Population Growth Models
Part 5.1: Constant-Rate Harvesting
Our logistic growth model is
where P(t) is the population at time t, r is the natural growth factor, and K is the maximum supportable population. If we harvest from the population at a constant rate H, then the model becomes
- To get a sense of how this added term affects solutions of the differential equation, experiment with changes of r, K, and (especially) H in the following applet. First observe how the slope field changes. Then, for each combination of parameters, click at various places to set starting populations P0. Try to find as many different shapes of solution curves as you can. Which starting populations lead to extinction of the population?
- Observe the graph of dP/dt versus P -- the phase plane -- in the lower right corner of the applet. How does this graph change when harvesting is added to the model? That is, how is the case for H > 0 different from the case for H = 0?
- Suppose H > 0, so extinction is possible. In terms of points in the phase plane, where would you locate the starting populations P0 that lead to extinction? Explain why extinction occurs in terms of the corresponding values of dP/dt.
- There is another region in the phase plane in which dP/dt is negative. Why doesn't extinction occur when the starting population is in that region?
- If H > 0 (and not too large, relative to r and K), then there are two equilibrium values of the population, both positive. Find expressions for those values in terms of the parameters r, K, and H. [Hint: What is the value of dP/dt when the population is at equilibrium?] For each of the equilibrium populations, determine whether the equilibrium is stable or unstable. (If necessary, refer back to Part 4.) How does the phase plane help you determine stability?
- The two equilibrium populations divide the range of possible populations into three regions. Describe in words the solutions of the differential equation for starting values in each of the three regions. [Your response should agree with what you determined from your experiments in step 1.]
- The value H = rK2/4 is called the critical harvesting rate. What is the significance of this value? [Hint: Refer to your symbolic expressions in step 5.] What happens if the harvesting rate is greater than the critical rate? What happens if H is less than the critical rate?
- Summary of constant-rate harvesting: For H > 0, there are three distinct families of solutions of
These distinct cases depend on the roots of the equation
r P (K - P) - H = 0
(where P is the variable). Describe each case in terms of roots of this equation. For each case, describe the equilibrium states and the shapes of non-equilibrium solutions.
