Discrete population models

Transcription

1 Discrete populatio odels D. Gurarie Ratioal: cclic (seasoal) tiig of reproductio ad developet, schroizatio Topics:. Reewal odels (Fiboacci). Discrete logistic odels (Verhulst vs. Ricker); cobwebs; equilibria, ccles, chaos 3. Discrete-dela odels: a. Delaed Ricker b. IWC whale odel c. Tuor growth Reewal/growth odels; Fiboacci Fiboacci sequece: x = x + x ; x = ; x = (.) + defies secod order liear recurrece, or fiite differece equatio. Like higher order DEs such differece equatios ca be coverted to a atrix sste xt Xt+ = AXt; with Xt= ; atrix A x = t It ca be viewed as two-stage populatio sste: xt - oug, x = ; = + x t+ t t+ t t i uit tie step. More atural discrete tie step -stage odel obes where b xt+ = bt; t+ = sxt + sat; with atrix A= s s a t () - adults, that grows (reews) as (.3) b > - proliferatio (growth) factor, < sa, < - survival fractios (of oug ad adults), so µ, = - ortalit factors. Stabilit of solutios () X = A X, depeds o a s a, eigevalues of atrix A s s λ = ± + = 4 a a, b? Show (i) ustable λ >, correspods to sb > µ a (survivig oug exceed adult death reoval); (ii) stable (deca) gives sb < µ a.

2 A geeral fiite differece equatio xt+ = ax t axt () ca also be solved b characteristic poloial ( ) a p λ = λ λ... a whose (coplex) roots give special power solutios { λ j }, ad geeral oe k = jλ, j= x c j λ λ 5 i particular, Fiboacci ubers: x = ; λ, = ±. λ λ Verhulst ad Ricker Discrete Logistic Models. The offer two differet was to discretize cotiuous logistic odel: i tie steps t, so t = t, ad = ( t ). ( / ) Verhults discretizes derivative: ( ) = r N (.5) + r / N t, whece ( ) r N r r t N + r t r t * * * * + = / ; with = +, = N Ricker solves (.5) over short tie [ t, t t] r( / N) r( / N), to get +, assuig ear costat rate coefficiet () * / N * r t = + r e ; r = e (.7) I both cases rescaled variable u = / N solves a oliear -st order recurrece: ( u) ru - quadratic (Verhulst) u+ = f ( u) ; where f ( u) = u rue - Ricker Verhulst populatio odel does ot allow values above threshold N (urealistic), while Ricker has o such liitatios (ore realistic). iclude ( ( )) Solutio of () are ade of iterates of ap f : u f f... f ( u ) ( ) (i) equilibria: u = f u - fixed poits of f ; =. Special cases ties ()

3 (ii) - ccles (periodic orbits): u t + ut (iii) chaotic trajectories. Stabilit of equilibriu u { }, or ccle So B < are stable, ad B > (ustable). paraeter ( )) = - fixed poits of ( )... ( f u = f f f u ; u, u,... u is deteried b liearized odel: ( ) ( ) ut+ = But; where B = f u, or f u (for ccle) (.9) Discrete logistic odels () exhibit a coplex chai of bifurcatios i ters of growth r >, that progresses fro stable equilibria to liit ccles of differet periodicities to chaos. The are suarized i the followig table: < r < 3 stable equilibriu = ê r 3 < r < r 4 stable period : orbit r 4 < r < r 8 period doublig: orbit r k < r < r k+ stable k orbit r 3 < r < r 3, stable 3 orbit r 3,k < r < r 3,k+ 3 k orbit r c < r chaos Two iportat tpes of bifurcatios that occur i iterated aps are period doublig illustrated i Fig., ad tagetial bifurcatio (e.g. period 3 ccle, or a other odd ). I period doublig a stable -ccle (i.e. fixed poit a ew stable pair { * * * } * of ( ) f ) looses stabilit for f ( ), but < < (-ccle of f ) coes i place. So period doublig serves as a discrete versio of pitchfork bifurcatio: stable equilibriu stable-ustable-stable triplet. Tagetial bifurcatio (triple ccle) is illustrated i Fig.: three stable fixed poits of f 3 coe out of the coplex doai at critical r 3 = 3.88, ad develop ito stable period-3 ccle of ( ) f i the rage r3 < r< r 6, whe aother period doublig occurs. Fig.3 shows critical case r 3 ad period-3 ccle. Proble: Show that -ccle {,..., } of ap f ( ) fixed poit of the -th iterate f ( ) is stable if ad ol if the correspodig is stable. Hit: liearize f about - ccle, ad show that the resultig liear sste is give b a Leslie tpe (cclic) atrix with etries aj = f ( j)

4 A... a a... = a... a Use characteristic poloial ( ) fixed poits of f. det λ A = λ ± a... a, ad lik stabilit of A to stabilit of r= tie (a) (b) r= tie Power ω Fig.: Period doublig: Plot (a) stable equilibriu ( ) ( ) * r r = / ; (< < 3) for f ( x ) ad = ( ), turs Plot (b) ito stable -ccle { * * * } f x f f x * * ( ) ( ) f > ; f i <. < <, so that

5 C B A Fig. : Tagetial bifurcatio at r 3 = 3.88 (a) r= tie Power (b) r=3.86. ω tie Power. ω Fig.3: Daics of quadratic ap i two cases: (a) critical r 3 = 3.88 ; (b) stable 3-ccle. Further details ad coputatios are give i Matheatica otebook.

6 Fourier Power Spectra Periodic or chaotic (tie series) solutio ca be exaied b Fourier ethods (power spectra) discussed i the otebook. We recall that a (geeralized) Fourier expasio () f t ˆ i kt ~ fke π ω k=, cosists of (aplitude) coefficiets { ˆk } f, ad frequecies { ω } periodic fuctios f with period T, all frequecies are ultiples (haroics) of the lowest oe: π ωk = kω; ω =, ad coefficiets T ˆ πikt () πikt fk = f t e dt = f e - (square-ea) ier product of f ad expoet. More geeral quasi-periodic fuctios have arbitrar set of ω s. Frequecies { ω k } defie power { } spectru ( ) k. For ˆf ω of sigal f () t. So spectral peaks i Fig.,3 (right botto) idicate the ost iportat frequecies that appear i log ter series if iterates { k }. Oe ca see frequecies ω = / ;/ 3;... correspodig to -ccles, 3-ccles, etc.