Question 3 Suppose that we are modeling the population of bacteria in a certain medium. Let N(t) denote the population d

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Question 3 Suppose That We Are Modeling The Population Of Bacteria In A Certain Medium Let N T Denote The Population D 1 (115.55 KiB) Viewed 16 times

Question 3 Suppose that we are modeling the population of bacteria in a certain medium. Let N(t) denote the population density of bacteria at time t. A common starting assumption is that the per-capita growth rate (i.e., the number of new bacteria per square meter per second divided by the total number of bacteria) is constant. That is, we assume that where r is constant, and so 1 dN N dt dN dt 1 dN N dt =rN. This is called an exponential growth model because, as we already saw in class, the population density is governed by the exponential function N(t) = Noert, where No = N(0) is the initial population. Exponential growth models are not particularly realistic, not least because they suggest that the population of bacteria in a medium will grow to arbitrarily large sizes. In reality, there is probably some maximum population density that a given medium can support. To get a more realistic description, we can assume that the per capita growth rate depends on the population density in some way. One common choice is to assume that the per capita growth rate is a linear function of N. In particular, we assume = r, N r (1-K) = r 1
or dN dt =rN|1 N K where K is the maximum population size. This is called a logistic growth model. 1. Find the general solution to the logistic differential equation. (Hint: You will need the technique of partial fractions.) 2. Find the particular solution to the logistic differential equation with N(0) = No, assuming that No 0 and No # K. 3. Show that if N(0) = 0 or if N(0) = K, then the solution N(t) is constant. Explain in a sentence or two why these cases have to be handled separately from the previous two parts. 4. Sketch the solution from parts (2) and (3) for several different initial conditions No. In particular, plot one solution with No = 0, at least one solution with 0< No < K, one solution with No = K and at least one solution with No > K. You should have a plot with t on the x-axis and N on the y-axis. All of your solutions can go in the same figure. (You can choose any positive numbers for K and r. The graphs will have essentially the same shape for any choice, as long as you choose a reasonable scale for the N and taxes.) 5. Explain in words what happens to the population density over a very long time. What does this tell you about bacteria? Does it seem realistic? Why/why not?