A simpler version of the Bali Model was designed to test the idea that the interactions of farmers (and subaks) could trigger a transition from local to global management of the pest-water tradeoff (Lansing et al. 2017). This began as some exploratory simulations in Lisp, later simplified in Matlab. The model imagines a square lattice of rice farmers, each of whom select a rice harvesting schedule from a set of possible options, later receiving a harvest according to this initial choice and the choices of surrounding farmers.
The model is iterated repeatedly, each iteration comprising three steps:
The model is iterated repeatedly, each iteration comprising three steps:
1) The harvest of each farm is calculated
This involves calculating the amount of water stress and pest stress experienced by a given farm. The level of water stress depends on the number of farmers in the entire lattice who follow the same harvesting schedule. If everyone plants their rice at the same time, water demand for flooding rice paddies is synchronized, there is insufficient water to go around and everyone experiences harvest losses as a result.The level of pest stress depends on the number of farmers that follow the same harvesting schedule within a pest zone, defined as a circular radius from each farm. When groups of nearby farms harvest and flood their rice paddies at the same or similar times, pest stresses are lower. The harvest H of farm i is calculated as
\[ H^{i}(t+1) = H_{0} - {\frac{\rho}{{0.1 + f^{i}_{p}(t) }}} + \delta f^{i}_{w}(t) \]
where $H_{0}$ is the base harvest before losses due to water stress or pest stress, $\rho$ is a parameter describing the intensity of pest stress, $\delta$ is a parameter describing the intensity of water stress, $f_{p}^{i}(t)$ is the proportion of farmers in the local area who follow the same harvesting schedule as farm i at time t, and $f_{w}^{i}(t)$ is the proportion of farmers in the entire lattice who follow the same schedule strategy as farm i.
2) Harvest schedules are updated
Farmers compare their own harvest to k immediate neighbors. Each farmer copies the harvesting schedule of the neighbor who achieved the highest harvest, or they keep their current harvesting schedule if no neighbor performed better.
3) Noise is added to the system
This step reflects the fact that information is never perfect and decisions are never simple. Noise is added to the system by repeatedly switching the harvesting schedule of small blocks of farmers on the lattice to a random (and frequently suboptimal) harvesting schedule. Blocks are of uniformly distributed size, and blocks of farmers are switched to a new harvesting schedule until the required noise level has been reached.
This involves calculating the amount of water stress and pest stress experienced by a given farm. The level of water stress depends on the number of farmers in the entire lattice who follow the same harvesting schedule. If everyone plants their rice at the same time, water demand for flooding rice paddies is synchronized, there is insufficient water to go around and everyone experiences harvest losses as a result.The level of pest stress depends on the number of farmers that follow the same harvesting schedule within a pest zone, defined as a circular radius from each farm. When groups of nearby farms harvest and flood their rice paddies at the same or similar times, pest stresses are lower. The harvest H of farm i is calculated as
\[ H^{i}(t+1) = H_{0} - {\frac{\rho}{{0.1 + f^{i}_{p}(t) }}} + \delta f^{i}_{w}(t) \]
where $H_{0}$ is the base harvest before losses due to water stress or pest stress, $\rho$ is a parameter describing the intensity of pest stress, $\delta$ is a parameter describing the intensity of water stress, $f_{p}^{i}(t)$ is the proportion of farmers in the local area who follow the same harvesting schedule as farm i at time t, and $f_{w}^{i}(t)$ is the proportion of farmers in the entire lattice who follow the same schedule strategy as farm i.
2) Harvest schedules are updated
Farmers compare their own harvest to k immediate neighbors. Each farmer copies the harvesting schedule of the neighbor who achieved the highest harvest, or they keep their current harvesting schedule if no neighbor performed better.
3) Noise is added to the system
This step reflects the fact that information is never perfect and decisions are never simple. Noise is added to the system by repeatedly switching the harvesting schedule of small blocks of farmers on the lattice to a random (and frequently suboptimal) harvesting schedule. Blocks are of uniformly distributed size, and blocks of farmers are switched to a new harvesting schedule until the required noise level has been reached.
The simple rules described above combine to generate complex system behaviors. When there is no water stress but pest stress is high, all farmers may eventually follow a single cropping pattern. Conversely, when pest stress is high but water stress is low or moderate, the theoretically optimum 'quadrant state' may arise, whereby the four possible strategies are represented on the lattice as four large blocks. When pest and water stresses are more evenly balanced, harvesting strategies remain patchily distributed for an extended period of time.
This is shown in the following animations, where parameter $\rho$ is pest stress and parameter $\delta$ is water stress (as in the equation above). The first animation (left) shows that when pest stress dominates ($\rho$ = 0.5, $\delta$ = 9.6 for 4000 steps, 5% noise), the system evolves to a quadrant state.
This is shown in the following animations, where parameter $\rho$ is pest stress and parameter $\delta$ is water stress (as in the equation above). The first animation (left) shows that when pest stress dominates ($\rho$ = 0.5, $\delta$ = 9.6 for 4000 steps, 5% noise), the system evolves to a quadrant state.
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Conversely, the second animation (right) shows that when water stress and pest stress are approximately equal ($\rho$ = 0.5, $\delta$ = 0.5 for 2000 steps, 5% noise), the system remains in a power-law state for a very long time, just as is observed in aerial photographs of Bali.
A power law is indicated when all the points on the scatter plot fall on a straight line (within the limitations of random noise). Note in particular how the power law arises – due to a small number of widely followed harvesting strategies and many more less common harvesting strategies.
Questions:
A power law is indicated when all the points on the scatter plot fall on a straight line (within the limitations of random noise). Note in particular how the power law arises – due to a small number of widely followed harvesting strategies and many more less common harvesting strategies.
Questions:
- How long does it take for the power law to appear?
- Does a power law always occur, or only with certain parameters?
References:
Lansing, JS, Thurner, S, Chung, NN, Coudurier-Curveurh, A, Karakaşh, Ç, Fesenmyeri, KA, Chew, LY. 2017. Adaptive self-organization of Bali’s ancient rice terraces. Proceedings of the National Academy of Sciences USA 114:6504–9.
Lansing, JS, Thurner, S, Chung, NN, Coudurier-Curveurh, A, Karakaşh, Ç, Fesenmyeri, KA, Chew, LY. 2017. Adaptive self-organization of Bali’s ancient rice terraces. Proceedings of the National Academy of Sciences USA 114:6504–9.