Spatial Ecology: Lecture 3, Reaction-diffusion models: spatial patterns

Transcription

1 Spatial Ecology: Lecte 3, Reaction-diffsion models: spatial pattens II Sothen-Smme school on Mathematical Biology Chistina Cobbold Uniesity of Glasgow

2 Oeiew Peiodic Taelling waes Patten fomation

3 Peiodic taelling waes Q: Pedato-pey systems often show cyclic dynamics, what happens when yo add andom moement to sch a system?

4 Peiodic taelling waes Q: Pedato-pey systems often show cyclic dynamics, what happens when yo add andom moement to sch a system? A: Peiodic Taelling Waes PTW can fom.

5 Peiodic taelling waes Q: Pedato-pey systems often show cyclic dynamics, what happens when yo add andom moement to sch a system? A: Peiodic Taelling Waes PTW can fom. Q: What ae peiodic taelling waes?

6 Peiodic taelling waes Q: Pedato-pey systems often show cyclic dynamics, what happens when yo add andom moement to sch a system? A: Peiodic Taelling Waes PTW can fom. Q: What ae peiodic taelling waes? A: Like the Waes of cheeing yo see in cowds in a socce stadim.

7 Examples of PTW in nate Fennoscandian oles Field oles in Kielde foest UK Lach bdmoth in the Eopean Alps Atmnal moth in Nothen Noway Spatial-tempoal pattens in cyclic poplations ae chaacteised by the way synchony in poplation dynamics change coss a landscape Coss site Coelation Wae length. x

8 Pedato-pey model dh dt dp dt h 1 akp 1 kh h h 0 bp ckph 1 kh

9 Pedato-pey model dh dt dp dt h 1 akp 1 kh h h 0 bp ckph 1 kh Lineaise abot coexistence eqilibim and examine the behaio close to the hopf bifcation. Change aiables so that the cycle is a cicle in the tansfomed phase space conet to Nomal Fom.

10 Pedato-pey model dh dt dp dt h 1 akp 1 kh h h 0 bp ckph 1 kh Lineaise abot coexistence eqilibim and examine the behaio close to the hopf bifcation. Change aiables so that the cycle is a cicle in the tansfomed phase space conet to Nomal Fom.

11 Lambda-Omega system dt d dt d 1 0, 1 whee,

12 Lambda-Omega system Change to pola coodinates / tan, 1 dt d dt d 1 0, 1 whee,, dt d dt d

13 Lambda-Omega system Change to pola coodinates One soltion of this eqation is the limit cycle: Limit cycle has adis R=1, feqency R / tan, 1 dt d dt d 1 0, 1 whee, t R R, 0, dt d dt d

14 Adding space: andom moement We eqie diffsion constants D h = D p fo the analysis close to the limit cycle to wok. Scale space sch that D h = D p =1., x t x t

15 Adding space: andom moement We eqie diffsion constants D h = D p fo the analysis close to the limit cycle to wok. Scale space sch that D h = D p =1. Change to pola coodinates:, x t x t

16 Looking fo PTW soltions In pola fom the PTW is: Sbstitting into the PDE gies: R, t kx R, k R

17 Looking fo PTW soltions In pola fom the PTW is: Sbstitting into the PDE gies: So the 1-paamete R family of soltions is : Wae speed Peiod in time Peiod in space x R t R R x R t R R sin, cos kx t R,, R k R k R R k c

18 PTW stability and wae selection Infinite domain: Koppel & Howad 1973

19 PTW stability and wae selection Infinite domain: Koppel & Howad 1973 Stable Stable nstable nstable Mechanisms of wae geneation and selection Inasion Bonday conditions Jonathan Sheatt Heiot-Watt

20 Field oles Expeiments: Wae length in Keilde Foest 56-76km; Wae speed km/yea Size of Keilde foest = 30km. So waelength lage than foest Bandwidth of nstable PTW is also a lot lage than Keilde foest, so we cold obsee PTWs 1000km!

21 PTW geneation by bondaies Each obstacle geneates waes, bt those fom the lagest dominate. BCs: Hostile at lake edges Zeo flx at the Domain edges

22 Uneqal diffsion coefficients Diffsion coefficients can significantly change the popeties the peiodic taelling wae. =D /D Gey lines: Stable waes Black lines: Unstable waes

23 Patten fomation Geneally diffsion plays a ole of inceasing stability xx 0 xx 0 xx 0 t xx x

24 Patten fomation Labyinth patten in bsy egetation in Nige Regla maze pattens of shbs and tees in Sibeia Spotted patten Of isolated tees in Nige Coal eef islands in Astalia Stiped patten of tee lines and snow deposition USA Labyinth patten of maine benthic diatomes Pattened mssel bank in the Nethelands Regla spaced Tssocks of the Sedge Caex sticta

25 Patten fomation Geneally diffsion plays a ole of inceasing stability xx 0 xx 0 xx 0 t xx BUT, this is not always the case: Diffsion dien instability x

26 Geneal idea in 1-D Assme thee exists a spatially nifom positie eqilibim *,*. i.e. F*,*=G*,*=0, which stable in the absence of diffsion. The Jacobian associated to the lineaisation abot this eqilibim is So stability means F +G <0 and F G -F G >0,,, 1 G x D t F x D t G G F F J

27 Geneal idea in 1-D Assme thee exists a spatially nifom positie eqilibim *,*. i.e. F*,*=G*,*=0, which stable in the absence of diffsion. The Jacobian associated to the lineaisation abot this eqilibim is So stability means F +G <0 and F G -F G >0 A stable ecosystem that is pefectly homogeneos wold contine indefinitely to be homogeneos. In pactice iegla and stochastic flctations in poplation size and the enionment continosly intodce small local petbations.,,, 1 G x D t F x D t G G F F J

28 Stability of the local petbations Lineaise the PDE abot the eqilibim Look fo soltions of the fom Q: What is the feqency of gowing petbations? We want non-zeo soltions 0, 0. So we hae an eigenale poblem. Petbation gowth means >0. This occs if G G F F D D x t 1 exp cos,, 0 0 t kx t x t x D k G G F D k F 0 det 1 D k G G F D k F k Q

29 Obseed pattens Qk Possible wae nmbes k k c satisfies Qk c =0 ae the fist petbations to gow in an infinite spatial domain, and this is what we obsee.

30 -D bonded domain In a finite domain, bonday conditions select the waelength of the patten that is obseed. In -D domain geomety also detemines the waelength of the obseed pattens.

31 Intepeting the patten fomation conditions The sign stcte of the Jacobian of the non-spatial model mch hae the following sign stcte Positie feedback Actiato-Inhibito Positie feedback Actiato-Inhibito Withot loss of geneality let F >0 then fo patten fomation we eqie D > D 1, dispeses fthe than. We cannot get patten fomation in a competition model, as the off diagonal enties of J hae the same sign.

32 Examples in nate Otbeaks of Doglas fi tssock moths emain spatially esticted despite the widespead and continos aailability of thei abndant host plant Coss coelation of Caex sticta biomass and soil moiste

33 A geneal pedato-pey model If g=constant then G =0 so no pattens, so g mst depend on and g >0 density dependent motality of the pedato If =constant then we also eqie f >0 e.g. an Allee effect in the pey If f <0 e.g. logistic then we need <0 satation pedation ate g f D D x t 1 g f J ' ' ' '

34 Key ingedients fo patten fomation 1. Pedato dispeses faste than the pey. At low densities, an incease in pey leads to an incease in net ate of pey poplation gowth Pey poplation gowth is atocatalytic e.g. Allee effect Incease in pey leads to a decease in pe capita pedation isk e.g. Type II fnctional esponse and density-dependent pedato motality 3. Incease in pedato density leads to a decease in pey and pedato gowth e.g. Geneally holds fo pedato-pey systems

35 Othe types of moement Pedato aggegation towad pey can eithe pomote aggegation incease pedato esponse to pey o peent pedato apidly aggegates to contol pey patten fomation Patten fomation in competitie systems eqies two competitos to aoid each othe In a single species system non-local aggegation is needed.

36 Refeences A.Okbo, S.A.Lein 001 Diffsion and ecological poblems, Spinge-elag Malchow, H, Petoskii, S. Ventino, E. Spatio-tempoal pattens in ecology and epidemiology: Theoy, models and simlations. Chapman and Hall, CRC. Ting, A.M. 195 The chemical basis of mophogenesis. Philosophical Tansactions of the Royal Society B, 37, Rietkek M, an de Koppel J, 008 Regla patten fomation in eal ecosystems. TREE, J.D.May 001 Mathematical Biology I and II, Spinge-Velag J.A. Sheatt, M.J. Smith 008 Peiodic taelling waes in cyclic poplations: field stdies and eaction-diffsion models. J. R. Soc. Inteface 5, J.A. Sheatt, X. Lambin, T.N. Sheatt 003 The effects of the size and shape of landscape feates on the fomation of taelling waes in cyclic poplations. Am. Nat. 16, M.J. Smith, J.A. Sheatt 007The effects of neqal diffsion coefficients on peiodic taelling waes in oscillatoy eactiondiffsion systems. Physica D 36,