Introduction

Using the previously derived expressions from our ODEs, we use the Mathworks Simulink package to derive solutions to our system and model our system for a range of parameters.

ODE Solution

Solving:

\begin{eqnarray} \frac{x_2}{dt} = \alpha - \gamma x_2 \\ \frac{x_2}{dt} + \gamma x_2 = \alpha \end{eqnarray}

Integrating Factor:

\begin{eqnarray} e^{\int \gamma dt} = e^{\gamma t} \end{eqnarray}

Multiplying both sides by our integrating factor:

\begin{eqnarray} (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \alpha e^{\gamma t}\\ \int (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \int \alpha e^{\gamma t} \\ x_2 = \frac{\alpha}{\gamma} + ce^{-\gamma t} \end{eqnarray}

R plots

Visit our GitHub repositoryfor our code.

R Analysis

R Analysis:
Call:
lm(formula = log(x) ~ c(time, time, time))

Residuals:
     Min       1Q   Median       3Q      Max
-0.58853 -0.15536  0.01303  0.19867  0.44055

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)
(Intercept)          2.87199    0.21773   13.19 1.47e-15 ***
c(time, time, time)  0.15267    0.01142   13.37 9.74e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2935 on 37 degrees of freedom
Multiple R-squared:  0.8285,	Adjusted R-squared:  0.8238
F-statistic: 178.7 on 1 and 37 DF,  p-value: 9.741e-16

Intercept represents the equilibrium value of LacILov, our intercept: