⑧ 3 { part MMBB.in i course - and :3 : annoihino.IT#sineranxFsti::a.ion4&fLouigi 12’th MSJ-SI { / } August 7, 2019 , s f ) } Sarah Julien Addario - Pennington Berestycki Berry ¥ g§y§
Branching Motion Brownian - Time Particle at 0 IR OE : ( X . X # t ) ) t at Particles Time , HI positions : - , . . . . t at time particles Nt # = . branch - Particles at for particle branch rate ( i 1 to Time - e a . - distributed : Exp ( D is or , - dt ) branches eft.tt dt ) ) PC fixed particle - dt Motion Brownian Particles as more - - another branch Particles . from and in dep move one - . 1¥
Competition i' Nt ) ( Mitt ) Nt ) have ' ( Xi ( t ) I Particles Isis masses • , , with A at particle at particle • x mass causes m a ( with x 14 ) to at rate lose ly M y mass - . if C- ( o , 1) - xl I y Write x - y • (t¥¥t_ total Let near mass x of Xi ft ) Osset ) ancestral ( Xi .tk ) the be Let trajectory • . , • Set NtiH=exp(-/t5(s,X%t(sDds
⇒ ⇒ ⇒ Competition , environmental B. M Picture For single resources a • : of cost motion ( food ) energetic balance exactly . if compete for Particles resources ° from other distal each . insufficient of Loss competition resources mass o . each ( children Branching doubles Mass NB . parent ) of inherit mass . - ( 2- dtg.EE , .nMifo% Milt ) tdt ) Milt parts • . = analysis { of • Branching Mass splits → work for :
facts Basic = et : TEN , Obj • . nd t : EIN t.at/Nt--n ] Proof n t - dt ) - ( It El Nt + at ] ' EN so , so ¥ E Nt E Nt = Ana . Particledensity • = Ef # Ii : Xictledx } ] . PCX E Nt x ) , HI Ed Mlt x ) : = . = et PCN lat ) Ed x ) . #j = et . expfxyzt )
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Results .tt#t=eP/ot5sXi.ds5ltik)=gi?*.q!Y : Front location m ) #*hThhµvµ C. ( tix ) D( t.mi-maxfxso.sft.sc ) > Sm ) dlt.mi-minfxsoidt.sc ) # m ) dlt Dft -9 -0 m ) ° , ,
Results -tePf5Xdf5Hix)=gi*?q! : Front location m ) #*hthhµY C. ( tix ) max Goo :S ( tix ) > Dft m ) , m Sm } minfxso.at m ) x ) dlt = : , , # m ) dlt Dft -9 -0 m ) ° , , 312*30 time to .m* ) that such There Theorem exists m* o > , . for > Dft m ) isnfdlt.IR/ogt+s.m ) all sufficiently large , that such t s a. , .
Results -teP/5df5Hix)=gi*%,! : Front location m ) #*hThhµvµ C. ( tix ) max Goo :S ( tix ) > Dft m ) , m : Ht min Goo Sm } m ) x ) dlt = : . , # m ) dlt Dft → -0 m ) ° , , I R* time > 0 to .m* ) that such There Theorem exists * m so , . for > Dft m ) isnfodlttRlogtts.in ) all sufficiently large , that such t s a. , . finfapltjm ) time E to .m* ) that such There Theorem exists * m so ,
Results -tePf5df5Hix)=gi*§µ! : Front location C. ( tix ) # E*hThhµmµ m ) max Goo :S ( tix ) > Dft m ) , Sm } minfxso.at m ) x ) dlt , , # m ) dlt Dft -9 -0 m ) ° , , I R* time > 0 to .m* ) that such There Theorem exists m* o > , . for > Dft m ) isnfodlt.IR/ogt+s.m ) all sufficiently large , that such t s a. , . finfapltjm ) time E to .m* ) that such There Theorem exists m* o > , V. with s a. me to .m* ) that such There c Theorem = exists m* , . o > , - Dlt - Dft Rt m ) Rt m ) liminf , Iim z , S sup c c - - TV3 TV3 c- → a + → •
Results N.li#t=exp-/ot5sXi.ts5ltik)=ei..x&t,!Y : Particle masses Theorem : , for t large , and all c C- Co , rz ) sit for There IN m* is ne so . inf 't ) E t ; p( inf Tls n Lm , > c) > t s DCI ECS and ' a.) st - n > Msg ! Paused . Theorem for t and sufficiently large , IN he For act any , Milt ) > ¥ ) ft - n Pima .
5lt¥5dtH:€ ⇒ *?i! ynamiciim . Theorem : . for , for t > times . D > Ost Clg ? TECO C all and there For T so c- IN is n , ' { s In or } x ) Is - Utes Pearl Soltis " , . , s , of Fisher solution - KPP the - local equation the utcs.sc ) where is non - utf ! UH = tout - y ) dy tut Fust x , initial condition with = 5ft , x ) ut fo , x ) . has derived behaviour about Pennington ( In solo precise paper a , for initial with this . ) PDE condition location front compact the
facts Basic Mlt,x)=(zittkexpft-xYzt - Ip log t fit µ ( t t when + OCH Fast x ) = I = : , - Vt o ( t ) Ex ) for : Mlt exp ( Cl toad , Vt x ) Fact x - = - - . . particle density expected exponentially decays Vt I. beyond e . - , before Vt exponentially grows . IntuitionfforsomepurposeDietindepBrownianMotion
Proof idea stabilizes Density I quickly al . - - 9ft , x ) x ) 5ft - idt = SHIN , - Milt ) ) ( Milttdt ) [ Nlt ) is IE A gift + Emilia filthy Mitt ) Mitt ) " ' - . c Branching ttdB Motion - floyd ! ,d" if pcxittdthxl YI ,t¥ filth ± , × . . negligible typically So B. C . - Milt ) - Milt ) ) dt.flt.X.CH ) . . Milt ) ( t - ¥y¥MilttdH . - dt ; SHAHI ) = " =-D ! . . Milt )=dt 9ft x ) Hittin 'd Emilia , .
Proofidea stabilizes b) Density I quickly . at I d-dtfct.sc ) ' set 2 x 't e- I -2 - I x x x . Xiltl ) . . 9ft Elt x ) - . i . ( fct min { 5ft : ly ' 443 ) , x ) ( I , y ) - - > fct { 5ft : ly ' 443 ) , x ) ( I , y ) Max - - - SHH ¥54 x ) e - sets 1 then V-ys.t.ly > 5ft , g) I - e If • , - M Sct ,xl - Hs ¥54 x ) s Etty ) > Mt 't 1 then V-ys.t.ly If • , 0 ( log to ) 1 Density time Density 0<1 in → - 011 ) time D 1 Density Density > I → in ~
⇒ ⇒ ⇒ Profiled Proof II contradiction by as . . I dltt-dct.tl Elt . xD Write tossed Ct ) so , e I - Dlt . I ) > Dlt ) Dlt ) Ect x ) V. - so x . idea ( d , set ) too big Csl , Sst ) behind spends ( Xi , ( s ) time ( d much , set ) Csi trajectory every small all have particles mass be d ( t ) must smaller , set ) A has ancestral ( X , ( s ) particle Xi ( t ) whose path - 44 has My lil stoups ) then th dish ) f , + ( sik ( a : Hi Leb ( { e ? s .
Proof idea Slowpokes II. miss bs supper . - if almost o ) threshold surely slowpoke Say is Cgctl , t > a , > to ' NCH St Vi 3- to Vt st Xiltlsgltl . . , , Leb ( { : Hi s .to/EgcsY)s.tIzIfdCs)zgcs7V-sc-fY4.t then Whp ) > I 'tq=tg I ! Malti , + Csl ) ds . Sls so ftp.gltl X. , Xi sit V. , i ~ , . unfair > g ( t ) } - t 's # { i :X ; ( t ) Mitt ) E E , e . ; Ctl 3gal } { i :X !# In this case - - . - - - - at time t > 42 si.x.fm?gicftf expats ) 12 NEED make to slowpokes .
threshold Proof idea The slowpoke Ici . - Lemma " glt ) rzt threshold Ct Ost There slowpoke C is is : > - a - . . - o ) " proof ! Let motion Batt ( be Brownian D= > . OH ! P ( Leb { i ) th ) Easy fat ] 1131513 e- se > : : = " ? > th ) e- 0ft Ii ) IP ( Leb { sefo.tl scaling Brownian Bslsl } = : I : iii ) Branching - fitted - l } > th ) El # { i://.lt ) Leb { sefo , t ) : Xi .tk ) > Rs , } > th ) = et - Fst Isl Pll Bt Leb { .tt/Bs-r2s1ae lo se . , Iet .pl/B+.rzt/sl).e-O-ftle4 q ret #ish . Elyse , - PH Bt o - - l ) t PCB There , - t ( t - l ) e here PCB , > v , in stay .
threshold Proof idea The slowpoke Ic ) . - - FL t Lemma " g ( t ) threshold Ct sit There 0 slowpoke C is is " > a - . . . - o ) " proof ! Let motion , t ( be Brownian Blt ) D= > . OH ! I } I ) P ( Leb { 1B sls th ) Easy fat ] e- se > : : = " ? - 0ft > th ) Ii ) scaling Brownian al } I Bsl IP ( Leb { , t ) fo se = e : : iii ) Branching - fitted - l } > th ) El # { i://.lt ) Leb { sefo . t ) : Xi .tk ) > Rs , al } > th ) = et - Fst Isl - Fist l Bt IP ( Leb { lo .tl lbs se : . , Iet .pl/B+-rzt/sl).e-O-ftle4=.et.erl-t.e--Oltll4=erze.e-O-ftll4 - tar - t 's when t ) l ⑤ ( l ; = in conclusion small For obtain enough so c " rt ! , 's } > th ) e ' do Ef # Ii - et - ? Leb { set : Xi Hl Rt i. Hsls :X cs some e > .
Bound Proof idea Upper II. . - " lemma - et in rat ? previous c as Let glt ) . = Lenya . . , t ) s gcs ) -11 softly dls ) t st V large F Then > o as . . Proof : § ! ,Mlt , x ) dx Note > get ) } # { i Xilt ) E - 9% = : 's rat ⇐ emt ⇒ Mlt ,gCt ) ) e . g ( t ) # d ( s ) > gcs ) -11 then f t ) I t If use > x - tis Mitt ) # { i :X ; ( t ) [ > g ( t ) } y ( t x ) s e . t > gtl } si :X its , Rt " + t ) 18 o ( e- = e x . . d CHE gets +1 So am
Proof bound idea Lower II : . - strict threshold feast pole : > his ) Xi .tk ) sst Vt V. 3- htt ) whp set set i . , . . " ? fast poke has E If s t then V 3 shes ) DCs ) mass any s s h ( s ) V. sat fast poke then has , Lt ) ④ DCs If 71/2 any mass " s ) htt ) hold Et O ( t Fact have Can and take = : - . " ' ) . It ) Either set I sit O ( s ¥1 DCs rzs > • - . " - Oct Rt ' 3) k ) Dlt > • or , . stabilizes that density quickly Now use , I ) . It ) to DCs -1040Gt ) from to DCs go 08
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