Predator-Prey System

Copyright (C) 2020 Andreas Kloeckner

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In [11]:
import numpy as np

import matplotlib.pyplot as plt

This is the "right-hand side" of the Lotka-Volterra predator-prey system

$$\begin{align*} y_1' &= y_1 (\alpha _1 - \beta _1 y_2)\\ y_2' &= y_2 (- \alpha _2 + \beta _2 y_1) \end{align*}$$

written in vector form, with somewhat arbitrarily chosen coefficients:

In [7]:
alpha1 = 0.5

beta1 = 0.5

alpha2 = 0.5

beta2 = 0.5



def predator_prey_rhs(t, y):

    y1, y2 = y

    return np.array([

        y1*(alpha1-beta1*y2),

        y2*(-alpha2+beta2*y1)

    ])

We will integrate this with the help of the fourth-order Runge-Kutta method, which we will meet in more detail later in the chapter:

In [2]:
def rk4_step(y, t, h, f):

    k1 = f(t, y)

    k2 = f(t+h/2, y + h/2*k1)

    k3 = f(t+h/2, y + h/2*k2)

    k4 = f(t+h, y + h*k3)

    return y + h/6*(k1 + 2*k2 + 2*k3 + k4)

Next, integrate the IVP for some time:

In [13]:
times = [0]

ys = [np.array([0.1, 0.9])]



dt = 0.1



while times[-1] < 100:

    y = ys[-1]

    ynext = rk4_step(y, times[0], dt, predator_prey_rhs)

    ys.append(ynext)

    times.append(times[-1] + dt)



ys = np.array(ys)

times = np.array(times)

Lastly, plot the result:

In [14]:
plt.plot(times, ys[:, 0], label="Prey")

plt.plot(times, ys[:, 1], label="Predator")
Out[14]:
[<matplotlib.lines.Line2D at 0x7f72999a1d80>]
In [ ]: