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)