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Math 1303 - Homework 5

Show all needed steps of reasoning and include all needed computer codes and figures!

1. The following are stated.

(a) (8 points) A certain bacterium is known to grow according to the Malthusian model, doubling itself every 4 hours. Find the reproductive rate.

(b) (10 points) Now assume that an initial P0 population of bacteria grows at a rate r, and h individuals are harvested every hour. Write the mathematical model for this situation and solve it.

(c) (9 points) Discuss the faith of the population over the long run (i.e. t → ∞) based on the relationships that can exist between P0, r, h.

(d) (3 points) At what minimal rate can the population be harvested such that the container will not be overwhelmed with bacteria? [Hint: Use your answers in part (c).]

2. (5 points) Fit a linear model ln(P) = ln(P0) + r · (t − t0) through the data points

(ti, ln(Pi)), i = 0, 1, 2, . . . , N

and the exponential model P = P0 · e^r·(t−t0)  through the data points

(ti, Pi), i = 0, 1, 2, . . . , N.

The points are given below and an example is attached.