Collaborators: Roger Nisbet, Sebastian Diehl, Scott. Jonathan Sarhad, Bob Carlson, Lee Harrison

Transcription

1 Modeling spatially-explicit ecological dynamics in streams and rivers Kurt E. Anderson Collaborators: Roger Nisbet, Sebastian Diehl, Scott Cooper, Ed McCauley, Mark Lewis, Frank Hilker, Jonathan Sarhad, Bob Carlson, Lee Harrison

2 Environmental Variation Ecology distribution and abundance Regulatory factors vary Time Space Forecasting responses to environmental change Intense changes from human activities

3 Directional biases in dispersal

4 Spatial variation typifies streams and rivers esource bio omass Data courtesy S. Diehl Population Distribution??? Egg mass location R Distance Downstream (2cm) Lancaster et al. 23, Journal of Animal Ecology 72:

5 Population response depends on scale N Births Deaths + Immigration Emigration dominate at large scales dominate at small scales CLOSED OPEN

6 Population response depends on scale N Births Deaths + Immigration Emigration dominate at large scales dominate at small scales CLOSED OPEN HOW LARGE IS LARGE?

7 Let s talk about rivers Steady state responses to spatial environmental variation Response length Spatio-temporal temporal dynamics Transient dynamics Population persistence River networks

8 Let s talk about rivers Steady state responses to spatial environmental variation Response length Spatio-temporal temporal dynamics Transient dynamics Population persistence River networks

9 Model of an idealized river mx Flow eg x Rx N xt, t change in population Location x x R x e,,, G x N x t m x N x t eg y N y t h xy dy recruitment emigration mortality immigration

10 Steady State Response to Spatial Heterogeneity N( xt, ) t x R x e x ( x, t) m x ( x, t) e y N( y, t) h( xy) dy N N G G * * Define Rx ( ) R1 rx ( ) ; mx ( ) m1 ( x) * * e ( x) e 1 ( x) ; N( x) N 1 n( x) G G ; Substituting and discarding products of small quantities yields: n mr( x) egn( x) ( x) e n( y) ( y) h( x y) dy t x G

11 Spatial scale and the Fourier transform Fourier transform of (say) n(x) is defined by: 2i x L E F nx ( ) nl ( ) nxe ( ) dx E Interpretation: weighted measure with highest weighting to perturbations over a range of order L. E Large L E Small L E large scale small scale

12 Solving the linearized equation n mr( x) egn( x) ( x) e n( y) ( y) h( x y) dy t x G Apply Fourier transform, solve for nl ( E ) nl ( ) T( L) rl ( ) T( L) ml ( ) T( L) ( L) E r E E m E E E E eg 1 hl ( ) 1 E T L T L T L m r( E) m( E) ; ( E) eg eg 1 1 hl ( E) 1 1 hl ( E) m m

13 Approximation to Transfer Functions If L E is large 2 i x L 2 i i E E E E D L E LE h L h L e dx h L x dx L with LD xh( x) dx = mean distance per jump.

14 Approximation to Transfer Functions If L E is large 2 i x L 2 i i E E E E D L E LE h L h L e dx h L x dx L with LD xh( x) dx = mean distance per jump. Transfer functions: LD eg 2 i 1 L E m T r L E T m L E and T L E LD eg LD eg 12i 12i L m L m E E

15 Approximation to Transfer Functions If L E is large 2 i x L 2 i i E E E E D L E LE h L h L e dx h L x dx L with LD xh( x) dx = mean distance per jump. Transfer functions: LD eg 2 i 1 L E m T r L E T m L E and T L E LD eg LD eg 12i 12i L m L m E Largest response to large scale (L ) variation in Largest response to large scale (L E ) variation in recruitment and mortality; small scale (L E ) variations in emigration rate. E

16 Impulse response function Impulse response function: Inverse Laplace transform of transfer function Describes steady state response to localized perturbation (delta function) For perturbation in R at x =, downstream population density has the form, x L R R n exp L R = Response length on Populati * n N H * N H Response length perturbation Location x

17 Interpretation of Response Length L R eg per cap. emigration eg LR LD, where m per cap. mortality m LD avg. dispersal length Response length average lifetime dispersal distance Examples Baetis in Kuparak River L R 2 km (Hershey et al. 1993, Ecology 74: ) Gammarus in Lake District L R 15km (Humphries & Ruxton 23, Limnol. Oceanogr. 48: ) Multiple species es in Sierra Nevada L R m (Diehl, Anderson et al. 28. Chpt. 7, Aquatic Insects: Challenges to Populations. CABI Publishing)

18 Parameterized Example Stonefly Leuctra nigra in Broadstone Stream Winterbottom et al 1997 Freshw Biol 38; Speirs et al 2 J Anim Ecol 69; Speirs & Gurney 21 Ecol 82 Average per capita emigration rate and average dispersal length per event vary with average stream current velocity Estimate response length over range of velocities

19 Parameterized Example Stonefly Leuctra nigra in Broadstone Stream Winterbottom et al 1997 Freshw Biol 38; Speirs et al 2 J Anim Ecol 69; Speirs & Gurney 21 Ecol 82 Average per capita emigration rate and average dispersal length per event vary with average stream current velocity Estimate response length over range of velocities Low velocity response length L R 13 m High velocity response length LR 76 m

20 Recruitment Rate Population, Short Response Length 13 m Population, Long Response Length 76 m T r L E 1 L 1 2i L R E flow

21 Extension and applications Consumer-resource interactions Adaptive predator, local resource Qualitative similarities in steadystate response* Application to Merced River restoration project How does salmon food track flow variability?

22 Let s talk about rivers Steady state responses to spatial environmental variation Response length Spatio-temporal temporal dynamics Transient dynamics Population persistence River networks

23 Spatio-temporal variability Temporal perturbations may have spatial signature Transient dynamics Photos Wikipedia Commons Lancaster et al. 23, Journal of Animal Ecology 72:

24 Spatio-temporal dynamics Transient Dynamics are a key component of many advective systems Measures of transient response for non-spatial systems include: - Characteristic return time (asymptotic return to equilibrium) - Reactivity (maximum growth rate of perturbation) - Amplification envelope (maximum size perturbation can grow) For ODE system of form d n dt - Return time from leading eigenvalue of J dt - Reactivity from leading eigenvalue of H = ½(J+J T ) - Amplification envelope from matrix norm of exp(jt) Jn In advective systems, these calculations can be performed on the Laplace or Fourier transformed equations, thereby relating transient response to spatial scale.

25 Two-compartment model Benthos Drift v L D N t N t B D R e m N N G B D ND e GNB N D v x emigration settlement advection Response length L R v eg m L D e G m

26 Dynamics of recast system d n Jn ˆ dt Jn 1, and R R D D B E L L L L n L t J n and, 2 ˆ ˆ 1 D E E n L t i L J n

27 Dynamics of recast system d n Jn ˆ dt Jn 1, and R R D D B E L L L L n L t J n and, 2 ˆ ˆ 1 D E E n L t i L J n

28 Two-phase model is reactive maximum amplification LE very large LE 8 L 2 E LE very small

29 Effect of response length L R Spatial Wavelength L E Characterist tic Return Time (1/λ1)

30 Effect of response length L R Cha aracterist tic Return Time (1/λ λ 1 ) Larger spatial scales mean longer return times Spatial Wavelength L E

31 Effect of response length L R Cha aracterist tic Return Time (1/λ λ 1 ) Longer response lengths mean faster return times Spatial Wavelength L E

32 Effect of response length L R Maximu um Amplif fication L R =2 L R =5 L R =1 L R =3 Longer response lengths mean smaller maximum amplification L R =1 Spatial Wavelength L E

33 How do these temporal and spatial scales interact? t?

34 How do these temporal and spatial scales interact? t? Response length characterizes spatial transition between small scale and large scale temporal transients

35 How do these temporal and spatial scales interact? t? Response length characterizes spatial transition between small scale and large scale temporal transients Rapid, small Slow, large scale dynamics Response length scale dynamics

36 Consumer-resource model 5 Consumer emigration unstable e ˆ G = 1 e ˆ G = 5 e ˆ G = 15 stable Stability e ˆ G = Reactivity no transient transient amplification amplification LOG Spatial Wavelength L E

37 Consumer-resource model 5 Consumer emigration unstable e ˆ G = 1 e ˆ G = 5 e ˆ G = 15 stable Stability e ˆ G = transient amplification over large range Reactivity transient amplification no transient amplification LOG Spatial Wavelength L E

38 Stability Consumer-resource model unstab ble stab ble 5 new characteristic spatial scale Consumer emigration e ˆ = 1 G e ˆG = 5 e ˆG = 15 e ˆ = 3 G React tivity t ransient am mplification no tra ansient amplif fication LOG Spatial Wavelength L E

39 Let s talk about rivers Steady state responses to spatial environmental variation Response length Spatio-temporal temporal dynamics Transient dynamics Population persistence River networks

40 Population persistence and the critical domain size Drift paradox (Müller 1954) Closed population with only advection will become extinct Resolutions: Compensatory upstream movement? Only surplus drift? Theory points to spatial scale Critical domain size

41 Population models Source x = Mouth x = L Spiers-Gurney (SG) model (21): 2 N N N rn f ( N ) v D for xl t x x 2 population growth rate advection diffusion N vn, td, and N L, t, for all t x x zero flux absorbing

42 Population models Drift-benthos (DB) model (Lutscher et al. 25, Pachepsky et al. 25): Benthos Drift t N B rn f ( N ) e G NB N D population growth rate emigration settlement 2 N D N D N en G B N D v D 2 t x x emigration settlement advection diffusion for x L N vn, td, and ND L, x t D x zero flux, for all t absorbing

43 Persistence in an infinitely long river For L, persistence in the SG model is guaranteed if v 2 Dr

44 Persistence in an infinitely long river For L, persistence in the SG model is guaranteed if v 2 Dr upstream propagation p speed

45 Persistence in an infinitely long river For L, persistence in the SG model is guaranteed if v 2 Dr upstream propagation p speed A necessary condition for persistence is that the tendency to A necessary condition for persistence is that the tendency to propogate upstream wins over washout by advection

46 Persistence in an infinitely long river For L, persistence in the DB model is guaranteed if r > e G or if v 2 Dr 2 Dr if r e G e r G e G,

47 Persistence in an infinitely long river For L, persistence in the DB model is guaranteed if r > e G or if v 2 Dr 2 Dr if r e G e r G e G,

48 Persistence in an infinitely long river For L, persistence in the DB model is guaranteed if r > e G or if v 2 Dr 2 Dr if r e G e r G e G,

49 Persistence in a finite river is determined by critical domain size In SG model, the critical domain size L C determines the boundary between washout by advection and persistence: L C Dr 1 2 4Dr 4Dr v arctan 2 2 4Dr v v Persistence is hurt by a reduction in river length and an increase in advection speed.

50 Let s talk about rivers Steady state responses to spatial environmental variation Response length Spatio-temporal temporal dynamics Transient dynamics Population persistence River networks

51 Future directions river networks

52 Future directions river networks

53 Future directions river networks Campbell-Grant et al. 27. Ecol. Lett. 1:

54 Modeling river networks with quantum graphs Start with metric graphs Edges have length, married at vertices Graphs have a distance function Use quantum graph theory Pairs a differential operator (e.g. advection-diffusion) with a metric graph Edge Vertex Differential operator defined on each edge and glued together with appropriate boundary conditions at vertices Global operator on the graph

55 Extend response length to graph Response neighborhood Scaling relationships Responses to continuous variability Project goals Similar dependence on response length as 1D? Transient solutions Invasion waves, persistence Radial and stochastic ti networks

56 Some results on persistence Infinite domain result holds Population must be able to propagate upstream Radial networks suggest distribution of domain in the network, rather than simple size, is key to persistence Bounds on the long-term growth rate for general networks This and much more at t445 4:45pm!

57 Acknowledgements Collaborators R Ni b t Roger Nisbet Sebastian Diehl Scott Cooper Ed McCauley Mark Lewis Frank Hilker Jonathan Sarhad Bob Carlson Lee Harrison Funding Thoughtful commentary Papers Mike Neubert Frithjof j Lutscher Leeza Pachepsky Keith Taulbee John Melack Dougie Speirs Bill Fagan Maggie Simon Anderson et al. 25. Ecol. Lett. Anderson et al. 26. Am. Nat. Nisbet et al. 28. Math. Biosci. Eng. Diehl et al. 28. CABI Publishing Anderson et al. 28. Bull. Math. Biol. Nisbet et al. 29. CRC Press. Anderson et al Ecol. Lett.