91Ó°ÊÓ

In Exercises 3 and 4 you are given the phase curve associated with a system of predator-prey equations, where \(x(t)\) denotes the prey (caribou) population, in hundreds, and \(y(t)\) denotes the predator (wolves) population, in tens, at time t. (a) Describe how each population changes over time t starting from \(t=0 .\) (b) Make a rough sketch of the graphs of \(x\) and \(y\) as a function of \(t\) on the same set of axes.

Consider the predator-prey equations $$ \begin{array}{l} \frac{d x}{d t}=a x-b x y \\ \frac{d y}{d t}=-r y+s x y \end{array} $$ where \(a, b, r\), and \(s\) are positive constants.

In the Lotka-Volterra model it was assumed that an unlimited amount of food was available to the prey. In a situation in which there is a finite amount of natural resources available to the prey, the Lotka-Volterra model can be modified to reflect this situation. Consider the following system of differential equations: $$ \begin{array}{l} \frac{d x}{d t}=k x\left(1-\frac{x}{L}\right)-a x y \\ \frac{d y}{d t}=-b y+c x y \end{array} $$ where \(x(t)\) and \(y(t)\) represent the populations of prey and predators, respectively, and \(a, b, c, k\), and \(L\) are positive constants. a. Describe what happens to the prey population in the absence of predators. b. Describe what happens to the predator population in the absence of prey. c. Find all the equilibrium points and explain their significance.

Gompertz differential equation, a model for restricted population growth, is obtained by modifying the logistic differential equation and is given by $$ \frac{d P}{d t}=c P \ln \left(\frac{L}{P}\right) $$ where \(c\) is a constant and \(L\) is the carrying capacity of the environment. a. Find the equilibrium solution of the differential equation. b. Illustrate graphically the solutions of the equation with initial conditions \(P(0)=P_{0}\), where (i) \(P_{0}>L\), (ii) \(P_{0}=L\), and (iii) \(0

Refer to Exercise \(20 .\) A population of 20 goldfish was introduced into a pond that has an estimated carrying capacity of 200 fish. After 1 month, the population of goldfish had grown to \(80 .\) If the pattern of growth of the population followed the Gompertz curve, how many goldfish were in the pond after 3 months?

Gompertz Growth Curves The differential equation \(P^{\prime}=P(a-b \ln P)\), where \(a\) and \(b\) are constants, is called a Gompertz differential equation. This differential equation occurs in the study of population growth and the growth of tumors. a. Take \(a=b=1\) in the Gompertz differential equation, and use a CAS to draw a direction field for the differential equation. b. Use the direction field of part (a) to sketch the approximate curves for solutions satisfying the initial conditions \(P(0)=1\) and \(P(0)=4 .\) c. What can you say about \(P(t)\) as \(t\) tends to infinity? If the limit exists, what is its approximate value?

Mixture Problem A tank initially holds 40 gal of pure water. Brine that contains \(2 \mathrm{lb}\) of salt per gallon enters the tank at the rate of \(1.5 \mathrm{gal} / \mathrm{min}\), and the well-stirred mixture leaves at the rate of \(2 \mathrm{gal} / \mathrm{min}\). a. Find the amount of salt in the tank at time \(t\). b. Find the amount of salt in the tank after \(20 \mathrm{~min}\). c. Find the amount of salt when the tank holds 20 gal of brine. d. Find the maximum amount of salt present.

Growth of Bacteria The population of bacteria in a culture grows at a rate that is proportional to the number present. Initially, there are 600 bacteria, and after \(3 \mathrm{hr}\) there are \(10.000\) bacteria. a. What is the number of bacteria after \(t\) hr? b. What is the number of bacteria after \(5 \mathrm{hr}\) ? c. When will the number of bacteria reach 24,000 ?

Parachute Jump A skydiver and his equipment have a combined weight of \(192 \mathrm{lb}\). At the instant that his parachute is deployed, he is traveling vertically downward at a speed of \(112 \mathrm{ft} / \mathrm{sec}\). Assume that air resistance is proportional to the instantaneous velocity with a constant of proportionality of \(k=12\). Determine the position and velocity of the skydiver \(t\) sec after his parachute is deployed. What is his limiting velocity?

Lambert's Law of Absorption According to Lambert's Law of Absorption, the percentage of incident light \(L\), absorbed in passing through a thin layer of material \(x\), is proportional to the thickness of the material. For a certain material, if \(\frac{1}{2}\) in. of the material reduces the light to half of its intensity, how much additional material is needed to reduce the intensity to one fourth of its initial value?