Population Growth Models We can use direction fields to obtain a great deal o...

Population Growth

Models We can use direction fields to obtain a great deal of information about population growth models. In this problem you can create direction fields by hand or use a computer algebra system to create detailed ones. At time t = 0 a thin sheet of water begins pouring over the concrete spillway of a dam. At the same time, 1000 algae are attached to the spillway. We will be modeling P(t), the number of algae (in thousands) present after t hours.

Exponential Growth Model: We assume that the rate of population change is proportional to the population present: dP/dt = kP. In this particular case take

(a) Create a direction field for this differential equation and sketch the solution curve.

(b) Solve this differential equation and graph the solution. Compare your graph to the sketch from part (a).

(c) Describe the equilibrium solutions of this autonomous differential equation.

(d) According to this model, what happens as ?

(e) In our model, P(0) = 1. Describe how a change in P(0) would affect the solution.

f) Consider the solution corresponding to P(0) = 0. How would a small change in P(0) affect that solution?

Logistic Growth Model: As you saw in part (d), the exponential growth model above becomes unrealistic for very large t. What limits the algae population? Assume that the water flow provides a steady source of nutrients and carries away all waste materials. In that case the major limiting factor is the area of the spillway. We might model this as follows: Each algae-algae interaction stresses the organisms involved. This causes additional mortality. The number of such possible interactions is proportional to the square of the number of organisms present. Thus a reasonable model would be