Lotka-Volterra equations(predator prey) using Runge-Kutta in Python

Question

I am trying to write a program using the Lotka-Volterra equations for predator-prey interactions. Solve Using ODE's:

dx/dt = a*x - B*x*y
dy/dt = g*x*y - s*y

Using 4th order Runge-Kutta method

I need to plot a graph showing both x and y as a function of time from t = 0 to t=30.

a = alpha = 1
b = beta = 0.5
g = gamma = 0.5
s = sigma = 2
initial conditions x = y = 2

Here is my code so far but not display anything on the graph. Some help would be nice.

#!/usr/bin/env python

from __future__ import division, print_function
import matplotlib.pyplot as plt
import numpy as np

def rk4(f, r, t, h):
        """ Runge-Kutta 4 method """
        k1 = h*f(r, t)
        k2 = h*f(r+0.5*k1, t+0.5*h)
        k3 = h*f(r+0.5*k2, t+0.5*h)
        k4 = h*f(r+k3, t+h)
        return (k1 + 2*k2 + 2*k3 + k4)/6

def f(r, t):
        alpha = 1.0
        beta = 0.5
        gamma = 0.5
        sigma = 2.0
        x, y = r[2], r[2]
        fxd = x*(alpha - beta*y)
        fyd = -y*(gamma - sigma*x)
        return np.array([fxd, fyd], float)


tpoints = np.linspace(0, 30, 0.1)
xpoints = []
ypoints = []

r = np.array([2, 2], float)
for t in tpoints:
        xpoints += [r[2]]
        ypoints += [r[2]]
        r += rk4(f, r, t, h)

plt.plot(tpoints, xpoints)
plt.plot(tpoints, ypoints)
plt.xlabel("Time")
plt.ylabel("Population")
plt.title("Lotka-Volterra Model")
plt.savefig("Lotka_Volterra.png")
plt.show()

Answer 1

A simple check of your variable tpoints after running your script shows it's empty:

In [7]: run test.py

In [8]: tpoints
Out[8]: array([], dtype=float64)

This is because you're using np.linspace incorrectly. The third argument is the number of elements desired in the output. You've requested an array of length 0.1.

Take a look at np.linspace 's docstring. You won't have a problem figuring out how to adjust your code.

Answer 2

1) define 'h' variable.

2) use

tpoints = np.arange(30) #array([0, 1, 2, ..., 30])

not

np.linspace()

and don't forget to set time step size equal to h:

h=0.1
tpoints = np.arange(0, 30, h)

3) be careful with indexes:

def f(r,t):
    ...
    x, y=r[0], r[1]
    ...

for t in tpoints:
    xpoints += [r[0]]
    ypoints += [r[1]]
    ...

and better use .append(x):

for t in tpoints:
    xpoints.append(r[0])
    ypoints.append(r[1])
    ...

Here's tested code for python 3.7 (I've set h=0.001 for more presize)

import matplotlib.pyplot as plt
import numpy as np

def rk4(r, t, h):                    #edited; no need for input f
        """ Runge-Kutta 4 method """
        k1 = h*f(r, t)
        k2 = h*f(r+0.5*k1, t+0.5*h)
        k3 = h*f(r+0.5*k2, t+0.5*h)
        k4 = h*f(r+k3, t+h)
        return (k1 + 2*k2 + 2*k3 + k4)/6

def f(r, t):
        alpha = 1.0
        beta = 0.5
        gamma = 0.5
        sigma = 2.0
        x, y = r[0], r[1]
        fxd = x*(alpha - beta*y)
        fyd = -y*(gamma - sigma*x)
        return np.array([fxd, fyd], float)

h=0.001                               #edited
tpoints = np.arange(0, 30, h)         #edited
xpoints, ypoints  = [], []
r = np.array([2, 2], float)
for t in tpoints:
        xpoints.append(r[0])          #edited
        ypoints.append(r[1])          #edited
        r += rk4(r, t, h)             #edited; no need for input f
plt.plot(tpoints, xpoints)
plt.plot(tpoints, ypoints)
plt.xlabel("Time")
plt.ylabel("Population")
plt.title("Lotka-Volterra Model")
plt.savefig("Lotka_Volterra.png")
plt.show()

You can also try to plot "cycles":

plt.xlabel("Prey")
plt.ylabel("Predator")
plt.plot(xpoints, ypoints)
plt.show()

https://i.stack.imgur.com/NB9lc.png