Adaptive evolution : a population approach Benoît Perthame

Transcription

1 Adaptive evolution : a population approach Benoît Perthame

2 OUTLINE OF THE LECTURE Interaction between a physiological trait and space I. Space and physiological trait II. Selection of dispersal (bounded domain) III. Selection of dispersal (full space) See also A. Arnold, L. Desvillettes, C. Prevost, talk of K.-Y. Lam

3 Setting the model Adaptation to the evironment in a spatial ecology model x Ω space variable θ Θ physiological trait dispersion/motility {}}{ mutations reproduction t n(x, θ, t) = D xx 2 {}}{ n(x, θ, t) + n(x, θ, t) + α 2 θθ ρ(x, t) = 0 n(x, θ, t) dθ This is still an advantage on reproductive rate. {}}{ n(x, θ, t) ( K(x, θ) ρ(x, t) ) Question : (Bouin, Mirrahimi) What is the speed of a traveling wave?

4 Evolution of Dispersal Question Selection without a proliferative advantage? The context of Hastings, Dockery, Lou, Kim : no advantage regarding the reproductive rate K(x) motility of individuals is subject to selection and mutations Called : Spatial sorting

5 Evolution of Dispersal We model it for x Ω, θ > 0 + Neuman t n(x, θ, t) dispersion/motility reproduction {}}{ = θ xxn(x, 2 {}}{ θ, t) + n(x, θ, t) ( K(x) ρ(x, t) mutations on motility ) {}}{ +ε 2 θθ 2 n(x, θ, t) ρ(x, t) = 0 n(x, θ, t) dθ Remark : Parameters as θ are not given they are selected Question : which dispersal rate θ is selected?

6 Evolution of Dispersal We can again ask the question of rare mutations ε t n ε (x, θ, t) = θ 2 xxn ε (x, θ, t) + n ε (x, θ, t) ( K(x) ρ ε (x, t) ) + ε 2 2 θθ n ε(x, θ, t) ρ ε (x, t) = 0 n ε(x, θ, t) dθ Can we argue with the same argument as for proliferative advantage?

7 Evolution of Dispersal We can again ask the question of rare mutations ε t n ε (x, θ, t) = θ 2 xxn ε (x, θ, t) + n ε (x, θ, t) ( K(x) ρ ε (x, t) ) + ε 2 2 θθ n ε(x, θ, t) For ε small, n ε (x, θ, t) 0 ρ ε (x, t) = 0 n ε(x, θ, t) dθ θ 2 xxn ε (x, θ, t) + n ε (x, θ, t) ( K(x) ρ ε (x, t) ) = 0 +Neuman boundary condition the first eigenvalue H ( θ, ρ ε (, t) ) vanishes. Therefore n ε (x, θ, t) N(x, t)δ(θ = θ(t)),

8 Evolution of Dispersal We can again ask the question of rare mutations ε t n ε (x, θ, t) = θ 2 xxn ε (x, θ, t) + n ε (x, θ, t) ( K(x) ρ ε (x, t) ) + ε 2 2 θθ n ε(x, θ, t) ρ ε (x, t) = n ε(x, θ, t) dθ 0 When a mutant has an advantage, it diffuses everywhere and invades the domain Ω Therefore we expect (as before) that n ε (x, θ, t) N(x, t)δ(θ = θ(t)),

9 Evolution of Dispersal The Gaussian approximation to n ε (x, θ, t) N(x, t)δ(θ = θ(t)), a corrector as in homogenization. n ε (x, θ, t) N ε (x, t)e ϕ ε(θ,t) ε, The dominant terms in the expansion are t ϕ(θ, t) = θ ϕ 2 + θ 2 xxn ε (x, θ, t) + N ε ( K(x) ρε (x, t) ) + O(ε) Define the effective Hamiltonian H(θ, t) as the principal eigenvalue θ 2 xxn ε (x, θ, t) + N ε (K(x) ρ ε (x, t)) = H ( θ, ρ ε (, t) ) N ε +Neuman boundary condition x Ω

10 Evolution of Dispersal We get t ϕ(θ, t) = θ ϕ 2 + θ 2 xx N ε(x, θ, t) + N ε ( K(x) ρε (x, t) ) + O(ε) θ 2 xxn ε (x, θ, t) + N ε (K(x) ρ ε (x, t)) = H ( θ, ρ ε (, t) ) N ε x Ω therefore the limit is t ϕ(t, θ) = θ ϕ 2 + H ( θ, ρ(, t) ) max θ ϕ(t, θ) = 0 = ϕ(t, θ(t)) To be compared to the usual constrained H.-J. equation t ϕ(t, θ) = θ ϕ 2 + R ( θ, ϱ(t) ) max θ ϕ(t, θ) = 0 = ϕ(t, θ(t))

11 Evolution of Dispersal We get t ϕ(θ, t) = θ ϕ 2 + θ 2 xx N ε(x, θ, t) + N ε ( K(x) ρε (x, t) ) + O(ε) θ 2 xxn ε (x, θ, t) + N ε (K(x) ρ ε (x, t)) = H ( θ, ρ ε (, t) ) N ε x Ω therefore the limit is t ϕ(t, θ) = θ ϕ 2 + H ( θ, ρ(, t) ) max θ ϕ(t, θ) = 0 = ϕ(t, θ(t)) To be compared to the usual constrained H.-J. equation t ϕ(t, θ) = θ ϕ 2 + R ( θ, ϱ(t) ) max θ ϕ(t, θ) = 0 = ϕ(t, θ(t))

12 Evolution of Dispersal How do we handle this? Along the dynamics H ( θ(t), ρ(, t) ) = 0 (pessimism principle) ρ ε Nε(x, θ(t), t) := N(x, t) we can identify the limit of N ε (x, θ, t) as the solution to θ(t) 2 xxn = N ( K(x) N(x, t) ), +Neuman boundary condition x Ω What useful information do we conclude from this analysis? d dt θ(t) = ( D2 ϕ) 1. θ H ( θ(t), ρ(, t) ) Which behaviour?

13 Evolution of Dispersal How do we handle this? Along the dynamics H ( θ(t), ρ(, t) ) = 0 (pessimism principle) ρ ε Nε(x, θ(t), t) := N(x, t) we can identify the limit of N ε (x, θ, t) as the solution to θ(t) 2 xxn = N ( K(x) N(x, t) ), +Neuman boundary condition x Ω What useful information do we conclude from this analysis? d dt θ(t) = ( D2 ϕ) 1. θ H ( θ(t), ρ(, t) ) Which behaviour?

14 Evolution of Dispersal Theorem When K Cst, θ H ( θ(t), ρ(, t) ) < 0.

15 Evolution of Dispersal Proof We can normalize the eigenvalue problem in x as θ xxn 2 ε (x, θ, t) + N ε (K(x) ρ ε (x, t)) = H ( θ, ρ ε (, t) ) N ε, Then θ x N ε(x, θ, t) 2 dx + N ε 2 dx 2θ N ε,θ N ε + 2 x x But at θ(t) one has θ N ε,θ N ε + x Therefore x N 2 ε dx = Cst x N 2 ε (K(x) ρ ε (x, t)) dx = H ( θ, ρ(, t) ) x N εn ε,θ (K ρ ε ) dx = H θ x N εn ε,θ (K ρ ε ) dx = H ( θ, ρ(, t) ) = 0. x N 2 dx = H θ ( θ, ρ(, t) ) < 0. ( θ, ρ(, t) )

16 Evolution of Dispersal d dt θ(t) = ( D2 ϕ) 1. θ H ( θ(t), ρ(, t) ) θ H ( θ(t), ρ(, t) ) < 0 Conclusion θ(t) decreases. Do not move in a bounded domain! Already known, but here we give the dynamics Intuition : A mutant with small dispersal diffuses less wins advantage by staying near the maximum of K(x)

17 Accelerating waves Conclusion Do not move in a bounded domain! One can ask the same question for invasion fronts : Ω = R d

18 Accelerating waves Example of the cane toads invasion in Australia

19 Accelerating waves In full space the solution is an invasion front à la Fisher/KPP for t n(x, θ, t) = θ xxn(x, 2 θ, t) + rn(x, θ, t) (1 ρ(x, t)) + α θθ 2 n(x, θ, t) ρ(x, t) = 0 n(x, θ, t) dθ, n(x = +, t) = 0 Many rescalling possible They always show that large values of θ are selected at the front of the wave.

20 Accelerating waves (scale 1) Scaling 1 : Front in x, θ (0, Θ) ε t n ε (x, θ, t) = ε 2 θ 2 xxn ε (x, θ, t) + rn ε (x, θ, t) (1 ρ ε (x, t)) + α 2 θθ n ε(x, θ, t) ρ ε (x, t) = 0 n(x, θ, t) dθ, n ε(x = +, t) = 0 On the front, n ε Q(θ)e λ(x ct)/ε [θλ 2 + cλ + r]q α 2 θθ Q = 0 In other words, the principal eigenvalue is cλ which gives both c = c (λ), Q(θ, λ) = eigenfunction

21 Accelerating waves (scale 1) It remains to compute λ by the standard approach through H.-J. equation (Barles, Evans, Souganidis) We find n ε (x, θ, t) e u(x,t)/ε N ε (x, θ, t) t u ε N ε (x, θ, t) + θ x u ε 2 N ε = r(1 ρ ε )N + α 2 θθ N ε Therefore in the front N ε Q and In other words max ( u, t u c ( x u) x u ) = 0 c ( x u) = effective Hamiltonian

22 Accelerating waves (scale 1) Conclusion We can compute the speed of the front thanks to this H.-J. equation With θ (0, Θ) Θ c ( x u) 2r 2 front is faster than the average c ( x u) α 0 2r Θ

23 Accelerating waves (scale 2) Scaling 2 : Front in x, small mutations ε t n ε (x, θ, t) = ε 2 θ 2 xxn ε (x, θ, t) + rn ε (x, θ, t) (1 ρ ε (x, t)) + αε 2 2 θθ n ε(x, θ, t) ρ ε (x, t) = 0 n(x, θ, t) dθ, n ε(x = +, t) = 0 Rationale behind this rescaling θ αrt, x front θ t (αr) 1/4 t 3/2 (not the hyperbolic scaling) (t, x, θ) (t/ε, x/ε 3/2, θ/ε).

24 Accelerating waves (scale 2) Scaling 2 : Front in x, small mutations ε t n ε (x, θ, t) = ε 2 θ 2 xxn ε (x, θ, t) + rn ε (x, θ, t) (1 ρ ε (x, t)) + αε 2 2 θθ n ε(x, θ, t) ρ ε (x, t) = 0 n(x, θ, t) dθ, n ε(x = +, t) = 0 Use the Hopf-Cole/WKB change of variable n ε = e u ε/ε t u = θ x u 2 + α θ u 2 + r(1 ρ(x, t)) max θ u(x, θ, t) 0 0 = u ( x, θ(x, t), t ) The constraint can be inactive (extinction) max θ u(x, θ, t) < 0 ρ(x, t) = 0

25 Accelerating waves (scale 2) Scaling 2 : Front in x, small mutations t u = θ x u 2 + α θ u 2 + r(1 ρ(x, t)) max θ u(x, θ, t) 0 0 = u ( x, θ(x, t), t ) What is the canonical equation! New phenomena : The canonical is a PDE t u = θ x u 2 + α θ u 2 + r(1 ρ(x, t)) := R(x, θ, t) θ u(x, θ(t), t) = 0. t θ(x, t) = θtu θθ u, t) = x θ(x, θxu θθ u

26 Accelerating waves (scale 2) Scaling 2 : Front in x, small mutations R θ = x u 2 + 2θ x u xθ u + 2α θ u θθ u t θ(x, t) = θtu θθ u, The canonical is a Burgers type equation t) = x θ(x, θxu θθ u d t) = dt θ(x, xu θ(x, t) x u xθ u θθ u d t) 2 x u θ(x, t) dt θ(x, x θ(x, t) = xu 2 θθ u > 0 Numerically shocks are observed on the fittest traits.

27 Accelerating waves (scale ) Scaling 3 : Traveling wave in x, small mutations ε t n ε (x, θ, t) = θ 2 xxn ε (x, θ, t) + rn ε (x, θ, t) (1 ρ ε (x, t)) + αε 2 2 θθ n ε(x, θ, t) ρ ε (x, t) = 0 n(x, θ, t) dθ, n ε(x = +, t) = 0? Other examples Selection of competitive/colonize phenotype in tumor growth (Orlando, Gatenby, Brown).

28 Collaborators for this programm Vincent Calvez Nicolas Meunier Emeric Bouin Sepideh Mirrahimi Gaël Raoul Raphaël Voituriez