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TRANSCRIPT
-
DO SPATIALLY HOMOGENIZING AND HETEROGENIZING PROCESSES AFFECT TRANSITIONS BETWEEN ALTERNATIVE STABLE STATES?
THOMAS A. GROEN,
CLAUDIUS. A.D.M. VAN DE VIJVER AND
FRANK VAN LANGEVELDE
-
ALTERNATIVE STABLE STATES
Critical condition
e.g. grazing pressure
Ecosyste
m S
tate
e.g
. am
ount
of gra
ss b
iom
ass
+
+
-
-
-
EFFECT OF HETEROGENEITY ON THESE DYNAMICS
HeterogeneousHomogeneous
-
IMPACT OF HOMOGENIZING PROCESSES
No exchange Moderate exchange Strong exchange
space
bio
mass
Homonegizing processes
e.g. diffusion
-
SPATIAL PROCESSES
But what about heterogenizing processes?
Heterogenizing processes Homogenizing processes
Fires
Grazing
Facilitation
Disturbances
Dispersal(Intraspecific)
Competition
-
WHAT HAPPENS WITH BOTH HETEROGENIZING AND HOMOGENIZING PROCESSES AT THE SAME TIME?
Weak
Hom
ogenis
ation
Str
ong
Hom
ogenis
ation
Strong
Heterogenisation
Weak
Heterogenisation
?
?
-
EXAMPLE ECOSYSTEM: SAVANNAS
Source: http://biology.unm.edu/litvak/Juniper%20Savanna/Juniper%20Savanna.html
-
EXAMPLE ECOSYSTEM: SAVANNAS
Wide variety in physiognomy
Mainly grass dominated (= homogeneous)
Mixture of both (=heterogeneous)
Mainly wood dominated (=homogeneous)
Heterogenizing processes
Fires
Grazing
Homogenizing process(es)
Plant dispersal
-
SAMPLE ECOSYSTEM
𝑑𝑊
𝑑𝑡= 𝑟𝑊 𝑤𝑡
𝑢𝑊
𝐻 + 𝑢𝑊 + 𝑝𝑤𝑠+ 𝑤𝑠 −𝑑𝑤𝑊 − 𝑐𝑤𝐵𝑊 − 𝑘𝑤𝑛𝑎𝐻𝑊
𝑑𝐻
𝑑𝑡= 𝑟𝐻𝑤𝑡
𝐻
𝐻 + 𝑢𝑊 + 𝑝𝑤𝑠−𝑑𝐻 𝐻 − 𝑐𝐻𝐺𝐻 − 𝑘𝐻𝑛𝐻
W = woody biomass
H = Herbaceous biomass
Growth Mortality Herbivory Fire
-
- +
+
-
POSITIVE FEEDBACK
Grass
Biomass
Fire
Intensities
Wood
Biomass
-
NON-SPATIAL MODEL
01
00
Grass biomass
Index
(g m
2)
Time
03
00
Woody biomass
Index
(g m
2)
Time
200 300 400 500
05
01
00
15
02
00
Phase plane
Woody biomass (g m2)
Gra
ss b
iom
ass (
g m
2)
-
BI-STABILITY WHEN GRAZING INCREASES
5 10 15 20 25
05
01
50
25
03
50
Grazer biomass (g m2)
Gra
ss b
iom
ass (
g m
2)
-
MAKE THE MODEL SPATIAL
Discretize the fire
n = [0,1] ↔ (0) V (1)
Make fire occurrence function of available grass
Add diffusion as representation of “ dispersion of grasses”
-
MAKE MODEL SPATIAL: DISCRETIZE FIRE
Discretize fire process
n = [0,1] ↔ (0) V (1)
Fire frequency was set to 0.5
Two implementations:
Regular: 01010101010101010101010101 (avg=0.5)
Random: 00111011100000111011001001 (avg=0.5)
𝑑𝑊
𝑑𝑡= 𝑟𝑊 𝑤𝑡
𝑢𝑊
𝐻 + 𝑢𝑊 + 𝑝𝑤𝑠+ 𝑤𝑠 −𝑑𝑤𝑊 − 𝑐𝑤𝐵𝑊 − 𝑘𝑤𝑛𝑎𝐻𝑊
𝑑𝐻
𝑑𝑡= 𝑟𝐻𝑤𝑡
𝐻
𝐻 + 𝑢𝑊 + 𝑝𝑤𝑠−𝑑𝐻 𝐻 − 𝑐𝐻𝐺𝐻 − 𝑘𝐻𝑛𝐻
-
DISCRETE FIRE: REGULAR PATTERN
01
00
Grass biomass
Index
(g m
2)
Time
03
00
Woody biomass
Index
(g m
2)
Time
200 300 400 500
05
01
00
15
02
00
Phase plane
Woody biomass (g m2)
Gra
ss b
iom
ass (
g m
2)
-
DISCRETE FIRE: RANDOM PATTERN
01
50
Grass biomass
Index
(g m
2)
Time
03
00
Woody biomass
Index
(g m
2)
Time
100 200 300 400 500
05
01
00
15
02
00
Phase plane
Woody biomass (g m2)
Gra
ss b
iom
ass (
g m
2)
-
MAKE MODEL SPATIAL:FIRE PATCHES AND GRASS DISPERSION
Have fires of various
patch sizes
Ensure always 0.5 total fire
chance
[Locations with high grass
biomass had higher chance to
“ignite”]
Dispersion of plant biomass
simulated with simple diffusion
approach
Diffusion coefficient determines
how fast dispersion goesspace
bio
mass
Grass biomass
Chance t
o ignite
Hete
roge
niz
ing
Hom
og
en
izin
g
-
EXAMPLE SIMULATION
-
dH
= 0
dH
= 1
e-0
7d
H =
1e
-06
dH
= 1
e-0
5d
H =
1e
-04
dH
= 0
.00
1
nr patches = 2 nr patches = 8 nr patches = 50 nr patches = 200 nr patches = 1250
PATTERN IN THE LAST TIME STEP
NNumber of patches
Rate
of d
ispers
ion
2 8 50 200 1250
1 1
0-3
1 1
0-4
1 1
0-5
1 1
0-6
1 1
0-7
0
-
DID THIS CHANGE THE HETEROGENEITY?
dH
= 0
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
dH
= 1
e-0
7
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
dH
= 1
e-0
6
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
dH
= 1
e-0
5
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
dH
= 1
e-0
4
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
dH
= 0
.001
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
010000
Time
me
an
+ s
d
nr patches = 2 nr patches = 8 nr patches = 50 nr patches = 200 nr patches = 1250NNumber of patches
2 8 50 200 1250
Rate
of dis
pers
ion
1 1
0-3
1 1
0-4
1 1
0-5
1 1
0-6
1 1
0-7
0LagS
em
i V
ariance
-
HOW DOES THIS RELATE TO OUR HYPOTHESISW
eak
Hom
ogenis
ation
Str
ong
Hom
ogenis
ation
Strong
Heterogenisation
Weak
Heterogenisation
?
?
-
RESULTING DYNAMICS
seq(0, 25, 0.5)
media
nN
o d
iffu
sio
n (d
H=
0)
2 large patches
Gra
ss b
iom
ass (
g m
2)
0100
200
300
400
0 5 10 15 20 25
seq(0, 25, 0.5)
media
n
0100
200
300
400
0 5 10 15 20 25
1250 small patches
seq(0, 25, 0.5)
media
nW
ith d
iffu
sio
n (d
H=
0.0
01)
0 5 10 15 20 25
0100
200
300
400
Grazer density (g m2)
Gra
ss b
iom
ass (
g m
2)
seq(0, 25, 0.5)
media
n
0 5 10 15 20 25
0100
200
300
400
Grazer density (g m2)
5 10 15 20 25
05
01
50
25
03
50
Grazer biomass (g m2)
Gra
ss b
iom
ass (
g m
2)
-
CONCLUDING REMARKS
In general “adding space” makes the
transitions more gradual
More complex responses than anticipated
Small “crashes” (at level of a system) are still
possible
Questionable whether these can be “predicted”
from first principles
Perhaps need to test if “crashes” remain at
n=0.25
-
THANK YOU
-
SIMULATIONS WOULD FIRST SETTLE PATTERNS, AND THEN CHANGE HERBIVORE DENSITY
01
00
Grass biomass
Index
(g m
2)
Time
03
00
Woody biomass
Index
(g m
2)
Time
0 100 200 300 400 500
05
01
00
15
02
00
Phase plane
Woody biomass (g m2)
Gra
ss b
iom
ass (
g m
2)
-
dH
= 0
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
dH
= 1
e-0
7
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
dH
= 1
e-0
6
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
dH
= 1
e-0
5
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
dH
= 1
e-0
4
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
dH
= 0
.001
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
040
80
Time
me
an
+ s
d
nr patches = 2 nr patches = 8 nr patches = 50 nr patches = 200 nr patches = 1250
Lag over time
-
dH
= 0
200 300 400 500
050
100
150
200
WH
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
dH
= 1
e-0
7
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
dH
= 1
e-0
6
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
dH
= 1
e-0
5
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
dH
= 1
e-0
4
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
dH
= 0
.001
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 500
050
100
150
200
W
H
200 300 400 5000
50
100
150
200
W
H
nr patches = 2 nr patches = 8 nr patches = 50 nr patches = 200 nr patches = 1250
Phase planes of woody biomass (X-
axis) and grass biomass (Y-axis)