nonlinear dynamics and bifurcation analysis in two models

logo

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

logo

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


logo

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)


logo



Conclusions

Contents

1 Background




5 Conclusions


logo



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)


logo



Conclusions

Contents

1 Background




5 Conclusions


logo



Conclusions

Manizales city

Manizales city


logo



Conclusions

General Map

Scheme of the general map


logo



Conclusions

Manizales bio-region

Manizales bio-region map


logo



Conclusions

Manizales bio-region

Scheme of the bio-region


logo



Conclusions

Manizales bio-neighborhoods

Manizales bio-neighborhood map


logo



Conclusions

Modeling

Our modeling system includes

A State-Space Complex Network (with dynamics)

An Indicators System based on the Network


logo



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.


logo



Conclusions

Contents

1 Background




5 Conclusions


logo



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.


logo



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


logo



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


logo



Conclusions

NR-P System

Thus we have the basic system

S = (ρ(S/k1 − 1)(1− S/k2)− αβL)S


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.


logo



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


logo



Conclusions

Contents

1 Background




5 Conclusions


logo



Conclusions

Development system (5-dimensional)

S = (ρ(S/k1 − 1)(1− S/k2)− αβL)S


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.


logo



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.


logo



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:


logo



Conclusions

Development system

The following interesting result has been obtained as one of thepatterns leading to forest exhaustion:


logo



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.


logo



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.


logo



Conclusions

Contents

1 Background




5 Conclusions


logo



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.


logo



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.


logo



Conclusions

Contents of this talk

1 Background




5 Conclusions


logo



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


nonlinear dynamics and bifurcation analysis in two models

Documents