Recall that a logistic growth model takes the form dP dt = rP(1 − P K) where r is some growth or decay constant, P is the number in the population, and K is the carrying capacity. Suppose data has been collected on the population counts of different types of fish in Horsetooth Lake, and is shown below: Scientists suspect that these populations follow a logistic growth model. They know that Horsetooth's carrying capacity for Walleye is 500 fish, the carrying capacity for Bluegill is 600 fish, and the carrying capacity for small-mouth bass is 1000 fish. (a) Determine the growth/decay constant (r) for one of the species of fish using the data provided. Specify which species you have chosen. (hint: recall the form that the solution to a logistic growth model takes, and use log rules to isolate r!)
![Definite integrals are useful to science because we can often describe natural phenomena using definite integrals. However, integrating functions arising in real-life science can be rather challenging. In this exercise, we see how having only the graph of some functions is enough to answer some questions in science. The physical interpretation of an integral is a prerequisite knowledge you learned in MATH 1013. For a refresher, see the Net Change Theorem in Section 5.4 of the textbook. A group of researchers studies the population of arctic foxes in Northern Canada. Initially, the researchers identified a population of 1,000 arctic foxes. Due to various factors, the population of arctic foxes changes over time. The graph of the birthrate B(t) and the death rate D(t), for the population, are shown below. Time t is measured in years. (1) In a sentence or two, describe the meaning of the function f(t) = t 0 B(x) - D(x)dx in the context of this problem. (2) According the graph, is there a time t between 0 and 20 at which the population is equal to the initial population? If so, provide a rough estimate of this time, and write one or two sentences explaining how you found it. Your estimate should be a single number, not an interval. (3) Recall from Quiz 1 the definition of the average value of a function f(x). The average value of a function f(x) over an interval [a, b] is denoted by favg and is defined to be favg = 1 b - a b a f(x)dx. Suppose you know that the average birth rate over the first 20 years is 90 arctic foxes per year and the average death rate over the first 20 years is 100 arctic foxes per year. What is the population of the arctic foxes at t = 20 ? Show your work.](https://www.doubtrix.com/uploads/editor/1288831183fhExDQGbmw.png)