School Mathematical modeling in Biology and Medicine. Equilibria of quantitative genetics models, asexual/sexual reproduction

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1 School Mathematical modeling in Biology and Medicine Equilibria of quantitative genetics models, asexual/sexual reproduction Thibault Bourgeron with: V. Calvez, O. Cotto, J. Garnier, T. Lepoutre, O. Ronce

2 Plan 1 Selection-Mutation Models: asexual/sexual Reproduction, Age Structure 2 3

3 Mutation-selection model Adaptation Adaptation to a local environment: result of a mutation/selection process acting on phenotypical traits x Adapted population: dominant trait equals optimal trait Population dynamics t f (t, x) + µ(x) f (t, x) = β B σ (f (t, ))(x) f (t, x) density of population, x R continuous trait x Selection: µ(x) heterogeneous mortality rate, symmetric: best adapted trait x = 0 (µ(x) non-decreasing for x 0) β birth rate, B σ reproduction operator

4 Asexual/sexual reproduction operators Asexual reproduction ( ) 1 x x B σ (f )(x) = σ K f (x ) dx, σ R K a probability measure Sexual reproduction: infinitesimal model 1 B σ (f )(x) = f (x ) f (x ) G σ (x x + x ) dx dx, f (x) dx R R = R f (x) dx G σ f (2 ) f (2 ), G σ a centered Gaussian function with variance σ 2. x offspring = 1 2 (x parent1 + x parent2 ) + normally distributed random variable

5 Age structure Ageing increases mortality. µ(a, x) = d + m(x) δ a=a ; µ(a, x) = d + m(x) 1 a>a. Figure : (Jones et al., Nature 2014) d intrinsic death rate a ageing impact threshold Model with ageing t f (t, a, x) + a f (t, a, x) + µ(a, x) f (t, a, x) = 0, ( ) f (t, 0, x) = B σ β(a)f (t, a, ) (x). 0

6 Time scale, changing environment evolutionary time scale ecological time scale changing environment: µ(x) µ(y) with y = x ct lhs: t f (t, x) + µ(x) f (t, x) t f (t, y) c y f (t, y) + µ(y) f (t, y) rhs: unchanged

7 Goals Is there a mutation-selection equilibrium? What are the adaptive dynamics induced by sexual reproduction? In particular, quantification of climate change impacts: load δλ = λ c λ 0, global adaptation of the population, lag y c = argmax f (y 0 = 0), dominant trait

8 Background Infinitesimal operator Bülmer, 1980 Turelli-Barton, 1994 Mirrahimi, Raoul, 2013 Cotto, Ronce, 2014 Cole-Hopf, concentration around the dominant trait Diekmann, Jabin, Mischler, Perthame, 2005 É. Bouin, V. Calvez, J. Garnier (asexual case)

9 Principal eigenelements, age structure, speed c = 0 f (t, a, x) = e λt F (a, x) λf (a, x) + a F (a, x) + µ(a, x)f (a, x) = 0, ( ) F (0, x) = G σ β(a)f (a, ) da (x), 0 F (0, x) dx = 1 (homogeneity). R Hypothesis. There exists λ 0 R: x R 1 0 β(a)e λ0a e a 0 µ(a,x) da da 0 β(a)e λ0a da < Theorem. There exists a solution (λ, F ).

10 Proof 1/2: Properties and difficulties Properties of G σ 1-homogeneous operator (non linear), not increasing (Krein-Rutman/Mahadevan) double convolution: regularizing effect G σ conserves moments of orders 0 and 1 (M k (f ) = R x k f (x) dx) (cf. B. Wennberg s lecture) If M 0 (f ) = 1, M 1 (f ) = 0: M 2 (G σ (f )) = M 2 (G σ ) M 2(f ). {f L 1 M 0 (f ) = 1, M 1 (f ) = 0, M 2 (f ) 2M 2 (G σ )} is mapped into itself by G σ Age structure breaks conservation of moments no dispersion relation

11 Proof 2/2: Sketch (transport in age) F (a, ) x) is determined by ϕ(x) = F (0, x): ϕ(x) = G σ (ν(λ, ) ϕ (x), with ν(λ, x) = 0 β(a)e λa e a 0 µ(a,x) da da (dispersion relation) If ϕ is positive, ϕ dx = 1, there exists a R unique λ = λ(ϕ) R such that: ν(λ, x)ϕ(x) dx = 1. R (Schauder theorem) Closed convex ) set of L 1 (R), mapped into itself by M : ϕ G σ (ν(λ(ϕ), ) ϕ : C = { f L 1 f 0, M 0 (f ) = 1, f ( x) = f (x), M 2 (f ) 2 M 2 (G σ ) }. Regularizing effect of G σ : M(C) relatively compact

12 Ansatz for F ε Small mutations, long time dynamics concentrated mutation kernel B ε t tε α, α > 0 ε α ( t f ε c y f ε ) + µ(y)f ε = β B ε (f ε (t, ))(y). Adaptive dynamics limit: spectral approach λ c f ε (t, y) = exp(λ c tε α )F ε (y) = 0 maximal speed c for which population persists Concentration around dominant trait U ε = ε α ln F ε λ ε + c x U ε (x) + µ(x) = β F ε (x) B ε(f ε )(x)

13 Asexual reproduction (λ ε, U ε ) = (λ 0, U 0 ) + ε (λ 1, U 1 ) + ε 2 (λ 2, U 2 ) P.E. Jabin s lecture: time scaling t t/ε ( 1 F ε (x) Bε(F ε 1 x x )(x) = R ε K ε ( U ε (x) U ε ) (x εy) = K(y) exp R ε ) ( exp Uε (x ) ε ) ( U dx ε ) (x) exp ε dy K(y) exp (y x U ε (x) + o(1)) dy R λ 0 + cu 0(x) + µ(x) = βh(u 0(x)) Hamiltonian H is the two-sided Laplace transform

14 Adaptive dynamics for sexual reproduction without ageing Time scaling: t t/ε 1/2 0 th order, equilibrium of G σ. 1 st order. U 0 (x) = 1 4σ 2 (x x c) 2, x c unknown λ 0 + cu 0(x) + µ(x) = β exp ( (LU 1 )(x)) ( ) x + xc Non local derivation: (LU)(x) = 2U U(x) U(x c ) 2 λ 0 + µ(x c ) = β, µ (x c ) + c 2σ 2 = 0 x c, λ 0, LU 1 Particular solution to LU = f : x R U 1 (x c + x) = U 1 (x c ) + xu 1(x c ) + 2 n f (2 n x), with f (x) = β (µ(x + x c) µ(x c ) xµ (x c )). 2 nd order. n=0 Determination of U 1 (x c), λ 1, LU 2 ; U 2 (x c) missing

15 Proof: 0 th order λ ε + c (U ε ) (x) + µ(x) = β F ε (x) G ε(f ε )(x) lhs: λ 0 + cu 0 (x) + µ(x) + ε (λ 1 + cu 1 (x)) rhs: ( exp [ ε 1 U ε (x ) + U ε (x ) + 1 (x β R 2 2σ 2 x + x ) 2 U (x))] ε dx 2 σε 1/2 2 π exp( ε 1 U ε (x)) dx x R inf f (x ) + f (x ) + 1 (x x,x R 2σ 2 x + x ) 2 f (x) = 0 2 ) Legendre transformation: f (s) = sup x R (sx f (x) R ( f (s) 2f s ) = σ2 s f (s) = σ 2 s 2 +x c s f (x) = 1 4σ 2 (x x c) 2

16 Proof: 1 st order U ε = U 0 + εu 1 Denominator concentrates at x = x c 2 πσε 1/2 exp( U 1 (x c )) x = x c + ε 1/2 z Numerator concentrates at x = x = x+xc 2 The form q has a minimum value (0) at x = x = x+xc 2 : q(x, x ) = (x x c ) 2 + (x x c ) (x x + x ) 2 (x x c ) 2 2 ( ( ( ) )) x + ε2 3/2 πσ 2 xc exp 2U 1 U 1 (x) 2 2 nd order. Determinations of U 1 (x c), then λ 1 : { βσ 2 U 1 (x c) + 2cU 1 (x c) + 2λ 1 = 0, U 1 (x c)u 1 (x c) U 1 (x c) + c βσ U 2 1 (x c) = 0

17 Numerics without age structure Scheme on proportions: p(t, x) = f ε (t,x) R f, q(t, ξ) = p. ε dx ε 1/2 ( t q + ciξq) + µ(x)p = β G ε (ξ) q (ξ/2) 2 q ω with q(0) = 1 Figure : Profiles f ε, ε = 0, 05; 0, 2. µ(x) = x 2, µ(x) = 0, 5(x 1) 2 (x + 1) 2. Exact solution. Gaussian approx. 1 st order correction. Mortality rate µ.

18 0 th and 1 st orders with ageing λ ε + c x U ε ε 1 ( a U ε + µ(a, x) = 0 1 ) 1 = F ε (0, x) G ε β(a)f ε (a, ) (x) U 1 (a, x) = U 1 (0, x) + V (a, x), V (a, x) = λ 0 a + U 0 (x) = 1 4σ 2 (x x c ) 2 B(x c ) B( x+xc 2 ) 2 = exp ( (LU 1 (0, ))(x)) with B(x) = 0 β(a) exp( V (a, x)) da Determinations of x c, then λ 0 : B(x c ) = 1, B (x c ) = 0 0 a 0 µ(a, x) da

19 Example: µ(a, x) = 1 2 x 2 δ a=a { e a λ 0 e a λ x 2 c + λ0 1 = 0, c(a λ 0 + 1)e a λ 0 (2σ 2 x c λ 0 + ca λ 0 + c)e a λ x 2 c c = 0 Figure : Maximum speed f (a ): c > f (a ) no principal eigenelements. The function f is decreasing.

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21 Uniqueness: discretized problem, traits 0, ±1 Discretized operator G ( f ) (1) = (x + y + z) ( 1 ax 2 + 2bxy + cy 2 + 2cxz + 2dyz + ez 2) G ( f ) (0) = (x + y + z) ( 1 cx 2 + 2bxy + ay 2 + 2axz + 2byz + cz 2) G ( f ) ( 1) = (x + y + z) ( 1 ex 2 + 2dxy + cy 2 + 2cxz + 2byz + az 2), with: x = f (1), y = f (0), z = f ( 1), a = G(0), b = G ( ± 1 2), c = G (±1), d = G ( ± 3 2), e = G (±2). Mass conservation: a = 1 2e, 2b = 1 d, c = e K nonnegative orthant, K + = int(k), K 0 = K {0}, standard simplex S = {x K x, 1 = 1} (, cip in R 3 ) Positivity: G : K + K + iff 0 d 1 and 0 e 1 2 mass conservation: if λ ev associated to ev x K 0 then λ = 1

22 Existence: Brouwer theorem fixed point in the convex compact set K of R 3 Uniqueness: Hilbert s projective metric, Birkhoff s theorem x, y K + x i y j d(x, y) = log max 1 i,j n y i x j metric on half-lines included in int(k) A linear positive: k(a) := inf{λ x, y K + d(ax, Ay) λd(x, y)} = th If A(K + ) is bounded, A is a contraction for d. Here k(g) > 1 but uniqueness. ( ) 1 4 diam(a(k + ))