Biological Invasions in Heterogeneous Environments
Tom Robbins
Department of Mathematics
University of Utah
Mark Lewis
Centre for Mathematical Biology,
Department of Mathematics
University of Alberta
Biological Invasions in Heterogeneous Environments – p. 1/32
Outline
❁
Background on biological invasions
Biological Invasions in Heterogeneous Environments – p. 2/32
Outline
❁
Background on biological invasions
❁
Model for plant species
Biological Invasions in Heterogeneous Environments – p. 2/32
Outline
❁
Background on biological invasions
❁
Model for plant species
❁
Homogeneous environments
Biological Invasions in Heterogeneous Environments – p. 2/32
Outline
❁
Background on biological invasions
❁
Model for plant species
❁
Homogeneous environments
❁
Heterogeneous environments
Biological Invasions in Heterogeneous Environments – p. 2/32
Outline
❁
Background on biological invasions
❁
Model for plant species
❁
Homogeneous environments
❁
Heterogeneous environments
❁
Multiple dispersal scales
Biological Invasions in Heterogeneous Environments – p. 2/32
Outline
❁
Background on biological invasions
❁
Model for plant species
❁
Homogeneous environments
❁
Heterogeneous environments
❁
Multiple dispersal scales
❁
Summay
Biological Invasions in Heterogeneous Environments – p. 2/32
Biological Invasions?
❁
What are biological invasions?
Biological Invasions in Heterogeneous Environments – p. 3/32
Biological Invasions?
❁
What are biological invasions?
❀ A biological invasion is the introduction and spread of an
exotic species within an ecosystem
Biological Invasions in Heterogeneous Environments – p. 3/32
Biological Invasions?
❁
What are biological invasions?
❀ A biological invasion is the introduction and spread of an
exotic species within an ecosystem
❁
Why are biological invasions important?
Biological Invasions in Heterogeneous Environments – p. 3/32
Biological Invasions?
❁
What are biological invasions?
❀ A biological invasion is the introduction and spread of an
exotic species within an ecosystem
❁
Why are biological invasions important?
❀ Economic impact
($100 billion by 1991)
Biological Invasions in Heterogeneous Environments – p. 3/32
Biological Invasions?
❁
What are biological invasions?
❀ A biological invasion is the introduction and spread of an
exotic species within an ecosystem
❁
Why are biological invasions important?
❀ Economic impact
($100 billion by 1991)
❀ Contribute to the loss of biodiversity
Biological Invasions in Heterogeneous Environments – p. 3/32
Biological Invasions?
❁
What are biological invasions?
❀ A biological invasion is the introduction and spread of an
exotic species within an ecosystem
❁
Why are biological invasions important?
❀ Economic impact
($100 billion by 1991)
❀ Contribute to the loss of biodiversity
❀ Threat to endangered species
Biological Invasions in Heterogeneous Environments – p. 3/32
Example of Loss in Biodiversity
European zebra mussel
(Dreissena polymorpha)
Biological Invasions in Heterogeneous Environments – p. 4/32
Example of Loss in Biodiversity
European zebra mussel
(Dreissena polymorpha)
❁
Introduced into the Great Lakes in the mid-1980s
Biological Invasions in Heterogeneous Environments – p. 4/32
Example of Loss in Biodiversity
European zebra mussel
(Dreissena polymorpha)
❁
Introduced into the Great Lakes in the mid-1980s
❁
Filter feeder, removing the zooplankton
and algae from the water column
Biological Invasions in Heterogeneous Environments – p. 4/32
Example of Loss in Biodiversity
European zebra mussel
(Dreissena polymorpha)
❁
Introduced into the Great Lakes in the mid-1980s
❁
Filter feeder, removing the zooplankton
and algae from the water column
❁
Competes with the native crustaceans
Biological Invasions in Heterogeneous Environments – p. 4/32
Example of Loss in Biodiversity
European zebra mussel
(Dreissena polymorpha)
❁
Introduced into the Great Lakes in the mid-1980s
❁
Filter feeder, removing the zooplankton
and algae from the water column
❁
Competes with the native crustaceans
❁
Decline in native crustaceans −→
decline in the native fish population
Biological Invasions in Heterogeneous Environments – p. 4/32
Conceptual Framework
❁
Arrival and Establishment
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
❀ Invasion (or propagule) pressure is important
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
❀ Invasion (or propagule) pressure is important
❀ All communities are invasible
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
❀ Invasion (or propagule) pressure is important
❀ All communities are invasible
❁
Spread
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
❀ Invasion (or propagule) pressure is important
❀ All communities are invasible
❁
Spread
❁
Equilibrium and effects
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
❀ Invasion (or propagule) pressure is important
❀ All communities are invasible
❁
Spread
❁
Equilibrium and effects
❀ Most invaders have only minor consequences
Biological Invasions in Heterogeneous Environments – p. 5/32
Conceptual Framework
❁
Arrival and Establishment
❀ Most invasions fail!
❀ Invasion (or propagule) pressure is important
❀ All communities are invasible
❁
Spread
❁
Equilibrium and effects
❀ Most invaders have only minor consequences
❀ Few have disastrous consequences!
Biological Invasions in Heterogeneous Environments – p. 5/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
❁
Assume spatial heterogeneity for local growth and dispersal ability
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
❁
Assume spatial heterogeneity for local growth and dispersal ability
❁
How is invasibility and spread rates of the population affected by:
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
❁
Assume spatial heterogeneity for local growth and dispersal ability
❁
How is invasibility and spread rates of the population affected by:
❀ local deposition of seeds?
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
❁
Assume spatial heterogeneity for local growth and dispersal ability
❁
How is invasibility and spread rates of the population affected by:
❀ local deposition of seeds?
❀ the ratio of short to long-distance seed dispersal?
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
❁
Assume spatial heterogeneity for local growth and dispersal ability
❁
How is invasibility and spread rates of the population affected by:
❀ local deposition of seeds?
❀ the ratio of short to long-distance seed dispersal?
❀ local growth rates?
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological Problem
❁
Consider the invasion of terrestrial plant species
❁
Assume spatial heterogeneity for local growth and dispersal ability
❁
How is invasibility and spread rates of the population affected by:
❀ local deposition of seeds?
❀ the ratio of short to long-distance seed dispersal?
❀ local growth rates?
❁
Can spread rates be increased or slowed down (possible stalled) by habitat
heterogeneity?
Biological Invasions in Heterogeneous Environments – p. 6/32
Biological model: terrestrial plant species
Biological Invasions in Heterogeneous Environments – p. 7/32
Biological model: terrestrial plant species
❁
Infinite, one-dimensional environment
Biological Invasions in Heterogeneous Environments – p. 7/32
Biological model: terrestrial plant species
❁
Infinite, one-dimensional environment
❁
Assume that growth and dispersal occur in
distinct, nonoverlapping stages
Biological Invasions in Heterogeneous Environments – p. 7/32
Biological model: terrestrial plant species
❁
Infinite, one-dimensional environment
❁
Assume that growth and dispersal occur in
distinct, nonoverlapping stages
❁
Plant survival is limited to one cycle,
with nonoverlapping generations
Biological Invasions in Heterogeneous Environments – p. 7/32
Mathematical model
Integrodifference equation (IDE) model for the population density
Nτ +1 (x) =
Z
+∞
f (Nτ (y); y)k(x, y)dy
−∞
where
Biological Invasions in Heterogeneous Environments – p. 8/32
Mathematical model
Integrodifference equation (IDE) model for the population density
Nτ +1 (x) =
Z
+∞
f (Nτ (y); y)k(x, y)dy
−∞
where
❁
Nτ (x) is the population density
Biological Invasions in Heterogeneous Environments – p. 8/32
Mathematical model
Integrodifference equation (IDE) model for the population density
Nτ +1 (x) =
Z
+∞
f (Nτ (y); y)k(x, y)dy
−∞
where
❁
Nτ (x) is the population density
❁
f is the nonlinear growth function
Biological Invasions in Heterogeneous Environments – p. 8/32
Mathematical model
Integrodifference equation (IDE) model for the population density
Nτ +1 (x) =
Z
+∞
f (Nτ (y); y)k(x, y)dy
−∞
where
❁
Nτ (x) is the population density
❁
f is the nonlinear growth function
❁
k(x, y) dispersal kernel
Biological Invasions in Heterogeneous Environments – p. 8/32
Growth Function
Nτ + 1 (x) = f (Nτ (x); x)
2
Biological Invasions in Heterogeneous Environments – p. 9/32
Growth Function
Nτ + 1 (x) = f (Nτ (x); x)
2
❁
Nτ (x) is the seed density at the start of generation τ
Biological Invasions in Heterogeneous Environments – p. 9/32
Growth Function
Nτ + 1 (x) = f (Nτ (x); x)
2
❁
Nτ (x) is the seed density at the start of generation τ
❁
Nτ + 1 (x) is the seed density at the end of generation τ , before dispersal
2
Biological Invasions in Heterogeneous Environments – p. 9/32
Growth Function example
Beverton-Holt stock-recruitment curve
r0 N
f (N ) =
1 + [(r0 − 1)N ]
where r0 is the per capita reproductive ratio.
r0 = 2.0
Biological Invasions in Heterogeneous Environments – p. 10/32
Dispersal Kernel
❁
k(x, y) is a pdf for a seed released from a location y and
being deposited at a location x
Z
+∞
k(x, y)dx = 1
−∞
Biological Invasions in Heterogeneous Environments – p. 11/32
Dispersal Kernel
❁
k(x, y) is a pdf for a seed released from a location y and
being deposited at a location x
Z
❁
+∞
k(x, y)dx = 1
−∞
The IDE is the sum of all seeds dispersed to x
Z
+∞
−∞
Nτ + 1 (y)k(x, y)dy
2
Biological Invasions in Heterogeneous Environments – p. 11/32
Model for seed dispersal
❁
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − ac
∂x
=
ac
=
δ(x − y)
=
0
Biological Invasions in Heterogeneous Environments – p. 12/32
Model for seed dispersal
❁
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − ac
∂x
=
ac
=
δ(x − y)
=
0
❀ c is the airborne seed density
Biological Invasions in Heterogeneous Environments – p. 12/32
Model for seed dispersal
❁
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − ac
∂x
=
ac
=
δ(x − y)
=
0
❀ c is the airborne seed density
❀ d is the seed density on the ground
Biological Invasions in Heterogeneous Environments – p. 12/32
Model for seed dispersal
❁
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − ac
∂x
=
ac
=
δ(x − y)
=
0
❀ c is the airborne seed density
❀ d is the seed density on the ground
❀ a is the deposition rate
Biological Invasions in Heterogeneous Environments – p. 12/32
Model for seed dispersal
❁
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − ac
∂x
=
ac
=
δ(x − y)
=
0
❀ c is the airborne seed density
❀ d is the seed density on the ground
❀ a is the deposition rate
❀ k(x − y) = lim d(x, t; y)
t→∞
Biological Invasions in Heterogeneous Environments – p. 12/32
Laplace Dispersal Kernel
The dispersal kernel is given by the Laplace distribution
k(x − y) =
r
r
a
a
exp −
|x − y|
4D
D
D = 1 and a = 1
Biological Invasions in Heterogeneous Environments – p. 13/32
Homogeneous Environment
❁
Infinite flow, one-dimensional homogeneous environment
Biological Invasions in Heterogeneous Environments – p. 14/32
Homogeneous Environment
❁
Infinite flow, one-dimensional homogeneous environment
❁
Growth function of the form
f (N ; y) = f (N )
Biological Invasions in Heterogeneous Environments – p. 14/32
Homogeneous Environment
❁
Infinite flow, one-dimensional homogeneous environment
❁
Growth function of the form
f (N ; y) = f (N )
❁
The dispersal kernel is translation invariant, i.e.,
k(x, y) = k(x − y)
Biological Invasions in Heterogeneous Environments – p. 14/32
Homogeneous Environment
❁
Infinite flow, one-dimensional homogeneous environment
❁
Growth function of the form
f (N ; y) = f (N )
❁
The dispersal kernel is translation invariant, i.e.,
k(x, y) = k(x − y)
❁
The IDE model reduces to the convolution integral
Nτ +1 (x) =
Z
+∞
f (Nτ (y))k(x − y)dy
−∞
Biological Invasions in Heterogeneous Environments – p. 14/32
Traveling Wave Solution
For the initial population density
N0 (x) = δ(x)
the solution of the IDE approaches a traveling wave
r0 = 6, D = 1 and a = 10
Biological Invasions in Heterogeneous Environments – p. 15/32
Traveling Wave Solution
For the initial population density
N0 (x) = δ(x)
the solution of the IDE approaches a traveling wave
r0 = 6, D = 1 and a = 10
Biological Invasions in Heterogeneous Environments – p. 15/32
Traveling Wave Solution
For the initial population density
N0 (x) = δ(x)
the solution of the IDE approaches a traveling wave
r0 = 6, D = 1 and a = 10
Biological Invasions in Heterogeneous Environments – p. 15/32
Traveling Wave Solution
For the initial population density
N0 (x) = δ(x)
the solution of the IDE approaches a traveling wave
r0 = 6, D = 1 and a = 10
Biological Invasions in Heterogeneous Environments – p. 15/32
Traveling Wave Solution
For the initial population density
N0 (x) = δ(x)
the solution of the IDE approaches a traveling wave
r0 = 6, D = 1 and a = 10
Biological Invasions in Heterogeneous Environments – p. 15/32
Traveling Wave Result
❁
Homogeneous, one-dimensional environment
Biological Invasions in Heterogeneous Environments – p. 16/32
Traveling Wave Result
❁
Homogeneous, one-dimensional environment
❁
N0 (x) has compact support
Biological Invasions in Heterogeneous Environments – p. 16/32
Traveling Wave Result
❁
Homogeneous, one-dimensional environment
❁
N0 (x) has compact support
❁
f is monotonic, bounded above and no Allee effect
Biological Invasions in Heterogeneous Environments – p. 16/32
Traveling Wave Result
❁
Homogeneous, one-dimensional environment
❁
N0 (x) has compact support
❁
f is monotonic, bounded above and no Allee effect
❁
The dispersal kernel has a moment generating function
U (s) =
Z
+∞
k(x) exp{|x|s}dx
−∞
Biological Invasions in Heterogeneous Environments – p. 16/32
Traveling Wave Result
❁
Homogeneous, one-dimensional environment
❁
N0 (x) has compact support
❁
f is monotonic, bounded above and no Allee effect
❁
The dispersal kernel has a moment generating function
U (s) =
❁
Z
+∞
k(x) exp{|x|s}dx
−∞
The asymptotic rate of spread is given by
1
ln (r0 U (s))
c = min
0<s
s
∗
[Weinberger - 1982]
Biological Invasions in Heterogeneous Environments – p. 16/32
Dispersion Relation
❁
A traveling wave, with positive wave speed c, is of the form
Nτ +1 (x) = Nτ (x − c)
Biological Invasions in Heterogeneous Environments – p. 17/32
Dispersion Relation
❁
A traveling wave, with positive wave speed c, is of the form
Nτ +1 (x) = Nτ (x − c)
❁
Near the front, population densities are low, so the IDE can be approximated by
Nτ (x − c) = r0
Z
+∞
k(x − y)Nτ (y)dy
−∞
where r0 = f 0 (0)
Biological Invasions in Heterogeneous Environments – p. 17/32
Dispersion Relation
❁
A traveling wave, with positive wave speed c, is of the form
Nτ +1 (x) = Nτ (x − c)
❁
Near the front, population densities are low, so the IDE can be approximated by
Nτ (x − c) = r0
Z
+∞
k(x − y)Nτ (y)dy
−∞
where r0 = f 0 (0)
❁
TW ansatz: near the leading edge of the wave
Nτ (x) ∝ e−sx
for s > 0
Biological Invasions in Heterogeneous Environments – p. 17/32
Dispersion Relation
❁
From this assumption
e−sx esc = r0
Z
+∞
k(x − y)e−sy dy
−∞
Biological Invasions in Heterogeneous Environments – p. 18/32
Dispersion Relation
❁
From this assumption
e−sx esc = r0
❁
Z
+∞
k(x − y)e−sy dy
−∞
Make the change of variables u ≡ x − y
Biological Invasions in Heterogeneous Environments – p. 18/32
Dispersion Relation
❁
From this assumption
e−sx esc = r0
Z
+∞
k(x − y)e−sy dy
−∞
❁
Make the change of variables u ≡ x − y
❁
We have the characteristic equation
esc = r0
Z
+∞
k(u)esu du
−∞
Biological Invasions in Heterogeneous Environments – p. 18/32
Dispersion Relation
❁
From this assumption
e−sx esc = r0
Z
+∞
k(x − y)e−sy dy
−∞
❁
Make the change of variables u ≡ x − y
❁
We have the characteristic equation
esc = r0
❁
Z
+∞
k(u)esu du
−∞
Solving for c as a function of s
1
c(s) = ln (r0 U (s))
s
[Kot et al. - 1996]
Biological Invasions in Heterogeneous Environments – p. 18/32
Dispersion Relation: Laplace Kernel
❁
For the Laplace dispersal kernel, the dispersion relation is
1
a
1
c(s) = ln r0
s
D a/D − s2
Biological Invasions in Heterogeneous Environments – p. 19/32
Heterogeneous Environment
❁
Growth function of the form f = f (N ; x), e.g.,
f (N ; y) =
r0 (x)N
1 + [(r0 (x) − 1)N ]
Biological Invasions in Heterogeneous Environments – p. 20/32
Heterogeneous Environment
❁
Growth function of the form f = f (N ; x), e.g.,
f (N ; y) =
❁
r0 (x)N
1 + [(r0 (x) − 1)N ]
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − a(x)c
∂x
=
a(x)c
=
δ(x − y)
=
0
Biological Invasions in Heterogeneous Environments – p. 20/32
Heterogeneous Environment
❁
Growth function of the form f = f (N ; x), e.g.,
f (N ; y) =
❁
❁
r0 (x)N
1 + [(r0 (x) − 1)N ]
The dispersal model:
∂c
∂t
∂d
∂t
c(x, 0; y)
d(x, 0; y)
=
∂2c
D 2 − a(x)c
∂x
=
a(x)c
=
δ(x − y)
=
0
Z
+∞
The IDE model is of the form
Nτ +1 (x) = ANτ =
f (Nτ (y); y)k(x, y)dy
−∞
Biological Invasions in Heterogeneous Environments – p. 20/32
Periodic Heterogeneity
Biological Invasions in Heterogeneous Environments – p. 21/32
Periodic Heterogeneity
❁
Periodically divide the environment into good (r0 > 1) and bad (r0 < 1) patches:

 r
if 0 ≤ x < xa
1
r0 (x) =
 r2 if xa ≤ x < l
r0 (x + l)
=
r0 (x)
Biological Invasions in Heterogeneous Environments – p. 21/32
Periodic Heterogeneity
❁
Periodically divide the environment into good (r0 > 1) and bad (r0 < 1) patches:

 r
if 0 ≤ x < xa
1
r0 (x) =
 r2 if xa ≤ x < l
r0 (x + l)
❁
=
r0 (x)
Divide the environment into high (a = 1) and low (a < 1) deposition patches:

 a
if 0 ≤ x < xa
1
a(x) =
 a2 if xa ≤ x < l
a(x + l)
=
a(x)
Biological Invasions in Heterogeneous Environments – p. 21/32
Dispersal Kernel
❁
k(x, y) for the homogeneous environment and heterogeneous environment:
D = 1, a1 = 1, a2 = 0.4, l = 1, xh = 0.4
Biological Invasions in Heterogeneous Environments – p. 22/32
Colony Persistence
❁
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
Biological Invasions in Heterogeneous Environments – p. 23/32
Colony Persistence
❁
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
❁
Consider when N ∗ ≡ 0 is unstable, where
N ∗ = AN ∗ ,
i.e., for all ξ0 (x) such that kξ0 k < , does kAτ ξ0 k → 0?
Biological Invasions in Heterogeneous Environments – p. 23/32
Colony Persistence
❁
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
❁
Consider when N ∗ ≡ 0 is unstable, where
N ∗ = AN ∗ ,
i.e., for all ξ0 (x) such that kξ0 k < , does kAτ ξ0 k → 0?
❁
This is equivalent to studying the spectrum of the linearization of A near N ∗
Biological Invasions in Heterogeneous Environments – p. 23/32
Colony Persistence
❁
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
❁
Consider when N ∗ ≡ 0 is unstable, where
N ∗ = AN ∗ ,
i.e., for all ξ0 (x) such that kξ0 k < , does kAτ ξ0 k → 0?
❁
This is equivalent to studying the spectrum of the linearization of A near N ∗
❁
For the linear IDE, we analyze the eigenvalue problem
Biological Invasions in Heterogeneous Environments – p. 23/32
Colony Persistence
❁
A colony is said to persist in an environment if when initially introduced at low
densities, the population density eventually increases
❁
Consider when N ∗ ≡ 0 is unstable, where
N ∗ = AN ∗ ,
i.e., for all ξ0 (x) such that kξ0 k < , does kAτ ξ0 k → 0?
❁
This is equivalent to studying the spectrum of the linearization of A near N ∗
❁
For the linear IDE, we analyze the eigenvalue problem
❁
N ∗ is stable if λ1 < 1 and unstable if λ1 > 1, where λ1 is the dominant eigenvalue
Biological Invasions in Heterogeneous Environments – p. 23/32
Colony Persistence
The resulting condition for λ1 = 1 is:
0 = Fi (a2 , r1 , r2 , xa , l)
p
:= 1 − cos xa [r1 − 1] ×
p
cos (l − xa ) a2 [r2 − 1]
[r1 − 1] + a2 [r2 − 1]
p
×
+ p
2 [r1 − 1] a2 [r2 − 1]
p
sin xa [r1 − 1] ×
p
sin (l − xa ) a2 [r2 − 1]
Biological Invasions in Heterogeneous Environments – p. 24/32
Colony Persistence: Example
❁
high deposition in good patches
Biological Invasions in Heterogeneous Environments – p. 25/32
Colony Persistence: Example
❁
high deposition in good patches
❁
Deposition rate in bad patch (a2 ) vs. relative fraction of good patches (xa ):
fixed growth rate in the good patch (r1 )
Biological Invasions in Heterogeneous Environments – p. 25/32
Colony Persistence: Example
❁
high deposition in good patches
❁
Deposition rate in bad patch (a2 ) vs. relative fraction of good patches (xa ):
fixed growth rate in the good patch (r1 )
a1 = 1, l = 1, r2 = 0.5
Biological Invasions in Heterogeneous Environments – p. 25/32
Colony Persistence: Example
❁
high deposition in good patches
❁
Deposition rate in bad patch (a2 ) vs. relative fraction of good patches (xa ):
fixed growth rate in the good patch (r1 )
a1 = 1, l = 1, r2 = 0.5
❁
Stable region is below the curve
Biological Invasions in Heterogeneous Environments – p. 25/32
Traveling Periodic Wave
❁
N0 (x) has compact support
Biological Invasions in Heterogeneous Environments – p. 26/32
Traveling Periodic Wave
❁
N0 (x) has compact support
❁
N ∗ ≡ 0 is unstable
Biological Invasions in Heterogeneous Environments – p. 26/32
Traveling Periodic Wave
❁
N0 (x) has compact support
❁
N ∗ ≡ 0 is unstable
❁
Nτ (x)/a(x) for τ large
0.8
Nτ(x)/a(x)
ns
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
a2 = 0.4, R1 = 1.2, R2 = 0.5, l = 1
Biological Invasions in Heterogeneous Environments – p. 26/32
Dispersion Relation for the TPW
Assume a traveling periodic wave and derive the dispersion relation:
0 = Gi (s, c ; a2 , r1 , r2 , xa , l)
p
:= cosh(sl) − cos xa [r1 e−sc − 1] ×
p
cos (l − xa ) a2 [r2 e−sc − 1]
[r1 e−sc − 1] + a2 [r2 e−sc − 1]
p
+ p
×
2 [r1 e−sc − 1] a2 [r2 e−sc − 1]
p
sin xa [r1 e−sc − 1] ×
p
sin (l − xa ) a2 [r2 e−sc − 1]
Biological Invasions in Heterogeneous Environments – p. 27/32
Dispersion Relation: Example
❁
high deposition in good patches, low deposition in bad patches
Biological Invasions in Heterogeneous Environments – p. 28/32
Dispersion Relation: Example
❁
high deposition in good patches, low deposition in bad patches
❁
Dispersion relation for TPW:
fixed fraction of good patches (xa )
a2 = 0.4, r1 = 1.2, r2 = 0.5, l = 1
Biological Invasions in Heterogeneous Environments – p. 28/32
Multiple Dispersal Scale Model
❁
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
Dη
− a(x)c1
∂x2
=
a(x)c1
=
δ(x − y)
=
0
Biological Invasions in Heterogeneous Environments – p. 29/32
Multiple Dispersal Scale Model
❁
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
Dη
− a(x)c1
∂x2
=
a(x)c1
=
δ(x − y)
∂c2
∂t
∂d2
∂t
c2 (x, 0; y)
=
0
d2 (x, 0; y)
=
∂ 2 c2
D
− a(x)c2
∂x2
=
a(x)c2
=
δ(x − y)
=
0
Biological Invasions in Heterogeneous Environments – p. 29/32
Multiple Dispersal Scale Model
❁
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
Dη
− a(x)c1
∂x2
=
a(x)c1
=
δ(x − y)
∂c2
∂t
∂d2
∂t
c2 (x, 0; y)
=
0
d2 (x, 0; y)
=
∂ 2 c2
D
− a(x)c2
∂x2
=
a(x)c2
=
δ(x − y)
=
0
❀ Dη D
Biological Invasions in Heterogeneous Environments – p. 29/32
Multiple Dispersal Scale Model
❁
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
Dη
− a(x)c1
∂x2
=
a(x)c1
=
δ(x − y)
∂c2
∂t
∂d2
∂t
c2 (x, 0; y)
=
0
d2 (x, 0; y)
=
∂ 2 c2
D
− a(x)c2
∂x2
=
a(x)c2
=
δ(x − y)
=
0
❀ Dη D
❀ Define the dispersal ratio

 w
1
w(x) =
 w2
if 0 ≤ x < xa
if xa ≤ x < l ;
w(x + l) = w(x)
Biological Invasions in Heterogeneous Environments – p. 29/32
Multiple Dispersal Scale Model
❁
The dispersal model:
∂c1
∂t
∂d1
∂t
c1 (x, 0; y)
d1 (x, 0; y)
=
∂ 2 c1
Dη
− a(x)c1
∂x2
=
a(x)c1
=
δ(x − y)
∂c2
∂t
∂d2
∂t
c2 (x, 0; y)
=
0
d2 (x, 0; y)
=
∂ 2 c2
D
− a(x)c2
∂x2
=
a(x)c2
=
δ(x − y)
=
0
❀ Dη D
❀ Define the dispersal ratio

 w
1
w(x) =
 w2
if 0 ≤ x < xa
if xa ≤ x < l ;
w(x + l) = w(x)
❀ Let k(x − y) = lim w(y)d1 (x, t; y) + (1 − w(y))d2 (x, t; y)
t→∞
Biological Invasions in Heterogeneous Environments – p. 29/32
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 )
a1 = 1.0, r1 = 1.2, r2 = 0.5, l = 1
❁
Local dispersal increases species persistence
Biological Invasions in Heterogeneous Environments – p. 30/32
Invasion Model: Traveling Periodic Wave
❁
0 ≤ w(x) ≤ 1 and high deposition in good patches
❁
Dispersion relation for the speed of the wave:
a2 = 0.4, r1 = 1.2, r2 = 0.5, w2 = 0, l = 1
❁
Local dispersal can slow the invasion
Biological Invasions in Heterogeneous Environments – p. 31/32
Summary
❁
Use of the periodicity assumption to derive conditions for species persistence
Biological Invasions in Heterogeneous Environments – p. 32/32
Summary
❁
Use of the periodicity assumption to derive conditions for species persistence
❁
Dispersion relation for the traveling periodic wave without explicit knowledge
of the dispersal kernel
Biological Invasions in Heterogeneous Environments – p. 32/32
Summary
❁
Use of the periodicity assumption to derive conditions for species persistence
❁
Dispersion relation for the traveling periodic wave without explicit knowledge
of the dispersal kernel
❁
Invasibility:
❀ Bad patches decrease invasibility
❀ Local dispersal increases invasibility
Biological Invasions in Heterogeneous Environments – p. 32/32
Summary
❁
Use of the periodicity assumption to derive conditions for species persistence
❁
Dispersion relation for the traveling periodic wave without explicit knowledge
of the dispersal kernel
❁
Invasibility:
❀ Bad patches decrease invasibility
❀ Local dispersal increases invasibility
❁
Spread rate:
❀ Bad patches decrease spread rates
❀ Local dispersal decreases spread rates
Biological Invasions in Heterogeneous Environments – p. 32/32