Solution of Lotka Volterra Problem using Runge Kutta 4

This project demonstrates the graphical solution for the Lotka-Volterra Equations using Runge-Kutta-4 method. MATLAB and Python both have been used to find the graphical solution.

Runge-Kutta-4 (RK4) method Basics:

Runge-Kutta-4 is the most widely known member of Runge-Kutta family, used for the approximate solution of ordinary differential equations (ODE).

Mathematical Equations:

x_(n+1) = x_n + 1/6*(k1+k2+k3+k4)
t_(n+1) = t_n + h    for n=0,1,2,3,...

where,
1) k1 = f(t_n,x_n)
2) k2 = h*f(t_n + h/2, x_n + k1/2)
3) k3 = h*f(t_n + h/2, x_n + k2/2)
4) k4 = h*f(t_n + h, x_n + k3)

Representation in MATLAB Code:

1. t_(n+1) is reperesented by t
2. t_n is reperesented by t(k)
3. x_(n+1) is represented by x(:,k+1)
4. x_n is represented by x(:,k)
5. 1/6*(K1+K2+K3+K4) is represented by dx
6. k1+k2+k3+k4 is represented by is represented by k1,k2,k3,k4

Representation in python Code:

1. t_(n+1) is reperesented by t
2. t_n is reperesented by t[k]
3. x_(n+1) is represented by x[:,k+1]
4. x_n is represented by x[:,k]
5. 1/6*(K1+K2+K3+K4) is represented by dx
6. k1+k2+k3+k4 is represented by is represented by k1,k2,k3,k4

NOTE: All other important notes are written alongside the code itself in their respective files.

Lotka-Volterra Basics:

The Lotka-Volterra equations or prey-predator equations, are a pair of first-order non-linear differential equations, used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.

Mathematical Equations:

dx/dt = α*x - β*x*y
dy/dt = δ*x*y - γ*y

Representation in MATLAB Code:

xdot = [alpha*x(1) - beta*x(1)*x(2)
            delta*x(1)*x(2) - gamma*x(2)];

Representation in Python Code:

xdot = np.array([alpha*x[0] - beta*x[0]*x[1], delta*x[0]*x[1] - gamma*x[1]])

NOTE: Please see the coding files for more information regarding the application part of Lotka-Volterra

Graphical Results:

MATLAB Plot Result:

Python Plot Result:

Both plots are approximately same as both python and MATLAB codes have same initial conditions