Invasion slowing and pinning due to spatially heterogeneous selection

Transcription

1 Invasion slowing and pinning due to spatially heterogeneous selection Georgetown University May 2016

2 Cane toad Evolving: leg length

3 Purple loosestrife Evolving: flowering time, vegetative size at first flower Latitudinal gradient (1200 km) wikimedia

4 How important is rapid adaptation in introduced species? Prevalence? Buswell et al. J. Ecology 2011 Role at different stages of invasion: establishment, spread? Which traits are under selection?

5 How does adaptation in introduced species work? Sources of genetic variation? Adaptive processes (standing variation, hybridization, polyploidy)? How to identify key loci/genes? Coevolution of native and introduced species?

6 Mendelian genetics; Fisher-KPP u t = u xx +u(1 u) Derived to describe spatial spread of an allele Traveling waves Fisher Ann Eugenics 1937

7 Mutation surfing Neutral or selected locus Position relative to wave front is key

8 Quantitative traits Continuous random variable Contributions from many genes (QTL) Fruit size, flowering time, critical photoperiod

9 ψ(z) mean - 3 std mean -2 std mean -1 std trait mean mean + 1 std mean + 2 std mean + 3 std z Key quantities Trait distribution ψ(z) Trait mean z Phenotypic variance V P Additive genetic variance V A Heritability h 2 = V A V P (narrow-sense) Quantitative trait distribution

10 Selection on quantitative traits Directional vs. stabilizing Breeder s equation: z t+1 z t = h 2( z parents z t ) wikimedia

11 Goals Introduce Kirkpatrick/Barton system Describe analytical results, proof methods Present numerical results: wave speed, pinning Place in context

12 Environmental gradients Latitudinal gradients Altitudinal gradients

13 Why are there treelines? Not all range boundaries are seashores Why can t the invader adapt just a bit more?

14 Haldane s hypothesis Proc. Roy. Soc. London B 1953 Genetic swamping: verbal model Higher population density at range center Migration from center overwhelms adaptation at edge N Z opt N(x), Z opt (x) x Genetic swamping in an environmental gradient

15 Model environmental gradient Continuous 1-dimensional habitat Trait optimum Z opt (x) varies linearly with x Latitudinal gradients common: flowering time Will maladaptation set range limits?

16 Assumptions Continuous time (overlapping generations) Constant V P = V A (100% narrow-sense heritability) Distribution ψ(z;x,t); mean Z(x,t) Population density N(x, t)

17 Dimensional equations Kirkpatrick/Barton Am Nat 1997 n t = σ2 2 n xx +n (r max [1 nk (z opt(x) z) 2 V ]) P 2V s 2V s z t = σ2 2 z xx +σ 2n x n z (z opt (x) z) x +V A V s V s strength of stabilizing selection

18 Nondimensional equations Z now difference between trait mean and optimum Garcia-Ramos/Rodriguez Evolution 2002 A = B = V A 2V s(r max V P /2V s) N t = N xx +N (1 N 12 ) Z2 Z t = Z xx +2 N x N (Z x +B) AZ. scaled strength of selection bσ (r max V P /2V s) 2V s scaled trait optimum gradient

19 Pinned states Kirk-Bart first to model

20 Traveling waves García-Ramos/Rodriguez Evolution 2002: more detail

21 Waves or no waves? A = 1: waves for B 1.4 (G-R/R) or B 1.05 (K/B)? 1 N x t =28 1 Z x A = 1, B = 2, N(x,0) = sechx, Z(x,0) = Z opt

22 Competition Can gene swamping alone pin ranges? Case/Taper Am Nat 2000: Only in extreme parameter regimes Adding competition pins ranges with much milder trait optimum gradient

23 Unconstrained variance Barton in Antonovics (ed.) 2001 V A (x,t) allowed to vary (3rd PDE) Numerics: No stable pinned states Traveling wave develops when V A crosses threshold

24 Stochasticity Bridle et al. Ecol. Letters 2010 Compare PDE with individual-based simulations Low patch carrying capacity can pin where PDE predicts spread

25 Goals Existence of pinned states for small A/B Existence of traveling waves for small B/A Geometrical singular perturbation theory Numerical explorations

26 Difficulties N t = N xx +N (1 N 12 ) Z2 Z t = Z xx +2 N x N (Z x +B) AZ. System with no comparison principle Unbounded domain N x /N can t be transformed away No general existence theorem

27 Pinned state existence Theorem If 0 < A/B < 2 and A > 0 is sufficiently small, then the K/B PDE system has stationary solutions satisfying N 0 as x ±.

28 Traveling wave existence Theorem 2 If B/A > 0, c > and A > 0 is sufficiently small, then 1+2(B/A) 2 the K/B PDE system has a traveling wave solution with speed c whose profile has exactly one local maximum.

29 Ansatz and ODEs N = N(x ct), Z = Z(x ct) R = N /N, Z = Z/ 2, W = W/ 2 (drop tildes) Obtain N = RN R = cr (1 N Z 2 ) R 2 W = cw 2R(W + B)+AZ Z = W. R = N /N eliminates zero denominator as x ±

30 Pinned state scaling Steep environmental gradient x = ǫ x A = αǫ 2 B = ǫ 2 W = ǫ 1 W

31 Traveling wave scaling Weak selection Possibly steep environmental gradient A = ǫ W = ǫ W B = βǫ

32 Singularly perturbed slow system (A/B small) = d dξ with ǫx = ξ Pinned state: Ṅ = RN Ṙ = R 2 ( 1 Z 2 N ) ǫ W = 2R(ǫW +1)+αZ ǫż = W,

33 Singularly perturbed slow system (B/A small) Traveling wave : ǫṅ = RN ǫṙ = cr (1 N Z 2 ) R 2 ǫ W = cw 2R(W +β)+z Z = W

34 Geometric singular perturbation theory Goal: heterocline connecting equilibria E, E + ǫ = 0: slow and fast systems each with reduced dimension Construct template by concatenating slow and fast ǫ = 0 solutions ǫ > 0: manifolds of (fast) equilibria perturb to (slow) invariant manifolds Track unstable (stable) manifold of E (E + ) using fibrations of invariant manifolds Tracking tangent spaces when ǫ = 0 gives orbit near template when ǫ > 0

35 The critical (ǫ = 0) manifold M 0 ǫ = 0: slow differential-algebraic system Ṅ = RN Ṙ = R 2 ( 1 Z 2 N ) 2R +αz = 0 W = 0 Makes sense only on { M 0 = (N,R,Z,W) : W = 0, Z = 2R } α

36 Dynamics on the critical manifold Slow ODES reduce to planar system Ṅ = RN ( ) 4 Ṙ = α 2 1 R 2 1+N Poincaré-Bendixson, symmetry give heteroclines joining two equilibria E, E + with N = 0

37 Proof via Fenichel s first theorem M 0 normally hyperbolic when A/ B < 2 Fenichel: compact submanifolds of M 0 perturb to invariant manifolds M ǫ of full flow when ǫ > 0 small Proof (Implicit function theorem) E, E + M ǫ Linearization near equilibria + symmetry heteroclines persist

38 Traveling wave profile ODEs: Equilibria Goal: heterocline from N = 1 to N = 0 E (1,0,0,0) E + 1 E + 2 0, c + c 2 4(1 4β 2 ) 2(1 4β 2,0, ) 0, c c 2 4(1 4β 2 ) 2(1 4β 2,0, ) ( β c + c 2 4(1 4β 2 ) 1 4β 2 ( β c c 2 4(1 4β 2 ) 1 4β 2 ) )

39 Critical manifolds E M 0 {(N,R,W,Z) = N = 1 Z 2, R = 0, W = Z } c E 1,2 + M 0 {(N,R,W,Z) + = N = 0, R 2 +cr +1 Z 2 } = 0, (c +2R)W +2βR Z = 0 0 M 0 + M Z R N

40 Stable and unstable manifolds of M ± 0 Subgoal: W u (M 0 ) intersects Ws (M + 0 ) transversally dimw u (M 0 ) = 1+dimM 0 dimw s (M + 0 ) = 3+dimM+ 0 Normal hyperbolicity on relevant submanifolds; Fenichel s First applies

41 Fenichel s second theorem Have E M ǫ, E± 1 M+ ǫ Normal hyperbolicity stable and unstable manifolds of M ± 0 perturb to W u (M ǫ ), W s (M + ǫ ) Transversality will imply intersection persists But we need stable/unstable manifolds of equilibria, not manifolds

42 Fenichel s third theorem Forward evolution of A D restricted to D: A D t = {x t : x A and x [0,t] D}. W u (M ǫ ) is fibrated by noninvariant manifolds W u (v ǫ ) satisfying W u (v ǫ ) D t W u (v ǫ t) if v ǫ s D for all s [0,t] for useful D M 0 As trajectory in W u (M ǫ ) passes from fiber to fiber, base points move away from E within M 0

43 Fenichel s third theorem M 0 trajectory fibers

44 Critical slow dynamics E unstable within M 0 E + 1 unstable within M+ 0 0 M 0 + M Z R N

45 Critical fast N-R dynamics Phase plane: N = RN R = cr (1 N Z 2 ) R 2 Heterocline T 0 (Z) joins equilibria with R = 0, N = N R

46 Full critical fast dynamics Along T 0 (Z) have [ x ( W(x) = exp c +2R(x ) ) ] dx x [ 0 x ( Z 2βR(x ) ) [ x ( exp c +2R(x ) ) ] dx x 0 dx ], L Hôpital heterocline joins equilibria on M 0, M+ 0 (Z constant)

47 Complete template Start at E Travel slowly down M 0 until Z = Z+ 1 Fast jump to E M 0 + M Z R N

48 Unstable manifold of E Suppose ǫ > 0 small, β > 0, c > 2/ 1+4β 2, P W u (M ǫ ) Define φ x (P) = image of P after time x Then lim x φ x (P) = E Proof: Fenichel fibration Corollary: W u (M ǫ ) = W u (E ) M 0 trajectory fibers

49 Stable manifold of E + 1 Suppose ǫ > 0 small, β > 0, c > 2/ 1+4β Then dim ( W s ǫ(e + 1 )) = 3 Also W s ǫ (E+ 1 ) not tangent to line through E+ 1 parallel to Z-axis ǫ = 0 transversality W u (E ) W s (E + 1

50 Monotonicity Linearization and reparametrization near equilibria N lim Z 0 Z = lim 2 (RN)/ Z 2 Z 0 2 (ǫw)/ Z 2 = 1 W < 0 on heterocline near E N decreasing there Invoke phase plane away from equilibria

51 Numerical notes N x N : so fast spatial decay problematic Fourth-order centered differences in x 3/8 Runge-Kutta/Crank-Nicolson in t Periodic boundary conditions (irrelevant)

52 Where did we use c > 2 1+2(B/A) 2? Instability of E dimw s (E + 1 ) Monotonicity

53 Wave speed; pinning Steeper environmental gradient gives slower waves, promotes pinning Weaker selection/greater V A gives faster waves, inhibits pinning Heavier tails give faster waves; Fisher-KPP similar Sherratt Dyn. Stab. Systems 1998 Conjecture: Compact initial data yields traveling wave with speed c = 2 1+2(B/A) 2 (1)

54 c Does conjecture pan out? A = 0.01, initial population density N(x,0) = sechx 2 A=ǫ = β=b/2 1/2 A Initial population density N(x, 0) = sechx

55 Narrow initial data pins range 1 A = 1, B = N x N init =sech(x/2), c=0 Z x Initial population density N(x, 0) = sech(x/2)

56 Wider initial data sets off range expansion 1 N init =sech(x/4), c= N x A = 1, B = 2 1 Z x Initial population density N(x, 0) = sech(x/4)

57 Hysteresis? Temporary disturbance/climate change/transportation can start range expansion But will expanded range persist?

58 Summary First rigorous results for full Kirkpatrick-Barton system Numerics: bistability previously overlooked

59 What s next mathematically? General existence theory Stability Bifurcation diagram

60 What s missing? Long-distance dispersal Genetic detail Data!