Figure 4.1 A great tit is a species of bird whose foraging behavior was studied by biologist Richard Cowie and whose behavior can be predicted by optimal foraging models.

Preview

“‘If one way be better than another, that you may be sure is nature's way.’ Aristotle clearly stated the basic premise of optimization in biology, yet it was almost 2,000 years before the power of this idea was appreciated. The essence of optimization is to calculate the most efficient solution to a given problem, and then to test the prediction. The concept has already revolutionized some aspects of biology, but it has the potential for much wider application.”

William Sutherland, on “The best solution” in Nature (2005) 435:569

One of the central ideas in physics, chemistry, and biology is that processes act to optimize some physically or biologically meaningful quantity. For example, from physics we know that light in a vacuum travels along a path that is the shortest distance between two points (taking into account that gravity “bends” space), and from biochemistry we know that proteins fold in a way that minimizes the energy of their constituent amino acid configuration.

Differential calculus is an important tool for analyzing optimization (maximization or minimization). In this chapter we show how optimization applies to various biological problems and processes. Before we do this, however, we study how calculus can be used to construct the graphs of a variety of functions; in particular, we identify where the graph has turning points corresponding to local minimum or maximum values. We then develop procedures for modeling and solving optimization problems, including how to draw the best-fitting line through data plotted on a graph. After considering a number of biological applications, we study how calculus provides insight into dynamic processes such as the growth of populations and the spread of deleterious or mutant genes (e.g., the gene that causes sickle cell anemia) within populations. We end the chapter with an application of difference equations that is at the heart of many numerical methods used by current technologies for finding solutions to nonlinear equations.

4.1 Graphing Using Calculus

In this section, we combine many of the tools that we have studied so far (e.g., limits involving infinity, first and second derivatives) to graph a function. In graphing a function, envision walking along the graph and indicating all the highlights of your walk. For instance, vertical asymptotes are places with such a rapid ascent or descent that they make climbing Mount Everest seem like a stroll in a park. Horizontal asymptotes are places where the landscape levels out into a never-ending plain. Where the derivative is positive, the graph is ascending; and where the derivative is negative, the graph is descending. Switches in the sign of the slope correspond to either hilltops or valley bottoms along your walk. On ascents where the second derivative is positive, the walk is getting harder. On descents where the second derivative is negative, the descent becomes faster.

Properties of graphs

When graphing the function y = f(x) by hand, follow these procedures to find the highlights of the function shape:

Vertical asymptotes: Determine at what points the function is not well defined (e.g., division by zero). At each of these points, say x = a, evaluate the one-sided limits, f(x) and f(x), to determine what the graph looks like near x = a. If either of these one-sided limits is +∞ or −∞, then there is a vertical asymptote at this point.

Intervals of increase and decrease: Compute the first derivative f′(x) of f(x) and determine on which intervals f′(x) > 0 and on which intervals f′(x) < 0. On these intervals, f is increasing and decreasing, respectively.

Intervals of concavity: Compute the second derivative f″(x) and determine on which intervals f″(x) > 0 and f″(x) < 0. On these intervals, f is concave up and concave down, respectively.

The x and y intercepts Find the x intercepts (i.e., where f(x) = 0) and the y intercept (i.e., y = f(0)). These points help pin down the placement of the graph.

After identifying these highlights of a function, you are ready to sketch the function.

Many functions have no horizontal asymptotes. Nonetheless, understanding the limits as x approaches ±∞ may help graph the function.

Sometimes just using limits and first derivatives is enough to get a good sense of the graph.

In the previous example, we note that because the function has a maximum above zero and then approaches 0 as x → ∞, the graph necessarily has a point of inflection where C″(t) switches from being positive to negative. In Problem Set 4.1, we ask you to discuss the sign of the second derivative and to solve for the value of x where this point of inflection occurs.

Graphing families of functions

The shape of some functions, such as f(x) = 3x + a, does not depend in any critical sense on the value of the parameter a. All a does is move the line of slope 3 up and down the xy plane. As we see in this subsection, however, in more complicated functions the value of a parameter can have a surprising effect, and we can use calculus to discover such effects.

Level 1 DRILL PROBLEMS

In Problems 1 to 14, graph the functions by hand by finding asymptotes and using first and second derivatives. Compare your graphs to what you get using technology.

In Problems 15 to 20, graph the families of functions by finding asymptotes and using first and second derivatives. In particular, determine how the graph of the functions depends on the parameter a > 0.

In Problems 21 and 22, sketch the graph of a function with the given properties.

21. x = 2, x = −2 are vertical asymptotes f is increasing for 0 < x < 2 and x > 2 f is decreasing for x < −2 and −2 < x < 0 graph is concave down on (−∞, −2) and (2, ∞) intercepts are (−1, 0), (−3, 0), (3, 0) and (1, 0)

22. y = 1, y = −1 are horizontal asymptotes f is increasing for x < − and for x > f is decreasing for −1 < x < 1 graph is concave down for x < −1 and for 0 < x < 1 graph is concave up for x > 1 and for −1 < x < 0

Level 2 APPLIED AND THEORY PROBLEMS

23. Consider the graph of y = ax² + bx + c for constants a, b, and c. Use second derivatives to determine what happens to the graph as a changes.

24. Consider the graph of y = . Use limits and first derivatives to determine how the shape of this curve depends on the parameter a.

25. In Example 6, we saw that the dose-response curve

has asymptotes y = a and y = b, respectively, as x approaches ±∞. Find the second derivative y″(x) and discuss its properties including an equation that can be used to identify any points of inflection, if they exist.

26. In Example 4 of Section 2.7, we consider patterns of local species richness of ants along an elevational gradient. A function that best fits the data is

where x is elevation measured in kilometers and S is the number of species. Plot this function using information about first derivatives.

27. In Example 6 of Section 1.5, we develop the Michaelis-Menton model for the rate at which an organism consumes its resource. For bacterial populations in the ocean, this model was given by

where x is the concentration of glucose (micrograms per liter) in the environment. Use asymptotes and first derivatives to sketch this function by hand.

28. In Example 5 of Section 2.4 we found that the rate at which wolves kill moose can be modeled by

where x is measured in number of moose per square kilometer. Use asymptotes and first derivatives to sketch this function.

29. In Problem 42 in Problem Section 2.4, we examined how wolf densities in North America depend on moose densities. We found that the following function provides a good fit to the data:

where x is number of moose per square kilometer.

1. Find the horizontal and vertical asymptotes.
2. Determine on which intervals f is increasing and decreasing.
3. Determine on which intervals f is concave up and concave down.
4. Use the information from parts a–c to sketch the graph of f(x).

30. Two mathematicians, W. O. Kermack and A. G. McKendrick, showed that the weekly mortality rate during the outbreak of the plague in Bombay (1905–1906) is reasonably well described by the function

where t is measured in weeks. Sketch this function using information about asymptotes and first derivatives. Recall that

31. In Example 4 we modeled the rate at which acetaminophen diffuses from the stomach and intestines to the blood stream using this equation:

Calculate the second derivative and discuss its behavior. Identify if the function C(t) has any points of inflection for t > 0.

32. In an experiment, a microbiologist introduces a toxin into a bacterial colony growing in an agar dish. The data on the area of the dish covered by living colony members at time t minutes after the introduction of the toxin are given by this equation:

Sketch the graph of A(t) showing its salient features.

33. Let f be a function that represents the weight of a fish at age t. Write a function that satisfies the following properties.

• The weight of the fish at birth must be positive.
• As the fish ages, the weight increases at decreasing rate.
• No fish can grow bigger than 2 kg.

34. Aerobic rate is the rate of a person's oxygen consumption and is sometimes modeled by the function A defined by

for x ≥ 10. Graph this function.

35. As a project for their mathematical biology class, three college students developed a model of how acetaminophen levels varied in the blood stream of a child after taking a dose of 325 milligrams. Using FDA data, they found that

where t is hours after taking the dose.

1. Use information about asymptotes and first derivatives to sketch this function.
2. Discuss the meaning of your graph. In particular, address when the maximum concentration is achieved and what the maximum concentration is.

36. A naturalist at an animal sanctuary determined that the function

provides a good measure of the number of animals in the sanctuary that are x years old. Sketch the graph of f for x > 0.

4.2 Getting Extreme

When viewing the graph of a function as a landscape of hilltops and valley bottoms, each top and each bottom corresponds to a place where the function in question has an extremum. Methods to identify extrema play an important role in applications. For instance, if the function of interest represents how profits due to harvesting a crop depend on the amount of seeds planted, then the farmer would like to know how many seeds per acre yield the greatest profits. In other words, the farmer would like to identify the largest hilltop of the function. Alternatively, if a northwestern crow minimizes the amount of energy required to break whelk shells, then the optimal behavior for a crow corresponds to the deepest valley of a function. In this section, we develop methods to find these hilltops and valleys.

Local extrema

The previous example suggests that either extrema occur at end points of the domain, points where f is not differentiable, or points where the derivative of f equals zero. The following theorem verifies these observations.

Fermat's theorem shows that we can find possible local maxima and local minima by finding points where f′(x) = 0 or f′ is not defined. Such points have a special name.

Although all local extrema are critical values, not all critical values are local extrema. Consider, for example, y = x³ plotted on the left for −1 ≤ x ≤ 1.

The derivative is = 3x². Hence, x = 0 is the only critical point. However as y = x³ increases over all the reals, x = 0 is neither a local maximum nor a local minimum.

Example 2 illustrates one method for identifying local maxima and local minima; this method is called the first derivative test.

Another possibility for identifying local maxima and local minima is using the second derivative. Suppose f has a critical point at x = a and has second-order derivatives at x = a. In Section 3.6, we saw that a second-order approximation of f(x) is given by

Since a is a critical point, f′(a) = 0 and the second-order approximation reduces to

Figure 4.5 The second derivative test determines whether a critical point x = a where f′(a) = 0 is a local minimum (a) or a local maximum (b).

Provided that f″(a) ≠ 0, the graph of this approximation is given by a parabola whose vertex is at x = a. Furthermore, if f″ (a) > 0, then this parabola is facing up, which suggests that there is a local minimum at x = a as shown in Figure 4.5a. Alternatively, if f″(a) < 0, then this parabola is facing down, which suggests that there is a local maximum at x = a as shown in Figure 4.5b. What is suggested by the second approximation can be proven, thereby providing an alternative test for classifying extrema.

Global extrema

Often in applied problems, we want to find the largest or small value of a function on its domain.

Example 5 illustrates that global extrema may occur at critical points or end points for a continuous function on a closed interval. Thus, we have the following procedure for finding global extrema.

Figure 4.7 Codling moth search period.

Data Source: P. L. Shaffer, and H. J. Gold, “A Simulation Model of Population Dynamics of the Codling Moth, Cydia pomonella,” Ecological Modeling 30 (1985): 247–274.

In many problems we need to find the global extrema on open intervals, half-closed intervals, or intervals involving infinity. For each of these cases, we deal with the limits as we approach the end points of the intervals, as illustrated with the open interval method. (The other cases appear as exercises in Problem Set 4.2.)

Level 1 DRILL PROBLEMS

In Problems 1 to 4, identify the local and global extrema.

In Problems 5 to 12, find the critical points and use the first derivative test to classify them.

In Problems 13 to 16, find the critical points and use the second derivative test to classify them.

In Problems 17 to 20, use the closed interval method to find the global extrema on the indicated intervals.

17. f(x) = x² − 4x + 2 on [0, 3]

18. f(x) = x³ − 12x + 2 on [−3, 3]

19. f(x) = x + on [0.1, 10]

20. f(x) = xe^−x on [0, 100]

In Problems 21 to 24, use the open interval method to find the global extrema on the indicated intervals.

21. f(x) = x² − 4x + 2 on (−∞, ∞)

22. f(x) = x³ − 12x + 2 on (0, ∞)

23. f(x) = x + on (0, ∞)

24. f(x) = xe^−x on (−∞, ∞)

25. Let f be continuous on the half-open interval [a, b) with b possibly equal to +∞. Devise a method to find the global extrema of f on this interval.

26. Let f be continuous on the half-open interval (a, b] with a possibly equal to −∞. Devise a method to find the global extrema of f on this interval.

In Problems 27 to 30, use the half-open interval methods you developed in Problems 25 and 26 to find the global extrema on the indicated intervals.

27. f(x) = x² − 4x + 2 on [0, ∞)

28. f(x) = x³ − 12x + 2 on [1, 10)

29. f(x) = x + on [−1, ∞)

30. f(x) = xe^−x on (−∞, −1]

Level 2 APPLIED AND THEORY PROBLEMS

31. Let f be defined on (a, b) and c (a, b). Prove that if x = c is a local maximum and f is differentiable at x = c, then f′(c) = 0.

32. In Example 4 of Section 1.2, we examined how CO₂ concentrations (in ppm) have varied from 1974 to 1985. Using linear and periodic functions, we found that the following function gives an excellent fit to the data:

where x is months after April 1974. Using the closed interval method, find the global maximum and minimum CO₂ levels on the interval [12, 24].

33. In the previous problem, use the closed interval method and find the global maximum and minimum CO₂ levels between April 2000 and April 2001.

34. A close relative of the codling moth is the pea moth, Cydia nigricana, which is a pest of cultivated and garden peas in several European countries. If its search period in one of the regions where it is a pest is given by the function

then graph s(T) using information about the first derivative over the domain 20 ≤ T ≤ 30. Be sure that your graph indicates the largest and smallest value of s over this interval.

35. In Problem 30 of Problem Set 4.1, we saw that the weekly mortality rate during the outbreak of the plague in Bombay (1905–1906) can be reasonably well described by the function

where t is measured in weeks. Find the global maximum of this function. Recall that

36. Some species of plants (e.g., bamboo) flower once and then die. A well-known formula for the average growth rate r of a semelparous species (a species that breeds only once) that breeds at age x is

where s(x) represents the proportion of plants that survive from germination to age x, n(x) is the number of seeds produced at age x, and p is the proportion of seeds that germinate.

1. Find the age of reproduction that maximizes r in terms of the parameters a, b, c, and p where

   

   and

   

   0 < c < 1.

2. Sketch the graph of y = r(x) for the case where a = 0.2, b = 3, c = 0.8, and p = 0.5.

37. The production of blood cells plays an important role in medical research involving leukemia and other so-called dynamical diseases. In 1977, a mathematical model was developed by A. Lasota. (See W. B. Gearhart and M. Martelli, “A Blood Cell Population Model, Dynamical Diseases, and Chaos,” in UMAP Modules 1990:Tools for Teaching [Arlington, MA: Consortium for Mathematics and Its Applications, 1991].) The model involved the cell production function

where A, s, and r are positive constants and x is the number of granulocytes (a type of white blood cell) present.

1. Find the granulocyte level x that maximizes the production function P. How do you know it is a maximum?
2. Graph this function.

38. When you cough, the radius of your trachea (windpipe) decreases, thereby affecting the speed of the air in the trachea. If r is the normal radius of the trachea, the relationship between the speed S of the air and the radius r of the trachea during a cough is given by a function of the form

where a is a positive constant. (Philip M. Tuchinsky, “The Human Cough,” UMAP Modules 1976: Tools for Teaching [Lexington, MA: Consortium for Mathematics and Its Applications, 1977].) Find the radius r for which the speed of the air is the greatest.

39. Research indicates that the power P required by a bird to maintain flight is given by the formula

where v is the relative speed of the bird, w is its weight, ρ is the density of air, and S and A are constants associated with the bird's size and shape. (See C. J. Pennycuick, “The Mechanics of Bird Migration,” IBIS III (1969): 525–556.) What speed will minimize the power? You may assume that w, ρ, S, and A are all positive and constant.

40. An epidemic spreads through a community in such a way that t weeks after its outbreak, the number of residents who have been infected is given by a function of the form

where A is the total number of susceptible residents. Show that the epidemic is spreading most rapidly when half the susceptible residents have been infected.

4.3 Optimization in Biology

One of the most important applications of calculus to biology in particular, and science and technology in general, involves finding the extrema of functions. Consider, for example, the problem of determining the most effective treatment regimen for a malignant tumor using chemo- or radiation therapy. Such treatments are toxic to the body, so physicians want to prescribe the minimum dosage that will do the job. Typically, a single treatment will not destroy the tumor. Instead, the tumor will initially shrink in size after treatment and sometime later will begin to regrow. Ideally, therapy should be readministered immediately—when the tumor is at its smallest—before this regrowth phase. Using calculus in conjunction with tumor growth data, we can estimate the time when therapy should be reapplied. As another example, think of a farmer planting a corn crop. The farmer might be interested in knowing the planting density of seeds that would maximize his or her profit. To find out, the farmer could formulate a function that describes how profits depend on planting density, and maximizing this function. In this section, we consider these problems as well as the behavior of dogs fetching balls, sustainable harvesting of arctic fin whales, and vascular branching. More examples appear in the problem set. We end this section with applications to finding best-fitting functions to empirically collected data.

Steps to solve an optimization problem

Optimization problems typically require that we develop an appropriate model for the problem being considered, then analyze this model to find the optimal solution for the problem. To tackle these problems successfully, use the following steps:

1. Read, understand, and visualize. Take the time to carefully read the problem so that you fully understand what is being asked. In particular, ask yourself: What am I trying to maximize or minimize? What information am I given? Is it sufficient to solve the problem at hand? When appropriate, draw a picture or figure that summarizes the problem.
2. Identify key variables and quantities. Ask yourself: What are the important quantities in the problem? What quantity is being optimized? This is the dependent variable. Which of the variables is the one whose value I can control to obtain my optimal solution? This is the independent variable. What additional quantities presented in the problem do I need to obtain the sought after relationship between the dependent and independent variable? Associate units with each of these variables.
3. Write the function. In this step, you need to determine how the dependent variable is determined by the quantities that you identified in the previous step. Think carefully about this crucial step. Make sure that units on both sides of your equation agree.
4. Optimize. Determine whether you need to minimize or maximize the function and over what interval you need to perform the optimization. To find the optimal value, it suffices to find the critical points and evaluate the function at these critical points and at the end points of the interval. Whichever of these values is the largest (smallest) yields the maximum (minimum).
5. Interpret your answer. Interpret the results of your optimization. Ask yourself whether your answer makes sense. If not, check your work.

In this next example, we demonstrate how these steps are applied to the problem of determining the density of seedlings that a farmer needs to plant to optimize profits from a particular crop. In the remaining examples in this section, we do not stress the steps involved, but these steps are implicit in our approach. Remember to you these steps whenever you get stuck in solving an optimization problem. The examples relate to behavior, physiology, resource management, and fitting functions to data. You won't necessarily have time to study them all, so you can pick and choose those of greatest interest to you.

In many problems in population biology, a variable x is used to represent the density (or number) of individuals in a population and the function G(x) is used to represent the population growth rate. For instance, for the discrete logistic equation presented in Example 7 of Section 2.5, we modeled the growth using the function.

where r > 0 has the interpretation of the maximum per capita growth rate and K > 0 has the interpretation of the environmental carrying capacity. Here, we use the function G(x) to determine the optimal rate at which a population of whales should be harvested, if not illegal to do so.

The Arctic fin whale Balaenoptera physalus, which at fifty to seventy tons for adults of both sexes is second in size to the blue whale, was a highly desirable catch during the whaling heydays of the nineteenth and twentieth centuries. As many as 30,000 individuals were slaughtered each year from 1935 to 1965. This level of exploitation could not be sustained for long, so today population levels are estimated to be an order of magnitude (i.e., a factor of 10) below historical highs of around a half million individuals. Some individuals are still taken each year for purposes of subsistence by aboriginal people in Greenland. A moratorium on whale hunting is needed, however, to allow this species to recover to levels where the populations can be safely exploited on a sustainable basis.

Figure 4.9 Arctic fin whale.

Sometimes when solving a problem it is useful to sketch a figure, as illustrated in the next example.

So what was the outcome of Professor Pennings' experiment? When Professor Pennings measured the point at which Elvis entered the water, he found that Elvis ran 15 − x = 14.1 meters along the shore (i.e., x = 0.9). Does this dog know calculus? Well, he could have been lucky on this one throw. So Professor Pennings performed thirty-five throws, with the ball landing different distances d from the shoreline. Professor Pennings measured the point x where Elvis entered the water on each throw. In the problems at the end of this section, you will be asked to show that the optimal place to enter the water as a function of the distance d the ball lands from the shore is

A scatter plot of the data and the line is shown in Figure 4.11. This figure illustrates that Elvis is on average acting pretty optimal, which is quite remarkable.

Figure 4.11 Scatter plot of distance of ball from shore (in the horizontal direction) and Elvis's point of entry 15 − d in the water (in the vertical direction). The best-fitting line x = 0.144d passes through the center of the scatter plot.

The next example concerns treating tumors with chemotherapy (Figure 4.12), which was mentioned in the introduction to this section. Refer to Problems 44 and 45 in Problem Set 1.4 and Problem Set 44 in Problem Set 1.5 for further insights into modeling tumor growth.

Figure 4.12 Chemotherapy is a treatment for cancer patients in which powerful drugs are used to kill cancer cells. Often it is used in conjunction with other treatments, including radiation therapy and surgery.

In the next example, we consider the vascular system, which consists of arteries and veins that branch in different directions to pump blood through all parts of the body. Ideally, the body is designed to minimize the amount of energy it expends in pumping blood. According to one of Poiseuille's laws, the resistance blood experiences by traveling down the center of a blood vessel with radius r and length L is proportional to

Without loss of generality, we assume that this proportionality constant equals one, and use this law to determine optimal branching angles in the vascular system of animals.

Regression

Regression, loosely speaking, is the process of drawing a graph through a set of data, where the graph represents the information in the data after process and measurement errors have been “averaged out.” The term regression dates back to Sir Francis Galton's work on measuring heritable traits in humans, such as height, and his publication of this work in 1886 under the title “Regression Towards Mediocrity in Hereditary Status” (Journal of the Anthropological Institute of Great Britain and Ireland, 15 (1886): 246–263). To simplify our notation in developing some of the ideas associated with regression theory, we introduce the summation notation, a notation that we revisit in Chapter 5 when developing integral calculus.

The following three properties of this summation notation are easily verified.

In our development of regression, we use the following definition that we informally introduced in Section 1.2 and depicted in Figure 1.14 for the case of linear regression. Here we provide a definition for any function f(x), not just the linear function f(x) = mx + c.

Recall Example 5 of Section 1.7. We plotted the data in Figure 1.46 of that example on the decline of the Glued gene in a species of fruit fly. The data, which exhibit an exponential decline from 0.5 to 0.0036 over six generations, are reported in Table 4.2. To model how rapidly the Glued gene is lost from the experimental fruit fly population in each generation, we used the function 0.5e^−rt for some appropriately chosen constant r > 0. Taking the logarithm of this equation yields the linear function ln 0.5 − rt of t. In the next example, we use this observation to find the best-fitting line to the logarithmically transformed data.

Table 4.2 Changes in the frequency x of the Glued gene in a species of fruit fly over seven (generation 0 is also experimentally generated) experimental generations t(values to two significant digits)

Level 1 DRILL PROBLEMS

1. In Example 1, the selling price of corn was $1.50 per bushel and the seeds cost $3 per thousand seeds. Determine the density of seeds that maximize profit if the selling price and cost are both doubled.

2. In Example 1, determine the density of seeds that maximize profit if the selling price is $5 per bushel and the seeds cost $2 per thousand seeds.

3. In Example 1, determine the density of seeds that maximize profit if the selling price is $2.20 per bushel and the seeds cost $2.50 per thousand seeds.

4. In Example 2, we estimated that the maximum per capita growth rate of the Arctic fin whale is r = 0.08. Suppose a better estimate is r = 0.1. Determine the maximum sustainable harvesting rate for this value of r.

5. In Example 2, we estimated that the long-term abundance of the Arctic fin whales in the absence of harvesting the Arctic fin whale is K = 500,000. Suppose a better estimate is K = 400,000. Determine the maximum sustainable harvesting rate for this value of K.

6. In a species of fish the growth rate function is given by G(x) = 1.4x(1 − x/K), where K = 5 million metric tons (i.e., the population of fish is measured in metric tons rather than number of individuals). If the harvest rate is a function of the harvesting effort h and the total amount of fish x, that is H = hx, find the harvesting effort value h that corresponds to the maximum sustainable yield.

7. In a species of fish, the growth rate function is given by G(x) = 2.1x(1 − x/K), where K = 8 million metric tons. If the harvest rate is H = hx, find the harvesting effort value h that corresponds to the maximum sustainable yield.

In Problems 8 to 12, find the optimal angle for the following vascular branching problems, as considered in Example 5.

8. A larger artery has a radius of 0.05 mm, and a smaller artery of radius 0.025 mm branches from the larger artery with branching angle θ.

9. A larger artery has a radius of 0.06 mm, and a smaller artery of radius 0.04 mm branches from the larger artery with branching angle θ.

10. The radius of the main blood vessel is r₁ = 2 and the radius of the branching vessel is r₂ = 1.

11. The radius of the main blood vessel is r₁ = 4 and the radius of the branching vessel is r₂ = 3.

12. In a general case, the radius of the main blood vessel is r₁ and the radius of the branching vessel is r₂. Assume that r₁ > r₂.

13. In Example 3, calculate at what point x along the shore Elvis should enter the water if the distance of the ball from the shore is 20 meters rather than 6.

14. In Example 3, calculate at what point x along the shore Elvis should enter the water if the distance of the ball from the shore is 10 meters rather than 6.

15. In Example 3, calculate at what point x along the shore Elvis should enter the water if the distance of the ball from the shore is d meters.

In Problems 16 to 22, calculate the residual sum-of-squares of the listed function through this data set:

Draw a graph of the function and the data on the same plot. In Problems 21 to 22, plot the residual sum-of-squares graph as a function of the free parameter and selected range for this parameter. Note that you are not being asked to fit the exponential functions in the relevant examples below on a semi-log plot, but to use the function itself directly in the calculations of the residuals with respect to the data.

16. f(x) = 2x

17. f(x) = 2x + 1

18. f(x) = 3x − 1

19. f(x) = 0.4e^0.8x

20. f(x) = 0.2e^0.9x

21. f(x) = mx, 0 ≤ m ≤ 4

22. f(x) = 0.2e^rx, 0 ≤ r ≤ 1

Level 2 APPLIED AND THEORY PROBLEMS

23. Find a general formula for which Example 3 is a specific case that describes how to calculate at what point x along the shore Elvis should enter the water if the distance of the ball from the shore is d meters (rather than 6) and the point on the shore to which this distance d holds is k meters (rather than 15) from where Tim is standing.

24. In a species of fish, the growth rate function is given by G(x) = 1.5x(1 − x/K), where K = 6,000 metric tons (i.e., the population of fish, x, is measured in metric tons rather than number of individuals). The price a fisher can get is p = $600 per metric ton. If the amount the fisher can harvest is determined by the function H = hx, where each unit of h costs the fisher c = $100, what is the maximum amount of money the fisher can expect to make on a sustainable basis? (Hint: The fisher's sustainable income is given by pH − ch, where H is a sustainable harvesting rate.)

25. In the tumor growth study described in Example 4, the tumor consisted of proliferating cells (clonogenic cells) that grew exponentially with a doubling time of approximately 2.9 days. Suppose that each mouse was given a dose of 25 mg/kg of cisplatin per treatment with the following results: At the time of the therapy, the average tumor size was approximately 0.44 cm³. After treatment, 99.73% of the proliferating cells became quiescent cells and decayed with a half-life of approximately 6.24 days.

1. Write a function V(t) that represents the size of the tumor (proliferating plus quiescent cells) t days after therapy.
2. Determine at what point in time the tumor starts to regrow and therapy should be readministered.
3. Compare your answer to the data figure in Example 4.

26. In a follow-up study to the tumor growth study described in Example 4, mice were infected with a relatively aggressive line of proliferating clonogenic cells that grew exponentially with a doubling time of approximately 1.8 days. Each mouse was given a dose of 20 mg/kg of cisplatin per treatment with the following results: At the time of the therapy, the average tumor size was approximately 0.6cm³. After treatment, 99.10% of the proliferating cells became quiescent cells and decayed with a half-life of approximately 4.4 days.

1. Write a function V(t) that represents the size of the tumor (proliferating plus quiescent cells) t days after therapy.
2. Determine at what point in time the tumor starts to regrow and therapy should be readministered.

27. In certain tissues, cells exist in the shape of circular cylinders. Suppose such a cylinder has radius r and height h. If the volume is fixed (say, at v), find the value of r that minimizes the total surface area (S = 2πr² + 2πrh) of the cell.

28. Farmers regularly use fertilizers to enhance the productivity of their crops. Determining the appropriate amount of fertilizer to use requires balancing the costs of fertilization with the increases in yield. In a 2004 study, Baker and colleagues studied the relationship between nitrogen fertilization and yield of hard red spring wheat. (See Dustin A. Baker, Douglas L. Young, David R. Huggins, and William L. Pan, “Economically Optimal Nitrogen Fertilization for Yield and Protein in Hard Red Spring Wheat,” Agronomy Journal 96 (2004): 116–123.) For conventional tillage practices in eastern Washington in the late 1980s, they found that the grain yield (in kilograms per hectare) as a function of nitrogen (in kilograms per hectare) is well approximated by

These researchers suggested that a high selling price for wheat would be $191.1/kg and low cost for nitrogen would be $0.49/kg. Determine the amount of nitrogen that maximizes profits per hectare.

29. Baker and colleagues suggested that a low selling price for wheat would be $139.65/kg and a high cost for nitrogen would be $0.71/kg. Using the same yield function as in the previous problem, determine the amount of nitrogen that maximizes profits per hectare.

30. If the effects of density dependence in a whale population set in less rapidly closer to the final carrying capacity, K, than the logistic equation used in Example 2, then the equation should be replaced by an asymmetric growth model

for some α (0, 1). For the case α = 0.5, calculate the stock level x that provides the maximum sustainable.

31. If the effects of density dependence in a whale population set in less rapidly or more rapidly closer to the final carrying capacity, K, than the logistic equation used in Example 2, then the equation should be replaced by an asymmetric growth model

For α > 0, r > 0, and K > 0, calculate the stock level x that provides the maximum sustainable yield. Discuss whether rapid onset of density dependence (i.e., large α) or gradual onset of density dependence (i.e., small α) leads to larger sustainable yields.

32. During the winter, a species of bird migrates from the coast of a mainland to an island 500 miles southeast. If the energy the bird requires to fly one mile over the water is twice more than the amount of energy it requires to fly over the land, determine what path the species should fly to minimize the amount of energy used.

33. The Statue of Liberty is 92 meters high, including the 46 meter pedestal upon which it stands. How far from the base should an individual stand to ensure that the view angle, θ, is maximized?

34. In the northeastern part of Sweet Water County, a large dam is being constructed on the Shuga River to produce hydroelectricity (i.e., the generation of electricity through water pressure). An important part of this project is running power lines from the power stations at the downstream side of the dam to various parts of the county, including Pickle City, the largest city in the county. On the recommendation of a number of other counties, Sweet Water County officials have hired you as a consultant to resolve cost issues for running these power lines.

County officials have informed you that the Shuga River runs due east, and on its southern side lies an expanse of federally protected wetlands. Pickle City lies several miles to the south of these wetlands, as shown in the map below.

The federally protected wetlands are divided into two regions. In the western region, county officials expect that due to federal regulations it will cost 40% more to run conduit here than it does through non-wetland ground. The eastern region of the wetlands is a habitat for the endangered Brown Barbaloots. Consequently, federal law prevents the county from running conduits through this region.

As the county officials intend to submit a budget proposal for the project to the county council in the next week, they would like you to determine the path from the power station to downtown Pickle City that minimizes the cost of installing the conduit.

35. An oil spill has fouled 200 miles of Pacific shoreline. The oil company responsible has been given fourteen days to clean up the shoreline, after which a fine will be levied in the amount of $10,000/day. The local cleanup crew can scrub five miles of beach per day at a cost of $500/day. Additional crews can be brought in at a cost of $18,000, plus $800/day for each crew. Determine how many additional crews should be brought in to minimize the total cost to the company and how much the cleanup will cost.

36. Consider a spherical cell with radius r. Assume that the cell gains energy at a rate proportional to its surface area (i.e., nutrients diffusing in from outside of the cell) and that the cell loses energy at a rate proportional to its volume (i.e., all parts of the cell are using energy). If the cell is trying to maximize its net gain of energy, determine the optimal radius of the cell. Note: Your final expression will depend on your proportionality constants.

37. Consider a cylindrical cell with radius r and height r/2. Assume that the cell gains energy at a rate proportional to its surface area (i.e., nutrients diffusing in from outside of the cell) and that the cell loses energy at a rate proportional to its volume (i.e., all parts of the cell are using energy). If the cell is trying to maximize its net gain of energy, determine the optimal value of r. Note: Your final expression will depend on your proportionality constants.

38. A dune buggy is in the desert at a point A located 40 km from a point B, which lies on a long, straight road, as shown in Figure 4.14. The driver can travel at 45 km/hour on the desert and 75 km/hour on the road. The driver will win a prize if she arrives at the finish line at point D, 50km from B, in 85 minutes or less. Set up and analyze a model to help her decide on a route to minimize the time of travel. Does she win the prize?

Figure 4.14 Path traveled by a dune buggy.

39. The question of whether an optimal body size exists for different kinds of animals is one that is of great interest to biologists. The reproductive power P of an individual can be modeled, following the ideas of ecologist James H. Brown (see his book Macro-ecology, University of Chicago Press, 1995), as the harmonic mean of two limiting rates. The harmonic mean of two numbers a and b is the reciprocals of the average of the inverses of the two numbers: 1/(1/a + 1/b) = ab/(a + b). The two rates are a per-unit-mass rate R₁ at which individuals acquire resources, and a per-unit-mass rate R₂ at which individuals convert those resources into new individuals; that is,

Assuming both R₁ and R₂ are the following allometric functions of body mass measure in kilograms

find the body mass M that maximizes the reproductive power P, and show that this extremum is a maximum for the case b₁ = 0.75 and b₂ = −0.25.

40. Suppose that we express the two rate functions in Problem 39 using the general form R₁ = c₁M^b1 and R₂ = c₂M^−b2. Show in this case that the maximum body size is given by the expression

41. In Example 6, we determined the slope a of the linear component of a type I functional response for the rate (cells/hour) at which copepods feed on diatoms as a function of diatom density (cells/milliliter). Suppose now that only the following partial set of data (Table 4.3) is available for estimating the parameter a.

Table 4.3 Diatom densities x (cells/milliliter) and copepod feeding rates y (cells/hour)

1. Find the slope a of the line f(x) = ax through the origin that minimizes the residual sum-of-squares through the data.
2. Plot these data and the best-fit line on the same graph.

42. In Table 1.4, the CO₂ concentrations at the Mauna Loa Observatory in Hawaii are listed by month, starting at month 1 (May 1974) until month 140 (December 1985). The data for the month of May, for the twelve periods spanning May 1974 to May 1985, are shown in Table 4.4.

Table 4.4 CO₂x(cell/milliliter) in year t

1. Find the slope m of the regression line f(x) = mx + 333.2 through the point (t, x_t) = (0, 333.2) that minimizes the residual sum-of-squares through the remaining eleven data points.
2. Plot these data and the regression line on the same graph.

43. In Example 7, we found the value of the decay rate parameter r > 0 using linear regression to fit a semi-log plot of the frequency of Glued genes in an experimental fruit fly population. This experiment was repeated by the same group of researchers and the following data were obtained:

Use these data to find the value r of the best-fitting exponential decay function x_t = 0.59e^−rt through the starting point (t, ln x_t) = (0, −0.53). Present the solution on a semi-log plot together with a plot of the data.

44. In Section 1.4, we presented the following data on the growth of the United States from 1815 until 1895.

Year	Population (in millions)
1815	8.3
1825	11.0
1835	14.7
1845	19.7
1855	26.7
1865	35.2
1875	44.4
1885	55.9
1895	68.9

Use linear regression on a semi-log transformation of the above data to find the best estimate of the annual growth rate r > 0 in the populationmodel x(t) = 8.3e^rt (million individuals) for t [0, 80] (years), with t = 0 corresponding to the year 1815.

4.4 Decisions and Optimization

Optimal decisions

The behavior of animals has been honed by natural selection to maximize the reproductive potential of individuals. Thus, from an individual point of view, individuals should act in ways that maximize the number of offspring they rear to sexual maturity. This number is referred to as an individual's fitness. From a genetic point of view, a gene encoding for a behavior that maximizes an individual's fitness will have a greater representation in the gene pool of future generations than a gene that encodes for a behavior that is detrimental to an individual's fitness (e.g., a gene that causes an individual to be excessively reckless, making it likely that the individual will die before reaching sexual maturity). Theories of optimal behavior are based on the premise that organisms maximize their fitness by behaving in a certain way. Using models, researchers can develop hypotheses about these optimal behaviors. These hypotheses can be tested experimentally or through comparative studies.

In our first example in this section, we obtain insights into the reason why Northwestern crows consistently drop whelks from a specific height to break them open. If the crows fly too low, the shells require too many drops to get them to break open. If they fly too high, the crows waste energy. Assuming that crows have evolved to minimize energy expenditures, scientists might be interested in testing this hypothesis by formulating a suitable function to minimize. This function would characterize the number of of drops, and hence work, required to break open a whelk as a function of the height from which the shell is dropped. In addition to modeling the dropping behavior of Northwestern crows, this section investigates optimal foraging in a patchy environment, optimal timing of seed production, and optimal time to harvest crops. In addition to these examples, we present a key theorem called the marginal value theorem, which has applications to problems maximizing or minimizing average rates of change.

The Northwestern crow, illustrated in Figure 4.15, feeds on whelks, a type of mollusk. To get the meat from inside the whelk's shell, individual crows lift whelks into the air and drop them onto a rock to break open the shell. The biologist Reto Zach (see Example 6 of Section 3.2) observed that individual crows typically drop the shells from a height of five meters. This led him to ask this question: Does the height from which Northwestern crows drop whelk shells minimize the amount energy required to open a shell?

Figure 4.15 A Northwestern crow.

The next example explores the optimal time for a plant to produce seeds. This is just one in a class of optimal time-to-reproduction problems. Others include the optimal time for a honey bee colony to swarm and the optimal time for salmon to return from the ocean to lay eggs upriver.

Optimal foraging and marginal value

Very often, food is not distributed homogeneously over the environment; rather, it occurs in discrete patches in the environment. For fruit bats, a patch may correspond to a fruit tree or a stand of fruit trees. For a hummingbird, which feeds on the nectar of flowers, a patch may correspond to a single flower or a field of flowers. In optimal foraging theory, we want to know how long an animal should continue to collect resources in a patch when it has the choice of traveling to another resource-rich patch. The question of when to leave a patch as resources in the patch are being depleted is known as the optimal residence time problem.

In Example 3, the average rate of change was being maximized over a time interval. A fundamental result for problems of this type is the marginal value theorem.

The marginal value theorem has a simple graphical interpretation, which is explored in the next example.

Our final example in this section introduces the concept of discounting when optimizing a sustainable stream of revenue calculated for all time in the future. Discounting arises if someone promises to pay you D dollars next year, and the current interest rate compounded continuously is r%; then this person should be willing to receive De^−r/100 dollars now. Namely, if this person took the De^−r/100 dollars now and invested it, then a year later the investment would yield e^r/100 dollars for each dollar invested. Hence, a year later the person would have e^r/100De^−r/100 = D dollars. As a result of this reasoning, economists use the discount factor e^−δt, where δ = r/100 to reduce D dollars needed at time t in the future to their current value De^−δt dollars now.

Level 1 DRILL PROBLEMS

In Problems 1 to 6, the amount of energy a hummingbird gains after remaining in a patch for t seconds is given. For each problem, find how long a hummingbird should stay in a patch if it wants to maximize its average energy intake rate.

1. The travel time between patches is 15 seconds and

2. The travel time between patches is 5 seconds and

3. The travel time between patches is 10 seconds and

4. The travel time between patches is 5 seconds and

5. The travel time between patches is 5 seconds and

6. The travel time between patches is 10 seconds and

In Problems 7 to 10, rework Example 5 with the given graphs.

Assume the house martins in Example 3 can choose between two patches. In Problems 11 to 16, the time to fly to a patch and the energy yield as a function of patch residence time (t minutes) are given for two patches. If an individual can visit only one patch and wants to maximize the average amount of calories it receives, then which patch of each pair should it choose?

11. B(t) = calories with a travel time of 2 minutes or B(t) = calories with a travel time of 3 minutes.

12. B(t) = calories with a travel time of 1 minute or B(t) = calories with a travel time of 2 minutes.

13. B(t) = calories with a travel time of 3 minutes or B(t) = calories with a travel time of 2 minutes.

14. B(t) = calories with a travel time of 2 minutes or B(t) = calories with a travel time of 3 minutes.

15. B(t) = calories with a travel time of 2 minutes or B(t) calories with a travel time of 3 minutes.

16. B(t) = calories with a travel time of 2 minutes or B(t) = calories with a travel time of 15 seconds.

In Example 5 (optimal time to harvest), assume that the profit function P(t) has the form specified in Problems 17 to 22. For these profit functions, find the optimal age at which to harvest the stands of trees to maximize profit where t is measured in decades.

Level 2 APPLIED AND THEORY PROBLEMS

27. At the National Council of Teachers of Mathematics (NCTM) illuminations website, students are encouraged to collect data on how many drops are required to break a blanched peanut in two pieces. The sample data provided at this website are shown in the following graph.

The data can be modeled by the function

where h is the height in centimeters. Suppose that the “peanut hummingbird” collects peanuts and wants to minimize the amount of work required to break a peanut into two halves. Determine the height which minimizes the amount of work to break open the peanuts.

28. In Example 4, we found how the optimal residence time in a patch for a great tit depended on the travel time between patches. Although our prediction described the data reasonably well, more than half of the data points lay above the optimal curve. Cowie proposed that part of the reason for this result was that the birds expend energy traveling between patches and searching for food within a patch. In this problem, determine how these expenditures of energy influence the optimal residence time. Let

denote the amount of energy gained by a bird after residing in a patch for t seconds. Assume that the bird requires T seconds to travel the patch. Cowie found that great tits expend approximately 0.697 calories per second while traveling between patches and expend approximately 0.155 calories per second while searching for food in a patch.

1. Write a function V(t) that represents the average gain in energy in a patch after residing there for t ≥ 0 seconds.
2. Use the marginal value theorem to find an expression relating the optimal residence time t to the travel time T.
3. Compare your solution to the solution found in Example 4.

29. Suppose the crop developed by plant geneticists, as discussed in Example 2 — that is, the weight of the crop t days after planting satisfies the growth equation w (t) = 5 + 400t − t²—is grown in a location that is relatively pest free, so that the proportion of germinating seeds is

The crop, however, must be harvested on or before the first frosty day of fall. Suppose the crop has relative value 1 when harvested at its optimum time of maturity, as represented by the day T on which the yield Y = a w(t)P(t) is maximized and that this value is reduced by 10t_e%, where t_e is the number of days prior to T that harvest actually occurs, so that the relative crop value is 0 if harvested at time T − 10 or earlier.

1. What is the value of T?
2. If the expected number of days t_s in the growing season—that is the number of frost-free days plus 1—is equally likely to fall on any day from day 165 to day 190, then what is the expected value of the harvest in any year?

30. When a codling moth larva hatches from its egg, it goes looking for an apple. The period between hatching and finding an apple is called the search period. The search period S seems to be a function of the temperature, as shown in Table 4.5.

Table 4.5 Search period for the codling moth

Temperature	S, in days
20°C	0.129
21°C	0.122
22°C	0.116
23°C	0.112
24°C	0.109
25°C	0.106
26°C	0.105
27°C	0.104
28°C	0.104
29°C	0.105
30°C	0.106

Source: P. L. Shaffer and H. J. Gold, 1985. “A simulation model of population dynamics of the codling moth, Cydia pomonella” Ecological Modeling 30:247–274.

Following the lead of Shaffer and Gold (see Section 4.2, Example 8), find 1/S for each data value and then use technology to fit a quadratic function to the data. Find the largest and smallest value of this fitted function S.

31. In a plantation of a particular species of trees, a forest economist estimated the number of board feet that can be harvested as a function of the age of the plantation. Data are given in Table 4.6. By using your technology to fit a quadratic function to the data, estimate at what age the plantation should be harvested to maximize the yield of board per acre.

Table 4.6 Harvest yield for a lumber crop

Age (years)	Yield (board feet per acre)
15	6013
20	7021
25	8793
30	9411
35	9786
40	9958
45	9921
50	9766

32. By using your technology to fit a cubic equation to the data in Problem 31, find the age in [15, 50] at which the plantation represented by the data should be harvested to maximize the yield.

33. By using your technology to fit a quartic equation to the data in Problem 31, find the age in [15, 50] at which the plantation represented by this data should be harvested to maximize the yield.

34. By using your technology to fit a quintic equation to the data in Problem 31, find the age in [15, 50] at which the plantation represented by the data should be harvested to maximize the yield.

4.5 Linearization and Difference Equations

As we have seen in earlier chapters, difference equations x_n+1 = f(x_n) are useful for describing biological dynamics. The simplest dynamics occur at an equilibrium, because by definition, these equilibria are the solutions of the difference equation that remain constant for all time. Specifically, if a given first value x₀ satisfies x₀ = f(x₀), then our difference equation implies x₁ = x₀. Repeated application results in x_n+1 = x_n = ··· = x₀ for all n > 0.

While equilibria may be easily identified, as discussed in Section 1.7, by solving the equation x = f(x), their biological relevance depends on their stability. Many biological systems, when perturbed, naturally return to their equilibrium state. The temperature of the human body is a case in point. If a person's temperature is perturbed because of an infection, it returns to its equilibrium value of 98.6°F once he or she is well again. Not all equilibria, however, are stable. If we stand up a six-month-old child, the child may stay upright for a second or two, but until the child is around a year old, he or she will soon fall over. Standing vertically without feedback control from muscles constantly moving to correct the tendency to fall over is an unstable situation.

Thus, when a biological system is perturbed away from equilibrium, it may do one of two things. A system may return to the equilibrium state, in which case the equilibrium is considered stable. Alternatively, even if the perturbation is small, a system may continue to drift away from the equilibrium. In this case, the equilibrium is unstable. In this section, we make the notion of stability precise and provide a simple algebraic method for checking stability—a method that relies on linearizing the difference equation near the equilibrium. These ideas and methods are applied to models of population growth and population genetics.

We conclude the section by considering another application of linearization and difference equations, namely, numerically solving for the roots of a nonlinear equation. This numerical method is an important alternative to the bisection method presented in Example 10 of Section 2.3.

Equilibrium stability

We begin with the following example, which motivates the notion of a stable equilibrium.

Example 1 illustrates that some solutions starting near the equilibria approach the equilibrium, while other solutions starting near an equilibrium move away. These observations suggest the following definitions.

Stated more simply, stability of an equilibrium means that if the solution starts near the equilibrium, then it remains nears the equilibrium and asymptotically approaches the equilibrium. Alternatively, solutions starting near (but not at) an unstable equilibrium eventually move further away from the unstable equilibrium.

Stability via linearization

Stability of an equilibrium can be verified directly using the definition, but this method can be challenging. To make things easier, we take advantage of linearization and our work in Example 3.

Suppose x* is an equilibrium of x_n+1 = f(x_n) and f is differentiable at x*. If we approximate f by its tangent line at x = x*, we get

Using this linear approximation and setting r = f′(x*), we get

Equivalently,

Using the change of variables y_n = x_n − x*, we get

Example 3 suggests that if |r| < 1 and x₀ is sufficiently close to x*, then we expect that y_n approaches zero at a geometric rate. Equivalently, since we defined y_n = x_n − x*, it follows that x_n approaches x* at a geometric rate. Alternatively, if |r| > 1 and x₀ is sufficiently close to x*, then we expect that y_n increases initially at a geometric rate. Equivalently, x_n initially moves away from x* at a geometric rate. As it turns out, all of these statements hold provided that x_n is sufficiently close to x*, as the following theorem states.

Stability of monotone difference equations

A special, and important, class of difference equations x_n+1 = f(x_n) was introduced in Section 2.5. For these difference equations, f is a continuous and increasing function over some domain of interest. Within this domain, solutions to this difference equation are monotone (i.e., either increasing for all n or decreasing for all n). As a consequence of this monotonicity, it is possible to provide a simple graphical approach to stability for these difference equations.

Combined with the monotone convergence theorem in Section 2.5, this stability theorem allows us to determine the fate of solutions to difference equations where f is a continuous, increasing function.

As we saw in Example 5 of Section 1.7, we can construct models that allow us to trace the fate of alleles that code for genes affecting the biological fitness (i.e., the ability to survive and reproduce) of individuals. Recalling our discussion in Section 1.7 regarding genetic models of diploid organisms, we consider an allele that codes for a particular trait. If the frequency of this allele in the population is x [0, 1], then a well-known model describing how the frequency of this trait changes from generation n to generation n + 1 in a very large (essentially infinite) randomly mating population is

where w₁ and w₂ are the fitness of individuals who, respectively, have two or no copies, relative to individuals who have only one copy, of the allele in question. Referring back to the equation given in Example 5 of Section 1.7 regarding the spread of a deleterious mutant allele, we see that the equation is the same as the above equation for the special case w₁ = 0 and w₂ = 1. This case is equivalent to the statement that the allele a in question is recessive (heterozygous Aa individuals are not affected) and lethal (aa individuals die before reproducing). Despite the drastic effect of this lethal allele a, we found that it can take a very long time for it to be eliminated from the population. In the next example, we consider a variant of this model in which the allele that is lethal in the homozygous state actually confers a benefit on an individual when combined with the other allele (i.e., when in the heterozygous state).

Newton's method

The final application of linearization to difference equations is to illustrate the inner workings of an algorithm called Newton's method. This method is used to find the roots of nonlinear algebraic equations of the form f(x) = 0 that are too difficult or impossible to solve algebraically. The algorithm is based on the Newton-Raphson difference equation, which we now describe. Suppose our initial guess for the solution of f(x) = 0 is x = x₀. Assuming that this guess is not the solution, we need to find an improved guess for the root. Since the nonlinear function in question is too hard to manipulate by hand, we consider the linear approximation of y = f(x) at x = x₀:

To get our next guess, x₁, for the solution to f(x) = 0, we set x = x₁ and y = 0 in the linear approximation and solve for x₁:

To get the next guess, x₂, we can proceed similarly to get the equation

Proceeding inductively, we get the following difference equation:

This difference equation is illustrated in Figure 4.21. In this figure, r is a root of the equation f(x) = 0.

Figure 4.21 Graphical representation of Newton's method.

One of the key requirements of the method is to start with a reasonable guess x₀ for the root x* because the closer x₀ is to x* the more likely the solution will converge to x*. The following theorem implies that if the sequence converges, then it converges to a root.

When applying Newton's method, we choose a number > 0 that determines the allowable tolerance for estimated solutions. Given an appropriate initial guess, x₀, we iteratively compute x_n until |f(x_n)| < . This procedure is shown in the following flowchart.

Implementation of Newton's method for finding roots is widespread, as a quick search of the Web will reveal. Several websites will turn up that allow you to input a function, an initial condition, and the number of iterations, and you will obtain the corresponding sequence from Newton's method.

Newton's method may not converge to a solution, as shown by the following example.

Level 1 DRILL PROBLEMS

In Problems 1 to 4, find the equilibria of x_n+1 = f(x_n) and determine their stability using cobwebbing.

In Problems 5 to 10, find the equilibria of the difference equation. Moreover, use the definitions of an unstable equilibrium and a stable equilibrium to determine their stability.

In Problems 11 to 16, find the equilibria of the difference equation. Moreover, use linearization to determine their stability.

17. Consider the following alternative formulation of the logistic difference equation, which is different from the formulation presented in Example 1: x_n+1 = rx_n(1 − x_n/100) with r > 0.

1. Find the equilibria.
2. Determine under what conditions the origin is stable.
3. Determine under what conditions the nonzero equilibrium is positive.
4. Determine under what conditions the nonzero equilibrium is stable.

18. Consider the logistic difference equation x_n+1 = rx_n(1 − x_n/50) with r > 0.

1. Find the equilibria.
2. Determine under what conditions the origin is stable.
3. Determine under what conditions the nonzero equilibrium is positive.
4. Determine under what conditions the nonzero equilibrium is stable.

19. Consider the Beverton-Holt difference equation x_n+1 = with r > 0.

1. Find the equilibria.
2. Determine under what conditions the origin is stable.
3. Determine under what conditions the nonzero equilibrium is positive.
4. Determine under what conditions the nonzero equilibrium is stable.

20. Consider the Beverton-Holt difference equation x_n+1 = with r > 0.

1. Find the equilibria.
2. Determine under what conditions the origin is stable.
3. Determine under what conditions the nonzero equilibrium is positive.
4. Determine under what conditions the nonzero equilibrium is stable.

Following the approach laid out in Example 7 (i.e., graphing and using Theorem 4.4) investigate the fate of an allele in a large, randomly mating population when the fitnesses of individuals with two and zero copies of the allele relative to those that have one copy are given in Problems 21 to 24.

21. w₁ = 1/2 and w₂ = 1/2

22. w₁ = 2 and w₂ = 1

23. w₁ = 1/2 and w₂ = 2

24. w₁ = 2 and w₂ = 2

Use Newton's method to estimate a root of the equations in Problems 25 to 32. Use x₀ as a starting value and iterate twenty times.

25. x² − 2 = 0, x₀ = 1

26. x² + 2 = 0, x₀ = 1

27. x³ − x + 1 = 0, x₀ = −1

28. x⁴ + 2x − 1 = 1, x₀ = 1

29. cos x = x, x₀ = 1

30. sin x + 0.1 = x², x₀ = 0

31. e^x − 5x = 0, x₀ = 0

32. e^x + x = 0, x₀ = −1

33. Let f(x) = −2x⁴ + 3x² +

1. Show that the equation f(x) = 0 has at least two solutions.
2. Use x₀ = 2 and Newton's method to estimate a root of the equation f(x) = 0.
3. Show that Newton's method fails if you choose x₀ = as the initial estimate.

34. Let f(x) = x⁶ − x⁵ + x³ − 3

1. Show that the equation f(x) = 0 has at least two solutions.
2. Use x₀ = 2 and Newton's method to find a root of the equation f(x) = 0.
3. Show that Newton's method fails if you choose x₀ = 0 as the initial estimate.

Level 2 APPLIED AND THEORY PROBLEMS

35. For the beetle species Lasioderma serricorne, Bellows found that the fraction f(x) of eggs surviving as a function of their initial density x is well described by

A graph of this function and the corresponding data are shown below:

If we assume that each adult produces two eggs, then the dynamics of the population is given by

1. Find the equilibria and determine their stability.
2. Simulate the model with x₀ = 0.1

36. For the flour beetle species Tribolium castaneum, Bellows found that the fraction f(x) of eggs surviving as a function of their initial density x is well described by

A graph of this function and the corresponding data are shown below:

If we assume that each adult produces r eggs, then the dynamics of the population is given by

1. Find the equilibria and determine their stability for r = 2, 4, 6.
2. Simulate the model with x₀ = 0.1 for r = 2, 4, 6.

37. Consider the genetic model

Show that the following statements are true:

1. This model has three equilibrium solutions:
2. p = 1 is stable and p = 0 is unstable when w₁ > 1 > w₂ > − 1.
3. (Harder problem) p* as defined in part a is the only stable equilibrium when w₁ < 1 and w₂ < 1 (a condition known as heterozygote superiority).
4. (Harder problem) p* as defined in part a is the only unstable equilibrium when w₁ > 1 and w₂ > 1 (a condition known as inbreeding depression).

38. It can be shown that the volume of a spherical cap is given by

where R is the radius of the sphere and H is the height of the cap, as shown in Figure 4.23.

Figure 4.23 Spherical segment is the portion of a sphere between two parallel planes

If V = 8 and R = 2, use Newton's method to estimate the corresponding H.

39.

The Greek geometer Archimedes is acknowledged to be one of the greatest mathematicians of all time. Ten treatises (as well as traces of some lost works) of Archimedes have survived the rigors of time and are masterpieces of mathematical exposition. In one of these works, On the Sphere and Cylinder, Archimedes asks where a sphere should be cut in order to divide it into two pieces whose volumes have a given ratio.

Show that if a plane at distance d from the center of a sphere with R = 1 divides the sphere into two parts, one with volume five times that of the other, then

where H = 1 − d. Find d by using the Newton-Raphson method to estimate H. (Hint: see Problem 38.)

40. Suppose the plane in Problem 39 is located so that it divides the sphere in the ratio of 1:3. Find an equation for d, and estimate the value of d using Newton's method.

41. In Example 4 of Section 4.3, we considered the growth of a tumor in a mouse after the mouse was given a drug treatment. To model the volume of the tumor, we used the function

where x is measured in days after the drug was administered. Using Newton's method, solve within a tolerance of 0.01 for the time x at which the tumor volume has doubled in volume. For an initial guess, use x = 25 days.

42. In Example 4 of Section 4.3, we considered the growth of a tumor in a mouse after the mouse was given a drug treatment. To model the volume of the tumor, we used the function

where x is measured in days after the drug was administered. Using Newton's method, solve within a tolerance of 0.01 for the time x at which the tumor volume has quadrupled in volume. For an initial guess, use x = 30 days.

43. In Problem 25 in Problem Set 4.3, you found that the volume of a tumor for mice under a different drug regimen was

where x is days after treatment. Using Newton's method, solve within a tolerance of 0.01 for the time x at which tumor volume has regrown to its original volume. For an initial guess, use x = 20 days.

44. In Problem 25 in Problem Set 4.3, you found that the volume of a tumor for mice under a different drug regimen was

where x is days after treatment. Using Newton's method, solve within a tolerance of 0.01 for the time x at which tumor volume has doubled in volume. For an initial guess, use x = 25 days.

45. Show that for different initial values Newton's method converges to a unique solution for the function

but yet converges to one of three solutions for the function

Why is this the case?

46. According to an online article in the New Scientist (Catherine Brahic, “Carbon Emissions Rising Faster Than Ever,” p. 9), recent research suggests that stabilizing carbon dioxide concentrations in the atmosphere at 450 parts per million (ppm) could limit global warming to 2°C. In Section 1.2, we modeled carbon dioxide concentrations in the atmosphere with the following function (which we now present to higher precision to make more transparent the numerical details of the convergence process):

where x is months after April 1974. In Example 11 of Section 2.3, we used the bisection method to estimate the first time that the model predicts carbon dioxide levels of 450 ppm. Use Newton's method to estimate this time with a stopping value of = 0.001.

47. Repeat Problem 46 except estimate the first time until reaching 400 ppm.

1. Use the first derivative test and the second derivative test to find and classify all the extrema of g(x) = x³ − 3x − 4.
2. Find the global maximum and global minimum of f(x) = on [0, 6].
3. Using asymptotes and first derivatives, graph f(x) = by hand and then check it using a calculator.
4. Consider the family of curves

   

   Using calculus, graph the curves for the given values of b.

   1. b = 0
   2. b = 0.05
   3. b = 0.01
   4. b = −0.05
   5. b = −0.1
5. The canopy height (in meters) of a tropical grass may be modeled by (for 0 ≤ t ≤ 30)

   

   where t is the number of days after mowing.

   1. Sketch the graph of h(t).
   2. When was the canopy height growing most rapidly? Least rapidly?
6. Find the value of r in the function f(x) = e^−rx that provides the best fit through a semi-log plot of the points{(0, 1), (1, 0.6), (2, 0.4), (3, 0.3)}.
7. A travel company plans to sponsor a tour to Africa. There will be accommodations for no more than forty people, and the tour will be canceled if no more than ten people book reservations. Based on past experience, the manager determines that if n people book the tour, the profit (in dollars) may be modeled by the function

   

   What is the maximum profit?

8. Two towns, A and B, are 12.0 miles apart and are located 5.0 and 3.0 miles, respectively, from a long, straight highway, as shown in Figure 4.24.

   

   Figure 4.24 Highway construction project.

   A construction company has a contract to build a road from A to the highway and then to B. Analyze a model to determine the length (to the nearest tenth of a mile) of the shortest road that meets these requirements.

9. A particular plant is known to have the following growth and seed production characteristics: At time of planting (t = 0), the seedling has a mass of 3 grams. At time t > 0 days after planting, the seedling has grown into a plant that weighs

   

   grams. The plant has a gene that can be manipulated to control the age t at which the plant matures. At maturity the number of seeds S(t) produced by the plant is given by

   

   A farmer asks a geneticist to genetically engineer a plant line that accounts for the fact that on his farm, because of losses from pests, drought, and disease, a proportion

   

   of germinating seeds can be expected to develop and survive to age t as plants. What age of maturity should the geneticist select for the plants to maximize the seed production of the mature crop for the farmer?

10. If the value of a forest stand (units are dollars per square meter) is given by the function

    

    where t represents the number of years after the stand has been clear-cut, and the discount is 0.02 per year, solve for the optimal rotation period, assuming that V(t) applies every time the stand is clear-cut.

11. On a particular island in the tropics, scientists have determined that individuals who are not protected by the sickle cell allele will have, on average, 20% fewer progeny than individuals who have one sickle cell allele, while individuals who have two sickle cell alleles will not reproduce.
    1. Write a difference equation for the proportion x_n of non-sickle-cell alleles in the population in generation n, assuming that initially x₀ > 0, under assumptions of random mating and random segregation of alleles.
    2. Find the equilibria proportions for x on the interval (0, 1] and determine if they are stable.
    3. Interpret your results.
12. Let C(t) denote the concentration in the blood at time t of a drug injected into the body intramuscularly. In a now classic paper by E. Heinz (“Probleme bei der Diffusion kleiner Substanzmengen inner-halb des menschlichen Körpers,” Biochem. Z, 319 (1949): 482–492), the concentration was modeled by the function

    

    where a, b (with b > a), and k are positive constants that depend on the drug. At what time does the largest concentration occur? What happens to the concentration as t → +∞?

13. Consider a bird that has arrived at a wooded patch with two trees. If the bird spends x minutes foraging for insects on the first tree, she gains E₁(x) = 200(1 − e^−x) Calories from insects. If the bird spends x minutes on the second tree, she gains E₂(x) = 100(1 − e^−x) Calories of insects. Assuming the bird has five minutes to spend in the patch, determine the time she should spend on each tree to optimize her energy intake.
14. For the flour beetle species Tribolium confusum, Bellows found that the fraction f(x) of eggs surviving as a function of their initial density x is well described by

    

    A graph of this function and the corresponding data are shown below:

    

    If we assume that each adult produces r eggs, then the dynamics of the population is given by

    

    1. Find the equilibria and determine their stability for r = 2, 4, 6.
    2. Simulate the model with x₀ = 0.1 for r = 2, 4, 6.
15. A model for the population growth rate of Eastern Pacific yellowfin tuna is given by

    

    where N is measured in thousands of tons, t is measured in years, and θ > 0 is a parameter that determines the strength of density dependence. Find the population size at the maximum sustainable harvesting rate. Discuss what effectθhas on this population size.

16. The energy gain from nectar for a hummingbird flying at t = 0 to a flower is shown in the graph below:

    

    Ignoring the cost of flying constantly, find the time at which the hummingbird's energy gain per unit time is maximal and, consequently, the bird should fly to another flower. Discuss what effect including a fixed energetic cost per unit time would have on your answer.

17. A simple model of a biochemical switch is given by

    

    where x_n is the concentration of a biochemical at time n.

    1. Find the equilibria of this model.
    2. Verify that f(x) = is an increasing function of x for x > 0.
    3. Discuss what you can say about all nonnegative solutions to this difference equation.
18. Public awareness of a new drug is modeled by

    

    where t is the number of months after FDA approval and P(t) is the fraction of people who are aware of the drug and its possible uses.

    1. Find the critical points for P(t).
    2. Sketch the graph of P(t).
    3. At what time, t, during the time interval 0 ≤ t ≤ 36 is P(t) the largest?
19. Suppose that systolic blood pressure of a patient t years old is modeled by

    

    for 0 ≤ t ≤ 60, where P(t) is measured in millimeters of mercury.

    1. Sketch the graph of P(t).
    2. At what rate is P(t) increasing at age t?
20. During the time period 1905–1920, hunters virtually wiped out all large predators on the Kaibab Plateau near the Grand Canyon in northern Arizona. This, in turn, resulted in a rapid increase in the deer population P(t), until food supplies were exhausted and famine led to a steep decline in P(t). A study of this ecological disaster determined that during the time period 1905–1920, the rate of change of the population, P′(t), could be modeled by the function

    

    0 ≤ t ≤ 20, where t is the number of years after the base year of 1905.

    1. In what year during this period was the deer population the largest?
    2. In what year did the rate of growth P′(t) begin to decline?