nonlinear dynamics and bifurcation analysis in two models
TRANSCRIPT
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Nonlinear Dynamics and Bifurcation Analysisin Two Models of Sustainable Development
Fabiola Angulo, Gerard Olivar, Gustavo A. Osorio andLuz S. Velasquez
IDEA - CeiBA ComplexityUniversidad Nacional de Colombia, Sede Manizales
TerrassaNovember 2009
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 1 /31
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Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 2 /31
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Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 2 /31
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Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 2 /31
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Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 2 /31
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Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 2 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
The team working in this project
Gerard OlivarFabiola Angulo
Gustavo A. OsorioLuz S. Vel asquez
IDEA and CeiBA ComplexityUniversidad Nacional de Colombia
Sede Manizales (Colombia)
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 3 /31
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 4 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Objectives of this project
We pretend a new development modeling with the followingproperties
Better scenarios forecast
Possibilities for control actions (sustainability actions)
Institutional planning
Link with the actual system (IDEA Manizales)
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 6 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Manizales city
Manizales city
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
General Map
Scheme of the general map
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Manizales bio-region
Manizales bio-region map
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 9 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Manizales bio-region
Scheme of the bio-region
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 10 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Manizales bio-neighborhoods
Manizales bio-neighborhood map
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Modeling
Our modeling system includes
A State-Space Complex Network (with dynamics)
An Indicators System based on the Network
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Complex Network (with dynamics)
Nodes are associated to space locations (bio-cities,bio-neighborhoods)
Links are mainly based on traffic roads
At each node, we have a basic ODE system (parameters arenode-dependent), and the links couple some variables of eachnode pair.
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 14 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Natural resources
We begin considering that the region economy is based on theexploitation of two natural resources
An inexhaustible resource: land (coffee, for example)
An exhaustible renewable resource: forest S (to producewood)
Then the productions are
Coffee: λLδc , (λ is the land fertility)
Wood: αSLm (α is a technology parameter)
We can think that each personal income comes from coffee andwood. Let β be the percentage of the income on wood.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Natural resources
The change in the stock in the forest S, is the natural growthminus the harvest. We assume that the forest has a maximumcarrying capacity k1, and that below some threshold k2, theforest is not able to regenerate. Thus
S = (ρ(S/k1 − 1)(1− S/k2)− αβL)S
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Population
With regards to the population L, we assume that there isgrowth only if the production level (coffee and wood) is abovean average threshold σ. Some algebra leads to the followingequation
L = γ(λ(1− β)δLδ−1 + φαβS − σ)L
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
NR-P System
Thus we have the basic system
S = (ρ(S/k1 − 1)(1− S/k2)− αβL)S
L = γ(λ(1− β)δLδ−1 + φαβS − σ)L
where L is the population and S is the exhaustible resource(forest). We have the trivial equilibrium points:
L = 0 S = 0
L = 0 S = k1
L = 0 S = k2
L = (λ(1−β)δ
σ )1
1−δ S = 0
plus another two nontrivial equilibrium points.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
NR-P System
Generic bifurcation diagram for the nontrivial equilibrium points:
label = H x = (5018, 2847, 0,000105)
First Lyapunov coefficient = −5,055880
label = LP x = (3555, 2144, 0,000112)
a = −1,641549e − 006
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 20 /31
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Development system (5-dimensional)
S = (ρ(S/k1 − 1)(1− S/k2)− αβL)S
L = γ(λ(1− β)δLδ−1 + φαβS − σ)L
α = kLαLδ2(L− Lmin
L2min + (L− Lmin)2
)
λ = λ(ka(α0 − α)− kbLδ3)
σ = (r1(S − S2) + r2(λ− λ2))(r3α + r4σ)− r5L
where S is the renewable (exhaustible) resource, L is thepopulation, α is a variable of technological development, λ is avariable of environmental quality and σ is a variable ofeconomical wellfare.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Basic analysis
First analysis shows that depending on the parameters,there can be up to 12 equilibrium points, and sometimes,there are manifolds of equilibriums.
Thus even at the node level there can be some sort ofcomplexity.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Development system
Several models for α, λ and σ have been tried, but there issome robustness to the specific model, since we obtaingenerically the following diagrams in the state space:
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Development system
The following interesting result has been obtained as one of thepatterns leading to forest exhaustion:
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Development system
The following interesting result has been obtained as one of thepatterns leading to forest exhaustion:
Phase 1: Population growth and partial forest deterioration.
Phase 2: First (small) oscillations of forest and population.
Phase 3: Convergence to a seemingly equilibrium point.
Phase 4: Second (big) oscillations of forest and population.
Phase 5: Final forest deterioration (to exhaustion).
This pattern depends strongly on small variation of the initialconditions.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Development system
Even more interesting, the oscillations occur in a relative smalltime period. Thus, if any control action is going to be applied, itmust be done very fast, when the oscillations are initiallydetected. If they are detected or aplied too late, the forestprobably will be exhausted.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Conclusions
Even with simple models, bifurcation analysis is able toshow non-trivial dynamics, which cannot be predicted withlinear systems.
With the variation of some parameters, the system displaysvery different dynamics after bifurcation values.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
References and links
Center of Excellence of Basic and Applied InterdisciplinaryStudies on Complexity
CeiBA Complexity http://www.ceiba.org.coCeiBA Complexity (wikispace) http://ceiba.wikispaces.comGlobal System Dynamics & Policies Networkhttp://www.globalsystemdynamics.eu
Simone D’Alessandro.
Non-linear dynamics of population and natural resources: The emergence of different patterns ofdevelopment.Ecological Economics, Vol. 62, pp. 473 - 481, 2007.
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BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents of this talk
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 30 /31
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents of this talk
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 30 /31
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents of this talk
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 30 /31
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents of this talk
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Contents of this talk
1 Background
2 Modeling Development with Complex Networks
3 A (very) Simple Model of Sustainable Development
4 A (not so) Simple Model of Sustainable Development
5 Conclusions
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 30 /31
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logo
BackgroundModeling Development with Complex Networks
A (very) Simple Model of Sustainable DevelopmentA (not so) Simple Model of Sustainable Development
Conclusions
Nonlinear Dynamics and Bifurcation Analysisin Two Models of Sustainable Development
Fabiola Angulo, Gerard Olivar, Gustavo A. Osorio andLuz S. Velasquez
IDEA - CeiBA ComplexityUniversidad Nacional de Colombia, Sede Manizales
TerrassaNovember 2009
IDEA - CeiBA Complexity Nonlinear Dynamics and Bifurcation Analysis 31 /31