A model of a harvested fish population

Note: this is very similar to a problem in Homework 4.

Here is a fish-population model incorporating the effects of

• fish not finding mates if the population is too small,
• the death rate going up as the population increases, and
• the population being harvested with "effort" e:

(dP)/(dt) = f(P) = P(1 − P)(4P − 1) − eP

P is the population size (number of fish). The parameter e, which might for example represent the number of boats allowed in the fishing fleet, is to be considered constant in time, but we'll be interested in various values for that constant.

Our goal in this lab is to understand how the dynamics of the fish population depends on the harvesting effort.

(1a) Sketch the phase line for this equation when there is no fishing at all (e = 0). Use all you know about the graphs of cubic functions to help you. Or use Python to sketch the graph of f when e = 0.

Here's some code you could use:

from resources306 import *
P = sp.symbols('P')
plt.plot([0,1.1],[0,0],'k') # draw the horizontal P-axis
f = P*(1-P)*(4*P-1) - 0*P
expressionplot(f,P,0,1.1)  # draw f(P)
plt.xlabel('P')
plt.ylabel('f(P)',rotation=0);

(1b) Use your phase line to help you describe the possible things that can happen to the fish population in this case:

• in words (write them down)
• in a qualitative sketch of the solution curves: rotate your phase line vertical and use the horizontal axis for time.

(2a) Now figure out what happens when e is greater than 0. Start by thinking what the effect is on the graph of the RHS of the DE when you subtract eP first for tiny positive values of e, then for larger values. Use Python if you like, varying the value of e and seeing what it does to the graph of f.

(2b) Sketch the phase line for this equation when there is fishing with effort e = 1.

Use your phase line to help you describe the possible things that can happen to the fish population in this case:

• in words (write them down)
• in a qualitative sketch of the solution curves: rotate your phase line vertical and use the horizontal axis for time.

(3) The phase lines in parts 1 and 2 are qualitatively different. Try to illustrate how the phase line in 1 "morphs" into the phase line in 2, by drawing a sequence of phase lines for various e-values side by side. This is called a "bifurcation diagram", which will have e on the horizontal axis and P on the vertical.