18 Biological models with discrete time

Transcription

1 8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so on), and is successful applicaion o many fundamenally linear problems in he physical sciences, ends o dominae even moderaely advanced Universiy courses in mahemaics and heoreical physics. The mahemaical inuiion so developed ill equips he suden o confron he bizarre behavior exhibied by he simples of discree nonlinear sysems, such as equaion (3). Ye such nonlinear sysems are surely he rule, no he excepion, ouside he physical sciences. I would herefore urge ha people be inroduced o, say, equaion (3) early in heir mahemaical educaion. This equaion can be sudied phenomenologically by ieraing i on a calculaor, or even by hand. Is sudy does no involve as much concepual sophisicaion as does elemenary calculus. Such sudy would grealy enrich he suden s inuiion abou nonlinear sysems. o only in research, bu also in he everyday world of poliics and economics, we would all be beer off if more people realized ha simple nonlinear sysems do no necessarily possess simple dynamical properies. Rober M. May Simple mahemaical models wih very complicaed dynamics. aure, 26(556), 976, A cerain man pu a pair of rabbis in a place surrounded on all sides by a wall. How many pairs of rabbis can be produced from ha pair in a year if i is supposed ha every monh each pair beges a new pair which from he second monh on becomes producive? 8. Inroducion Leonardo Pisano Bigollo (7 25), known as Fibonacci Liber Abaci, 22 Up ill now I discussed mahemaical models of biological process ha are characerized by coninuous ime; his means ha a every ime insan i is possible o have only one elemenary even, and he parameers of my models specify he raes of he evens in he sysem, i.e., he number of evens per ime uni. For populaion models his, for insance, means ha he generaions of our models are overlapping, and birh and deah evens can occur a every ime insan. A number of biological sysems, however, can be characerized by discree ime, which means ha here are specific ime momens a which he elemenary evens in our sysem can occur, and i is no required ha a hese discree ime insans only a unique even happens. For example, such discree ime is reasonable o inroduce in some models of fish populaions, which reproduce a specific ime momens, or for insec populaions, for which quie ofen non-overlapping populaions are wha is acually observed in realiy. Anyway, for many siuaions (hink also abou observable ime series) I need a modeling ool o describe sequences of he variables ha are of ineres o me, and hance I am naurally led o consider discree maps or discree dynamical sysems in he form + = f( ), f : U U, U R, Z, () Mah 484/684: Mahemaical modeling of biological processes by Arem ovozhilov arem.novozhilov@ndsu.edu. Fall 25. Equaion (3) is given by X + = ax ( X )

2 where he index noaion emphasizes he discree characer of he ime variable in he sysem. Someimes, however, I will use he usual nuaion (), when i is more convenien. There is an equivalen noaion for sysem ():, U R. (2) Quie ofen he maps ha I consider are non-inverible, and in his case Z + = {,, 2,...}. If I am given an iniial condiion hen discree dynamical sysem () defines an orbi γ( ) = {, f( ), f 2 ( ),...}, where I use he noaion f k := f f... f f, for k imes composiion of funcion f (i.e., k imes successive applicaions of f). I.e., f 2 (x) = f(f(x)) and f 4 (x) = f(f(f(f(x)))). Example. Consider a simple example of a populaion growh. By definiion, a relaive populaion growh a ime momen is defined by r := +, where is he populaion size a ime. Assuming ha r = r = cons for any, I find + = ( + r) = w, w := + r. This equaion linear and can be easily solved explicily: = w = ( + r). Therefore, I obain an imporan conclusion ha he populaion is growing exponenially wihou bounds if w > (r > ), declining o zero if < w < ( < r < ) and says consan if w = (r = ). I is insrucive o compare hese hree phases of populaion behavior wih he soluions of he coninuous ime Malhus model Ṅ = m. In general, of course, linear models canno be used on long ime inervals, since hey predic eiher unbounded growh or exincion. To guaranee ha he orbi is bounded, I should consider a nonlinear model. Example 2 (Discree logisic equaion). Consider he discree dynamical sysem ( + = r ), K where r, K are posiive parameers. ow he model is nonlinear, and he populaion canno grow o infiniy. However, here is anoher drawback of his model: For exceeding K, + <, which conradics biological inerpreaion of he model. Example 3 (Ricker s equaion). To make sure ha he populaion is bounded for all and a he same ime is nonnegaive, I can consider he so-called Ricker model, which is widely used for modeling fish populaions: Here, obviously, for all > if. + = e r ( 2 K ).

3 Example 4. Concluding his shor secion, consider he second epigraph o his lecure, which mahemaically can be formulaed as + = +, where is he number of he pairs of rabbis capable of reproducion a he -h monh. Given he iniial condiions =, =, i is easy o see ha he soluion should be given by he sequence of Fibonacci numbers:,,, 2, 3, 5, 8, 3, 2, 34, 55, 89,... bu he quesion is how o find a general soluion o his equaion. Anoher hing o poin ou here is ha he equaion wrien in his example is no a discree dynamical sysem according o my definiion, since he populaion size a + is deermined hrough wo prior poins. This can, however, be fixed by considering an addiional variable and hen M =, M + =, + = + M, which is a wo-dimensional discree dynamical sysem. In general, a d-dimensional discree dynamical sysem is defined as a map of subse U of R d o iself: 8.2 Cobweb diagram x f(x), x U R d, f : U U. There is a simple and efficien way o ge an idea of he general behavior of he orbis of () by looking a he graph of funcion f. Before describing his mehod, I noe ha poin is called a fixed poin of () if f( ) =. Geomerically his means ha fixed poins are he poins of he inersecion of he graph of f and he bisecrices of he firs and hird quadrans on he coordinae plane. ow consider a discree map wih f shown in he figure below. 2 2 Figure : Cobweb diagram 3

4 Le be an iniial poin, herefore, = f( ) is he poin of inersecion beween he graph of f and he verical line passing hrough. ow I can use he diagonal + = o find he locaion of on he -axis: This can be done simply by finding he inersecion of he horizonal line wih he ordinae and he diagonal. Afer his I can projec o -axis o find. 2 = f( ) and so on. The whole orbi is simply is a series of reflecions from he diagonal (see he figure). The picure I obain is someime called a cobweb diagram. Consider now he siuaion as in Figure 2. The large black dos show he locaions of he fixed poins, small black do is he iniial condiion, and ogeher wih he cobweb diagram I presen he ime series ( ) k = on he righ. A lile playing wih cobweb diagram and choosing differen iniial condiions should convince you ha he picure presened in he righ panel of Fig. 2 is universal: For any iniial condiion he orbi approaches he fixed poin, and he convergence o his poin (excep for maybe several few seps) is monoonous. I is naural o call such fixed poin asympoically sable or a sink Figure 2: Cobweb diagram Using cobweb diagram I can ge an idea wha kind of phenomena can be expeced in discree dynamical sysems. For example, in Figure 3 one can see ha an asympoically sable fixed poin Figure 3: Cobweb diagram. on-monoonous convergence can arac orbis in an oscillaory way. This fac alone should convince you ha one dimensional discree dynamical sysems possess a richer behavior in comparison wih scalar ordinary differenial 4

5 equaions. Due o he fac ha he sae space for he scalar equaions is one dimensional, and orbis canno inersec, any non-monoonous behavior of soluions o ODE is prohibied. In scalar discree dynamical sysems a periodic soluions can be observed (see Fig. 4). I can be seen in he figure ha here are wo poins and 2 such ha 2 = f( ) and = f( 2 ), hence = f 2 ( ) and 2 = f 2 ( 2 ). The orbi {, 2 } is called a 2-periodic orbi, or orbi wih period Figure 4: Cobweb diagram. Periodic soluions A remarkable fac is ha if sysem () has a 3-periodic soluion hen i has k-periodic soluion for any k. An example of a 3-periodic soluion is given in Fig Figure 5: Cobweb diagram. A 3-periodic soluion Finally, aperiodic orbis can be observed. In Fig. 6 an example of such an orbi is given. In he op row firs 5 (lef panel) and 25 (righ panel) poins respecively of he orbi are shown. Such orbis are called chaoic, and he sysem iself is chaoic. I will need a few mahemaical preliminaries o define wha chaoic means exacly, bu here you can imagine wriing down every ime he coordinae of he orbi is below.5, and when he coordinae is above.5. As a resul you will ge a sequence like... You can produce a similar sequence by ossing a coin and recording if i is a head and if i is a ail. You will ge anoher sequence. Boh sequences look (whaever his means) random. Moreover, here exiss no saisical es o deermine which sequence is produced by a random experimen (ossing a coin) or by a deerminisic discree dynamical sysem of he form (). 5

6 Figure 6: Cobweb diagram. A chaoic orbi 6