a. State Variables and State Space:
1.Cell Density (N): The number of yeast or bacterial cells present in the chemostat at a given time. The state space for N is the set of non-negative real numbers (N ≥ 0).
2.Concentration of Substrate (S): The concentration of the nutrient (e.g., glucose) in the liquid medium. The state space for S is the set of non-negative real numbers (S ≥ 0).
3.Dilution Rate (D): The rate at which medium is added to the chemostat relative to the volume of the chemostat. The state space for D is the set of non-negative real numbers (D ≥ 0).
4.Effluent Concentration (S_out, N_out): The concentration of substrate and cell density in the effluent leaving the chemostat. The state space for S_out and N_out is the set of non-negative real numbers (S_out ≥ 0, N_out ≥ 0).
b. Parameters:
1.Maximum Specific Growth Rate (μ_max): The maximum growth rate of cells under ideal conditions (maximal nutrient availability and absence of inhibitory factors). It is a positive real number (μ_max > 0).
2.Half-Saturation Constant (K_s): The concentration of substrate at which the specific growth rate is half of μ_max. It is a positive real number (K_s > 0).
3.Yield Coefficient (Y): The amount of biomass (cells) produced per unit of substrate consumed. It is a positive real number (Y > 0).
4.Dilution Rate (D): This is both a state variable and a parameter. As a parameter, it represents the rate at which medium is added to the chemostat, and it can vary within the state space (D ≥ 0).
5.Inlet Concentration (S_in): The concentration of substrate in the incoming medium. It is a positive real number (S_in > 0).
6.Effluent Flow Rate (Q): The rate at which medium and cells exit the chemostat through the effluent tube. It is a positive real number (Q > 0).
7.Cell Death Rate (μ_death): The rate at which cells die in the chemostat due to factors such as predation or aging. It is a positive real number (μ_death > 0).
c. Justification for Model Type:
This should be a continuous time model because the growth and dynamics of yeast and bacterial populations in a chemostat occur continuously over time. Cells divide continuously, and changes in cell density, substrate concentration, and other state variables are continuous and smooth. Discrete time models, which operate in discrete time steps, may not capture the nuances of these continuous processes accurately. Therefore, a continuous time model, possibly using differential equations, would better represent the system's behavior in a chemostat.