How to explore replicator equations? G.P. Karev

Transcription

1 How o explore replcor equos? GP Krev Locheed r SD Nol Isue of Helh Bldg 38 R 5N5N 86 Rocvlle Pe Behes D 2894 US E-l: rev@clhgov src Replcor equos RE) re og he sc ools hecl heory of seleco d evoluo We develop ehod for reducg wde clss of he RE whch geerl re syses of dfferel equos Bch spce o escor syses of ODEs h y cses c e explored lyclly The ehod hs poel for dffere pplcos soe exples re gve Iroduco Cosder syse fored y vrees ech of whch s chrcerzed y he specfc vlue of he vecor-preer ) I geerl vecor c e cosdered s crose of he syse he preers y hve dffere org Le l e he desy of dvduls v he se he oe so h l s he uer of dvduls wh vlues of he phse volue v d N) l s he ol populo sze oe The dycs of he syse s defed y he followg equos dl / d l F ) P ) l / N ) The reproduco re fess ) F s supposed o e sooh fuco of d esurle fuco of y deped o soe exesve vrles whch re soe verges over P The l dsruo P ) d he l populo sze N) re supposed o e gve I s ow h N ) ssfes he equo dn / d NE [ F] here d elow we use he oo E [ ϕ ] ϕ ) P I s lso ow [] h solves he replcor equo dp / d P F E [ F )]) d h he soluo of RE gve l dsruo P ) s uque f exss) I he ls decdes ws dscovered h replcor equos pper o oly populo geecs d seleco heory [2] u lso very dffere res such s heorecl ecology [9] dycl ge heory [4] d eve soe physcl proles see he survey [3] os of hese pplcos ssue h he fess depeds lerly o he frequeces Here we show h wde clss of replcor equos cludg hose wh he ler fess c e solved P 2)

2 explcly d he soluo hs for of e-depede Bolz dsruo The oed resuls re ppled o soe prculr seleco syses d correspodg replcor equos 2 The ehod If he reproduco re F s ow explcly s fuco of he he RE c e esly solved: P exp ) Φ P ) Z ) where Φ ) F u du d Z ) E [exp Φ )] Geerlly he reproduco re F s o gve s explc fuco d should e copued depedg o he curre populo chrcerscs For exple wdely used logsc odels hve he reproduco re of he for F ϕ N ) / B) where B s he upper oury of he populo sze So we should explore he seleco syses wh he reproduco re h c deped o soe egrl chrcerscs of he syse We ccou for exesve chrcerscs he for G ) g l N ) E [ g ] 2) whch deped o he ol syse sze d populo deses d esve chrcerscs he for H ) h P E [ h ] 22) whch do o deped o he syse sze u oly o he populo frequeces We wll refer o oh of he s o regulors for revy Flly we hve he followg geerl verso of he ser odel: dl / d l F 23) F u G ) ϕ + v H ) ψ P ) l / N ) where u v re ppropre fucos The l pdf P d he populo sze N) eed e gve The syse dsruo P solves he replcor equo dp ) / d P F E [ F )]) 24) where ow E [ F )] u G ) E [ ϕ ] + v H ) E [ ψ ] d ll regulors G ogeher wh E [ ϕ ] E [ ψ ] re o gve fucos of e d should e deered odel 23) ws suded [8] see lso [7] for dscree e verso) The developed heory yelds effecve lgorh for vesgo of seleco syses wh frewors of odel 23) d for solvg of replcor equo 24) Le us descre he seps of he lgorh Cosder he proly spce { P } d defe he fucol H

3 r exp λϕ + r λ δ) δ ψ ) P for esurle fucos r o he spce supposed o e esurle o hs spce Defe he uxlry vrles q s 25) { P } ll he fucosϕ ψ re lso y he escor syse of ODE dq / d u G * )) q ) 26) ds / d v H * )) s ) where we deoe G * ) N) g q ) s )) H * ) h q ) s )) / q ) s )) Le K exp q ) ϕ + s ) ψ ) he he soluo o syse 23) l l K G ) G * ) H ) H * ) N ) N ) q ) s )) P ) P K / E [ K ] 27) Forul 27) whch gves he soluo of replcor equo 24) s he cerl resul of he heory The geerl ehod s splfed por cse of he reproduco re F f ) φ wh he regulors S of he fors N ) E φ ] N ) E [ φ ] oly I hs cse we c use he oe geerg fuco of he jo l dsruo of he vrles S [ { φ } oly λ) E [exp λ φ )] sed of geerl fucol 25) The escor syse reds dq / d f S )) q ) where S ) re defed wh he help of foruls N ) N) q )) E [ ϕ ] l q )) Here we deoed δ ) δ) / δ for revy The soluo of correspodg replcor equo P ) P K / E [ K ] E [ K ] q )) The followg exples deosre he lgorh wor 3 pplcos d exples Ihoogeeous lhus odel d he odel of erly evoluo The sples replcor equo wh sgle couous preer reds dp / d P E [ ]) The correspodg seleco syse s he hoogeeous lhus odel dl / d l

4 Le λ) exp λ P The he soluo of he odel l exp l N ) N ) d he soluo o he replcor equo P P exp ) / ) Soluos of hoogeeous lhus d logsc odels d her pplcos were suded [5 6 8] I ws show h eve hese sples cses he replcor equos possess vrey of soluos depedg o he l dsruo whch y hve y eresg d eve coueruve peculres Le us deosre soe of he o he followg hoogeeous lhus odel wh lg fcors odel of erly ologcl evoluo ws suggesed [2] Ech orgs s chrcerzed y he vecor where he copoe s he herodyc proly h proe s s ve coforo I order o sudy he coeco ewee oleculr evoluo d populo he uhors supposed h orgs deh re d depeds o he sly of s proes s d d ) d cos Hece eglecg possle uos ccoued for y he uhors her sulos) he odel c e forlzed s he syse dl ) / d l B )) 3) [ ] B / where ) s he rh re s he ve se proly of proe I wh follows we pu re depede of ech oher we c cosder B for splcy Followg [2] suppose h he vlues s he -h relzo of rdo vrle wh coo pdf f The s well ow h he pdf of [ ] where re depede declly dsrued rdo vrles s g ) G )) f ) where G ) f s he cuulve dsruo fuco The equo dl ) / d l ) ) s verso of hoogeeous lhus equo d c e solved explcly I prculr f f exp / T ) / Z Z exp / T ) 32) s he Bolz dsruo wh > he g G ) f exp x / T )) x> )exp / T ) / x exp x / T )) 33) For dsruo 32) wh couous rge of vlues of ) Z T G ) exp / T ) d g ) exp ) / T )exp / T ) / T / T exp / T ) 34) exp E ) / T ) If E) he Z T exp E / T )) G ) d exp E / T ) exp / T ) exp E ) / T ) g ) [ ] T exp E / T )) exp E / T ) 35) Le λ ) E [exp λ)] For l dsruo 34) λ) λt / l ) )exp ) ) N ) N) exp ) T / P P T / ) exp )

5 he oe / T he populo lows up : N ) d l x ed o fy x Le us deoe p P { : }) The < x p / T exp / T + ) T / ) / T ) exp / T )) The proly P { : < }) eds o for y fe x Loosely speg he ol proly ss goes o fy fer fe e ervl So we should coclude h odel 3) 32) whch llows rrry lrge vlues of he preer wh ozero proly hs o physcl sese Ths prole c e eled y g he l dsruo 35) whch llows oly ouded vlues of he preer For pdf 35) he egrl λ) exp λx) g x dx s well defed for y λ u o expressed qudrures Neverheless we c o uch foro ou he syse dsruo d s dycs The curre pdf exp p exp E / T )exp E / T ) ) T exp E / T ) ) ) E ) where ) s fe for ll So he pdf s well deered y e oe cors o he prevous cse The ol dsruo coceres wh e he po E whch provdes he xl reproduco re 2 The Fsher-Hle-Wrgh equo I sees h oe of he frs replcor equos ws roduced y R Fsher 93 [] for geoype evoluo: dp p W W ) 36) d where W W p W pw p Here p s he frequecy of he gee W s he solue fess of he zygoe I hecl geecs hs equo s ow s he Fsher- Hle-Wrgh equo FHWe) d soees s referred o s he equo of hecl geecs see []) The rx { W } s syerc d hece hs he specrl represeo W ω h h ) where ω re o-zero egevlues d h re correspodg orhoorl egevecors of W s he r of W } The { W ) W p ) ω h h ) p ) ω E [ h ] h The FHW-equo ow reds dp 2 p ω h E [ h ] E [ h ]) ) d Cosder he ssoced seleco syse: 37)

6 dl / d l ω h E [ h ] 38) The rge of vlues of he preer s ow fe se ule he prevous exples Defe he gf of he l dsruo of he preer : δ ) exp δ h ) P E [exp δ h ))] Copose d solve he escor syse of ODE ds / d ω E [ h ) exp s h ))]/ E [exp s h ))] These equos c e wre ore copc for ds / d ω l s ) / s The he soluo o he seleco syse 38) l l K where K exp s ) h ) he populo sze N ) N) s )) he vlues of regulors oe H ) E [ h ] l s )) / s d he curre syse dsruo P P K / E [ K )] wh E [ K )] s )) The ls forul gves he soluo of FHW-equo 36) Techclly he descred pproch s useful oly f he r of he fess rx W s sgfcly sller he s deso < The pproch s especlly useful for fely desol syse 36) Le us rer h geerl he fess rx c o e ow excly u s elees c e well pproxed y expresso 37) wh sll For exple f W ww for ll he he l y-deso or eve fe-deso) syse 36) s reduced o sgle ODE Ths cse correspods o well-ow exple of populo he Hrdy-Weerg equlru 4 Dscusso I hs pper we forule d pply ehod h llows us o effecvely solve wde clss of replcor equos d correspodg odels of seleco syses os of hese odels hve for of y or fely) -desol syses of dfferel equos Soe heores of exsece d uqueess d sypoc ehvor of soluos o prculr clsses of such equos were eslshed erler d y prculr odels were suded however o he es of our owledge o geerl ehods for solvg he RE lyclly excep for ler cses) were ow The suggesed lgorh s sed o recely developed heory of hoogeeous populo odels d seleco syses wh dsrued preers [8] The odel ehvor y e dffere d eve couer uve depedg o he l dsruo eve for sples lhus d logsc odels We hve ppled he ehod o soe replcor equos ow fro lerure such s fuel Fsher-Hle-Wrgh geec equo We hope h hs pper y e useful for

7 udersdg dyc peculres of soluos of replcor equos d he crucl role of he l dsruos we lso hope h he geerl ehod d prculr exples preseed here c help sudy replcor equos whch pper dffere res of hecl ology Refereces [] R Fsher The Geecl Theory of Nurl Seleco: Coplee Vroru Edo Oxford Uv Press Oxford 999 [2] N Gor RG Khleopros Deo of Drw: Ide of oply d url seleco Nu FzGz) oscow 988 Russ) [3] N Gor Seleco Theore for Syses wh Iherce h odel N Pheo 2 4) 27) -45 [4] J Hofuer K Sgud Evoluory Ges d Populo Dycs Crdge Uversy Press 998 [5] GP Krev Ihoogeeous odels of ree sd self-hg Ecol odel 6 23) [6] GP Krev Dyc heory of o-ufor populo d glol deogrphy odels J of Bologcl Syses 3 25) 83-4 [7] GP Krev Ihoogeeous ps d hecl heory of seleco JDE 4 28) 3-58 [8] GP Krev O hecl heory of seleco: Couous-e populo dycs JB 28) sued) [9] FN Seevsy S Seeov hecl odelg of ecologcl processes Gdroeeoz Legrd 982) Russ) [] P Schuser K Sgud Replcor dycs J Theor Bology 983) [] Svrezhev Y Psseov VP Fuels of hecl Evoluory Geecs Dordrech: Kluwer cd Pul 99 [2] KB Zeldovch P Che BE Shhovch EI Shhovch Frs-Prcples odel of Erly Evoluo: Eergece of Gee Fles Speces d Preferred Proe Folds PLoS Copu Bol 3 7) 27)