Seed Dispersal and Biological Invasion
Tom Robbins
Department of Mathematics
University of Utah
Seed Dispersal and Biological Invasion – p.1/25
Outline
♦
Biology background
♦
Seed Dispersal Model
♦
Biological Invasions in Heterogeneous Environments
Seed Dispersal and Biological Invasion – p.2/25
Biology background
Population dynamics: Nτ +1 (x) =
30
1.0
25
0.8
20
0.6
10
0.2
5
5
10
x
15
20
0
25
1.2
−3
Fat−tailed kernel
Laplace distribution
15
0.4
0.0
0
f (Nτ )k(x − y)dy
k(x)
1.2
k(x)
Nτ(x)
♦
R
−0.2
−0.1
0
x
x 10
1.2
0.8
0.4
0
1
0.1
1.5
x
2
0.2
20
1.0
15
x(τ)
Nτ(x)
0.8
0.6
I
10
0.4
5
0.2
0.0
0
♦
5
10
x
15
20
25
0
0
2
4
τ
6
8
Shape of the tail defines the rate of spread
Seed Dispersal and Biological Invasion – p.3/25
Biology background
♦
Understanding biological invasions with long-distance
dispersal in a heterogeneous environment
♦
Need for a mechanistic model to predict the shape of
the tail
♦
A tractable model for the population dynamics that
includes heterogeneity
Seed Dispersal and Biological Invasion – p.4/25
Biology background
6
2
Log seed density (m )
10
4
10
2
10
0
10
−2
10
0.5
1
2
4
8
16
Log distance (m)
32
64
128
Bullock and Clarke [2000]
♦
Long-Distance dispersal events are difficult to measure
♦
Dispersal is not accurately described by a simple
decay process
♦
Local and long-distance dispersal
Seed Dispersal and Biological Invasion – p.5/25
Biology background
♦
Mechanistic model for seed dispersal:
✦
Start from first principles
✦
Turbulent transport and inertial effects
✦
Estimate the tail of the dispersal kernel
Seed Dispersal and Biological Invasion – p.6/25
Biology background
♦
♦
Mechanistic model for seed dispersal:
✦
Start from first principles
✦
Turbulent transport and inertial effects
✦
Estimate the tail of the dispersal kernel
For a spatially heterogeneous environment, how are
persistence and spread rates affected by:
✦
Local growth rates
✦
Seed deposition rates
✦
Local vs. long-distance dispersal
Seed Dispersal and Biological Invasion – p.6/25
Seed Dispersal Model
2
d
d
m 2 y = mGz + µs uw − y
dt
dt
♦
m is the seed mass
♦
y is the seed location
♦
Gz is the acceleration of gravity
♦
µs coefficient of friction for the seed
♦
uw is the wind velocity
Seed Dispersal and Biological Invasion – p.7/25
Seed Dispersal Model
♦
Inertial scaling parameter: ǫ =
1
Tc
(nondimensional)
d
y1
dt
d
ǫ v1
dt
d
y2
dt
d
ǫ v2
dt
m
µs
= v1
= [uw − v1 ]
= v2
= [ww − 1 − v2 ]
Seed Dispersal and Biological Invasion – p.8/25
Seed Dispersal Model: SDE
dY1,t = V1,t dt
ǫdV1,t = [û(Y2 ) − V1,t ]dt +
√
2Ddωt
dY2,t = V2,t dt
ǫdV2,t = [ŵ(Y2 ) − 1 − V2,t ]dt +
q
2D̂z (Y2 )dωt
♦
(Y1,t , Y2,t ), (V1,t , V2,t ) are random variables
♦
dωt is Brownian noise
Seed Dispersal and Biological Invasion – p.9/25
Seed Dispersal Model: SDE
0.30
2.0
Ws = 1.0, u* = 0.5
W = 1.0, u = 0.005
s
*
Ws = 5.0, u* = 0.05
0.25
1.5
1
k(x )
x3
0.20
1.0
0.15
0.10
0.5
0.05
0.0
0
2
4
x1
6
8
10
0.00
0
5
10
x
15
20
25
1
♦
(left) 5 realizations of the SDE model
♦
(right) Frequency plot of first hitting locations
Seed Dispersal and Biological Invasion – p.10/25
Seed Dispersal Model: PDE (ǫ = 0)
2
∂
∂ ρ
∂ρ
= D 2+
∂t
∂x1 ∂x2
∂ρ
∂ρ
+
−û
∂x1 ∂x2
♦
∂
ρ
D̂z
∂x2
Initial condition:
ρ(x, 0) = δ(x1 )δ(x2 − 1)
♦
Boundary condition at x2 = 0:
∂ρ
+ ρ = Vd ρ
D̂z
∂x2
Seed Dispersal and Biological Invasion – p.11/25
Seed Dispersal Model
0.30
Rounds model
ε = 0.35
0.25
−3
8
0.20
6
1
k(x )
k(x1)
x 10
0.15
4
2
0
14
0.10
16
18
x
20
22
24
1
0.05
0.00
0
5
10
x
15
20
25
1
♦
Logarithmic wind profile for û
♦
The SDE model predicts a fast drop off in the tail
♦
The parabolic model predicts a “fat tail”
Seed Dispersal and Biological Invasion – p.12/25
Seed Dispersal Model: PDE (ǫ > 0)
2
2
∂
∂ρ
∂ ρ
∂ ρ
∂ρ
D̂z
+ǫ 2 = D 2 +
∂t
∂t
∂x1 ∂x2
∂x2
∂ρ
∂ρ
+
−û
∂x1 ∂x2
2
∂ 2 ∂2
+ǫ
{û (ŵ − 1) ρ}
û ρ + 2
2
∂x1
∂x1 ∂x2
2 ∂
2 + 2 (ŵ − 1) ρ
∂x2
♦
Limited velocity of seed propagation
♦
Cross diffusion terms
∂2ρ
(ǫ ∂t2 )
Seed Dispersal and Biological Invasion – p.13/25
Seed Dispersal Model
♦
Mechanistic basis for existing PDE models
♦
Derived a new, velocity limited, PDE model
♦
SDE model predicts a “thin tail”?
♦
PDE model predicts a “fat tail” (ǫ = 0 )
♦
Analysis on the new PDE model to determine the
shape of the tail (ǫ > 0)
Seed Dispersal and Biological Invasion – p.14/25
Invasion Model
♦
Consider the invasion of terrestrial plant species
Seed Dispersal and Biological Invasion – p.15/25
Invasion Model
♦
Consider the invasion of terrestrial plant species
♦
How is invasibility and spread rates of the population
affected by:
✦
local deposition of seeds
✦
ratio of local to long-distance seed dispersal
✦
local growth rates
Seed Dispersal and Biological Invasion – p.15/25
Invasion Model
♦
Consider the invasion of terrestrial plant species
♦
How is invasibility and spread rates of the population
affected by:
♦
✦
local deposition of seeds
✦
ratio of local to long-distance seed dispersal
✦
local growth rates
Can environmental heterogeneity increase or
decrease/stall an invasion?
Seed Dispersal and Biological Invasion – p.15/25
Invasion Model
♦
Infinite, one-dimensional environment
Seed Dispersal and Biological Invasion – p.16/25
Invasion Model
♦
Infinite, one-dimensional environment
♦
Growth and dispersal occur in distinct, non-overlapping
stages
Seed Dispersal and Biological Invasion – p.16/25
Invasion Model
♦
Infinite, one-dimensional environment
♦
Growth and dispersal occur in distinct, non-overlapping
stages
♦
Integrodifference model for the population density
Z +∞
Nτ +1 (x) =
f (Nτ (y); y)k(x, y)dy
−∞
✦
Nτ (x) is the population density
✦
f is the nonlinear growth function
✦
k(x, y) dispersal kernel
Seed Dispersal and Biological Invasion – p.16/25
Invasion Model
♦
Beverton-Holt growth dynamics
f (N ; x) =
r0 (x)N
1 + [(r0 (x) − 1)N ]
Seed Dispersal and Biological Invasion – p.17/25
Invasion Model
♦
Beverton-Holt growth dynamics
f (N ; x) =
♦
r0 (x)N
1 + [(r0 (x) − 1)N ]
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
− a(x)c1
Dη
∂x2
=
a(x)c1
=
w(y)δ(x − y)
=
0
✦ c, d are seed densities
✦ a is the deposition rate
✦ w is the dispersal ratio
Seed Dispersal and Biological Invasion – p.17/25
Invasion Model
♦
Beverton-Holt growth dynamics
f (N ; x) =
♦
r0 (x)N
1 + [(r0 (x) − 1)N ]
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
− a(x)c1
Dη
∂x2
=
a(x)c1
=
w(y)δ(x − y)
∂c2
∂t
∂d2
∂t
c2 (x, 0; y)
=
0
d2 (x, 0; y)
=
∂ 2 c2
− a(x)c2
D
∂x2
=
a(x)c2
=
(1 − w(y))δ(x − y)
=
0
✦ c, d are seed densities
✦ a is the deposition rate
✦ w is the dispersal ratio
Seed Dispersal and Biological Invasion – p.17/25
Invasion Model
♦
Local and long-distance dispersal scales: Dη ≪ D
♦
The dispersal kernel is the steady-state seed concentration on the ground:
k(x, y)
=
=
♦
k1 (x, y; Dη ) + k2 (x, y)
lim {d1 (x, t; y) + d2 (x, t; y)}
t→∞
The IDE model is
Nτ +1 (x)
=
+
Z
+∞
−∞
Z +∞
f (Nτ (y); y)k1 (x, y; Dη )dy
f (Nτ (y); y)k2 (x, y)dy
−∞
Seed Dispersal and Biological Invasion – p.18/25
Invasion Model
♦
Periodically divide the environment into good (r0 > 1) and bad (r0 < 1) patches:
r0 (x)
=

 r
1
 r2
if 0 ≤ x < xa
if xa ≤ x < l ;
r0 (x + l) = r0 (x)
Seed Dispersal and Biological Invasion – p.19/25
Invasion Model
♦
Periodically divide the environment into good (r0 > 1) and bad (r0 < 1) patches:
r0 (x)
♦
if 0 ≤ x < xa
if xa ≤ x < l ;
r0 (x + l) = r0 (x)
Divide the environment into high (a = 1) and low (a < 1) deposition patches:
a(x)
♦
=

 r
1
 r2
=

 a
1
 a2
if 0 ≤ x < xa
if xa ≤ x < l ;
a(x + l) = a(x)
Local (w = 1) and long-distance (w < 1) dispersal:
w(x)
=

 w
1
 w2
if 0 ≤ x < xa
if xa ≤ x < l ;
w(x + l) = w(x)
Seed Dispersal and Biological Invasion – p.19/25
Invasion Model: Colony Persistence
♦
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
Seed Dispersal and Biological Invasion – p.20/25
Invasion Model: Colony Persistence
♦
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
♦
w(x) = 0 and high deposition in good patches
♦
Deposition rate in bad patch (a2 ) vs. relative fraction of good patches (xa )
1.0
0.8
xa
0.6
r1=1.01
0.4
r1=1.20
r1=4.00
0.2
0.0
0
♦
0.2
0.4
a2
0.6
0.8
1
Region of persistence is located above the curve
Seed Dispersal and Biological Invasion – p.20/25
Invasion Model: Colony Persistence
♦
0 ≤ w(x) ≤ 1 and high deposition in good patches
♦
Deposition rate in bad patch (a2 ) vs. relative fraction of good patches (xa )
1.0
w(x) ≡ 0.000
w(x) ≡ 0.680
w(x) ≡ 0.825
0.8
x
a
0.6
0.4
0.2
0.0
0.0
♦
0.2
0.4
a2
0.6
0.8
1.0
Local dispersal increases species persistence
Seed Dispersal and Biological Invasion – p.21/25
Invasion Model: Traveling Periodic Wave
For a persistent species: Nτ (x) for τ large
0.8
N (x)/a(x)
τ
n(x)/a(x)
0.6
0.4
τ
N (x)/a(x)
♦
τ = 70
τ = 75
0.2
0.0
0
10
20
30
x
40
50
60
70
Seed Dispersal and Biological Invasion – p.22/25
Invasion Model: Traveling Periodic Wave
♦
For a persistent species: Nτ (x) for τ large
0.8
N (x)/a(x)
τ
n(x)/a(x)
0.4
τ
N (x)/a(x)
0.6
τ = 70
τ = 75
0.2
0.0
0
♦
10
20
30
x
40
50
60
70
Dispersion relation for the speed of the wave
Seed Dispersal and Biological Invasion – p.22/25
Invasion Model: Traveling Periodic Wave
♦
For a persistent species: Nτ (x) for τ large
0.8
N (x)/a(x)
τ
n(x)/a(x)
0.4
τ
N (x)/a(x)
0.6
τ = 70
τ = 75
0.2
0.0
0
10
20
30
x
40
50
60
70
♦
Dispersion relation for the speed of the wave
♦
How does heterogeneity affect the wave speed?
Seed Dispersal and Biological Invasion – p.22/25
Invasion Model: Traveling Periodic Wave
♦
w(x) = 0 and high deposition in good patches
♦
Dispersion relation for the speed of the wave:
5
xa = 1.0
xa = 0.6
xa = xc
4
c
3
2
1
0
0.0
♦
0.3
s
0.6
0.9
The rate of spread is the minimum c(s)
Seed Dispersal and Biological Invasion – p.23/25
Invasion Model: Traveling Periodic Wave
♦
w(x) = 0 and high deposition in good patches
♦
Dispersion relation for the speed of the wave:
1.0
5
xa = 1.0
xa = 0.6
xa = xc
4
x = 1.0
a
0.8
0.6
*
c
a
c (x )
3
0.4
2
0.2
1
x =x
0
0.0
0.3
s
0.6
0.9
0.0
0.0
a
c
0.1
0.2
s*(x )
0.3
0.4
0.5
a
♦
The rate of spread is the minimum c(s)
♦
Existence of bad patches can halt the invasion
Seed Dispersal and Biological Invasion – p.23/25
Invasion Model: Traveling Periodic Wave
♦
0 ≤ w(x) ≤ 1 and high deposition in good patches
♦
Dispersion relation for the speed of the wave:
5
w(x) ≡ 0.00
w(x) ≡ 0.65
w(x) ≡ 0.82
4
c(s)
3
2
1
0
0.0
♦
0.3
s
0.6
0.9
Local dispersal can slow the invasion
Seed Dispersal and Biological Invasion – p.24/25
Invasion Model:
♦
♦
Invasibility
✦
Bad patches lower invasibility
✦
Local dispersal increases invasibility
Spread rate
✦
Bad patches lower spread rate
✦
Local dispersal lowers spread rates
Seed Dispersal and Biological Invasion – p.25/25