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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)