MATH 372: Differece Equatios G. de Vries February 23, 2003 1 Itroductio I this chapter, we use discrete models to describe dyamical pheomea i biology. Discrete models are appropriate whe oe ca thik about the pheomeo i terms of discrete time steps, or whe oe wishes to describe experimetal measuremets that have bee collected at fixed time itervals. I geeral, we are cocered with a sequece of quatities, x 0, x 1, x 2, x 3, x 4,..., where x i deotes the quatity at the i-th measuremet or after i time steps. For example, x i may represet the size of a populatio of mosquito isects i year i; the proportio of idividuals i a populatio carryig a particular gee i the i-th geeratio; the umber of cells i a bacterial culture o day i; the cocetratio of oxyge i the lug after the i-th breath; the cocetratio i the blood of a drug after the i-th dose. You ca udoubtedly thik of may more such examples. Note that the time step may or may ot be costat. I the example of the bacterial culture, the time step is fixed to be a day, but i the example of the oxyge cocetratio i the lug, the time step is variable from breath to breath. Also, time steps ca be aywhere from millisecods to years, depedig o the biological problem at had. We ca ow ask ourselves, what does it mea to build a discrete model? I the cotext of our sequece of quatities x i, a discrete model is a rule describig how the quatities chage. I particular, a discrete model describes how x +1 depeds o x (ad perhaps x 1, x 2,... ). Restrictig ourselves to the case where x +1 depeds o x aloe, a model the is a sort of updatig fuctio [1], of the form x +1 = f(x ), (1) 1
x : iput or measuremet after time steps updatig fuctio x +1 : output or measuremet after +1 time steps Figure 1: This is the updatig fuctio as show i Figure 1. The equatio x +1 = f(x ) ofte is referred to as a map. Fidig the precise fuctio f that describes a set of experimetal data or that gives a certai desired type of behaviour is ot always straightforward. I fact, it is ofte said that modellig (fidig the right fuctio f) is more of a art tha a sciece. I ay case, it geerally is a iterative process. Oe starts with a particular fuctio f, ad the makes adjustmets. Isight ito how a fuctio f should be adjusted ca ofte be obtaied from kowledge of the behaviour of the curret model. The remaider of this chapter discusses techiques that ca aid i the aalysis of a discrete model. 2 Simple populatio models We begi by aalyzig oe of the simplest maps, with f(x ) = rx, givig the liear map x +1 = rx. (2) For cocreteess, let us thik of x as the size of a isect populatio after years, or durig the -th geeratio. Here, r > 0 is a model parameter represetig the et reproductio rate of the populatio. The model thus says that ext year s populatio is proportioal to this year s populatio. If we let x 0 be the populatio of isects at the start of the study, the x 1 = rx 0, x 2 = rx 1 = r(rx 0 ) = r 2 x 0,. x = r x 0. The resultig sequece x 0, x 1, x 2,... is called a orbit of the map (startig with differet iitial populatios x 0 gives differet orbits). 2
x x (a) (b) Figure 2: (a) Geometric growth for r > 1; (b) Geometric decay for 0 < r < 1. x +1 x +1 x +1 0 x 0 x 0 K/2 K (a) Liear map (b) Verhulst process (c) Ricker curve x Figure 3: (a) Liear map; (b) Verhulst process; (c) Ricker curve. It should be clear that the isect populatio grows without boud if r > 1 (for example, if r = 2, we obtai the sequece x 0, 2x 0, 4x 0, 8x 0,...), as show i Figure 2(a). Similarly, the isect populatio shriks ad evetually goes extict if 0 < r < 1 (for example, if r = 1 2, we obtai the orbit 1 x 0, 2 x 1 0, 4 x 1 0, 8 x 0,...), as show i Figure 2(b). Whe the value of r is precisely 1, the populatio either grows or shriks, but stays costat at its iitial value x 0. Of course, this simple model is ot very realistic for describig the logterm growth of ay populatio. There is ot a populatio whose growth will ot stagate evetually due to competitio for limited resources amog the idividuals. The problem with this simple model is that it is based o the assumptio that the reproductio rate always is the same, o matter whether the populatio is large or small. Figure 3(a) shows the graph of x +1 versus x for our curret model, (2). The graph is a straight lie with slope r. To take ito accout the effects of overcrowdig, we require a graph with a local maximum. A possibility is show i Figure 3(b), correspodig to the followig equatio: ( x +1 = rx 1 x ), (3) K with r > 0 ad K > 0. The graph of this equatio is a cocave-dow parabola, with roots at 3
x = 0 ad x = K, ad the maximum at x = K/2, as show i Figure 3(b). This model of populatio growth is referred to as the Verhulst process, or the discrete logistic equatio (we will ecouter the cotiuous versio of the logistic equatio i Chapter XXX). It should be clear that if the populatio durig ay year exceeds K/2, the populatio a year later is less tha K/2. That is, the populatio is self-regulatig. Oe complicatio with this model is that if the populatio durig ay year exceeds K, the populatio a year later is egative. To esure that the populatio always remais oegative, aother adjustmet ca be made to the model. Figure 3(c) shows the Ricker curve [2], give by [ ( x +1 = x exp r 1 x K )], (4) with r > 0 ad K > 0. We ca thik of the factor exp(r) as a costat reproductio factor, ad of the factor exp ( rx /K) as a mortality factor. The larger the populatio x, the more severe the mortality factor. As before, this populatio is self-regulatig. Both the Verhulst process ad the Ricker model are examples of oliear models. I cotrast to the liear map from the begiig of this sectio, the dyamical behaviour of these models is much more iterestig, ad ofte uituitive. There are may tools that allow us to predict the dyamical behaviour of these models, ad most of the rest of this chapter is devoted to developig ad explorig such tools i the cotext of the Verhulst process. The Ricker model will be ecoutered agai i the Maple sectio of this book, i Chapter XXX. 3 Cobwebbig, Fixed Poits, ad Stability Aalysis For all the models we have looked at so far, if we kow the values of the model parameters, plus a iitial coditio x 0, the we ca sequetially fid the etire orbit, that is, the iterates x 1, x 2, x 3,..., oe after the other. With the fast computers of today, it is easy to geerate may orbits ad deduce the dyamics of the model. However, it is easy to miss some subtle behaviour. We ofte ca gai valuable isight ito the model dyamics from sophisticated, but easy to lear, mathematical techiques. We will examie a few of such techiques i this sectio. Oe of the questios we would like to address is the logterm dyamics of the model. I the case of the liear map 2, we ca easily predict the -th iterate, amely x = r x 0, ad deduce that there is expoetial growth whe r > 1 ad expoetial decay if 0 < r < 1. I geeral, it is ot possible to write dow a formula for x if the model is oliear. However, we ca obtai a aswer to our questio from cobwebbig. Cobwebbig is a graphical solutio method, a method for visualizig the orbits ad their logterm behaviour without explicitly calculatig each ad every iterate alog the way. We demostrate the cobwebbig techique i Figure 4. Figure 4 shows the graphs of a fuctio x +1 = f(x ) ad the straight lie x +1 = x. We choose our first iterate, x 0, o the horizotal axis. The ext iterate is x 1 = f(x 0 ), which we ca just read off the parabola. Visually, this is show by a vertical lie from x 0 o the horizotal axis to the poit (x 0, x 1 ) o the parabola. The ext iterate, x 2, ca be obtaied i a similar way from x 1. We first eed to locate x 1 o the horizotal axis. We already have x 1 o the vertical axis, ad the easiest way to get it oto the horizotal 4
x +1 y = rx (1 x /K) = f(x) x 2 y = x x 0 x 3 x 3 x 0 x 1 x 2 x Figure 4: Cobwebbig for the logistic equatio, (3). This figure should be chaged slightly - choose r betwee 1 ad 3 so that the otrivial fixed poit is stable, ad the cobwebbig procedure gives a ice staircase as i the ext figure. axis is to reflect it through the diagoal lie x +1 = x. Visually, this is show by a horizotal lie from x 1 o the vertical axis to the poit (x 1, x 1 ) o the diagoal lie, ad the a vertical lie from the poit (x 1, x 1 ) o the diagoal lie to x 1 o the horizotal axis. I summary, oe starts by travellig from x 0 vertically to the parabola, the horizotally to the diagoal lie, vertically to the parabola, ad so o, as idicated by the solid portio of the vertical ad horizotal lies o the cobwebbig diagram i Figure 4. I this particular case, the orbit coverges to the itersectio of the parabola ad the diagoal lie. Ay itersectio of the parabola ad the diagoal lie represets a special poit. Let x be such a poit. The f(x ) = x. We call ay such poit a fixed poit or a equilibrium poit of the model. If ay iterate is x, the all subsequet iterates also are x. A questio of iterest is what happes whe a iterate is close to, but ot exactly at, a fixed poit. Do subsequet fixed poits move closer to the fixed poit or further away? I the former case, the fixed poit is said to be stable, whereas i the latter case, the fixed poit is said to be ustable. Examples of both stable ad ustable fixed poits are show i Figure 5. The three fixed poits show are x 1, x 2, ad x 3. Choosig a iitial coditio x 0 just to the left of x 2, we see that the orbit moves away from x 2, ad towards x 1. Similarly, choosig a iitial coditio x 0 just to the right of x 2, we see tha the orbit agai moves away from x 2, but ow towards x 3. We say that x 1 ad x 3 are stable fixed poits, ad x 2 is a ustable fixed poit of the model x +1 = f(x ). From Figure 5, it appears that the slope of f at the fixed poit has somethig to do with the stability of that fixed poit. I particular, the slope of f at the stable fixed poits x 1 ad x 3 is less tha 1 (the slope of the straight diagoal lie), whereas the slope of f at the ustable fixed poit x 2 is greater tha 1. We ca formalize these ideas via a liear stability aalysis, as follows: We choose the -th iterate to be close to a fixed poit x of (1), x = x + y, (5) 5
x +1 x +1 = x x +1 = f (x ) x* x 0 x* x 1 0 x* 2 3 x Figure 5: Illustratio of stable ad ustable fixed poits of the differece equatio x +1 = f(x ). The fixed poits x 1 ad x 3 are stable, ad the fixed poit x 2 is ustable. with y small, so that x ca be thought of as a perturbatio of x. The questio of iterest ow is what happes to y, the deviatio of x from x, as the map is iterated. If the deviatio grows, the the fixed poit x is ustable, ad if the deviatio shriks, the it is stable. We ca fid the map for the deviatio by substitutig (5) ito (1) to obtai x + y +1 = f(x + y ). (6) We expad the right had side usig a Taylor series about x to obtai x + y +1 = f(x ) + f (x )y + O(y 2 ). (7) Sice x is a fixed poit, we ca replace f(x ) o the right had side by x. If, i additio, we ca safely eglect all the terms i the Taylor series that have bee collected i the term O(y 2 ), the we are left with a map for the deviatio, y +1 = f (x )y. (8) We recogize that f (x ) is some costat, λ say. The map for the deviatio thus is the liear map y +1 = λy. (9) We ve already looked at the liear map i the cotext of populatio growth, (2). There, we restricted the growth parameter to be > 0. Here, f(x ) ca take o ay value, depedig o the map f, ad so there are o restrictios o λ. The behaviour of the deviatio y, ad the subsequet coclusio regardig the stability of the fixed poit x, ca be summarized as follows: λ > 1: geometric growth, fixed poit x is ustable; 0 < λ < 1: geometric decay, fixed poit x is stable; 1 < λ < 0: geometric decay with sig switch, fixed poit x is stable; λ < 1: geometric growth with sig switch, fixed poit x is ustable. 6
x x (a) (b) x x (c) (d) Figure 6: Behaviour of the geeral liear map, (9), for the cases (a) λ > 1; (b) 0 < λ < 1; (c) 1 < λ < 0; (d) λ < 1. The four cases are illustrated i Figure 6. Note that o coclusio ca be reached whe λ = ±1. These two cases require advaced treatmet, ivolvig a careful examiatio of the eglected terms collectig i the term O(y 2 ) i (7), which is beyod the scope of this book. For treatmet of these cases, the reader is referred to a textbook o discrete equatios, for example CITE SOMETHING HERE. More geerally, we ca summarize the results of the aalysis as follows: x is stable whe λ = f (x ) < 1, x is ustable whe λ = f (x ) > 1, there is o coclusio about the stability of x whe λ = f (x ) = 1. That is, the liear stability of a fixed poit x is determied by the slope of the map at the fixed poit, λ = f(x ). The parameter λ is geerally referred to as the eigevalue of the map at x. 4 Aalysis of the Discrete Logistic Map We ow retur to the logistic map, (3), ad apply the tools discussed i the previous sectio to this map. We begi with applyig the trasformatio x = x K to elimiate the parameter K, ad 7
_ x ustable 2/3 _ x = (r 1)/r stable stable ustable 0 1 2 3 4 case 1 case 2 case 3 r Figure 7: Partial bifurcatio diagram for the rescaled logistic map, (10). Show are the fixed poits ad their stability as a fuctio of the model parameter r. Solid lies idicate stability of the fixed poit, ad dashed lies idicate istability. obtai x +1 = f(x ) = rx (1 x ) (10) after droppig the overbars. Note that if we have x > 1, the x +1 < 0. To avoid such situatios, we impose the restrictio 0 r 4, so that x [0, 1]. The fixed poits of the map ca be foud exactly by settig f(x ) = x ad solvig for x. There are two fixed poits. The trivial fixed poit, x = 0, always exists, while the otrivial fixed poit, x = r 1 r, exists oly whe r > 1. To determie the stability of the fixed poits, we eed f (x), which is f (x) = r(1 2x). (11) At the trivial fixed poit, x = 0, the derivative is f (0) = r. That is, the trivial fixed poit is stable for 0 r < 1, ad ustable for 1 < r 4. At the otrivial fixed poit, x = r 1 r, we have f ( r 1 r ) = 2 r. That is, the otrivial fixed poit is stable for 1 < r < 3, ad ustable for 3 < r 4. The existece ad stability of the fixed poits is summarized i the bifurcatio diagram of the fixed poits versus the parameter r, show i Figure 7. Readig the diagram from left to right, ote that the trivial fixed poit becomes ustable as soo as the otrivial fixed poits come oto the scee at r = 1. The otrivial fixed poit is stable iitially, but loses its stability at r = 3. The two poits r = 1 ad r = 3 are kow as bifurcatio poits. A bifurcatio poit is a parameter value at which there is a qualitative chage i the dyamics of the map. The bifurcatio at r = 1 is called a trascritical bifurcatio. There are may other types of bifurcatios. A detailed discussio of bifurcatio theory is beyod the scope of this course, as is a detailed discussio of the dyamics of the logistic model for the case 3 < r <= 4. The iterested reader is referred to [3]. 8
Refereces [1] F. Adler, Modelig the Dyamics of Life: Calculus ad Probability for Life Scietists, Brooks/Cole, Pacific Grove, CA, 1998. [2] W. E. Ricker, Stock ad recruitmet, J. Fisheries Res. Board of Caada, 11 (1954), pp. 559 623. [3] S. Strogatz, Noliear Dyamics ad Chaos: with Applicatios to Physics, Biology, Chemistry, ad Egieerig, Addiso-Wesley Publishig Compay, Readig, MA, 1994. 9