BIFURCATION EXERCISE
Practical Complexity-Science by Patrick Bogaart and Stefan Dekker
Wednesday 26th August 13:30-14:30
For extra reading you can read the paper by Dekker et al. GCB (2007).
1. Run the model bifurcation.m. For that open Matlab and open the file bifurcation.m. Then
click on run.
2. You now see an opening screen. You can choose either the spatially uniform or spatially
distributed model.
3. The goal of this exercise is to make a bifurcation diagram (bottom-right panel). With
different precipitation rates you can model how the vegetation will react on that.
Questions to address:
1. At which tipping point of the precipitation regime is there no biomass left over for the
uniform model (LP2) with this parameter setting? The figure bellows gives an idea how the
bifurcation can look like
2. At which tipping point of the precipitation regime (LP1) is there no biomass left over for the
spatially uniform model?
3. At which precipitation rate is there no difference between the uniform solution and the
distributed solution? This is called the Turing instability point (T)
4. Describe in your own language how the positive and negative feedbacks work in the
spatially distributed model
5. Describe in your own words the emerging properties of the spatial patterns.
6. Can you understand the dotted line reflecting hystereses.
Z (ITCZ)
0
0.2
IPMod
With
0.6
0.8
1
hystereses
PMod
No
space
spac
30
Biomass (gr/m2)
0.4
20
ma
LP1
T
10
LP2
0
0.1
0.2
0.3
mi
0.4
Pa (m/y)
0.5
0.6
RESILIENCE EXERCISE
Friday 28th August, Stefan Dekker
For extra reading, read Dekker et al. GCB 2007, Rietkerk et al. 2004 Science
1. Now run the resilience.m model. Again open resilience.m and run the model.
2. As initialization the model starts with a patterned model. The model is now run with a
perturbation on the forcing. The precipitation is reduced from 1.4 mm/d to 1.3 mm/d
during 500 days with a transition lag time of 10 days. The model produces two different
realizations. The dotted lines are min/max of biomass the solid lines are the mean vlues
3. Although there was perturbation, the system returns to their original state.
4. The goal of this exercise is to find the resilience of the system
Questions to address
1. What is the Rdry tipping point limit before the system shifts from the patterned state
to the bare state?
2. Explore the connection between duration and strength of the perturbation to the
resilience of the system.
3. Why is it that the tipping point of Rdry, before the system shifts to the bare state, is
higher than the Tipping point found under the first exercise last Wednesday with the
bifurcation?