Adaptive evolution : A population point of view Benoît Perthame

Transcription

1 Adaptive evolution : A population point of view Benoît Perthame t 50 t 10 x x

2 Population formalism n(x, t) t }{{} variation of individuals = birth with mutations {}}{ b(y)m(x, y)n(y, t) +n(x, t) R ( x, I(t) ) }{{} growth rate n(x, t) = number of indivuduals with trait x x = phenotypical trait I(t) = (I 1 (t),...i J (t)) = environmental unknowns R(x, I) of Lotka-Volterra type, can be negative Standard : Calsina, Cuadrado, Desvillettes, Raoul, Jabin, Mirrahimi,...

3 Population formalism The variable x can be Size of the adult individuals Cannibalism rate (and evolutionary suicide) Cooperative behaviour Dispersal rate Resistance to therapy Chisholm,... Clairambault, Emergence of reversible drug tolerance..., Cancer Research, 2015

4 Population formalism n(x, t) t }{{} variation of individuals = birth with mutations {}}{ b(y)m(x, y)n(y, t) +n(x, t) R ( x, I(t) ) }{{} growth rate n(x, t) = number of indivuduals with trait x I(t) = (I 1 (t),...i J (t)) = environmental unknowns interplay between population and environment

5 Rescaling ε t n ε(x, t) = b(y)m ε (x, y)n ε (y, t)dy + n ε (x, t)r ( x, I ε (t) ), M ε (x, y) means mutations are rare/have small effect

6 Rescaling ε t n ε(x, t) = b(y)m ε (x, y)n ε (y, t)dy + n ε (x, t)r ( x, I ε (t) ), M ε (x, y) means mutations are rare/have small effect ε t n ε(x, t) means we consider a long time scale

7 Rescaling ε t n ε(x, t) = b(y)m ε (x, y)n ε (y, t)dy + n ε (x, t)r ( x, I ε (t) ), M ε (x, y) means mutations are rare/have small effect ε t n ε(x, t) means we consider a long time scale M ε (x, y) = 1 ) ε dm( x y ε

8 Examples of behaviors Branching can occur for more general right hand sides (convolution)

9 Examples of behaviors Branching for a Gaussian convolution

10 t 4 Darwin s divergence principle Examples of behaviors Morphological trait Time x Branching for a non-gaussian convolution Figure 2: Evolutionary branching. Periodic boundary conditions. Parameters are a =1,K =1,b =3,d =0.05, L =40. Under the conditions of self-organization presented in section 3 an initially monomorphic population will undergo several successive branchings to

11 Rescaling ε t n ε(x, t) = b(y)m ε (x, y)n ε (y, t)dy + n ε (x, t)r ( x, I ε (t) ), M ε (x, y) means mutations are rare/have small effect ε t n ε(x, t) means we consider a long time scale Simple case I ε (t) is reduced to the knowledge of ϱ ε (t) = R d n ε(x, t)dx

12 Concentration phenomena ε t n ε(x, t) ε 2 n ε = n ε (x, t)r ( x, ϱ ε (t) ), x R, ϱ ε (t) = R d n ε(x, t)dx. ϱ M > 0 s.t. max x R(x, ϱ M ) = 0 R ϱ < 0 R x > 0 Theorem (d=1) For well-prepared initial data, we have n ε (x, t) εk 0 ϱ(t)δ(x = x(t)), x(t), ϱ(t) BV loc(0, ) R ( x(t), ϱ(t) ) = 0 for a.e. t > 0 x(t) is the fittest trait

13 Concentration phenomena ε t n ε(x, t) ε 2 n ε = n ε (x, t)r ( x, ϱ ε (t) ), x R, ϱ ε (t) = R d n ε(x, t)dx. ϱ M > 0 s.t. max x R(x, ϱ M ) = 0 R ϱ < 0 R x > 0 Theorem (d=1) For well-prepared initial data, we have n ε (x, t) εk 0 ϱ(t)δ(x = x(t)), x(t), ϱ(t) BV loc(0, ) R ( x(t), ϱ(t) ) = 0 for a.e. t > 0 as t R ( x, ϱ ) = 0 = max x R( x, ϱ )

14 Concentration phenomena, d 1 R ϱ < 0 ε t n ε(x, t) ε 2 n ε = n ε (x, t)r ( x, ϱ ε (t) ), ϱ ε (t) = R d n ε(x, t)dx. ϱ M > 0 s.t. max x R(x, ϱ M ) = 0 x R d D 2 xr KId, Theorem (Any dim.) For well-prepared initial data, we have n ε (x, t) ϱ(t)δ(x = x(t)), x(t), ϱ(t) C 1( [0, ) ) R ( x(t), ϱ(t) ) = 0 for all t > 0 as t R ( x, ϱ ) = 0 = minρ max x R ( x, ϱ )

15 Concentration phenomena Why is mathematics interesting? Nonlocal nonlinearity drastically changes the picture Control in L 1 only Constrained Hamilton-Jacobi eq. Is there a simple rule for the dynamics of x(t)?

16 Related approaches Evolutionary game theory Blue (stronger), Orange (middle size), Yellow (smaller) compensate by mating strategies... from B. Sinervo. http : //bio.research.ucsc.edu/barrylab. J. Hofbauer- M. Nowak- K. Zigmund..

17 Related approaches Evolutionary game theory Blue (stronger), Orange (middle size), Yellow (smaller) compensate by mating strategies... from B. Sinervo. http : //bio.research.ucsc.edu/barrylab The relation can be seen by max S R(x, ϱ ) = 0 = R( x, ϱ ). min ϱ max S R(x, ϱ) = 0 = R( x, ρ )

18 Related approaches Dynamical systems. H. Metz, S. Geritz, G. Meszena,. S. Kisdi, O. Diekmann Can a mutant invade the resident population?

19 t Related approaches Stochastic models, Individual Based Models N individuals, rescale mutation, birth, death rates U. Dieckmann-R. Law, R. Ferriere S. Billard, N. Champagnat S. Méléard, V. C Tran x

20 Asymptotics with mutations ε t n ε(t, x) ε 2 n ε = n ε (t, x)r ( x, ϱ ε (t) ), ϱ ε (t) = R d n ε(t, x)dx.

21 Asymptotics with mutations ε t n ε(t, x) ε 2 n ε = n ε (t, x)r ( x, ϱ ε (t) ), ϱ ε (t) = n ε(t, x)dx. R d In the limit one can expect 0 = n(t, x)r ( x, ϱ(t) ), n(t, x) = ρδ Γ(t), Γ(t) { R(, ρ(t)) = 0 }.

22 Proof Step 1. ϱ ε (t) b L, n ε b L t (L1 x) Step 2. A BV estimate Step 3. Represent n ε (t, x) = exp ϕ ε(t, x) ε the fittest trait x(t) is characterised by the Eikonal equation with constraints t ϕ(t, x) = R( x, ϱ(t) ) + ϕ(t, x) ) 2 max x ϕ(t, x) = 0 ( = ϕ(t, x(t)) ).

23 Proof Uniqueness R(x, ϱ) = b(x)a(ϱ) d(x) J.-M. Roquejoffre et S. Mirrahimi P. E. Jabin Approach by modulation

24 Canonical equation Any concentration point x i (t) will satisfy (i) R ( x i (t), Ī(t) ) = 0 (ii) dt x d i(t) = ( D 2 ϕ( x i (t), t) ) 1. R ( xi (t), Ī(t) )

25 Canonical equation Any concentration point x i (t) will satisfy (i) R ( x i (t), Ī(t) ) = 0 (ii) Conclusions : dt x d i(t) = ( D 2 ϕ( x i (t), t) ) 1. R ( xi (t), Ī(t) ) Gause s competitive exclusion principle n ε = exp(ϕ/ε) the shape of ϕ plays a role

26 Canonical equation

27 Canonical equation dt x(t) d = ( D 2 ϕ( x(t), t) ) 1 ). R ( x(t), ϱ(t) Effect of the matrix ( D 2 ϕ( x(t), t) ) (microstructure of the Dirac)

28 Space-trait concentration Let y R a space variable ε t n ε (y, x, t) = [ r(x)c ε (y, t) d(x)ϱ ε (y, t) µ(x) ] n ε (y, x, t) y c ε (y, t) + [ ϱ ε (y, t) + λ ] c ε (y, t) = λc B, ϱ ε (y, t) = n ε (y, x, t)dx!! r=0!!!!!!!!!!r=1!! Interpetation Nutrients/drugs are diffused and consumed by cells Local conditions select space-dependent traits

29 Space-trait concentration Let y R a space variable ε t n ε (y, x, t) = [ r(x)c ε (y, t) d(x)ϱ ε (y, t) µ(x) ] n ε (y, x, t) y c ε (y, t) + [ ϱ ε (y, t) + λ ] c ε (y, t) = λc B, ϱ ε (y, t) = n ε (y, x, t)dx Theorem : For well-prepared initial data, as ε k 0, we have n ε (y, x, t) ρ(y, t)δ ( x X(y, t) ) Difficulty : Space works well with L. Traits with L 1 Outcome : Explains heterogeneity

30 Space-trait concentration Let x = resistance to drugs ε t n ε (y, x, t) = [ r(x)c ε (y, t) d(x)ϱ ε (y, t) µ(x) ] n ε (y, x, t) y c ε (y, t) + [ ϱ ε (y, t) + λ ] c ε (y, t) = λc B, ϱ ε (y, t) = n ε (y, x, t)dx 8 n(t,r,x) dr / T (t) at t=t 8 n(t,r,x) dr / T (t) at t=t x Without cytotoxic drug High heterogeneity x With cytotoxic drug Lower heterogeneity

31 Evolution of Dispersal Selection without a proliferative advantage? motility/dispersal of individuals is subject to variability no advantage regarding their reproductive rate R(x, ρ) = Operator acting on the space variable Hastings, Theor. Popul. Biol. 1983

32 Evolution of Dispersal We model it for y Ω, x > 0 + Neuman BC t n(t, x, y) dispersion/motility {}}{ = x 2 yyn(t, x, y) + reproduction {}}{ n(t, x, y) (K(y) ρ(t, y)) x 2 yyn(t, x, y) + n(t, x, y) (K(y) ρ(t)) }{{} =R(x, ) mutations on motility {}}{ +ε 2 2 xxn(t, x, y) ρ(t, y) = 0 n(t, x, y) dy

33 Evolution of Dispersal t n(t, x, y) dispersion/motility {}}{ = x 2 yyn(t, x, y) + ρ(t, y) = reproduction {}}{ n(t, x, y) (K(y) ρ(t, y)) 0 n(t, x, y) dy mutations on motility {}}{ +ε 2 2 xxn(t, x, y) Theorem (P. E. Souganidis, BP and K. Y. Lam, Y. Lou) The ESS are of the form n(t, x, y) ρ (y)δ ( x = 0 ) ε 0, t and the constrained H.-J. eq. tu(x, t) = Λ(x, ϱ(, t)) + u 2 max x u(x, t) = 0 = u( X(t), t),

34 Evolution of Dispersal Same question for traveling waves Accelerating waves Example cane toads invasion in Australia J. Berestycki, E. Bouin, V. Calvez, C. Mouhot, G. Raoul, L. Ryzhik., C. Henderson

35 Turing (dentritic) patterns

36 Thanks to my collaborators O. Diekmann, P.-E. Jabin, S. Mischler, M. Gauduchon, J. Clairambault, A. Escargueil, G. Barles, S. Mirrahimi, P. E. Souganidis, A. Lorz, T. Lorenzi, THANK YOU ALL

37 Thanks to my collaborators O. Diekmann, P.-E. Jabin, S. Mischler, M. Gauduchon, J. Clairambault, A. Escargueil, G. Barles, S. Mirrahimi, P. E. Souganidis, A. Lorz, T. Lorenzi, THANK YOU ALL