Consider the competition model defined by

Question

Consider the competition model defined by

dx/dt = x(2 - 0.4x -0.3y)

dy/dt = y(1 -0.1y - 0.3x),

where the populations x(t) and y(t) are measured in thousands and t is measured in years. Analyze the populations over a long period of time for each of the following cases:

(a) x(0) = 1.5, y(0) = 3.5

(b) x(0) = 1, y(0) = 1

(c) x(0) = 2, y(0) = 7

(d) x(0) = 4.5, y(0) = 0.5

Answer 1

This system does not admit an explicit solution. Here is an example of solving it numerically in Python.

from scipy.integrate import odeint

import matplotlib.pyplot as plt

import numpy as np

t = np.linspace(0, 10, 100)

sol = odeint(lambda z, t : [z[0]*(2 - .4*z[0] - .3*z[1]),

z[1]*(1 - .1*z[1] - .3*z[0])],

[2, 7], t)

plt.plot(t, sol[:, 0], 'b', label='x(t)')

plt.plot(t, sol[:, 1], 'g', label='y(t)')

plt.legend();