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1 Suppleenary Onlne Maeral In he followng secons, we presen our approach o calculang yapunov exponens. We derve our cenral resul Λ= τ n n pτλ ( A pbt λ( = τ, = A ( drecly fro he growh equaon x ( = AE x ( (2 ( We hen use perurbaon heory o show how he lag es (expressed n ers of egenvecor proecons can be calculaed o arbrary precson n ers of he swchng raes. We gve a ore general forulaon han n he an ex, allowng he nuber of envronens (n o be dfferen fro he nuber of phenoypes (. We use noaon nroduced n Fgure, an ex.. yapunov Exponens for Srucured Populaon Growh In order for our general approxaon o be vald, we requre he op egenvalue of he arces A o be real, and for here o be a gap n he egenvalue specru beween he wo egenvalues of larges agnude. Furherore, we wll resrc our dscusson o cases when he op egenvecor s non-negave. In he case of responsve swchng, hese condons hold, as seen n secon 3. In he case of sponaneous sochasc swchng, hese condons are guaraneed by he Perron-Frobenus heore. We apply he heore o he nonnegave arx A + γ I, where I s he deny arx and γ s a large posve consan. The egenvecors of hs arx are dencal o hose of A. As

2 long as each phenoype can gve rse o any oher phenoype, va soe sequence of swchngs, he arx A + γ I s prve, and he heore ay be appled. Prvy holds rvally f all swchng raes are srcly posve. We subdvde e no consecuve nervals n whch he envronen varable E( does no change. The duraon of he l-h such nerval s denoed T l, and he oal e elapsed by he end of he -h nerval s denoed of he envronen durng nerval l s denoed ε(l. = Tl l=, where 0 = 0. The sae We nroduce he generalzed egenvecors vr whch brng he arx A o s Jordan-bloc for. Tha s, f we defne he arx M = v v, hen he arx M AM s n Jordan-bloc dagonal for (. The egenvalue assocaed wh he r-h egenvecor s denoed λ ( A, and hese are decreasng wh ncreasng r, and r appear wh ulplcy (he op egenvalue s non-degenerae. If duraons T l are suffcenly long (we wll explan wha hs eans shorly, he drecon of he populaon vecor a he end of he (l--h nerval s very close o ha of he op egenvecor of envronen ε(l-. Thus x( N( v, where we choose ε ( l l l he op egenvecors noralzed so her enres su o one: ( v s = for all (hs s always possble due o he posvy of he op egenvecors. When he envronen changes, we ay hen sply proec hs op egenvecor ono he new egenbass o s= descrbe he dynacs n he new envronen. Proecng x( ono he egenbass of l envronen ε(l, he coponen along he new op egenvecor v ε ( l s gven by 2

3 q N( ε( l ε( l l, where q e M M e and e = (,0,0. Noe ha s q ndependen of he agnudes of all egenvecors oher han he op egenvecors, and hus our prescrbed noralzaon of v unquely deernes q. The e evoluon of N ( for l < l s hen gven by ( λ ( Aε ( l ( l ( l ε( l l l N ( = q e + G ( N ( (3 ε λ ( A T ε ( l where GT ( s a funcon ha grows slower han e, and G(0 = q ε ( l ε ( l. The exac for of ( depends on he lower egenvalues of A ε and her ulplces, and GT ( l on he proecon of v ε ( l ono her correspondng egenvecors. The yapunov exponen s hen gven by he followng l: λ ( Aε ( l Tl ( qε( l ε( l e λ ( Aε ( l Tl ( qε( l ε( l e G Tl log N ( Λ= l = l log + ( l l= log l= = l Tλ ( A + l log q τ l ε( l ε( l ε( l l= τ l= (4 where τ = l (he average duraon of envronens. The approxaon n (4 s vald when λ ( A T ε ( l l qε( l ε( l e G( Tl. Ths can be acheved f Tl are suffcenly long. We have an explc bound on GT ( when he oher egenvalues are non-degenerae: λ ε( l ε( l 2 ( A ( T ε l GT ( K e, where K = ax ( r M ( r s r s= e v v and are he sandard Eucldean bass vecors. Thus he e r followng condon s suffcen: 3

4 T l K λ ( Aε( l λ2 ( Aε( l qε( l ε( l ε( l ε( l log (5 If we defne Tn K = ax log, λ( A λ2( A q, hen our approxaon s vald for envronenal duraons Tl T n. The rue yapunov exponen approaches he approxaon exponenally fas n T, for T > T, due o he exponenal decay of he l correcon er negleced n (4, so he approxaon s a very good one n hs rege. l n We can furher splfy equaon (4 f we assue ha envronenal changes follow he Marov chan b. We denoe he duraon of he -h occurrence of envronen usng he rando varable ( T (hs s us a regroupng of he rando ( varables Tl. For fxed, we assue ha he varables are ndependen, dencallydsrbued varables, wh eanτ. If s he nuber of nervals elapsed, hen for large, he nuber of occurences of envronen approaches p, and of he envronen par followed by approaches p b, so he yapunov exponen s T τλ= T A + p b q n p n ( l λ ( log = =, = n = pτλ( A + p b logq =, = n (6 Ths s our cenral resul allowng copuaon of yapunov exponens for srucured populaon growh. I can be nerpreed n ers of delay es as follows. The quany q s he proecon of he populaon vecor a he end of envronen ono he op egenvecor of he new envronen (he proecon operaon uses he new egenbass, va he arx M. If he populaon sze s N, hen N q s he sze of he 4

5 subpopulaon nally growng a he rae λ ( when he envronen changes. Defnng he delay e T A as he aoun of e aes hs subpopulaon o reach a sze of N, we fnd T = (log q / λ ( A, and subsung hs no he above equaon, we oban equaon (. Rewrng condon (5 usng hs defnon, we fnd T ( log( K + λ ( λ( A λ2( A T A, where T s he envronenal duraon. Copuaon of yapunov exponens s now reduced o copuaon of log q and λ ( A, whch s done for sponaneous and responsve swchng n he nex wo secons. I s soees easer o drecly copue he delay es T, and hs gves dencal resuls, as shown n secon yapunov Exponen for Sochasc Swchng Assung ha he sochasc swchng raes are sall copared o he growh raes, we can use perurbaon heory o wre he egenvalues and egenvecors of A o frs order n (, and use hese expressons n equaon (6 o copue he long-er growh rae. The forulae for he egenvecors, v, and egenvalues, λ ( A, of he arx are as r r A follows (2: ( ( ( v e e, and λ ( A = f, where r = r + C r r r r rr Δ f f, ( ( ( C = / Δ, for, and ( ( ( ( C = 0 for all. We wll use C o refer o he arx ( C ( wh enres. 5

6 Reurnng o he defnon of q, and usng α o denoe he fases-growng phenoype n envronen, we can wre q M = M α α e e. Ths expresson gves he correc value for q when he order of he egenvecors appearng n he arces M and M s arbrary. We expand M M o frs order n he swchng raes: ( ( ( ( M M = I + C I + C I C I + C I + C C ( ( ( ( ( ( where I s he deny arx. I follows ha q = + C C α. If α = α, we ( ( δ α α αα α have log( q = 0 o frs order, and f α α ( (, log( q log( Cαα C αα. Usng hese expressons n equaon (7 gves he yapunov exponen o frs order: ( ( τλ = pτ ( fα p b log αα + + ( ( αα αα S ( (, : α α Δαα Δα α In he case ha ( =, and α =, we recover he expresson gven n he an ex. If he perurbaon expanson were carred ou o second order, ers lnear n would appear n he second su above, wh coeffcens ha are ndependen of τ. Snce he lnear ers n he frs su are proporonal o τ, for large τ we are safe n gnorng any lnear conrbuon cong fro he second su. Usng hgher order perurbaon heory for he egenvalues and egenvecors, he yapunov exponen can be calculaed as a seres expanson n, f desred. Noce ha he use of α above allows one o exane he general case n whch he nuber of phenoypes and envronens are no necessarly equal. We noe ha he case of n envronens and phenoypes, wh n >, can be apped, by approprae choce of paraeers, o a proble of 6

7 phenoypes n envronens, wh he yapunov exponen ang he sae general for as he expresson gven n he an ex. 3. yapunov Exponen for Responsve Swchng In he case of responsve swchng, he arces A ae he followng for: A f ( ( f 0 ( = α 0 f 0 ( 0 0 f f ( The arx of egenvecors of A and s nverse are as follows, where r ( = +Δ ( α : M r r r r r r r r = = ( Snce he leadng egenvecor n envronen s e α, we fnd q = rα =. ( +Δ ( ( ( ( ( ( ( ( α M α + α α+ α α Usng equaon (6, he yapunov exponen s gven by R n n ( ( pτfα pb α α =, = τ Λ = log( +Δ / Tang α = and replacng ex. f by ( ( f c, we oban he expresson gven n he an 7

8 4. Mehod of Delay-Tes for Copung yapunov Exponens The resuls of he prevous wo secons can also be derved by calculang he delay es nroduced n secon, and usng he relaon log q = λ ( A T n equaon T (6 o copue he yapunov exponen. As explaned prevously, he delay e s he aoun of e aes, upon a change of envronen fro o, for he subpopulaon growng a he fases rae n envronen o reach a populaon sze equal o he oal populaon sze ha was reached a he end of he envronen. These es can herefore be copued drecly by soluon of dfferenal equaons, as follows. We wll ae repeaed use of he followng par of equaons x = γ x x = γ x + δ x ( (0 (0 γ T δ (0 T x T = x + x e + x e γ. δ whose soluon s ( γ γ γ γ T For responsve swchng, assue he populaon has reached a sze N a he end of envronen. The populaon s alos enrely coposed of phenoype. When he envronen changes o, phenoypes swch o phenoype a a rae. T s he e aes for phenoype o reach a sze of N. In he above equaons, hs corresponds o ( γ = f c, f γ = c, and δ =, and s found by solvng ( T x T ( = N. For large, γ γ, we can sply solve ( ( + ( c T Ne ( f N f f =, fndng ( ( R T = f c log( +Δ /. Noe ha hs expresson holds for sall as well, snce sall ples T s large, so we are agan usfed n gnorng er e γ T. Snce 8

9 he bes phenoype n envronen does no swch o any oher phenoype, ( ( λ A = f c. For sochasc swchng wh sall swchng raes, he fases phenoype n envronen has reached a populaon sze N, and here s a sall aoun of phenoype due o swchng a rae fro phenoype. Ths aoun s found by ang γ = f (, γ = f, and δ = n he above equaons, and assung ( ha T s long enough and swchng raes are sall, ( f T ( ( f f x ( T = x (0 e. Snce f ( T x (0 e N, we fnd he aoun of phenoype s equal o N f f ( ( /(. When he envronen swches o, T sze N. Ths s found by solvng s approxaely he e aes phenoype o reach x T ( = N wh γ = f ( (, f γ =, δ =, x (0 T ( ( N, and x (0 N/( f f, gvng S ( ( = log + = log Δ / o lowes order n. Snce s sall, ( ( ( ( ( ( f f f f f f T can be wren as n he responsve case: log ( ( S T / +Δ. The op egenvalue f ( o frs order n swchng raes s ( λ A = f. 5. yapunov Exponens and Envronenal Flucuaons The yapunov exponen deernes he long-er growh rae of a populaon characerzed by growh/swchng arces A when presened wh a changng envronen gven by he process E(. Rearably, equaon ( shows ha, provded he envronens rean consan for long enough perods, he long-er growh rae wll 9

10 depend only on he ean duraons of he envronens, τ, and on he parwse correlaons beween he, b and wll no depend on oher characerscs of he envronenal flucuaons. To see hs drecly, we underoo he followng nuercal exercse. We consdered wo envronens and wo phenoypes, descrbed by a par of 2-by-2 arces, A and A 2. In hs case, b s rval, as envronen always follows envronen 2, and vce versa. The only non-rval envronenal flucuaon s due o he process generang he duraons of each envronen. For splcy, we ep he duraon of envronen 2 consan, ha s, f ( T s he duraon of he -h occurrence of envronen, we oo T (2 (2 = τ for all. The rando varables hus have he dela 2 T funcon probably dsrbuon, cenered a he value τ 2. For he duraon of envronen we exaned hree dfferen dsrbuons: unfor, exponenal, and dela funcon. For each dsrbuon, we generaed any realzaons of he envronen, and calculaed he yapunov exponen nuercally. In Fgure SA, we plo he yapunov exponen as a funcon of τ. The exponen depends srongly on he dsrbuon ha deernes he envronenal flucuaons. Accordng o equaons ( and (5 we expec he exponen o becoe ndependen of he exac ( dsrbuon f s always larger han soe cuoff value. Tang hs cuoff o be 0, T and usng he sae hree dsrbuons, we fnd n Fgure SB ha hs s ndeed he case: he yapunov exponens calculaed usng dfferen dsrbuons for ( T are dencal, and depend only on he ean duraon, τ. 0

11 ( Ths concluson holds only f he envronenal duraons are all suffcenly T large, ore precsely, f T ( T n for all and. In hs case, equaon ( also gves a way o calculae he exponen. The resul of hs exac calculaon s gven by he sold lne n Fgure SB, showng ha equaon ( s n excellen agreeen wh he nuercal resuls. 6. yapunov Exponens for Fne Populaons So far we have allowed he populaon sze o grow whou bound. Suppose nsead ha a axu populaon sze, N, s posed, for exaple by perodc resaplng. The yapunov exponen, as defned by Λ l log N (, would be zero, because he oal populaon sze asypocally would no grow. If wo dfferen srans of organs were copeng whn a fxed populaon sze, however, one would evenually go exnc. A ore general defnon of he long-er growh rae us herefore exs, whch we now descrbe. Fro he orgnal populaon growh gven n (2, we see ha he oal populaon N ( f ( x ( f ( n ( N (, where f ( and sze, N(, obeys he equaon = = ( n ( x (/ N( are he growh rae and frequency of phenoype a e. Soluon of hs equaon yelds N ( N(0exp f( n( d, suggesng he followng = 0 defnon of he long-er growh rae: Λ l ( ( 0 f n d (8

12 Ths defnon has he advanage of dependng only on he frequences of phenoypes, so s eanngful even when populaon sze s fxed. For unled growh, Λ= l log N (, so agrees wh he prevous defnon. Calculaon of Λ for fne-sze populaons usng sochasc sulaons was perfored and was n excellen agreeen wh he calculaed value gven by equaon (, for populaon szes N / q. For saller populaon szes, devaons fro hs value of Λ were observed, and could be accouned for by replacng T by he approprae expecaon of he delay e (o be descrbed elsewhere, provded N > / q. When populaon sze s so sall ha slower phenoypes are no suffcenly represened ( N < q, he heory presened here ay no hold. / 7. Phenoypc Meory Suppose he probably of a ranson o envronen depends boh on he curren envronen,, and on he prevous envronen,. We defne copose ndces I = (, and J = (,, and wre b IJ o ean he probably of a ranson o envronen, gven ha he par of envronens conssng of followed by has occurred. In a slar anner, we can consder he phenoypc hsory of an ndvdual,.e. he seres of phenoypc ransons ha occurred n s ancesral lneage gong bacwards n e. By phenoypc eory we ean he ably o reeber a fne nuber of hese ransons, ncludng he curren phenoype. We sress ha such eory s long-er n he sense ha he gven ndvdual reebers no only s edae ancesor s phenoype (hs 2

13 wll usually be dencal o s own phenoype, f swchng raes are sall, bu also he phenoypc saes of he las few phenoypc ransons n s ancesral lneage. We use IJ o denoe he rae of swchng o phenoype fro phenoypes ha were prevously (n he ancesral lneage. The generalzaon o eory of phenoypes, and -pon envronenal correlaons, s by copose ndces of he for I = ( 0,,, - and J = (, 2,. ere our convenon s ha he ably o reeber only he curren phenoype, n a flucuang envronen whose ranson probables depend only on he curren envronen, corresponds o =,.e. he case consdered n he an ex. Wh hs noaon, he expresson for ΛS gven n he an ex holds, for sall swchng raes, when all ndces are replaced by copose ndces. To see hs, for exaple f = 2, consder he dervaon of equaon ( and suppose ha an envronenal ranson J = (, 2 o I = ( 0, occurs. The leadng phenoype a he end of envronen J s of he ype (ore precsely, s an ha cae fro 2, or sply phenoype J. There s also a subpopulaon of 0 a he end of envronen J, specfcally 0 ha cae fro, or sply phenoype I. When he envronen swches o 0, hs phenoype I wll be aplfed unl donaes he populaon. The opal swchng raes are agan gven by ( opal = b / τ. As n he IJ IJ J case of sensors, here s a axal cos for whch eory s benefcal. If we le I env ( be he envronenal enropy when -pon correlaons are consdered, and Λ ( S be he correspondng yapunov exponen, hen a basc heore n nforaon heory (3 saes ha I = I ( I (2 I (3... ec. The dfference n growh rae beween env env env env organss wh -pon eory vs. -pon eory s gven by 3

14 ( τ ΛS( Λ S( = Ienv ( Ienv (, assung ha τ and f are unchanged and ( depend only on he curren envronen. If he cos of such eory s c, eory s benefcal for c < ( Ienv ( Ienv (. τ We sress ha our analyss and resuls peranng o phenoypc eory are usfed only n he l of very large populaons, and sall swchng raes. For fne populaons, a ore delcae reaen s necessary, whch we wll no underae here. To see why, consder a sall populaon and a very long-lasng envronen J = (, 2. Evenually, he populaon wll be donaed by phenoype, bu because of he sall populaon sze and long envronenal duraon, we canno neglec he fac ha oher phenoypes wll evenually swch o phenoype. Thus, he populaon a he end of envronen J ay be donaed by a xure of phenoypes (,, for varous values of, raher han only he phenoype J. 8. Naural Selecon and Phenoype Swchng Mechanss Naural selecon can anan a phenoype swchng echans, as follows. Suppose ha n envronen a uaon arses (wh frequency ω abolshng swchng. The new genoype grows a a rae ( f, whle he swchng genoype grows a a rae f (. The e o fxaon s log(/ ω / = τ log(/ ω, for opal swchng raes. Snce ω s sall (, hs e s longer han τ. If ypcal duraons of he envronen are close oτ, he uaon wll no reach fxaon. The sae holds for nonopal swchng raes, provded ha are sall. The echans can be los, 4

15 however, f a very long envronenal duraon occurs. The wdh of he dsrbuon of envronenal duraons and he behavor of s al hus play a role n deernng wheher a swchng echans can be ananed. Ceran srans of Candda albcans, for exaple, do no swch phenoypes, and ay have los he ably o do so. References. S. Roan, Advanced near Algebra, Graduae Texs n Maheacs (Sprnger- Verlag, New Yor, 992, vol R.. boff, Inroducory Quanu Mechancs (Addson-Wesley, New Yor, ed. 2nd, T. M. Cover, J. A. Thoas, Eleens of Inforaon Theory (John Wley & Sons, Inc., New Yor, A. Crsan, G. Paladn, A. Vulpan, Producs of Rando Marces (Sprnger- Verlag, Berln,

16 Fgure Capon Fgure S: Dependence of he yapunov exponen on envronenal flucuaons Nuercal copuaon used he arces A = and. 6 A2 = To calculae each pon, we generaed a realzaon of he changng envronen n whch each envronen occurred es, he duraon of envronen 2 was fxed a 20 hours, and he duraon of he -h occurrence of envronen was a rando varable ( ( T A 20 A2 copued he arx produc G = e e, and hen esaed he yapunov = ( T. We exponen usng he forula τ Λ= log(tr( G / 2, ang = 00 (see (4. We averaged hs value over 00 separae runs. A. The yapunov exponen ploed for ( T havng an exponenal (large open squares, a unfor (edu open squares, or a dela (sall flled squares dsrbuon. The unfor dsrbuon exended fro 0 o 2τ, he rae of he exponenal dsrbuon was /τ, and he dela dsrbuon was cenered a τ. B. The yapunov exponen ploed for he hree dsrbuon fro panel A, bu each ( 20 A2 T A 0 A dsrbuon was shfed by 0 hours. Ths was done usng G ( e e e = The sold lne s he copuaon of τ Λ usng equaon (6. =. 6

17 Fgure S A B 4 25 τλ τλ τ τ