Spatial Ecology: Lecture 4, Integrodifference equations. II Southern-Summer school on Mathematical Biology

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1 Spaial Ecology: Lecure 4, Inegrodifference equaions II Souhern-Summer school on Mahemaical Biology

2 Inegrodifference equaions Diffusion models assume growh and dispersal occur a he same ime. When reproducion and dispersal occur a discree inervals an inegrodifference equaion is a more relevan formulaion. E.g. annual plans, Many insecs, Migraing bird species

3 Inegrodifference equaion H Dynamics H Dispersal H 1 One generaion (e.g.1 year) The model: The model: K ( x, y) ( y) dy Prob. of Dynamics 1 from dispersing y o x and non-fores)

4 Inegrodifference equaion H Dynamics H Dispersal H 1 One generaion (e.g.1 year) The model: The model: Dynamics K ( x, y) ( y) dy Prob. of Dynamics 1 from dispersing y o x and non-fores) ( y) f ( ( y)) Kernels k(x,y)=k(x-y) when dispersal depends on disance only

5 Mechanisic derivaion of dispersal kernels u k( x) Du xx a ( ) u, Seling rae a( ) u( x, ) d 0 Toal seled u(0, x) Gaussian: a()=(-t), sops a ime T 2 1 x k G exp 4DT 4DT Laplacian: a()=a>0, consan seling rae k L a 4D exp a D x

6 Dispersal kernels from daa

7 Travelling wave speeds Assume f is linearly bounded, f()<=f (0) (o Allee effec) f is monoone K(x) has a momen generaing funcion (no fa-ailed dispersal kernels) Then we can linearly deermine he asympoic wave speed. Look a behaviour near * =0 (linearise here) K( x, y) f ( ( y)) dy f '(0) K( x, y) ( y) dy 1 1

8 Travelling wave speeds A ravelling wave soluion moves wih consan shape and speed c, so ( x ) 1 c

9 Travelling wave speeds A ravelling wave soluion moves wih consan shape and speed c, so 1( x) ( x c) Then (assume disance dependen dispersal) ( x c) f '(0) K( x y ) ( y) dy

10 Travelling wave speeds A ravelling wave soluion moves wih consan shape and speed c, so 1( x) ( x c) Then (assume disance dependen dispersal) ( x c) f '(0) K( x y ) ( y) dy Look for soluions a he edge of he ravelling wave which decay exponenially, so exp( sx) exp( sc) f '(0) K( y)exp( sy) dy f '(0) M ( s) M(s) is he momen generaing funcion for k(y)

11 Asympoic wave speed Differeniaing wih respec o s, and noing ha iniial condiions wih compac suppor lead o a minimum speed give c * 1 c * min ln M ( s) f '(0) s s

12 Examples of wave speeds Gaussian M G ( s) exp( s / 2), where 2D c 2 2ln f '(0) oe if r=ln f (0) and D= 2 hen he wave speed is he same as he PDE case: c 2 Dr Laplacian 1 2 M L ( s), where 2D / a s / 2 We can find c explicily, bu since M L (s)>=m G (s) hen c Laplace >c Gaussian

13 Shape of he kernel grealy affecs speed.

14 Fa ailed kernels

15 Spaial exen Fa ailed kernels can give acceleraing waves, we can calculae he speed, bu we can measure he spaial exen of he wave a a given ime. Spaial exen= disance from source where populaion firs falls below a hreshold. 1 f '(0) K( x, y) ( y) dy, 0 0 Use Fourier Transforms ˆ ( w) e iwx dx, ˆ ( w) e iwx dw Hence, ˆ ( w) ( f '(0)) ( kˆ( w)) 0

16 Spaial exen In he case of he Cauchy Kernel: Is easy o find he inverse of he Fourier ransform in his case so More generally ) exp( ) ˆ(, ) ( ) ( 2 2 w w k x x k ) ( ) (, ) ( ) ( R x x R x f 1 x provided, ) ( ), ( ) ( f R k x x k R x

17 Populaion spread and invasion Muskra Linear expansion wih ime Slow iniial spread followed be linear expansion (e.g.allee effecs) Spread rae coninually increases wih ime (e.g long disance dispersal) House finch Chea grass

18 House finch model Reproducion Spring, Survival Form breeding pairs J Juveniles (9 12 monhs old in spring) A Aduls J A Breeding pairs Summer Survival Fall Disperse

19 Reproducion Average number of offspring produced 2 c f ( ) 2 4 /( T ) 2 / f 2 c ) 4 /( T ) 2 / ( 2 Compeiion for nesing sies C= average number of offspring born ha survive summer, rae or pair formaion T, Time for pair formaion densiy of nes sies Allee efffec!

20 Dispersal J 1 A (1 pj ) f ( ) pj K J ( y x) f ( ( y)) dy 1 on-dispersing juveniles s(1 p A ) f ( ) p A K A ( y x) ( y) dy on-dispersing Aduls which survive Dispersing juveniles Dispersing aduls Add he equaions ogeher o ge an equaion for breeders 1 (1 p J ) f ( ) s(1 p A ) f ( ) K J ( y x) p J f ( ( y)) dy p A K A ( y x) ( y) dy Expeced densiy of birds a he Chrismas Bird Couns in successive years

21 Dispersal kernels

22 Resuls: Range expansion Slow iniial spread followed be linear expansion

23 Invasion summary Shape of he kernel significanly affecs speed. Travelling waves may exhibi acceleraing spread if he dispersal kernels have fa ails (no exponenially bounded) Populaions escaping an Allee effec may ermporally accelerae before achieving a consan speed

24 References M. Ko, M.A. Lewis, P van den Driessche. (1996) Dispersal daa and he spread of invading organisms, Ecology 77: J.A.Powell,(2009)Spaioemporal Models in Ecology:An Inroducion o Inegro-Difference Equaions hp:// euber, M.G., M. Ko and M.A. Lewis, Dispersal and paern formaion in a discree-ime predaor-prey model. Theoreical Populaion Biology 48: R.R. Vei, M.A. Lewis(1996) Dispersal populaion growh and he Allee effec: Dynamics of he house finch invasion of Easern orh America, American auralis, 148(2),