a) Numerically solve the Lotka-Volterra (ie, predator-prey) equations using
- Euler method
- 4th order runge kutta method (you can find its algorithm on the internet)
\begin{eqnarray}
\frac{dx}{dt} &=& \alpha x – \beta xy \\
\frac{dy}{dt} &=& \delta xy – \gamma y
\end{eqnarray}
with parameter values.
\begin{eqnarray}
\alpha = 1,\quad \beta = 1,\quad \gamma = 1,\quad \delta = 1
\end{eqnarray}
for the following four initial conditions: (x,y) = (1.1, 1), (2, 1), (4, 1), (4, 0.2). Solve from t=0 to t=40 in all four cases and use \(h=0.01\) as a time step.
For each initial condition, plot both \(x(t)\) and \(y(t)\) against time to see the populations evolve in response to each other.
b) Using only Runge-Kutta 4, plot the phase portrait of the system.
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You should generate a pdf which includes the eight solutions (4 with RK4, 4 with Euler) and a phase portrait.
send your results to semih.ay@marmara.edu.tr untill 21 oct, 11:59pm.