Non-linear dynamics of population and natural resources: The emergence of different patterns of development

Model Description

This model explores the long-term dynamic interaction between the exploitation of natural resources and population growth. This is a variant of Brander and Taylor (1998). The reference article, D'Alessandro (2007), gives the following description of the model. "Two new assumptions are introduced: i) the disaggregation of the ecological complex into two different resources; ii) irreversibility —namely, an inexorable tendency to exhaustion when the renewable resource stock is below a certain threshold". Unlike the model here, Brander and Taylor (1998) and other variants assume one ecological complex and no critical depensation. In such cases, the long-term equilibrium always approaches a positive population level and some level of resource due to resource regeneration. In reality however, a resource stock (i.e., trees in Easter Island)  can completely disappear while some positive population level is still lingering. The model presented here reflects this reality by introducing the two assumptions above".

 

Reference

D’Alessandro, S. (2007). Non-Linear Dynamics of Population and Natural Resources-The Emergence of Di?erent Patterns of Development. Ecological Economics, 62(3), 47 3 – 481.

Scenarios

Three cases are explored here. In the first case, one stable fixed point exists (zero resource level but some positive population). To see this, set alpha=0.0001, lambda=20, delta=0.7, beta=0.3, rho=0.025, KL=700, KH=12000, gamma=0.1, phi=3, sigma=1.8. For initial condition, set L(0)=200 (poulation level) and S(0)=12000 (resource level).

 

In the second case, two basins of attraction exist: (1) a limit cycle around an internal fixed point and (2) an asymptotic convergence to a stable corner fixed point (zero resource level but some positive population). To see this, set set alpha=0.0001, lambda=12.95, delta=0.7, beta=0.3, rho=0.025, KL=700, KH=12000, gamma=0.1, phi=3, sigma=1.4.

 

In the third case, two basins of attraction exist: (1) a convergence to a stable internal fixed point and (2) another convergence to a stable corner fixed point (zero resource level but some positive population). To see this, set set alpha=0.0001, lambda=18.5, delta=0.7, beta=0.3, rho=0.025, KL=700, KH=12000, gamma=0.1, phi=3, sigma=1.8.

$\Large \frac{dS}{dt}=\rho S \left (\frac{S}{K_{L}} \right ) \left (1-\frac{S}{K_{H}} \right )-\alpha \beta LS$
Rate of change in stock of trees in forest
$\Large \frac{dL}{dt}=\gamma \left [\lambda(1-\beta)^{\delta} L^{\delta-1}+\phi \alpha \beta S -\sigma \right ]L$
Rate of change in population 
#=====define parameters


par alpha=0.0001, lambda=20, delta=0.7, beta=0.3, rho=0.025
par KLO=700, KHI=12000, gamma=0.1, phi=3, sigma=1.8


#par alpha=0.0001, lambda=12.95, delta=0.7, beta=0.3, rho=0.025
#par KLO=700, KHI=12000, gamma=0.1, phi=3, sigma=1.4

#par alpha=0.0001, lambda=18.5, delta=0.7, beta=0.3, rho=0.025
#par KLO=700, KHI=12000, gamma=0.1, phi=3, sigma=1.8

#=====define some hidden variables=====



#============economic functions


#==================demographicfunctions



#=====auxiliary quantities=================


#======right hand sides

dS/dt=rho*S*(S/KLO-1)*(1-S/KHI)-alpha*beta*L*S
dL/dt=gamma*(lambda*(1-beta)^delta*L^(delta-1)+phi*alpha*beta*S-sigma)*L


#=============initial data

init S=12000, L=200
@ meth=qualrk
@ bounds=10000
@ total=5000
@ dt=0.1
@ xp=L, yp=S
@ xlo=0, xhi=13000, ylo=0,yhi=12000


done

Yu JHD,  Arizona State University. 

Bozicevic M,  Arizona State University. 

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