Integrodifference Models with Temporally Varying Environments

Transcription

1 Integrodifference Models with Temporally Varying Environments Jon Jacobsen Harvey Mudd College Claremont, CA 91711

2 Integrodifference Models with Temporally Varying Environments Jon Jacobsen Harvey Mudd College Claremont, CA Yu Jin Mark Lewis

3 Introduction Motivation: Stream population I local density dependent growth I nonlocal dispersal in presence of unidirectional flow I bounded habitat I flow and growth regimes could depend on time Jon Jacobsen EDM 2012

4 Background Consider n t (x) modeled by n t+1 (x) = K(x, y)f(n t (y)) dy (1) = (0, L) density dependent growth f(n) dispersal kernel K(x, y)

5 Advective Kernel (Lutscher et al., Siam Review, 2005) Let z(x, t) be density of moving individuals z t + vz x = Dz xx αz, z(x, 0) = δ 0 (x y) (2)

6 Advective Kernel (Lutscher et al., Siam Review, 2005) Let z(x, t) be density of moving individuals z t + vz x = Dz xx αz, z(x, 0) = δ 0 (x y) (2) Density of settlers at x at time T is then k T (x) = α T 0 z(x, s) ds

7 Advective Kernel (Lutscher et al., Siam Review, 2005) Let z(x, t) be density of moving individuals z t + vz x = Dz xx αz, z(x, 0) = δ 0 (x y) (2) Density of settlers at x at time T is then k T (x) = α T 0 z(x, s) ds For T >> 1/α K(x; y) = α 0 z(x, s) ds

8 Advective Kernel (Lutscher et al., Siam Review, 2005) Let z(x, t) be density of moving individuals z t + vz x = Dz xx αz, z(x, 0) = δ 0 (x y) (2) Density of settlers at x at time T is then k T (x) = α T 0 z(x, s) ds For T >> 1/α Integrate (2) from 0 to with soln K(x; y) = α 0 z(x, s) ds D α K xx v α K x K = δ 0 (x y) (3)

9 Asymmetric Laplace Kernel (advective velocity v) In[41]:= where a 1,2 = v 2D ± K(x, y) = { A e a1(x y) A e a2(x y) x < y x y Plot K x, 4, v 2 1, 9 4D + α 4, K x, 4, 1, 16, x, 1, 1, a1a2 PlotStyle α 2 Thick, Black, D and A Thick, = Blue, a 2 a 1 = Dashed, AxesLabel "x", "K x, 0 ", BaseStyle FontSize v 2 +4αD 16, PlotRange Ticks 1, 1, 0.5, 0.5, 0.5, 0.5, 1, 1, 0.96, 1, 1.97, K x, 0 2 Out[41]= x advective flow v; local diffusion D; setting rate α

10 Linear Stability Assuming f(0) = 0, f (0) = R > 0 one can consider the linear stability of (1) for small solutions via n t+1 (x) = L [n t ] = R K(x, y) n t (y) dy Let λ 1 (K) = ρ(l ). For well-behaved K, f (e.g., Hardin et al. 88) λ 1 (K) > 1: trivial soln unstable, (1) admits stable n (x) λ 1 (K) < 1: trivial soln n (x) = 0 stable This allows study of relation between the organism s ability to persist and the biological parameters via the spectral radius of L

11 Principal Eigenvalue λ 1 Consider Differentiate λφ(x) = R K(x, y) φ(y) dy λφ (x) = R K xx (x, y) φ(y) dy One can use K s DE to derive SLP for φ of the form w/ associated Robin type BCs (n.b. c(0) = 0) φ + c(v)φ + d(λ)φ = 0 (4) facilitates study of dependence of λ 1 (K) on parameters (e.g., critical domain size)

12 Dispersal Success Function We can also approximate λ 1.

13 Dispersal Success Function We can also approximate λ 1. Consider s(y) = K(x, y) dx (5) In[210]:= Show, K x, y i s y 2 1 Out[210]= 1 Out[126]= y 0 y 1 y 2 1 x y 2 1 y s(y) indicates the probability an individual starting at y successfully settles in the habitat after the dispersal event;

14 Dispersal Success Approximation For principal eigenfunction φ we have λ 1 (K) φ(x) = R K(x, y)φ(y) dy Assume φ 1 = 1 and integrate

15 Dispersal Success Approximation For principal eigenfunction φ we have λ 1 (K) φ(x) = R K(x, y)φ(y) dy Assume φ 1 = 1 and integrate λ 1 (K) = R = R K(x, y)φ(y) dy dx s(y)φ(y) dy

16 Dispersal Success Approximation For principal eigenfunction φ we have λ 1 (K) φ(x) = R K(x, y)φ(y) dy Assume φ 1 = 1 and integrate λ 1 (K) = R = R R L K(x, y)φ(y) dy dx s(y)φ(y) dy s(y) dy

17 Dispersal Success Approximation For principal eigenfunction φ we have λ 1 (K) φ(x) = R K(x, y)φ(y) dy Assume φ 1 = 1 and integrate λ 1 (K) = R = R R L K(x, y)φ(y) dy dx s(y)φ(y) dy s(y) dy DSA tends to underestimate λ 1. The principle eigenvalue λ λ a,1 λ 1 (K) The stream length L

18 Dispersal Success Approximation For principal eigenfunction φ we have λ 1 (K) φ(x) = R K(x, y)φ(y) dy Assume φ 1 = 1 and integrate λ 1 (K) = R = R R L K(x, y)φ(y) dy dx s(y)φ(y) dy s(y) dy DSA tends to underestimate λ 1. We also have The principle eigenvalue λ 1 λ 1 (K) R s φ 1 < R λ a,1 λ 1 (K) The stream length L (dispersal loss will reduce growth rate from nonspatial model)

19 Temporal Variations Consider variations in growth and dispersal n t+1 (x) = K t (x, y)f t (n t (y)) dy (6) K t (x, y) kernel for dispersal event at step t f t (n) growth dynamics at step t

20 Example: Two kernel model Consider linearized two kernel model K 1, R 1 and K 2, R 2 in succession: n t+2 (x) = R 2 K 2 (x, y) n t+1 (y) dy

21 Example: Two kernel model Consider linearized two kernel model K 1, R 1 and K 2, R 2 in succession: n t+2 (x) = R 2 K 2 (x, y) n t+1 (y) dy ] = R 2 K 2 (x, y) [R 1 K 1 (y, z) n t (z) dz dy = R 1 R 2 K 2 (x, y)k 1 (y, z) n t (z) dz dy = R 1 R 2 K(x, z) n t (z)dz

22 Effective two-stage kernel Single kernel for two-stage succession n t+1 (x) = R K(x, y) n t (y) dy where R = R 1 R 2 and K(x, y) = K 2 (x, z)k 1 (z, y) dz Similar calculation for eigenfunction as before shows λ 1 (K) < R 1 R 2 λ1 (K) would be effective single rate for two stage process

23 BVP for advective kernels Consider eigenfunction Let Then by (7), λφ(x) = R 1 R 2 ψ(x) = R 1 φ(x) = R 2 λ K(x, y) φ(y) dy (7) K 1 (x, y) φ(y) dy K 2 (x, y)ψ(y) dy

24 BVP for advective kernels Consider eigenfunction Let Then by (7), Thus λφ(x) = R 1 R 2 ψ(x) = R 1 φ(x) = R 2 λ K(x, y) φ(y) dy (7) K 1 (x, y) φ(y) dy K 2 (x, y)ψ(y) dy ψ (x) = v 1 D 1 ψ (x) + α 1 D 1 ψ(x) α 1 D 1 R 1 φ(x) φ (x) = v 2 D 2 φ (x) + α 2 D 2 φ(x) α 2 D 2 R 2 λ ψ(x)

25 Sturm-Liouville Problem for two stage process Differentiating φ two more times and using ψ s equations yields: ( φ (4) Bφ (3) α1 + α 2 v ) 1v 2 φ (2) +Cφ + α ( 1α 2 1 R ) 1R 2 φ = 0 D 1 D 2 D 1 D 2 D 1 D 2 λ where B = [ ] [ ] v 1 D 1 + v2 v D 2 and C = 1α 2+α 1v 2 D 1D 2 and associated boundary conditions: φ (0) = a 1,2 φ (0) φ (L) = a 2,2 φ (L) ( φ (0) = a 1,1 + v ) 2 D 2 ( φ (L) = a 2,1 + v 2 D 2 where a i,j are the exponential coefficients for K i φ (0) a 1,1 v 2 D 2 φ (0) + α 2 D 2 (a 1,2 a 1,1 ) φ (0) ) φ (L) a 2,1 v 2 D 2 φ (L) + α 2 D 2 (a 2,2 a 2,1 ) φ (L)

26 Example: λ vs. flow rate Consider two flow rates v 1 and v 2, keeping fixed average Ev The priciple eigenvalue λ 1,two step max λ Possible values of λ 0.2 min λ 0 0 Ev 1 Ev Average flow velocity (Ev) For fixed Ev, as v 1 and v 2 vary, the principal eigenvalue λ 1 varies between two values Ev < Ev 1 implies λ 1 > 1 Ev > Ev 2 implies λ 1 < 1 Interesting regime for modest averages Ev 1 < Ev < Ev 2

27 Example: λ 1 vs. variation in flow rate Fix v 1 = 0.1, let v 2 vary The principle eigenvalue λ 1,two step The flow velocity v 2 λ 1 is a decreasing function of v 2 (the larger the second flow velocity, the harder it is for the population to persist)

28 Comparison of Eigenvalue Approximations The principle eigenvalue λ (λ ) 1/2 1,twostep The stream length L λ a,1 λ a,2 (λ a,1 λ a,2 ) 1/2 (λ a,12 ) 1/2 (λ a,1 λ a,2 ) 1/2 (λ 1,twostep ) 1/2 for small L (λ a,1 λ a,2 ) 1/2 > (λ 1,twostep ) 1/2 for larger L DSA λ a,12 underestimates λ 1,twostep for larger L

29 Random Kernels We now consider n t (x) = K αt (x y)f αt (n t 1 (y))dy (8) where kernel parameters come from some distribution K αt and f αt denote random growth and dispersal kernels at step t Linearized operator time dependent so no eigenvalue analysis, but one expects, at least with certain assumptions on the parameters, some asymptotic analogue of λ 1.

30 Abstract Result Hardin et al. 88 consider X t+1 = K(x, y)r(α t, y)f(x t (y)) = H αt (X t ), (9) Given various conditions of H α, such as,

31 Abstract Result Hardin et al. 88 consider X t+1 = K(x, y)r(α t, y)f(x t (y)) = H αt (X t ), (9) Given various conditions of H α, such as, (H1) H α : C + () C + () continuous (H2) If x, y C + () and x y then H α (x) H α (y) (H4) There exists a compact set D C + () such that H α (B b ) D for all α A. (H8) A α = H α(0) exists. (H10) (a) If x, y C() and x y then A αx A αy for all α A. (b) A α h for all α A. Theorem (Hardin et al 88) The limit r = lim t A αt A α1 1/t exists with probability one. (a) If r < 1, the population becomes extinct; (b) If r > 1, then population persists

32 Random Kernels The limit r = lim t A αt A α1 1/t is an analogue of the Gelfand formula for a bounded linear operator ρ(a) = lim n An 1/n The abstract result in Hardin et al. can be adapted to n t (x) = K αt (x y)f αt (n t 1 (y))dy w/ {α t } t 0 independent identically distributed from A = [α, ᾱ] with the following assumptions of f and K:

33 Assumptions (C1) (a) K α(x y) is strictly positive and continuous for x, y ; (b) There exist K, K > 0 such that K Kα(x y) K for all α A (C2) Then (a) f α : R R + continuous; f α(u) = 0 for all u 0. Moreover, f α(u) is continuous in α A uniformly in u R. (b) For any α A, (i) f α(u) is an increasing function in u; (ii) fα(u) < fα(v) if u > v > 0. u v (c) (i) f α is right differentiable at 0 (ii) fα(u) f u α(0) as u 0 + uniformly for α A, (iii) There exist f, f > 0 such that f = inf f α(0) f α(0) f α A (d) (i) There exists m > 0 such that 0 f α(u) m for all u C + () (ii) For b = m k dy, there exists f 1 = inf fα(b) > 0. α A

34 Assumptions (C1) (a) K α(x y) is strictly positive and continuous for x, y ; (b) There exist K, K > 0 such that K Kα(x y) K for all α A (C2) (a) f α : R R + continuous; f α(u) = 0 for all u 0. Moreover, f α(u) is continuous in α A uniformly in u R. (b) For any α A, (i) f α(u) is an increasing function in u; (ii) fα(u) < fα(v) if u > v > 0. u v (c) (i) f α is right differentiable at 0 (ii) fα(u) f u α(0) as u 0 + uniformly for α A, (iii) There exist f, f > 0 such that f = inf f α(0) f α(0) f α A (d) (i) There exists m > 0 such that 0 f α(u) m for all u C + () (ii) For b = m k dy, there exists f 1 = inf fα(b) > 0. α A Then r = lim t A αt A α1 1/t exists...

35 λ 1 analogue for random model Instead of r = lim t A αt A α1 1/t, consider where ñ t (x) = [ 1/t Λ = lim ñ t (x) dx] (10) t K t (x y)f α t (0)ñ t 1 (y) dy, Λ asymptotic growth rate of the population Conjecture: Λ and r cross 1 at same time

36 Example: convergence of Λ t = ( n t(x) dx ) 1/t Suppose K t {K 1, K 2 }, chosen with 50/50 probability (CFK) λ t 1.8 L=19,N 0 =π/19(sin(π x/19)) L=19,N 0 2/L L=19,N 0 1/L Time t Regardless of IC or flip sequence, Λ t Λ 1.22

37 Two-step vs. Coin-Flip For comparison, consider limit Λ for CFK model vs. exact λ 1 (K) for two-step model: Λ of the random model 1.1 The principle eigenvalue λ 1,twostep The stream length L Note: For L = 19, Λ 1.22 as in previous plot. (v 1 = 0.1, v 2 = 1, R 1 = 1.2, R 2 = 1.5, D 1 = 1, D 2 = 1, α 1 = 1, α 2 = 1)

38 Example: Random flow speed Consider Λ for random model with flow velocity chosen from log normal distribution with µ = 0.95, as a function of the variance: Mean (v)= Λ Variance of v Other parameters fixed with R = 1.2, D = 1, α = 1.

39 Example 2: Different means Similar scenario, with three different means: Λ mean(v)= mean(v)=0.95 mean(v)= Variance of v

40 Explicit calculation for Λ Let n 0 (x) = 1 L. Λ 1 = n 1 (x) dx = R 1 L Continuing, K 1 (x, y) dx dy = R 1 L s 1 (y) dy

41 Explicit calculation for Λ Let n 0 (x) = 1 L. Λ 1 = n 1 (x) dx = R 1 L K 1 (x, y) dx dy = R 1 L s 1 (y) dy Continuing, 1/2 ( ) 1/2 R1 R 2 Λ 2 = (R 2 K 2 (x, y) n 1 (y) dy dx) = s 2 (y) r 1 (y) dy L ( ) 1/3 R1 R 2 R 3 Λ 3 = s 3 (y) K 2 (y, z) r 1 (z) dy dz L ( ) R1 R 2 R 3 R 4 Λ 4 = s 4 (z 3 ) K 3 (z 3, z 2 ) K 2 (z 2, z 1 ) r 1 (z 1 ) dz 3 dz 2 dz 1 L. Λ n = R L }{{} n 1 terms s n (z n 1 ) n 1 i=2 K i (z i, z i 1 ) r 1 (z 1 ) dz n 1... dz 1 1/n

42 Asymmetric Laplace Kernels We can compute this exactly for random asymmetric Laplace kernels since, essentially, n t (x) = R α 1,t + L 2t+1 j=2 α j,t e γjx (11) where α j,t are certain computable coefficients that depend on the kernels and the γ j s are front/back kernel coefficients from the random kernels. Integrating (11) yields Λ t = [R] 1/t α 1,t + 2 L 2t+1 j=2 α j,t γ j sinh γ jl 2 1/t (12) Nice closed form solution, but proving computationally unstable due to small divisors; further complicated by the fact that we need t >> 1

43 References I D. Hardin, P. Takac, G Webb, Asymptotic properties of a continuous-space discrete time population model in a random environment, Journal of Mathematical Biology, 1988 I F.Lutscher, E. Pachepsky, M. Lewis, The effect of dispersal patterns on stream populations, Siam Review, 2005 Thanks for your attention! Jon Jacobsen EDM 2012