Team:Toronto/Analysis
Analysis
Introduction
Using the previously derived expressions from our ODEs, we use the Mathworks Simulink package to derive solutions to 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 repository for 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 ***
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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:
