# Carmen E. Castaneda: Week 5

### A simple chemostat model of nutrients and population growth

- Considering the nutrient/ cell population model

dn/dt = Du-Dn(t)-yV_{max}(n/(K+n))

dy/dt = yrV_{max}(n/(k+n))

we get that the state variables are: n, y
and the parameters are: K, V_{max}, r, u, D

After stimulating this system with different values for the constants and the initial concentrations of nutrients and cells. We begin to notice that the system behaves closer to a linear function. By this I mean that as time progress the yeast population increases as the nutirents gets used up. Looking at individual parameters we notice that each one affects the system in the same way just by different degrees, so they all get pretty linear pretty quickly.

- Adapting the system to a logistic growth model

dn/dt = Du-Dn(t)-yV_{max}(n/(K+n))

dy/dt = yr(1-(y/(an)))

we begin to notice that the system has an exponential tendency. When we change each of the parameters we see that there a growth of the yeast population until it reaches a sort of asymptote and levels out. However each parameter reaches this curve in a slightly different manner.

For the logistic growth model we definitely can adjust the system in such a way that we get rid of some toxin in the yeast. This would be recommended as seen by the plot that had a diffrent nutrient level because the population was definitely affected by the nutrient level and quickly leveled of.