Plotting trajectories and direction fields for a system of ODEs in Python

The aim of this post is to guide the reader through plotting trajectories and direction fields for a system of equations in Python. This is useful when investigating the equilibria and stability of the system, and to facilitate in understanding the general behavior of a system under study. I will use a system of predator-prey equations, that my most devoted online readers are already familiar with from my previous posts on identifying equilibria and stability, and on nondimensionalization. Specifically, I’ll be using the Lotka-Volterra set of equations with Holling’s Type II functional response:

\frac{\mathrm{d} x}{\mathrm{d} t}=bx\left ( 1-\frac{x}{K} \right )-\frac{axy}{1+ahx}

\frac{\mathrm{d} y}{\mathrm{d} t}=\frac{caxy}{1+ahx}-dy

where:

x: prey abundance

y: predator abundance

b: prey growth rate

d: predator death rate

c: rate with which consumed prey is converted to predator

a: rate with which prey is killed by a predator per unit of time

K: prey carrying capacity given the prey’s environmental conditions

h: handling time

This system has 3 equilibria: when both species are dead (0,0), when predators are dead and the prey grows to its carrying capacity (K,0) and a non-trivial equilibrium where both species coexist and is generally more interesting, given by:

y^*=\frac{b}{a}(1+ahx^*)\left(1-\frac{x^*}{K} \right)

x^*=\frac{d}{a(c-dh)}

The following code should produce both trajectories and direction fields for this system of ODEs (python virtuosos please excuse the extensive commenting, I try to comment as much as possible for people new to python):

import numpy as np
from matplotlib import pyplot as plt
from scipy import integrate

# I'm using this style for a pretier plot, but it's not actually necessary
plt.style.use('ggplot')

"""
This is to ignore RuntimeWarning: invalid value encountered in true_divide
I know that when my populations are zero there's some division by zero and
the resulting error terminates my function, which I want to avoid in this case.
"""
np.seterr(divide='ignore', invalid='ignore')

# These are the parameter values we'll be using
a = 0.005
b = 0.5
c = 0.5
d = 0.1
h = 0.1
K = 2000

# Define the system of ODEs
# P[0] is prey, P[1] is predator
def fish(P, t=0):
    return ([b*P[0]*(1-P[0]/K) - (a*P[0]*P[1])/(1+a*h*P[0]),
            c*(a*P[0]*P[1])/(1+a*h*P[0]) - d*P[1] ])

# Define equilibrium point
EQ = ([d/(a*(c-d*h)),b*(1+a*h*(d/(a*(c-d*h))))*(1-(d/(a*(c-d*h)))/K)/a])

"""
I need to define the possible values my initial points will take as they
relate to the equilibrium point. In this case I chose to plot 10 trajectories
ranging from 0.1 to 5
"""
values = np.linspace(0.1, 5, 10)
# I want each trajectory to have a different color
vcolors = plt.cm.autumn_r(np.linspace(0.1, 1, len(values)))

# Open figure
f = plt.figure()
"""
I need to define a range of time over which to integrate the system of ODEs
The values don't really matter in this case because our system doesn't have t
on the right hand side of dx/dt and dy/dt, but it is a necessary input for
integrate.odeint.
"""
t = np.linspace(0, 150, 1000)

# Plot trajectories by looping through the possible values
for v, col in zip(values, vcolors):
    # Starting point of each trajectory
    P0 = [E*v for E in EQ]
    # Integrate system of ODEs to get x and y values
    P = integrate.odeint(fish, P0, t)
    # Plot each trajectory
    plt.plot( P[:,0], P[:,1],
            # Different line width for different trajectories (optional)
            lw=0.5*v,
            # Different color for each trajectory
            color=col,
            # Assign starting point to trajectory label
            label='P0=(%.f, %.f)' % ( P0[0], P0[1]) )
"""
To plot the direction fields we first need to define a grid in order to
compute the direction at each point
"""
# Get limits of trajectory plot
ymax = plt.ylim(ymin=0)[1]
xmax = plt.xlim(xmin=0)[1]
# Define number of points
nb_points = 20
# Define x and y ranges
x = np.linspace(0, xmax, nb_points)
y = np.linspace(0, ymax, nb_points)
# Create meshgrid
X1 , Y1 = np.meshgrid(x,y)
# Calculate growth rate at each grid point
DX1, DY1 = fish([X1, Y1])
# Direction at each grid point is the hypotenuse of the prey direction and the
# predator direction.
M = (np.hypot(DX1, DY1))
# This is to avoid any divisions when normalizing
M[ M == 0] = 1.
# Normalize the length of each arrow (optional)
DX1 /= M
DY1 /= M

plt.title('Trajectories and direction fields')
"""
This is using the quiver function to plot the field of arrows using DX1 and
DY1 for direction and M for speed
"""
Q = plt.quiver(X1, Y1, DX1, DY1, M, pivot='mid', cmap=plt.cm.plasma)
plt.xlabel('Prey abundance')
plt.ylabel('Predator abundance')
plt.legend(bbox_to_anchor=(1.05, 1.0))
plt.grid()
plt.xlim(0, xmax)
plt.ylim(0, ymax)
plt.show()

This should produce the following plot. All P0s are the initial conditions we defined.
trajectories

We can also see that this parameter combination produces limit cycles in our system. If we change the parameter values to:

a = 0.005
b = 0.5
c = 0.5
d = 0.1
h = 0.1
K = 200

i.e. reduce the available resources to the prey, our trajectories look like this:

trajectories1

The equilibrium becomes stable, attracting the trajectories to it.

The same can be seen if we increase the predator death rate:

a = 0.005
b = 0.5
c = 0.5
d = 1.5
h = 0.1
K = 2000

trajectories2

The implication of this observation is that an initially stable system, can become unstable given more resources for the prey or less efficient predators. This has been referred to as the Paradox of Enrichment and other predator-prey models have tried to address it (more on this in future posts).

P.S: I would also like to link to this scipy tutorial, that I found very helpful and that contains more plotting tips.

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3 thoughts on “Plotting trajectories and direction fields for a system of ODEs in Python

  1. Hi Antonia! Nice code!
    I am still trying to understand all the math in it haha
    You might want to fix the indentation of the code on the code snippet though, as it is all lost.

    Thanks for sharing!

    • Hi Lucas, thanks for the comment! I’m sorry the math isn’t very clear, I had to refresh my calculus before embarking on these exercises too haha.
      I’m not sure what you mean about the indentation? On the left?

    • I just realized what you meant. I thought you meant the placement of the code on the page. I didn’t notice that when I pasted it the indentation in the loops was removed. It’s fixed now, thanks!

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