Figure 11.73 shows the spread of the Code-red computer virus during July 2001. Most of

(a) Show that P = 1/(1 + et ) satisfies the logistic equation dP dt = P(1 P). (b) What is the limiting value of P as t ?

A quantity P satisfies the differential equation dP dt = kP  1 P 100 . Sketch approximate solutions satisfying each of the following initial conditions: (a) P0 = 8 (b) P0 = 70 (c) P0 = 125

A quantity Q satisfies the differential equation dQ dt = kQ(1 0.0004Q). Sketch approximate solutions satisfying each of the following initial conditions: (a) Q0 = 300 (b) Q0 = 1500 (c) Q0 = 3500

A quantity P satisfies the differential equation dP dt = kP  1 P 250 , with k > 0. Sketch a graph of dP/dt as a function of P.

A quantity A satisfies the differential equation dA dt = kA(1 0.0002A), with k > 0. Sketch a graph of dA/dt as a function of A.

Figure 11.69 shows a graph of dP/dt against P for a logistic differential equation. Sketch several solutions of P against t, using different initial conditions. Include a scale on your vertical axis.

Figure 11.70 shows a slope field of a differential equation for a quantity Q growing logistically. Sketch a graph of dQ/dt against Q. Include a scale on the horizontal axis.

(a) On the slope field for dP/dt = 3P 3P2 in Figure 11.71, sketch three solution curves showing different types of behavior for the population, P. (b) Is there a stable value of the population? If so, what is it? (c) Describe the meaning of the shape of the solution curves for the population: Where is P increasing? Decreasing? What happens in the long run? Are there any inflection points? Where? What do they mean for the population? (d) Sketch a graph of dP /dt against P. Where is dP /dt positive? Negative? Zero? Maximum? How do your observations about dP /dt explain the shapes of your solution curves?

Exercises 910 give a graph of dP/dt against P. (a) What are the equilibrium values of P? (b) If P = 500, is dP/dt positive or negative? Is P increasing or decreasing?

Exercises 910 give a graph of dP/dt against P. (a) What are the equilibrium values of P? (b) If P = 500, is dP/dt positive or negative? Is P increasing or decreasing?

For the logistic differential equations in Exercises 1112, (a) Give values for k and for L and interpret the meaning of each in terms of the growth of the quantity P. (b) Give the value of P when the rate of change is at its peak.

For the logistic differential equations in Exercises 1112, (a) Give values for k and for L and interpret the meaning of each in terms of the growth of the quantity P. (b) Give the value of P when the rate of change is at its peak.

In Exercises 1316, give the general solution to the logistic differential equation

In Exercises 1316, give the general solution to the logistic differential equation

In Exercises 1316, give the general solution to the logistic differential equation

In Exercises 1316, give the general solution to the logistic differential equation

In Exercises 1720, give k, L, A, a formula for P as a function of time t, and the time to the peak value of dP/dt.

In Exercises 1720, give k, L, A, a formula for P as a function of time t, and the time to the peak value of dP/dt.

In Exercises 1720, give k, L, A, a formula for P as a function of time t, and the time to the peak value of dP/dt.

In Exercises 1720, give k, L, A, a formula for P as a function of time t, and the time to the peak value of dP/dt.

In Exercises 2122, give the solution to the logistic differential equation with initial condition.

In Exercises 2122, give the solution to the logistic differential equation with initial condition.

A rumor spreads among a group of 400 people. The number of people, N(t), who have heard the rumor by time t in hours since the rumor started is approximated by N(t) = 400 1 + 399e0.4t . (a) Find N(0) and interpret it. (b) How many people will have heard the rumor after 2 hours? After 10 hours? (c) Graph N(t). (d) Approximately how long will it take until half the people have heard the rumor? 399 people? (e) When is the rumor spreading fastest?

The Tojolobal Mayan Indian community in southern Mexico has available a fixed amount of land. The proportion, P, of land in use for farming t years after 1935 is modeled with the logistic function in Figure 11.72:18 P = 1 1+2.968e0.0275t . (a) What proportion of the land was in use for farming in 1935? (b) What is the long-run prediction of this model? (c) When was half the land in use for farming? (d) When is the proportion of land used for farming increasing most rapidly?

A model for the population, P, of carp in a landlocked lake at time t is given by the differential equation dP dt = 0.25P(1 0.0004P). (a) What is the long-term equilibrium population of carp in the lake? (b) A census taken ten years ago found there were 1000 carp in the lake. Estimate the current population. (c) Under a plan to join the lake to a nearby river, the fish will be able to leave the lake. A net loss of 10% of the carp each year is predicted, but the patterns of birth and death are not expected to change. Revise the differential equation to take this into account. Use the revised differential equation to predict the future development of the carp population.

Table 11.7 gives values for a logistic function P = f(t). (a) Estimate the maximum rate of change of P and estimate the value of t when it occurs. (b) If P represents the growth of a population, estimate the carrying capacity of the population.

Figure 11.73 shows the spread of the Code-red computer virus during July 2001. Most of the growth took place starting at midnight on July 19; on July 20, the virus attacked the White House, trying (unsuccessfully) to knock its site off-line. The number of computers infected by the virus was a logistic function of time. (a) Estimate the limiting value of f(t) as t increased. What does this limiting value represent in terms of Code-red? (b) Estimate the value of t at which f(t)=0. Estimate the value of n at this time. (c) What does the answer to part (b) tell us about Codered? (d) How are the answers to parts (a) and (b) related?

According to an article in The New York Times, 19 pigweed has acquired resistance to the weedkiller Roundup. Let N be the number of acres, in millions, where Roundup-resistant pigweed is found. Suppose the relative growth rate, (1/N)dN/ dt, was 15% when N = 5 and 14.5% when N = 10. Assuming the relative growth rate is a linear function of N, write a differential equation to model N as a function of time, and predict how many acres will eventually be afflicted before the spread of Roundup-resistant pigweed halts.

We define P to be the total oil production worldwide since 1859 in billions of barrels. In 1993, annual world oil production was 22.0 billion barrels and the total production was P = 724 billion barrels. In 2008, annual production was 26.9 billion barrels and the total production was P = 1100 billion barrels. Let t be time in years since 1993. (a) Estimate the rate of production, dP/dt, for 1993 and 2008. (b) Estimate the relative growth rate, (1/P)(dP/dt), for 1993 and 2008. (c) Find an equation for the relative growth rate, (1/P)(dP/dt), as a function of P, assuming that the function is linear. (d) Assuming that P increases logistically and that all oil in the ground will ultimately be extracted, estimate the world oil reserves in 1859 to the nearest billion barrels. (e) Write and solve the logistic differential equation modeling P

In Problem 29 we used a logistic function to model P, total world oil production since 1859, as a function of time, t, in years since 1993. Use this function to answer the following questions: (a) When does peak annual world oil production occur? (b) Geologists have estimated world oil reserves to be as high as 3500 billion barrels.21 When does peak world oil production occur with this assumption? (Assume k and P0 are unchanged

As in Problem 29, let P be total world oil production since 1859. In 1998, annual world production was 24.4 billion barrels and total production was P = 841 billion barrels. In 2003, annual production was 25.3 billion barrels and total production was P = 964 billion barrels. (a) Graph dP/dt versus P from Problem 29 and show the data for 1998 and 2003. How well does the model fit the data? (b) Graph the logistic function modeling worldwide oil production (P versus t) from Problem 29 and show the data for 1998 and 2003. How well does the model fit the data?

Use the logistic function obtained in Problem 29 to model the growth of P, the total oil produced worldwide in billions of barrels since 1859: (a) Find the projected value of P for 2010. (b) Estimate the annual world oil production during 2010. (c) How much oil is projected to remain in the ground in 2010? (d) Compare the projected production in part (b) with the actual figure of 26.9 billion barrels.

With P, the total oil produced worldwide since 1859, in billions of barrels, modeled as a function of time t in years since 1993 as in Problem 29: (a) Predict the total quantity of oil produced by 2020. (b) In what year does the model predict that only 300 billion barrels remain

The total number of people infected with a virus often grows like a logistic curve. Suppose that time, t, is in weeks and that 10 people originally have the virus. In the early stages, the number of people infected is increasing exponentially with k = 1.78. In the long run, 5000 people are infected. (a) Find a logistic function to model the number of people infected. (b) Sketch a graph of your answer to part (a). (c) Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point?

Policy makers are interested in modeling the spread of information through a population. For example, agricultural ministries use models to understand the spread of technical innovations or new seed types through their countries. Two models, based on how the information is spread, follow. Assume the population is of a constant size M. (a) If the information is spread by mass media (TV, radio, newspapers), the rate at which information is spread is believed to be proportional to the number of people not having the information at that time. Write a differential equation for the number of people having the information by time t. Sketch a solution assuming that no one (except the mass media) has the information initially. (b) If the information is spread by word of mouth, the rate of spread of information is believed to be proportional to the product of the number of people who know and the number who dont. Write a differential equation for the number of people having the information by time t. Sketch the solution for the cases in which (i) No one (ii) 5% of the population (iii) 75% of the population knows initially. In each case, when is the information spreading fastest?

In the 1930s, the Soviet ecologist G. F. Gause22 studied the population growth of yeast. Fit a logistic curve, dP /dt = kP(1 P/L), to his data below using the method outlined below. Time (hours) 0 10 18 23 34 42 47 Yeast pop 0.37 8.87 10.66 12.50 13.27 12.87 12.70 (a) Plot the data and use it to estimate (by eye) the carrying capacity, L. (b) Use the first two pieces of data in the table and your value for L to estimate k. (c) On the same axes as the data points, use your values for k and L to sketch the solution curve P = L 1 + Aekt where A = L P

The population data from another experiment on yeast by the ecologist G. F. Gause is given. Time (hours) 0 13 32 56 77 101 125 Yeast pop 1.00 1.70 2.73 4.87 5.67 5.80 5.83 (a) Do you think the population is growing exponentially or logistically? Give reasons for your answer. (b) Estimate the value of k (for either model) from the first two pieces of data. If you chose a logistic model in part (a), estimate the carrying capacity, L, from the data. (c) Sketch the data and the approximate growth curve given by the parameters you estimated.

The spread of a non-fatal disease through a population of fixed size M can be modeled as follows. The rate that healthy people are infected, in people per day, is proportional to the product of the numbers of healthy and infected people. The constant of proportionality is 0.01/M. The rate of recovery, in people per day, is 0.009 times the number of people infected. Construct a differential equation that models the spread of the disease. Assuming that initially only a small number of people are infected, plot a graph of the number of infected people against time. What

Many organ pipes in old European churches are made of tin. In cold climates such pipes can be affected with tin pest, when the tin becomes brittle and crumbles into a gray powder. This transformation can appear to take place very suddenly because the presence of the gray powder encourages the reaction to proceed. The rate of the reaction is proportional to the product of the amount of tin left and the quantity of gray powder, p, present at time t. Assume that when metallic tin is converted to gray powder, its mass does not change. (a) Write a differential equation for p. Let the total quantity of metallic tin present originally be B. (b) Sketch a graph of the solution p = f(t) if there is a small quantity of powder initially. How much metallic tin has crumbled when it is crumbling fastest?

The logistic model can be applied to a renewable resource that is harvested, like fish. If a fraction c of the population is harvested each year, we have dP dt = kP  1 P L  cP. Figure 11.74 assumes c

Federal or state agencies control hunting and fishing by setting a quota on how many animals can be harvested each season. Determining the appropriate quota means achieving a balance between environmental concerns and the interests of hunters and fishers. For example, when a June 8, 2007 decision by the Delaware Superior Court invalidated a two-year moratorium on catching horseshoe crabs, the Delaware Department of Natural Resources and Environmental Control imposed instead an annual quota of 100,000 on male horseshoe crabs. Environmentalists argued this would exacerbate a decrease in the protected Red Knot bird population that depends on the crab for food. For a population P that satisfies the logistic model with harvesting, dP dt = kP  1 P L  H, show that the quota, H, must satisfy H kL/4, or else the population P may die out. (In fact, H should be kept much less than kL/4 to be safe.)

In Problems 4244, explain what is wrong with the statement

In Problems 4244, explain what is wrong with the statement

In Problems 4244, explain what is wrong with the statement

In Problems 4548, give an example of:

In Problems 4548, give an example of:

In Problems 4548, give an example of:

In Problems 4548, give an example of:

There is a solution curve for the logistic differential equation dP/dt = P(2 P) that goes through the points (0, 1) and (1, 3).

For any positive values of the constant k and any positive values of the initial value P(0), the solution to the differential equation dP/dt = kP(L P) has limiting value L as t