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Robustness and Resilience across Scales: Migration and Resource Degradation in the Prehistoric U.S. Southwest

Model Type
SES Library XPP model

This is a simple model that integrates 1) resource-population dynamics, 2) population migration, and 3) spatial heterogeneity in biophysical conditions (i.e., soi fertility). The reference article, Anderies and Hegmon (2011), gives the following abstract of the model. \"Migration is arguably one of the most important processes that link ecological and social systems across scales. Humans (and other organisms) tend to move in pursuit of better resources (both social  and  environmental).  Such  mobility  may  serve  as  a  coping  mechanism  for  short-term  local-scale dilemmas and as a means of distributing organisms in relation to resources. Movement also may be viewed as a shift to a larger scale; that is, while it may solve short-term local problems, it may simultaneously have longer term and larger scale consequences. We conduct a quantitative analysis using dynamic modeling motivated by an archaeological case study to explore the dynamics that arise when population movement serves as a link between spatial scales. We use the model to characterize how ecological and social factors can lead to spatial variation in resource exploitation, and to investigate the circumstances under which migration may enhance or reduce the capacity of the system to absorb shocks at different scale\".

Reference

Anderies, J. M., & Hegmon, M. (2011). Robustness and Resilience across Scales : Migration and Resource Degradation in the Prehistoric U . S . Southwest. Ecology And Society, 16(2).

Scenarios

Here, we explore the changes in resource density in region 1 (R1) and region 2 (R2) as the parameter for treshold of migration transaction cost (bm) is varied. Region 1 is more fertile (i.e., higher resource regeneration rate) than region 2. An interesting phenomenon demonstrated by the model is that with extreme levels of migration (zero migration or infinite migration), more fertile and technologically-advanced regions (e.g., region 1) can have more severe environmental degradations than those regions with less fertile lands and back-dated technologies (e.g. region 2). However, with intermediate levels of migration, more fertile and advanced regions can spatially spread out the pressure for exploiting their resources to other regions and thus attain higher resource density than the less fertile regions. This is in essence shifting short-term local problems to a larger scale. In this model, region 1 has higher resource regeneration rate and harvest rate than region 2 (i.e., g1>g2 and q1>q2 ).

To observe this phenomenon, set the parameters as the following: g1=1, g2=0.33, gb=0.1, q1=0.1, q2=0.07, qb=0.03,  k1=100, k2=100, kb=100, am=4, bm=1, rg=0.01, ag=1, and bg=5. Set the y-axis to the auxiliary variable 'delta R12  (difference in resource density between region 1 and 2) and the x-axis to time. Under this setting (bm=1, i.e., migration is very easy), the long-run resource density of region 1 (R1) is less than that of region 2 (R1<R2) and so the difference is negative. Now increase 'migration transaction cost threshold' to 20 (bm=20). Under this setting (bm=20, i.e., migration is neither too easy nor too difficult), we get R1>R2 and so the difference is positive. Now increase 'migration transaction cost threshold' to 100 (bm=100). Under this setting (bm=100, i.e., strong resistence to migration), the difference (R1-R2) becomes negative again. 

To know more about other interesting phenomenon (i.e., how the system responds to external shocks such as droughts), please read Anderies and Hegmon (2011).

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$$\Large \frac{dR_{1}}{dt}=g_{1}(R)_{1}(1-\frac{R_{1}}{K_{1}})-q_{1}R_{1}H_{1}-\sigma_{1}s(t)R_{1}$$
$$\Large \frac{dR_{2}}{dt}=g_{2}(R)_{2}(1-\frac{R_{2}}{K_{2}})-q_{2}R_{2}H_{2}-\sigma_{2}s(t)R_{2}$$
$$\Large \frac{dR_{b}}{dt}=g_{b}(R)_{b}(1-\frac{R_{b}}{K_{b}})-q_{b}R_{b}H_{b}-\sigma_{b}s(t)R_{b}$$
$$\Large \frac{dH_{1}}{dt}=r_{g}\frac { anh {a_{g}(q_{1}R_{1}-b_{g})}}{\pi}H_{1}+M_{21}+M_{b1}-M_{12}+M_{1b}$$
$$\Large \frac{dH_{2}}{dt}=r_{g}\frac { anh {a_{g}(q_{2}R_{2}-b_{g})}}{\pi}H_{2}+M_{12}+M_{b2}-M_{21}+M_{sb}$$
$$\Large \frac{dH_{b}}{dt}=r_{g}\frac { anh {a_{g}(q_{b}R_{b}-b_{g})}}{\pi}H_{b}+M_{1b}+M_{2b}-M_{b1}+M_{b2}$$
$$\Large M_{ij}=0.5+\frac{ anh {a_{m}(d_{ji}-b_{m})}}{\pi}H_{i}$$
$$\Large d_{ji}=U(q_{i}R_{i})-U(q_{j}R_{j})$$
$$\Large W_{ci}=(q_{i}^{eq}R_{i}^{eq}-q_{i}R_{i})H_{i}$$
robustness_and_resilience_across_scales_migration_and_resource_degradation_in__IvGlMk2.ode
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