Lab 1: Population growth with harvesting

M242 Spring 2008

Introduction: A fish population is growing in a lake according to the model:

where P denotes the population in thousands of fish after t years from some initial time. This is logistic growth with carrying capacity b and inherent growth rate a. In addition there is fishing: d thousands of fish are being removed per year. For this lab, assume that a= 0.42 and b = 5.13 .

Experiment:

1) For d=0 there is no fishing. Find the equilibrium values of the population for this case. Determine the type of stability at each equilibrium point. Produce a phase portrait (slope field plus numerous solution curves) for and for , for values of P(0) ranging between –1 and 7 (include the equilibrium values from part 1). Choose your initial conditions carefully so as to get an accurate phase portrait. Discuss the phase portrait, both in the context of the fish population and in purely mathematical terms. In your discussion, describe the long-term behavior of the population for various initial conditions.

2) Repeat part 1) for d=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 . Thus, for each value of d you need to find the equilibrium values, and create a phase portrait. Be sure to include initial conditions corresponding to the equilibrium values for each case, so that the equilibrium solutions are plotted. Group the d values into groups that have similar behavior, and carefully describe what happens for various initial conditions within each group.

3) Based on your work in part 2), first estimate (based on the phase portraits and fixed points), and then find the exact value of the bifurcation point d*. Describe how the number and type of the equilibrium values changes at d*. Sketch a bifurcation diagram (by hand is OK).

Hint on finding the exact bifurcation point: Solve for the equilibrium values as a function of d (in other words, don't substitute in a numerical value for d when you set the right-hand side equal to zero). What would be the number of equilibrium values at the bifurcation point? Recall that quadratic equations have either two, one or no solutions depending on the quantity under the radical sign (called the discriminant) in the quadratic formula.

4) Find an exact general solution for the differential equation using the numerical values for d that you used in part 2). Does the solution have the same form for each d value? Use these formulas to determine the long-term behavior of the fish population in each case. To do this, eliminate terms that approach zero as . You may have to do some algebra first to get the expression into a form that has terms that approach zero, as discussed in class. Relate your results here to the phase portraits from parts 1) and 2).

5) A model that includes the seasonal nature of fishing (more in the summer, less in the winter) is given by

This model there is no fishing at t=0 or t=1 (winter), and the greatest amount of fishing occurs at t=0.5 (summer). Since this equation is nonautonomous, there are no equilibrium points. However, there may be periodic solutions, which we could call “equilibrium cycles”. These are solution curves similar to the equilibrium solutions in parts 1)-3), but they are cyclical (look like sin or cos curves) rather than constants. For each of the d values in part 2), find such equilibrium cycles, and plot them (include a few other solution curves as well to form a phase portrait). There should be two equilibrium cycles for some d values, and none for others, just as in the autonomous case. One of the equilibrium cycles will behave like a sink, and the other like a source (though they are not actually sinks or sources since they are not constant). The “source like” equilibrium cycles are hard to find: experiment with the grapher from my website. Try to identify a bifurcation value d* as in part 4). This would be the point where there is exactly one equilibrium cycle (periodic solution). Since you can't solve for equilibrium cycles as you do for equilibrium values, you can't get an exact value for d* so estimate, based on your phase portraits.

6) Bonus: Discuss what the model

might represent. Is the bifurcation value the same as in 5)?