##### Model Description

Most modeling exercises on resource-population dynamics of a socio-economic system assume that many growth-related phenomena are linearly related to population size. The model presented here departs from this linear thinking by exploring potential non-linear relationships, or power-law scaling behaviors, with population size. For example, twice as many people do not mean that twice as much resources are required to maintain existing population. Similarly, twice as many people do not necessarily mean that twice as many innovation and technological advances occur in a society. Further, the model presented here adds more realism by incorporating the effects of increased global inter-connectedness, i.e., accelerated immigration/emigration and imports/exports across boundaries. For simplicity, the well-known Gordon-Schaeffer model is extended to apply these different assumptions. The key insight to be gained from the model is that the incorporation of power law scaling behavior and connectivity to outside yields model behaviors that are qualitatively different from those with the usual linear assumptions.

Regarding the institutional aspects of the model, a possible hypothesis could be that it is the presence of well-functioning institutional arrangements themselves that give rise to such 'favorable' effects of non-linearity and connectedness. Through social institutions, transaction costs of human cooperation are mitigated—there is less uncertainty in human interactions and in levels of trust that others are also cooperating. Assuming that larger, complex social systems are more likely to carry advanced social institutions, we can expect that such social systems are characterized by increasing returns to scale on innovation and resource requirements with respect to population size.

**Reference**

Muneepeerakul, R., & Qubbaj, M. R. (2012). The effect of scaling and connection on the sustainability of a socio-economic resource system. Ecological Economics. Elsevier B.V. doi:10.1016/j.ecolecon.2012.02.017

##### Scenarios

First, we test the effects of power-law scaling. Set the parameters as the following: beta=1.1, alpha=0.9, h=0.5, m=0.32, mu=0, and xi=0 (mu=0 and xi=0 mean that cross-border trade and migration do not happen). Under this configuration, there are two stable fixed points in the system, one with zero population and the other with some positive population. Now, change beta to 1.2 and alpha to 0.8. Under this configuration, the system behavior is qualitatively different. Only one stable fixed point exists now with zero population, a clearly unsustainable system. You may also set beta and alpha to 1 (i.e., linear assumption) and see how the linear model is different from the ones with power-law scaling behavior.

Second, we test the effects of imports/exports. Set the parameters as the following: beta=1.2, alpha=0.8, h=0.5, m=0.32, mu=0.08, and xi=0. Observe that the presence of imports/exports turns the previously unsustainable system (only one stable fixed point with zero population level) into a qualitative different system with two stable fixed points. One of the two stable fixed points have a positive population level. Now, depending on initial conditions, the system can become sustainable.

Lastly, we test the effects of immigration/emigration. Set the parameters as the following: beta=1.2, alpha=0.8, h=0.5, m=0.32, mu=0, and xi=0.03 (note: vary xi among 0.008, 0.02, and 0.03) . Observe that the presence of immigration/emigration turns the previously unsustainable system into a qualitative different system with either one or two stable fixed points. At least one of the resulting stable fixed points has a population level above zero. Now, the system can become sustainable. In fact, if xi is set to 0.03, the system stays always sustainable (i.e., positive population levels) regardless of initial condition.

$\Large \textrm{Actual Model:}$ |

$\Large \frac{dY}{dt}=rY\left ( 1-\frac{Y}{K} \right )-HYN^{\beta}+T$ |

Rate of change in resource level |

$\Large \frac{dN}{dt}=\frac{H}{E}YN^{\beta}-\frac{M}{E}N^{\alpha}+I$ |

Rate of change in population level |

$\Large \textrm{Non-dimensionalized Model:}$ |

$\textrm{Variable/parameter descriptions:} \\
Y: \textrm{Resource level at time t}\\
N: \textrm{Population level at time t}\\
r: \textrm{Intrinsic renewal rate of resources}\\
K: \textrm{Carrying capacity of resources}\\
H: \textrm{Ability to harvest/extract resources, i.e., catchability}\\
M: \textrm{Rate of source usage required to maintain the existing populatin}\\
E: \textrm{Coefficient for converting resources into maintaining and growing population}\\
\alpha: \textrm{Scaling factor on the population size for the growth of the resource usage to maintain existing population}\\
\beta: \textrm{Scaling factor on the population size for the growth of creativity, innovation, technology,cooperation, or labor division}\\
T: \textrm{Flow of trade (>0 means import; <0 means export)}\\
I: \textrm{Flow of people (>0 means immigration; <0 means emigration)}\\$ |

$\Large \frac{dy}{d\tau}=y\left ( 1-y \right )-hyn^{\beta}+\mu$ |

Rate of change in resource level (non-dimensionalized) |

$\Large \frac{dn}{d\tau}=hyn^{\beta}-mn^{\alpha}+\xi$ |

Rate of change in population level (non-dimensionalized) |

$\textrm{Variable/parameter descriptions:} \\
y=Y/K \\
n=NE/K \\
\tau=rt \\
h=(H/r)(K/E)^{\beta} \\
m=(M/rK)(K/E)^{\alpha} \\
\mu=T/rK\\
\xi=IE/rK$ |

#=====define parameters par h=0.5, m=0.32, alpha=0.9, beta=1.1, mu=0, xi=0 #=====define some hidden variables===== #============economic functions #==================demographicfunctions #=====auxiliary quantities================= #======right hand sides dy/dt=y*(1-y)-h*y*n^beta+mu dn/dt=h*y*n^beta-m*n^alpha+xi #=============initial data @ meth=qualrk @ bound=1000 @ total=200 @ dt=0.1 @ xp=n,yp=y @ xlo=0,xhi=2,ylo=0,yhi=1.2 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.