Discussion Post

Date: June 23, 2020

Author: Joseph Simone

Part I

Equation 1:

\(Vt = Max \{ 0, V_{t-1} + C_t - D_t - O_t\}\)

Equation 2:

\(Vt = Min\{Max( 0, V_{t_1} + C_t - D_t), Cap)\}\)

In this paper, the design of the simulation is modeled after a on balance of equations. “Where the captured supply of rainwater from the roof (C), demand for purified water (D), and overflow (O) are the primary considerations for the volume (V) in the cistern.”

At any given moment, indexed with \(t \in T\), the volume in the water tank is represented in Equation 1. This is reflexed on the basis that t is a daily index rather than monthly or yearly. Equation 2 highlights a minimum & maximum approach for modeling tank capacity (Cap).

These equations were designed to optimize water volume within the systen, based on captured supply of rainwater, demand for purified water and the overflow of water.

Part II

def update_func_quad2(pop, t, system):
    """Compute the population next year with a quadratic model.
    
    pop: current population
    t: current year
    system: system object containing parameters of the model
    
    returns: population next year
    """
    r = system.alpha
    K = -system.alpha/system.beta
    
    net_growth = (r * pop * (1 -(pop/K)))
    return pop + net_growth