexample,wecouldmodfythelogstclawto get apple = rn(t) N(t T ) 1. (2.2) K It s mportant to recognse what a drastc change ths s. The models we dscussed n the frst lecture were the sort of thng you studed n the second term of the Frst Year: one can specfy a soluton unquely by settng up an ntal value problem, N(t =)=N, 2.1 = g(n).

2 If we want to know N(t) for<t<1 then, provded that g(n) sn ttoobadly behaved, all we would need to specfy s the ntal condton N(t =)=N. But for DDEs the stuaton s much more complcated more lke a partal dfferental equaton. To work out what happens on an nterval <t<t we need to specfy both the functon f(n(t), N(t T )) and a whole nterval worth of ntal data: we need to know N(t) for T < t <. To drve ths pont home, thnk about the data we need before we can compute / at t = T/2: we need both N(T/2) and N(T/2 T )=N( T/2). As you mght magne, ths make the problem consderably harder and, generally, one can t solve DDEs exactly. Despte ths added complexty, the delayed logstc model (2.2) has some behavour n common wth the orgnal, ODE verson. In partcular, the constant functons N(t) =an(t) =K are both stll solutons. But the DDE verson also supports a new possblty: oscllatory solutons. We won t generally be able to obtan exact expressons for N(t) andsobeforewebegntoanalysethedelayed logstc model I d lke to nvestgate an example where one can compute everythng explctly Malthusan growth wth mgraton and delays Recall that the very smplest growth law, / = rn, has solutons that are exponentals n tme, N(t) =N e rt, and so clearly doesn t permt oscllatons. But f we ntroduce a delay, new possbltes emerge. Suppose we consder the growth law = bn(t T ) (t) m (2.3) where b, d and m are all postve real numbers. Ths descrbes a populaton wth per-anmal brth rate b, death rate d and outward mgraton rate m (measured n beasts-per-unt-tme). The novel feature s that these beasts have a gestaton perod T, so that the number of o sprng beng born at tme t depends on the populaton at tme t T Brth, death and mgraton wthout delay The ODE verson of the model above s whch has an equlbrum at N(t) =N? = m/(b = bn m = (b d)n m (2.4) d) andgeneralsoluton N(t) =N? +(N N? )e rt where r =(b d), N? = m/r and N = N(t = ) s the ntal populaton. If b<d, so that r<, then N(t) tendston? for all ntal condtons, but f b>d, so that r>, then the behavour of N(t) ssomewhatcounter-ntutve: 8 < lm N(t) = t!1 : 1 f N >N? N? f N = N? 1 f N <N? 2.2

3 The last case, where the populaton becomes negatve, makes no bologcal sense, so t s nterestng to ask how ths problematc behavour arses. The ssue turns out to be the constant mgraton term m. It says that anmals keep mgratng away at, say, 1 beasts per week, even when the populaton has fallen to zero beasts. We can crcumvent ths problem by sayng that the model (2.4) only apples n cases where N N?. Ths restrcton rules out the pathologcal negatve populatons, though leaves open the possblty of a dvergng populaton Oscllatory solutons Let s seek solutons to (2.3), the DDE verson of the brth-plus-death-plus-mgraton model. Before we do so t s helpful to smplfy the problem by ntroducng dmensonless varables. One natural set of choces s x = N N? = N(b d) m and = bt and these lead, va calculatons smlar to those n Secton 1.4, to the DDE dx d = x( ˆT ) x( ) (1 ). (2.5) where = d/b and ˆT = bt s the dmensonless verson of the delay. Note that as we want there to be a postve net reproductve rate we need b>d>andso < < 1. It turns out that, for sutable values of the parameters, ths has oscllatory solutons of the form x( ) =A + B cos(! ). One can prove ths by computng dx/d n two ways: frst by drect d erentaton and then va the DDE (2.5). The drect route yelds dx d = d d The DDE, on other hand, says (A + B cos(! )) =!B sn(! ). (2.6) dx d = x( ˆT ) x( ) (1 ) = ha + B cos(!( ˆT )) [A + B cos(! )] (1 ) =(1 )A (1 ) h + B cos(! ˆT )cos(! )+sn(! ˆT )sn(! ) B cos(! ) =(1 )A (1 ) h + B cos(! ) cos(! ˆT ) h + B sn(! ) sn(! ˆT ). (2.7) 2.3

4 Condtons on! ˆT -1 1 cos( ) sn( ) /2 3 /2 2! ˆT Fgure 2.1: The curves above are graphs of the functon cos(! ˆT ) (green, sold) and sn(! ˆT ) (orange, dashed). They make t clear that one can satsfy both the condtons cos(! ˆT ) = and! = sn(! ˆT ) f 3 /2 <! ˆT < 2, whch ensures both cos(! ˆT ) > and sn(! ˆT ) <. Equatng expressons (2.6) and (2.7) and matchng up coe cos(!t) andsn(!t) termsyeldsthreecondtons: cents of the constant, (1 )A (1 )=, cos(! ˆT ) = and! =sn(! ˆT ). (2.8) The frst of these leads mmedately to A =1andfwelookatthesecondcondton and recall that < < 1, we see that ths mples cos(! ˆT ) >. Smlarly, the thrd condton mples that sn(! ˆT ) <. Fgure 2.1 makes t clear that both of these condtons can be satsfed provded 3 /2 <! ˆT <2. Ths means that we can fnd values of the parameters! and ˆT such that the DDE (2.3) has oscllatory solutons of the form x( ) =1+Bcos(! ). Proceed as follows: Use the second condton to obtan a value! ˆT that les n the range 3 /2 < < 2 and satsfes Ths s always possble as < < 1. = cos(! ˆT ) cos( ). Fnd! by rewrtng the thrd condton as Fnally, get ˆT by notng that! = sn(! ˆT ) = sn( ). ˆT =! ˆT! = sn( ). 2.4

5 Snusodal soluton to the DDE x(t) ! t 4! Fgure 2.2: Here s a snusodal soluton x( ) =1+B cos(! ) to the DDE (2.3) that has ampltude B =.35. The parameters are =.92, ˆT =15.4 and! =.38, whch correspond to puttng =! ˆT = 15 /8 nto the calculatons sketched n Secton Fgure 2.2 llustrates a soluton of the knd we ve just constructed. The ampltude B of the snusodal oscllaton does not enter nto the condtons (2.8) and so we are free to choose t almost arbtrarly, though we need B < 1toensurethatthe populaton never becomes negatve. Ths example may seem a bt contrved and ndeed, t s but t was desgned to produce easy calculatons and to make the pont that convertng an ODE model such as (2.4) nto a DDE can nduce oscllatons. We wll look at ways to establsh the same prncple n more general settngs, ncludng the delayed logstc model (2.2), n the remander of the lecture. 2.2 No oscllatons wthout delay I d lke to drve home the pont that delays can nduce oscllatons where none are otherwse possble. Here s a result that I learned from Murray, though I doubt that t s hs orgnally. It says that a large famly of ODE models for the populaton of a sngle-speces can never have oscllatory solutons. Defnton 2.1. A functon N : R! R s perodc wth perod T f: N(t + T )=N(t) for all t 2 R; T s the smallest value for whch the prevous condton holds and T>. The second requrement means that we do not regard constant functons as perodc and so a perodc functon s, necessarly, non-constant. Theorem 2.2 (No oscllatons wthout delay n sngle-speces models). If N(t) satsfes the d erental equaton = f(n) (2.9) for some contnuous functon f : R! R then N(t) cannot be perodc. 2.5

6 Proof. We ll prove ths by contradcton. Assume the ODE (2.9) has a perodc soluton N(t) wthperodt. Multply both sdes of the ODE by / to obtan 2 = f(n). Now ntegrate both sdes of ths expresson over a sngle perod of the oscllaton. An easy theorem (that I won t prove here) says that the result s ndependent of whch nterval we choose, so we mght as well ntegrate over apple t apple T to obtan Z T 2 Z T = f(n). (2.1) Consder frst the expresson on the left. We know that N(t) snon-constant(our defnton of a perodc functon ruled ths out explctly) and so we know / 6= somewhere n the regon of ntegraton. Thus (/) 2 s a strctly postve quantty somewhere n the nterval and so we can defne Z T 2 I = and conclude that I>. On the other hand, consder the rght-hand sde of (2.1). If we change the varable of ntegraton to N and ntroduce N = N() = N(T )weget: I = Z T f(n) Z N(T ) = f(n) = = N() Z N N f(n) where the last lne follows because we ended up ntegratng over an nterval of length zero. Ths provdes the contradcton we sought: t s mpossble that both I> and I =,sotherecannotbeanyperodcsolutonsto(2.9). 2.3 Oscllatons n the logstc law wth delay Let s conclude by returnng to the logstc DDE (2.2) apple 1 = rn(t) N(t T ) K. and makng a qualtatve argument (not aproof)sayngthattmayhaveoscllatory solutons. Suppose that at some tme t 1 we have the followng: N(t 1 )=K, t=t1 > and N(t) <K for t 1 T apple t<t 1. Ths stuaton s llustrated n Fgure 2.3 and prompts the followng observatons: 2.6

7 Logstc Growth wth Delay Populaton N(t) K 2K t 1 t 1 + T t 2 t 2 + T Tme Fgure 2.3: An llustraton of the qualtatve argument n Secton 2.3 that suggests one ought to expect oscllatory solutons to the logstc DDE (2.2). Here t 1 s a tme at whch N(t) s ncreasng and satsfes N(t) =K, whle t 2 s the next tme N(t) =K, at whch pont N(t) s decreasng. When t = t 1, N(t) =K, butn(t T ) <K, so the factor [1 N(t T )/K] that appears n the logstc DDE s postve. Ths means / > an(t) s an ncreasng functon. When t = t 1 + T, N(t T )=N(t 1 )=K, so the factor [1 N(t T )/K] vanshes, whch means that / =, and so N(t) has a crtcal pont. The crtcal pont s a local maxmum. To see ths, consder a slghtly later moment, say, t = t 1 + T + for some sutably small. Then t T>t 1 and N(t T ) >K, so the factor [1 N(t T )/K] s negatve and / <. Somewhat later, there s a tme t 2 >t 1 + T such that N(t 2 )=K, but now N(t T ) >K, so / < an(t) sadecreasngfuncton. When t = t 2 + T, N(t mnmum. T )=N(t 2 )=K, so / =an(t) hasalocal Ths oscllatory soluton N(t) ncreases from K to ts local maxmum n tme T and then also decreases from K to a local mnmum n T, so one mght expect the perod of the oscllaton to be around 4T. Although ths argument sn t especally precse n partcular, t says nothng very concrete about the length of the nterval between t 1 + T and t 2 t turns out to make a correct predcton: numercs show that the logstc DDE (2.2) does have oscllatory solutons for a wde range of values of the dmensonless quantty rt. Ths proves to be the relevant quantty because we can make a dmensonless verson of the problem wth the same change of varables that we used when analysng 2.7

8 the ODE verson of the logstc law: x = N K and = rt. The resultng dmensonless DDE s dx h d = x( ) 1 x( ˆT ) (2.11) where ˆT = rt s the dmensonless verson of the delay T. 2.8