[PPT] - Integrodifference equations for invasive species in heterogeneous PowerPoint Presentation

SLIDE 1

Integrodifference equations for invasive species in heterogeneous environments

F . Lutscher University of Ottawa Ottawa, Canada

Integrodifference equations for invasive species in heterogeneous environments – p. 1/5

SLIDE 2

Invasive Species

Economic and ecological damage from non-native invasive species (forest insect pests, plants, mussels...) Theory for homogeneous landscapes (SKELLAM,

WEINBERGER, KOT ET AL, LIEBHOLD...)

Sources of Heterogeneity Landscape features (SHIGESADA) Population structure (CASWELL) Temporal variation (NEUBERT, CASWELL, SCHREIBER)

Integrodifference equations for invasive species in heterogeneous environments – p. 2/5

SLIDE 3

Questions

How does habitat heterogeneity impact species persistence and spread? How can we manage biological invasions? What are the important spatial scales? How does fragmentation affect diversity?

Integrodifference equations for invasive species in heterogeneous environments – p. 3/5

SLIDE 4

Outline

Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 4/5

SLIDE 5

Discrete-time models and spatial spread

ut(x) : Density of individuals in generation t. ut

dynamics

− → f(ut)

dispersal

− → ut+1 ut+1(x) =

K(x − y)f(ut(y))dy =: K ∗ f(ut)(x)

f(u) : growth function K(z) : Dispersal kernel

KOT, LEWIS AND VAN DEN DRIESSCHE, 1996, Ecology

Integrodifference equations for invasive species in heterogeneous environments – p. 5/5

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Population growth

Discrete time, non-overlapping generations ut+τ = f(ut) Beverton-Holt, Ricker Allee

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

density ut density f(ut)

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

density ut density f(ut) Na

Integrodifference equations for invasive species in heterogeneous environments – p. 6/5

SLIDE 7

Dispersal kernel

Probability of moving from y to x: K(z) = K(x − y)

−4 −2 2 4 0.1 0.2 0.3 0.4 0.5 0.6

space density Gaussian Tent Exponential root Laplace Tophat

Integrodifference equations for invasive species in heterogeneous environments – p. 7/5

SLIDE 8

Spread in homogeneous landscape

Asymptotic speed of spread (WEINBERGER, 1982) c∗ = inf

s>0

1 s ln(RM(s)) f linearly bounded, R = f ′(0) ≥ f(u)/u f monotone Moment generating function, M(s) =

K(x)esxdx

For Gaussian kernel: cG =

2σ2 ln(R)

Integrodifference equations for invasive species in heterogeneous environments – p. 8/5

SLIDE 9

Example

Constant speed and accelerating traveling waves

KOT, LEWIS AND VAN DEN DRIESSCHE, 1996, Ecology

Integrodifference equations for invasive species in heterogeneous environments – p. 9/5

SLIDE 10

Outline

Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 10/5

SLIDE 11

Fragmented landscape

space 1−p p type 1 "good" type 2 "bad"

ut+1(x) =

K(x, y)f(ut(y), y)dy

Integrodifference equations for invasive species in heterogeneous environments – p. 11/5

SLIDE 12

Single species

ut+1(x) =

K(x, y)f(ut(y), y)dy

homogeneous, i.e., K(x − y),

KAWASAKI AND SHIGESADA, 2006

diffusive dispersal submodel

POWELL AND ZIMMERMANN, 2004; ROBBINS AND LEWIS

K(x + L, y + L) = K(x, y)

WEINBERGER, 2002; WEINBERGER ET AL, 2008

rigin dependent, e.g., variance: σ2(y)

write K(x − y, y). DEWHIRST AND LUTSCHER, 2009

Integrodifference equations for invasive species in heterogeneous environments – p. 12/5

SLIDE 13

Spread

Linearization: ut+1(x) =

K(x − y, y)r(y)ut(y)dy,

r(y) = ∂f ∂u(0, y) Traveling profile: ut+1(x) = ut(x − c) = e−s(x−c)v(x) escv(x) = ∞

−∞

K(x − y, y)es(x−y)r(y)v(y)dy

KAWASAKI AND SHIGESADA, 2006

Integrodifference equations for invasive species in heterogeneous environments – p. 13/5

SLIDE 14

Exact implicit formula

Laplace kernel K = 1 2d exp(−|x − y|/d) Piecewise constant r(x) cosh(µL) = cosh(q1L1) cosh(q2L2)+q2

1 + q2 2

2q1q2 sinh(q1L1)+sinh(q2L2), where q1 = 1 d

1 − r1e−µc,

q2 = 1 d

1 − r2e−µc.

Integrodifference equations for invasive species in heterogeneous environments – p. 14/5

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Global averaging

Assumption: variance(s) ≫ habitat period L escv(x) = 1

n

LK(L(x − y − n), y)esL(x−y−n)

r(y)v(y)dy

Riemann sum approximation

n

LK(L(x − y − n), y)esL(x−y−n) ≈ M(s, y) In the limit L → 0 c = min

s>0

1 s ln 1 M(s, y)r(y)dy

Integrodifference equations for invasive species in heterogeneous environments – p. 15/5

SLIDE 16

Piecewise constant habitat

r(y) = r1,2, r1 > 1, r1 ≥ r2 ≥ 0 K(x − y, y) = K1,2(x − y) Approximate spreading speed ˆ c = inf

s>0

1 s ln{r1pM1(s) + r2(1 − p)M2(s)} Averaged growth rate ¯ r = r1p + r2(1 − p) Invasion threshold (rule of thumb) ¯ r = 1: pmin = 1 − r2 r1 − r2

Integrodifference equations for invasive species in heterogeneous environments – p. 16/5

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Simulations: Gauss

0.2 0.4 0.5 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

percentage of good habitat p spread rate (c) (b) (a)

(a) r1 = 2, r2 = 0, σ2 = 2 (b) r1 = 2, r2 = 0, σ2 = 0.25 (c) r1 = r2 = 1.5, σ2

1 = 2, σ2 2 = 0.5

Integrodifference equations for invasive species in heterogeneous environments – p. 17/5

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Random landscapes

Gaussian kernel

0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1

spread rate percentage of good habitat

r1 = 1.8, r2 = 0.2, σ2

1 = 1, σ2 2 = 3

Integrodifference equations for invasive species in heterogeneous environments – p. 18/5

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Simulations: fat tails

The exponential square root kernel

5 10 15 20 25 30 10 20 30 40 50 60 70

generation t front location xf(t) p=0.5 p=0.75 p=1

r1 = 2, r2 = 0, σ2

1 = 1

Integrodifference equations for invasive species in heterogeneous environments – p. 19/5

SLIDE 20

Simulations: Laplace

0.2 0.4 0.5 0.6 0.8 1 0.5 1 1.5 2

percentage of good habitat p spread rate approximate true (c) (a) (b)

(a) r1 = 2, r2 = 0, σ2 = 2 (b) r1 = 2, r2 = 0, σ2 = 0.25 (c) r1 = r2 = 1.5, σ2

1 = 2, σ2 2 = 0.5

Integrodifference equations for invasive species in heterogeneous environments – p. 20/5

SLIDE 21

Outline

Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging - persistence Competing species Allee effect Density-Dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 21/5

SLIDE 22

Patch-level averaging

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

patch type I patch type II u(x) u1 u2 <u>

Integrodifference equations for invasive species in heterogeneous environments – p. 22/5

SLIDE 23

Single patch: persistence

Linearized equation: ut+1(x) = R L K(x − y)ut(y)dy, R = f ′(0) Persistence condition (critical patch size): λ(L) > 1/R λ(L) : leading eigenvalue of the integral operator

(KOT AND SCHAFFER, 1986)

Integrodifference equations for invasive species in heterogeneous environments – p. 23/5

SLIDE 24

Average dispersal success

Spatially averaged probability of successful settlement inside the patch (VAN KIRK AND LEWIS, 1997) Point-release experiments ADS(L) = 1 L L L K(x, y)dxdy Approximate persistence condition: ADS(L) > 1/R

LUTSCHER AND LEWIS, 2004, FAGAN AND LUTSCHER, 2006

Integrodifference equations for invasive species in heterogeneous environments – p. 24/5

SLIDE 25

Examples

Average dispersal success of different kernels

−1 1 0.5 1 1.5

space density

1 2 3 4 0.2 0.4 0.6 0.8 1

length ADS

Integrodifference equations for invasive species in heterogeneous environments – p. 25/5

SLIDE 26

Average dispersal success in fragmentation

Assume hostile bad habitat: r2 = 0 ADS(p) = average probability to land in a good patch Invasion threshold: ADS(pmin) = 1/r1

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5

variance σ2

1

invasion threshold pmin Gaussian kernel Laplace kernel Exponential square root kernel

Persistence and spread harder in fine grain landscapes

DEWHIRST AND LUTSCHER 2009

Integrodifference equations for invasive species in heterogeneous environments – p. 26/5

SLIDE 27

Comparison: Trees

Simulation model: Collingham and Huntley (2000) r1 = 1.02...1.09 gives invasion threshold p ≈ 1.

Integrodifference equations for invasive species in heterogeneous environments – p. 27/5

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Partial dispersal only

ut+1(x) = (1 − q)f(ut(x)) + q

K(x, y)f(ut(y), y)dy

f = 0 on bad patches, f ′(0) = r > 1 on good patches pmin = 1 − (1 − q)r qr ˆ c = inf

s>0

1 s ln{(1 − q)r + pqrM(s)} Example: r = 1.04, q = 0.05, σ2 = 94 gives pmin ≈ 0.23.

Integrodifference equations for invasive species in heterogeneous environments – p. 28/5

SLIDE 29

Outline

Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 29/5

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Competing species I

Non-spatial equations ut+1 = ruut 1 + (ru − 1)(ut + auvt) vt+1 = rvvt 1 + (rv − 1)(vt + avut) Mutual invasion implies coexistence Coexistence only if au, av < 1 Competitive exclusion if au > 1 > av Founder control if au, av > 1

Integrodifference equations for invasive species in heterogeneous environments – p. 30/5

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Competing species II

Spatial equations ut+1(x) =

Ku(x, y)

ruut(y) 1 + (ru − 1)(ut(y) + auvt(y))dy vt+1(x) =

Kv(x, y)

rvvt(y) 1 + (rv − 1)(vt(y) + avut(y))dy Coefficients can vary spatially.

Integrodifference equations for invasive species in heterogeneous environments – p. 31/5

SLIDE 32

Patchy landscape I: separation

space 1−p p type 1: v wins type 2: u wins

Spatially varying competition: au(x), av(x) No dispersal: spatial separation p = 1: v wins; p = 0: u wins Assume symmetry in interactions

Integrodifference equations for invasive species in heterogeneous environments – p. 32/5

SLIDE 33

Invasion conditions I

vt+1(x) =

Kv(x, y)α(y)vt(y)dy,

α(y) = rv 1 + (rv − 1)av(y). For Laplace kernel: exact but implicit L1 > 2d √α1 − 1Tan−1 √1 − α2 √α1 − 1 tanh √ 1 − α2 L2 2d

.

KAWASAKI AND SHIGESADA (2006)

Global averaging: approximate but explicit pα2 + (1 − p)α1 > 0.

Integrodifference equations for invasive species in heterogeneous environments – p. 33/5

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Patch-level averaging

ut+1(x) =

K(x, y)r(y)ut(y)dy,

Average per patch (type) ¯ ui,t = 1 Ωi

Ωi

ut(x)dx Approximate system for the averages ¯ u1,t+1 = r1s11¯ u1,t + r2s21¯ u2,t, ¯ u2,t+1 = r1s12¯ u1,t + r2s22¯ u2,t. sij: ADS from patch (type) j to i

Integrodifference equations for invasive species in heterogeneous environments – p. 34/5

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Invasion conditions II

a1=2, a2=0.6, r=e fraction of type 1 habitat scaled dispersal coefficient d/L

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5

u invades v coexistence v invades u a1=1.2, a2=0.8, r=e fraction of type 1 habitat scaled dispersal parameter d/L

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5

u invades v v invades u coexistence

Is coexistence possible for p = 1/2? How large is the range of p that allow coexistence?

Integrodifference equations for invasive species in heterogeneous environments – p. 35/5

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Coexistence region I

Assume symmetry: ru = rv, Ku = Kv. Coexistence condition at p = 1/2 4 ˜ d √α2 − 1Tan−1 √1 − α1 √α2 − 1 tanh √1 − α1 1 4 ˜ d

< 1,

As d → ∞ α1 + α2 = r 1 + (r − 1)a1 + r 1 + (r − 1)a2 > 2.

Integrodifference equations for invasive species in heterogeneous environments – p. 36/5

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Coexistence region: global averaging

Coexistence condition under global averaging

a2 a1

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10

r=3 r=2 r=1.5 a2 a1

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 4.5 5

no coexistence 0.2 0.4 0.6 0.8

ai : competition coefficients a2 < 1 < a1

Integrodifference equations for invasive species in heterogeneous environments – p. 37/5

SLIDE 38

Coexistence region: patch-level averaging

sij: ADS from patch j to i

a2 a1

0.2 0.4 0.6 0.8 2 3 4 5 6 7 8 9 10

s=0.7 s=0.6 s=0.5 invasion fails

Coexistence for p = 1/2, i.e., s11 = s22 = s.

Integrodifference equations for invasive species in heterogeneous environments – p. 38/5

SLIDE 39

Spreading speed I

Based on single-species linearization:

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7

p speed

Homogenization speed compared with exact speed for d=0.5 a1=2 and a2=0.6

u invading v v invading u

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25

p speed

Homogenization speed compared with exact speed for d=0.3 a1=1.2 and a2=0.8

BUT: Spreading speed is not necessarily linearly determined Lewis et al (2002)

Integrodifference equations for invasive species in heterogeneous environments – p. 39/5

SLIDE 40

Spreading speed II

Failure of linear determinacy...

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

σv

2/σu 2

relative spread rate p=0.5 p<0.5

... if resident spreads much farther than invader

Integrodifference equations for invasive species in heterogeneous environments – p. 40/5

SLIDE 41

Patchy landscape II: good and bad

space 1−p p type 1 "good" type 2 "hostile"

proportion of habitat p non-spatial: competitive exclusion dispersal-related loss of superior competitor coexistence for small enough invader dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 41/5

SLIDE 42

Coexistence

0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1

ADS for species u ADS for species v u invades v v invades u mutual invasion

Average Dispersal Success on landscape level

Integrodifference equations for invasive species in heterogeneous environments – p. 42/5

SLIDE 43

Spreading speed

Spreading speed of inferior invader

0.5 1 1.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

dispersal parameter d spread rate

Integrodifference equations for invasive species in heterogeneous environments – p. 43/5

SLIDE 44

Outline

Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 44/5

SLIDE 45

Allee effect I

Model: ut+1(x) =

K(x − y, y)f(ut(y), y)dy

f1(u) =    u < Na 1 u ≥ Na f2(u) = 0 Na : Allee threshold

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

density ut density f(ut) Na

Integrodifference equations for invasive species in heterogeneous environments – p. 45/5

SLIDE 46

Homogeneous landscape p = 1

Speed of spread c : c K(y)dy = 1 2 − Na provided Na < 1/2 Spreading speed cLaplace = −

σ2/2 ln(2Na)

KOT, LEWIS AND VAN DEN DRIESSCHE, 1996, Ecology

Integrodifference equations for invasive species in heterogeneous environments – p. 46/5

SLIDE 47

Averaging

Density u for fraction p Average density pu Replace Na by Na/p. Approximate speed of spread c : c K(y)dy = 1 2 − Na p Invasion threshold (rule of thumb): pmin = 2Na

Integrodifference equations for invasive species in heterogeneous environments – p. 47/5

SLIDE 48

Allee effect II

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2

percentage of good habitat p spread rate Gaussian Kernel Laplace Kernel

Na = 0.2, σ2 = 2

Integrodifference equations for invasive species in heterogeneous environments – p. 48/5

SLIDE 49

Near the invasion threshold

Persistence and spread are not the same Can enough individuals cross the gap to the next good patch?

−4+p −3 −3+p −2 −2+p −1 −1+p

space

Invasion threshold

−1

n=−∞

n+pmin

n

K(y)dy = Na

Integrodifference equations for invasive species in heterogeneous environments – p. 49/5

SLIDE 50

Allee invasion threshold

0.5 1 1.5 2 0.4 0.5 0.6 0.7 0.8 0.9 1

variance σ2 invasion threshold pmin Laplace kernel Gaussian kernel Exponential square root kernel

decreasing (no Allee: increasing) coarse grain to stop invasion (no Allee: fine) reverse the ordering of kernels

Integrodifference equations for invasive species in heterogeneous environments – p. 50/5

SLIDE 51

Outline

Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 51/5

SLIDE 52

Density-dependent dispersal: spreading speeds

ut+1(x) = gf(ut) + K ∗ [(1 − g)f(ut)] g = g(af(u)): probability to remain sedentary g′ ≤ 0 a: Sensitivity to crowding Not necessarily monotone, even if f is. Not necessarily order preserving. Existence of spreading speed for small enough a.

LUTSCHER 2008

Integrodifference equations for invasive species in heterogeneous environments – p. 52/5

SLIDE 53

Density-dependent dispersal: simulation

1 2 3 4 5 0.2 0.4 0.6 0.8 1

location x density u a=0.5 a=1 a=10 a=100

2 4 6 8 10 0.02 0.04 0.06 0.08

dispersal parameter a front speed

LUTSCHER 2008

Integrodifference equations for invasive species in heterogeneous environments – p. 53/5

SLIDE 54

Density-dependent dispersal: traveling waves

−5 5 0.5 1 1.5

location x density Profile after dispersal Profile before dispersal

−5 5 0.2 0.4 0.6 0.8 1 1.2 1.4

location x density Profile after dispersal Profile before dispersal

Non-monotone waves

LUTSCHER 2008

Integrodifference equations for invasive species in heterogeneous environments – p. 54/5

SLIDE 55

Density-dependent dispersal: fragmentation

1 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Sensitivity to crowding, log(a) Averaged front speed L1=0.8 L1=0.6 L1=0.4

spread rates in fragmented habitats are highest for intermediate sensitivity

LUTSCHER 2008

Integrodifference equations for invasive species in heterogeneous environments – p. 55/5

SLIDE 56

Summary

Approximations work and provide relevant scales ADS: Invasion thresholds, coexistence mechanisms Differences with Allee effect Density-dependent dispersal

Integrodifference equations for invasive species in heterogeneous environments – p. 56/5

SLIDE 57

Where to go from here?

Patch averaging for speed Existence of spreading speeds Individual movement response to landscape features

!10 !5 5 10

space

z = 0.9 z = 0.3 z = 0

Integrodifference equations for invasive species in heterogeneous environments – p. 57/5

SLIDE 58

Acknowledgements

Sebastian Dewhirst Yasmine Samia Mark Lewis Jeff Musgrave Funding: NSERC, MITACS , UOttawa

Integrodifference equations for invasive species in heterogeneous environments – p. 58/5