Exercise Numpy/Scipy

2. Solving ordinary differential equations (ODEs) using scipy

Consider the following hypothetical reaction system which was first proposed by Alfred J. Lotka (Lotka-Voltera System): $$ \begin{align} r1:& \;\; X \rightarrow 2 X \\ r2:& \;\; X + Y \rightarrow 2 Y \\ r3:& \;\; Y \rightarrow \varnothing \end{align} $$

1. set up the stoichiometric matrix of the system (this can be done without Python)
2. the rate of change of the species participating in a reaction system can be written as $$ \frac {dS} {dt} = N \cdot v(S,p) $$ with the vector of species concentrations $S$, the stoichiometric matrix N and the vector of reaction velocities $v(S,p)$ ($p$ denotes a vector of parmameters) define a function dSdt( S, t ) which computes the rate of change for each species, given the vector of current species concentrations $S$ and the current time $t$. Assume that all reactions follow irreversible mass-action kinetics with parameters $k_1 = 1.5$, $k_2 = 0.8 $ and $k_3 = 0.9$, each equal to
3. Use the scipy ODE integrator scipy.integrate.odeint to solve the reaction system numerically on the time interval $t \in [0, 30]$ using the following initial conditions: $X(0)= Y(0) = 1$