Global and local averaging for integrodifferential equations

Transcription

1 Global and local averaging for integrodifferential equations Frithjof Lutscher Miami December 2012 Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

2 Motivation Reaction-diffusion equation for individuals or genes u t = Du xx + F(u) Infinite homogeneous landscape: Invasion speed c = 2 DF (0) Fisher (1937), Weinberger (1982) Single patch: Minimal domain size L = π D/F (0) Skellam (1951), Kierstadt and Slobotkin (1953) Many patches? Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

3 Heterogeneous landscapes Reaction-diffusion equation in a periodic environment u t = (D(x)u x ) x + F(u, x) with D and F(u, ) of the same period. Persistence conditions and invasion speeds Exact conditions for piecewise constant landscapes Shigesada et al (1986) Abstract results for periodic landscapes Weinberger (2002), Berestycki et al (2005) Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

4 Homogenization Assume: Small and large spatial scale Average over the small scale u t = Du xx + F(u). where D = ( ) 1 1 L dy 1, F(u) = L 0 D(y) L L 0 F(u, y)dy, F (0) > 0 persistence D = 0 somewhere no spread No correlations between D and F enter the equation. Requires movement, not applicable to sedentary stages. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

5 Outline 1 Non-local dispersal 2 Persistence via Global Averaging 3 Persistence via Local Averaging 4 Invasion speeds via Global Averaging Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

6 Outline Non-local dispersal 1 Non-local dispersal 2 Persistence via Global Averaging 3 Persistence via Local Averaging 4 Invasion speeds via Global Averaging Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

7 Non-local dispersal Modeling movement Random walk on real line Time between moves has Poisson distribution with mean µ Move length distribution K u t (t, x) = µu(t, x) + K (x y)µu(t, y)dy = µu(t, x) + [K (µu)](t, x) K is symmetric and exponentially bounded Moment generating function M(s) = k(x)e sx dx Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

8 Non-local dispersal Including population dynamics 1 Continuous movement and reproduction (linear) u t = (b m)u µu + K (µu). b: birth rate, m: mortality rate 2 Mobile offspring, sessile adults (linear) u t = mu + γk (bu) γ: probability of successful establishment 3 Distributed contacts Mollison (1991) I t = β(n I)(K I) αi, 4 Distributed infectives Medlock and Kot (2003) I t = β(n I)I αi + µ(k I I) Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

9 Non-local dispersal Including population dynamics 1 Continuous movement and reproduction (linear) u t = (b m)u µu + K (µu). b: birth rate, m: mortality rate 2 Mobile offspring, sessile adults (linear) u t = mu + γk (bu) γ: probability of successful establishment 3 Distributed contacts Mollison (1991) I t = β(n I)(K I) αi, 4 Distributed infectives Medlock and Kot (2003) I t = β(n I)I αi + µ(k I I) Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

10 Non-local dispersal Including population dynamics 1 Continuous movement and reproduction (linear) u t = (b m)u µu + K (µu). b: birth rate, m: mortality rate 2 Mobile offspring, sessile adults (linear) u t = mu + γk (bu) γ: probability of successful establishment 3 Distributed contacts Mollison (1991) I t = β(n I)(K I) αi, 4 Distributed infectives Medlock and Kot (2003) I t = β(n I)I αi + µ(k I I) Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

11 Non-local dispersal Including population dynamics 1 Continuous movement and reproduction (linear) u t = (b m)u µu + K (µu). b: birth rate, m: mortality rate 2 Mobile offspring, sessile adults (linear) u t = mu + γk (bu) γ: probability of successful establishment 3 Distributed contacts Mollison (1991) I t = β(n I)(K I) αi, 4 Distributed infectives Medlock and Kot (2003) I t = β(n I)I αi + µ(k I I) Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

12 Non-local dispersal The linear model in a heterogeneous landscape We consider the model with 0 g 1, h = h(x) 0. Periodic Landscape: u t (t, x) = f (x)u(t, x) + g(x)[k (hu)](x) Patch type i of length L i Period L 1 + L 2 = L and fraction p = L 1 /L Parameter functions piecewise constant: f (x) = f i on patch i Dispersal kernel origin dependent: K i (z) on patch i Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

13 Outline Persistence via Global Averaging 1 Non-local dispersal 2 Persistence via Global Averaging 3 Persistence via Local Averaging 4 Invasion speeds via Global Averaging Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

14 Averaging I Persistence via Global Averaging Persistence condition λu(x) = f (x)u(x) + g(x) K (x y; y)h(y)u(y)dy Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

15 Averaging I Persistence via Global Averaging Persistence condition λu(x) = f (x)u(x) + g(x) Scale space x = Lz, y = Lw, ũ(z) = u(x/l) λũ(z) = f (z)ũ(z) + g(z) K (x y; y)h(y)u(y)dy L K (L(z w); w) h(w)ũ(w)dw Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

16 Averaging I Persistence via Global Averaging Persistence condition λu(x) = f (x)u(x) + g(x) Scale space x = Lz, y = Lw, ũ(z) = u(x/l) λũ(z) = f (z)ũ(z) + g(z) Periodicity λũ(z) = f (z)ũ(z) + g(z) 1 0 n K (x y; y)h(y)u(y)dy L K (L(z w); w) h(w)ũ(w)dw L K (L(z w n); w) h(w)ũ(w)dw. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

17 Averaging II Persistence via Global Averaging Riemann sum lim L 0 L K (L(z w n); w) = n K (v; w)dv = 1. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

18 Averaging II Persistence via Global Averaging Riemann sum lim L 0 L K (L(z w n); w) = Globally averaged equation n K (v; w)dv = 1. 1 λũ(z) = f (z)ũ(z) + g(z) h(w)ũ(w)dw, 0 Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

19 Averaging II Persistence via Global Averaging Riemann sum lim L 0 L K (L(z w n); w) = Globally averaged equation If only h depends on x: n K (v; w)dv = 1. 1 λũ(z) = f (z)ũ(z) + g(z) h(w)ũ(w)dw, 0 1 λ = f + g h(w)dw 0 Dispersal, via K, helps to average local growth. But f, g outside the integral have to be constant. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

20 Averaging III Persistence via Global Averaging Eigenvalue equation after scaling and limit 1 λũ(z) = f (z)ũ(z) + g(z) h(w)ũ(w)dw, 0 Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

21 Averaging III Persistence via Global Averaging Eigenvalue equation after scaling and limit 1 λũ(z) = f (z)ũ(z) + g(z) h(w)ũ(w)dw, 0 Piecewise constant parameter functions p 1 λu 1 (z) = f 1 u 1 (z) + g 1 h 1 u 1 (w)dw + g 1 h 2 u 2 (w)dw, 0 p 1 λu 2 (z) = f 2 u 2 (z) + g 2 h 1 u 1 (w)dw + g 2 h 2 u 2 (w)dw, System of two equations: u i is density on patch i 0 p p Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

22 Averaging Result Persistence via Global Averaging Consider the eigenvalue problem λu(x) = f (x)u(x) + g(x) K (x y; y)h(y)u(y)dy with piecewise constant, L-periodic coefficient functions. In the limit L 0, the dominant eigenvalue satisfies [ ] [ ] [ u1 f1 + g λ = 1 h 1 p g 1 h 2 (1 p) u1 u 2 g 2 h 1 p f 2 + g 2 h 2 (1 p) u 2 ]. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

23 Example Persistence via Global Averaging Continuous movement and growth u t = (b m)u µu + K (µu) b: birth rate, m: mortality rate, r = b m net growth Scale r 1 = 1, r 2 < 0. Persistence condition or µ 1 < 1 1 p µ 2 p r 2 < µ 1 (1 p) 1. µ 1 µ persistence p=0.2 p=0.1 p=0.75 p= Persistence boundary for zero mean growth Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

24 Outline Persistence via Local Averaging 1 Non-local dispersal 2 Persistence via Global Averaging 3 Persistence via Local Averaging 4 Invasion speeds via Global Averaging Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

25 Patch scale - idea Persistence via Local Averaging Eigenvalue equation as before 1 λũ(z) = f (z)ũ(z) + g(z) K (L(z w n); w) h(w)ũ(w)dw. 0 Now split into patch types p 1 λu 1 (z) = f 1 u 1 (z) + g 1 h 1 K 1 (z w)u 1 (w)dw + g 1 h 2 K 2 (z w)u 2 (w)dw, 0 p 1 λu 2 (z) = f 2 u 2 (z) + g 2 h 1 K 1 (z w)u 1 (w)dw + g 2 h 2 K 2 (z w)u 2 (w)dw, Take averages ū 1 = 1 p 0 p 0 u 1(z)dz, and ū 2 accordingly. p p Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

26 Patch scale - result Persistence via Local Averaging To lowest order, the averages satisfy [ ] [ ū1 f 1 + g 1 h 1 s 11 1 p 1 g 1 h 2 λ = p s12 2 p ū 2 g 2 h 1 1 f 2 + g 2 h 2 s p s21 ] [ ū1 ū 2 ]. Where the average dispersal success from patch j to patch i is s ij = 1 K (x y)dxdy. Ω j Ω i Ω j Global averaging results when s ij equals the fraction of type i patches. The s ij -terms contain local movement information Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

27 Persistence via Local Averaging Example Continuous movement and growth u t = (b m)u µu + K (µu) b: birth rate, m: mortality rate, r = b m net growth 3 Laplace Kernel persistence σ 2 =0.1 σ 2 =1 K (x) = 1 2d exp( x /d) variance σ 2 = 2d 2 µ 1 µ σ 2 =0.05 σ 2 = Persistence boundary for zero mean growth, p = 0.2 Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

28 Persistence via Local Averaging Example - continued Persistence boundary from dispersal distances d i : dispersal distance from patch i persistence µ 1 =2.5 µ 2 = 1 p = 0.2 d µ 1 =3 µ 1 = µ 1 =2 No persistence for global averaging d 1 Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

29 Outline Invasion speeds via Global Averaging 1 Non-local dispersal 2 Persistence via Global Averaging 3 Persistence via Local Averaging 4 Invasion speeds via Global Averaging Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

30 Invasion speeds via Global Averaging Traveling periodic wave - linear Ansatz Eigenvalue equation u(t, x) = e s(x ct) v(x) scv(x) = f (x)v(x) + g(x) As before (global averaging) scv(z) = f (z)v(z) + g(z) K (x y; y)e s(x y) h(y)v(y)dy 1 0 M(s; w)h(w)v(w)dw. M(s; w) = K (z ; w)e sz dz Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

31 Minimal speed Invasion speeds via Global Averaging Model equation u t (t, x) = f (x)u(t, x) + g(x)[k (hu)](x) Result: In the fine-grain limit L 0, the minimal TW speed is 1 c = min s>0 s ρ(s), where ρ(s) is the dominant eigenvalue of [ ] f1 + g 1 h 1 M 1 (s)p g 1 h 2 M 2 (s)(1 p) g 2 h 1 M 1 (s)p f 2 + g 2 h 2 M 2 (s)(1 p) Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

32 Example Invasion speeds via Global Averaging Continuous movement and growth u t = (b m)u µu + K (µu) b: birth rate, m: mortality rate, r = b m net growth 2.5 Laplace Kernel (mean d) K (x) = 1 2d exp( x /d) Moment generating function M(s) = 1 1 d 2 s 2 relative speed r 1 Speed with constant mean growth. Only r varies. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

33 Invasion speeds via Global Averaging Example - continued r 1 = 1, r 2 = 0 µ 2 = 1 relative speed µ 1 Maximum Speed for intermediate movement rate µ 1. Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

34 Example II Invasion speeds via Global Averaging Mobile offspring, sessile adults (linear) u t = mu + γk (bu) γ: probability of successful establishment 2 m 1 =0, m 2 =1 constant average net growth p(b 1 m 1 ) + (1 p)(b 2 m 2 ) = 1/2 spreading speed m 1 =m 2 =0.5 m i =0.5 b i b 1 Speeds for different scenarios Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27

35 Conclusions Invasion speeds via Global Averaging 1 Technique of homogenization deal with integrals deal with non-mobile compartment patch averaging retains movement information 2 Results for specific models Not just averaged growth and dispersal Correlations matter 3 Extensions Apply to kernels with movement behavior (Jeff Musgrave) patch averaging for RDE (with Christina Cobbold) Apply to reaction-diffusion equations with no-mobile stage F. Lutscher (2010) Nonlocal dispersal and averaging in heterogeneous landscapes Applicable Analysis 89(7): Frithjof Lutscher (University of Ottawa) Homogenization of IDEs December / 27