Invasive Spread of an Advantageous Mutation under Preemptive Competition

Transcription

1 Invasive Spread of an Advanageous Muaion under Preempive Compeiion Lauren O Malley, Joseph Yasi, James Basham, Balázs Kozma, and G. Korniss (Rensselaer) Andy Allsad and Thomas Caraco (SUNY Albany) Zolán Rácz (Eövös Univ.) Suppored by NSF, Research Corporaion

2 Moivaion invasive spread of a selecively favored muaion (advanageous allelle) R.A. Fisher, 37, Kolmogorov, Perovsky & Piscounov, 37 propagaing fron, iniially exiss and separaes he wo spaial regions occupied separaely by he wo alleles ( domain-wall moion, propagaion ino unsable saes) v + ( ) (FKPP) velociy selecion /marginal sabiliy: Aronson and Weinberger 78, Dee and Langer, 83, van Saarloos 87)

3 Model feaures Through muaions, an invasive allele appears in a habia originally dominaed by a common residen allele muaion is a rare sochasic process residens and invaders compee for common limiing resources hrough clonal propagaion (plans) compeiion is pre-empive (invader allele has an individual-level advanage, bu canno displace residens already presen, (Amarasekare, 003, Shurin e al., 004)

4 Laice Model 0: empy laice sie (available resource) allele : ( residen ) allele : ( invader ) n( x) 0 if allele is presen a sie x oherwise n( x) 0 if allele is presen a sie x oherwise common limiing resources excluded volume consrain A laice sie represens he minimum level of locally available resources required o susain an individual organism.

5 0: empy laice sie (available resource) : allele ( residen ) : allele ( invader ) local ransiion raes for an arbirary sie x: η( x η( x ϕ µ µ ) ) local spread of residens local spread of invaders forward-backward recurren muaion deah η ( x) i n 4 i x' nn( x) ( x') ϕ << µ < <

6 MF Equilibrium (, ) phase diagram 0 ϕ µ µ µ 0 µ ( ) ( ) ( ) ( ) φ µ φ µ + + ( ) ( ) µ µ ( µ) c

7 Saionary-Sae MC Simulaions (single-allele clonal plans: dispersal-limied exincion, Oborny e al., 005) (.65µ ) c * β ( β c )

8 Invasion ime (lifeime) ( ) i n (, ) i, i x L x ime-dependen global densiies ( ) τ / (firs-passage ime o a suiably chosen cu-off densiy) measable or quasi-equilibrium densiy of he residens

9 Single-cluser invasion L << R o! sochasic L L sysem wih p.b.c.

10 Single-cluser invasion L << R o ime 0 τ P no ( ) exp[ ( g ) / n ] for for > g g ( L I n ) g ~ L / v growh ime τ n + g n average ime beween nucleaion evens average lifeime Rikvold e al., 94 Richards e al., 95 Ramos e al., 99 Machado e al., 05 GK and Caraco, 05

11 Muli-cluser invasion L >> R o! self-averaging (near-deerminisic global densiies)

12 Muli-cluser invasion L >> R o 0 ime τ KJMA/Avrami s law (for homogeneous nucleaion, d): () e ln()( / τ ) 3 I( vτ ) τ ~ τ ~ τ ~ ( Iv /3 ) (average) lifeime R o ~ ( v / I / 3 ) average disance beween clusers

13 MC Resuls - Single-cluser invasion L µ 0.0 ϕ I n ~ ~ n ~ ϕ

14 MC Resuls - Muli-cluser invasion τ ~ ϕ 0.30 from Avrami s law: τ ~ I /3 ~ ϕ /3 From simulaions: τ ~ ϕ 0.30

15 Summary: Finie-size effecs slope slope 0.30

16 Surface/inerface properies

17 Curren work: Wave propagaion PDE approach vs. Mone Carlo ( ) ( ) ( ) ( ) 4 4 µ δ µ δ + + ) ( ),, ( µ µ v Propagaion ino unsable saes: Velociy selecion /marginal sabiliy (Aronson and Weinberger 78, Dee and Langer, 83, van Saarloos 87)

18 MC simulaion for he fron velociy 0 planar Circular wavefron

19 MC simulaion for he fron roughening 0

20 Summary and Oulook KJMA/Avrami s law applicable o a wide range of sysems: ferromagneic (Rikvold e al. 94, Ramos 99) and ferroelecric (Ishibashi & Takagi 7, Duiker & Beale 90) swiching flame propagaion in slow combusion (Karunen 98) chemical reacions (Machado e al. 04) invasive allele spread and ecological invasion (GK & Caraco, 04) asympoic linear spreading velociy v * (,,µ) comparison of coninuum PDE and discree Mone Carlo approaches properies of criical cluser R c (,,µ) G. Korniss and T. Caraco, J. Theor. Biol. 33, 37 (005). J.A. Yasi, G. Korniss, and T. Caraco, cond-ma/ hp://