Living in a network of scaling cities and finite resources

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Model Description

A modified Lotka-Volterra system in which multiple consumers and resource systems are connected by both consumer-consumer and consumer-resource links. Consumers represent cities, and thus exhibit nonlinear scaling behaviors as population increases w.r.t. harvest rate and harvest conversion efficiency. Populations can also migrate between cities along a welfare-dependent gradient.

Analysis of the model for a simple dyadic network reveals that the basic Lotka-Volterra formulation can lead to stable dynamics, but the inclusion of nonlinear scaling behaviors leads to marked instability. Equilibrium welfare is much more sensitve to scaling of the parameter governing resource use than the parameter for resource harvest intensity. 

Analysis of the model for networks of three or more nodes reveals that connections between cities through resource systems are often more important than direct connections between cities in shaping the dynamics of the system. Local network structures alone are not sufficient for predicting long run dynamics of a particular city, which emphasizes the importance of modeling nested feedbacks between local and global network structures.

$\frac{dY_i}{dt} = r_i Y_i \left(1-\frac{Y_i}{K_i}\right) - Y_i \sum{H_{ij}N^\beta_j}$
Change in resource biomass in a resource system over time, as a function of the intrinsic growth rate and carrying capacity of the resource, and the rate of harvest by connected cities. The rate at which a city harvests a resource is a function of the population of the city, a nonlinear scaling parameter, and the strength of the city-resource link.
$\frac{dN_j}{dt} = \frac{N^\beta_j}{E_j} \sum H_{ij}Y_i - \frac{M_j}{E_j}N^\alpha_j - \nu_j N_j + \sum_k \xi_{jk} \nu_k N_k \frac{W_j}{\sum_l \xi_{lk} W_l}$
The change in population at a city over time, as a function of efficiency at which resources from a connected resource system are converted to population, the amount of resources required to maintain the current population (with nonlinear scaling factor and constant of proportionality), the fraction of the population who migrate to a connected city, and welfare-dependent in migration from city-city connections. 
# modified Lotka-Volterra system with multiple consumers and resources

par r=.1,K=1,M=.0000004,H=.000001,E=.0001,v=.05

dY/dt = r*Y*(1-Y/K) - Y * sum(H * N^b)
dN/dt = (N^b)/E sum(H*Y) - M/E * N^a - v*N + sum(x*v*N)*(W/sum(x*W))

done

Gauthier N,  Arizona State University. 

Qubbaj MR, Shutters ST, Muneepeerakul R.  2014.  Living in a network of scaling cities and finite resources. Bulletin of Mathematical Biology. 77(2):390-407.