Part 3 - An application to population growth

As we have discussed in class, there are several varieties of equations we can use to model population growth.  In this part, we will investigate some further extensions to this.  In this situation, we will consider the population of fish in a pond.  We can use a logistic growth equation:

    Q ' = r Q (1 - Q/K)

Here:
    Q(t) = Quantity of fish at time t
    r = growth rate
    K = carrying capacity of the lake (or "Karrying Kapacity")
    
However, we need to also add in a term to take care of any fish "harvesting" that might occur (people do get hungry...).  We will use H(t) to represent the rate at which the fish are harvested over time:

    Q ' = r Q (1 - Q/K) - H(t)
    
In this case, assume that r = 4, K = 100, and T = 10.  For each of the following types of harvesting, answer the following questions (hint: make liberal use of copy and paste on this...):

a)  If the eqaution is autonomous, find the equilibrium points and describe the behavior between them.  If it isn't autonomous, graph the slope field to pick some good initial values.  (Alternately, you could just plug in a wide range of values, covering the region of interest, perhaps 10≤y≤150 or so.  Also, generally you only need to plot solutions for about 0≤t≤4.  If NDSolve give you a lot of errors, you could either use fewer initial points, use a smaller domain, or try a higher WorkingPrecision.)

b)  Pick several initial values to demonstrate the different possible behaviors of the population (you only need to consider cases where y≥0).

c)  Clearly explain (using actual words and sentences) what will happen to the fish population in this lake over time, given the different possible types of initial conditions.  Explain what effect the harvesting term has (compared to no harvesting).

Constant harvesting:  H(t) = 90

Constant harvesting:  H(t) = 110

"Constant effort" harvesting:  H(t) = 3Q

Periodic harvesting:  H(t) = 51 (1 - sin π t)

Periodic harvesting:  H(t) = 101 (1 - sin π t)


Created by Mathematica  (June 4, 2004)