Concept explainers
Logistic Growth Suppose a population can be modelled with the logistic function,
a. Find the growth function
b. Find the population and rate of growth of the population after 6 years.
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Chapter 4 Solutions
Calculus For The Life Sciences
- More on the Pacific Sardine This is a continuation of Example 5.1. In this exercise, we explore the Pacific sardine population further, using the model in Example 5.1. a. If the current level of the Pacific sardine population is 50,000 tons, how long will it take for the population to recover to the optimum growth level of 1.2milliontons? Suggestion: One way to solve this is to make a new logistic formula using K2.4, r0.338, and N(0)0.05. b. The value of r used in Example 5.1 ignores the effects of fishing. If fishing mortality is taken into account, then r drops to 0.215 per year with the carrying capacity still at 2.4milliontons. Answer the question in part a using this lower value of r. Note: The population estimate of 50,000 tons and the adjusted value of r are given in the paper by Murphy see footnote 3 on page 347. Murphy points out that factoring in the growth of the competing anchovy population makes the recovery times even longer, and he adds. "It is disconcerting to realize how slowly the population will recover to its level of maximum productivity ... even if fishing stops." Studies to fit a logistic model to the Pacific sardine population have yielded. N=241+239e0.338t where t is measured in years and N is measured in millions of tons of fish. Part 1 What is r for the Pacific sardine? Part 2 According to the logistic model, in the absence of limiting factors, what would be the annual percentage growth rate for the Pacific sardine? Part 3 What is the environmental carrying capacity K? Part 4 What is the optimum yield level? Part 5 Make a graph of N versus t. Part 6 At what time t should the population he harvested? Part 7 What portion of the graph is concave up? What portion is concave down?arrow_forwardModeling Human Height with a Logistic Function A male child is 21inches long at birth and grows to an adult height of 73inches. In this exercise, we make a logistic model of his height as a function of age. a. Use the given information to find K and b for the logistic model. b. Suppose he reaches 95 of his adult height at age 16. Use this information and that from part a to find r. Suggestion: You will need to use either the crossing-graphs method or some algebra involving the logarithm. c. Make a logistic model for his height H, in inches, as a function of his age t, in years. d. According to the logistic model, at what age is he growing the fastest? e. Is your answer to part d consistent with your knowledge of how humans grow?arrow_forwardEastern Pacific Yellowfin Tuna Studies to fit a logistic model to the Eastern Pacific yellowfin tuna population have yielded N=1481+36e2.61t where t is measured in years and N is measured in thousands of tons of fish. a. What is the r value for the Eastern Pacific yellowfin tuna? b. What is the carrying capacity K for the Eastern Pacific yellowfin tuna? c. What is the optimum yield level? d. Use your calculator to graph N versus t. e. At what time was the population growing the most rapidly?arrow_forward
- Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model n(t)=56000.5+27.5e0.044t where t is measured in years. a Find the initial bird population. b Draw a graph of the function n(t). c What size does the population approach as time goes on?arrow_forwardBird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model p(t)=73.26.1+5.9e0.02r Where t is measured in years. Find the initial bird population. Draw a graph of the function n(t) . What size does the population approach as time goes on?arrow_forwardA Population of Foxes A breeding group of foxes is introduced into a protected are and exhibits logistic population growth. After t years, the number of foxes is given by N(t)=37.50.25+0.76t foxes. a. How many foxes were introduced into the protected area? b. Calculate N(5) and explain the meaning of the number you have calculated. c. Explain how the population varies with time. Include in your explanation the average rate of increase over the first 10-year period and the average rate of increase over the second 10-year period. d. Find the carrying capacity for foxes in the protected area. e. As we saw in the discussion of terminal velocity for a skydiver, the question of when the carrying capacity is reached may lead to an involved discussion. We ask the question differently. When is 99 of carrying capacity reached?arrow_forward
- Population The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025. (a) Find the exponential growth or decay model y=aebt or y=aebt for the population of each country by letting t=15 correspond to 2015. Use the model to predict the population of each country in 2035. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y=aebt gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.arrow_forwardLogistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: p(t)=d1+kea Where c, d, and k are positive constants. For a certain fish population in a small pond d=1200, k=11, c=0.2, and t is measured in years. The fish were introduced into the pond at time t=0 . How many fish were originally put in the pond? Find the population after 10, 20, and 30 years. Evaluate p(t) for large values of t. What value does the population approach as t? Does the graph shown confirm your calculations?arrow_forwardSales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4 shows the number of games sold, in thousands, from the years 20002010. a. Let x represent time in years starting with x=1 for the year 2000. Let y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data. b. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.arrow_forward
- Logistic Growth Animal populations are not capable of unrestricted growth becomes of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: P(t)=d1+kect where c, d, and k are positive constants. For a certain fish population is a small pond d=1200,k=11,c=0.2, and t is measured in years. The fish were introduced into the pond at time t=0. a How many fish were originally put in the pond? b Find the population after 10, 20, and 30 years. c Evaluate P(t) for large values of t. What value does the population approach as t? Does the graph show confirm your calculations?arrow_forwardLogistic Population growth the table and scatter plot give the population of black flies in a closed laboratory container over an 18 day period. (a) Use the logistic command on your calculator to find a logistic model for these data. (b) Use the model to estimate the time when there were 400 flies in the containerarrow_forward
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