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[T]

  1. Using a calculator or computer program, find the best-fit cubic curve to the data.
  2. Find the derivative of the position function and explain its physical meaning.
  3. Find the second derivative of the position function and explain its physical meaning.
  4. Using the result from c. explain why a cubic function is not a good choice for this problem.
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The following problems deal with the Holling type I, II, and III equations. These equations describe the ecological event of growth of a predator population given the amount of prey available for consumption.

[T] The Holling type I equation is described by f ( x ) = a x , where x is the amount of prey available and a > 0 is the rate at which the predator meets the prey for consumption.

  1. Graph the Holling type I equation, given a = 0.5 .
  2. Determine the first derivative of the Holling type I equation and explain physically what the derivative implies.
  3. Determine the second derivative of the Holling type I equation and explain physically what the derivative implies.
  4. Using the interpretations from b. and c. explain why the Holling type I equation may not be realistic.

a.
The graph is a straight line drawn through the origin with slope 1/2.
b. f ( x ) = a . The more increase in prey, the more growth for predators. c. f ( x ) = 0 . As the amount of prey increases, the rate at which the predator population growth increases is constant. d. This equation assumes that if there is more prey, the predator is able to increase consumption linearly. This assumption is unphysical because we would expect there to be some saturation point at which there is too much prey for the predator to consume adequately.

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[T] The Holling type II equation is described by f ( x ) = a x n + x , where x is the amount of prey available and a > 0 is the maximum consumption rate of the predator.

  1. Graph the Holling type II equation given a = 0.5 and n = 5 . What are the differences between the Holling type I and II equations?
  2. Take the first derivative of the Holling type II equation and interpret the physical meaning of the derivative.
  3. Show that f ( n ) = 1 2 a and interpret the meaning of the parameter n .
  4. Find and interpret the meaning of the second derivative. What makes the Holling type II function more realistic than the Holling type I function?
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[T] The Holling type III equation is described by f ( x ) = a x 2 n 2 + x 2 , where x is the amount of prey available and a > 0 is the maximum consumption rate of the predator.

  1. Graph the Holling type III equation given a = 0.5 and n = 5 . What are the differences between the Holling type II and III equations?
  2. Take the first derivative of the Holling type III equation and interpret the physical meaning of the derivative.
  3. Find and interpret the meaning of the second derivative (it may help to graph the second derivative).
  4. What additional ecological phenomena does the Holling type III function describe compared with the Holling type II function?

a.
The graph increases from the origin quickly at first and then slowly to (10, 0.4).
b. f ( x ) = 2 a x n 2 ( n 2 + x 2 ) 2 . When the amount of prey increases, the predator growth increases. c. f ( x ) = 2 a n 2 ( n 2 3 x 2 ) ( n 2 + x 2 ) 3 . When the amount of prey is extremely small, the rate at which predator growth is increasing is increasing, but when the amount of prey reaches above a certain threshold, the rate at which predator growth is increasing begins to decrease. d. At lower levels of prey, the prey is more easily able to avoid detection by the predator, so fewer prey individuals are consumed, resulting in less predator growth.

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[T] The populations of the snowshoe hare (in thousands) and the lynx (in hundreds) collected over 7 years from 1937 to 1943 are shown in the following table. The snowshoe hare is the primary prey of the lynx.

Source: http://www.biotopics.co.uk/newgcse/predatorprey.html.
Snowshoe hare and lynx populations
Population of snowshoe hare (thousands) Population of lynx (hundreds)
20 10
55 15
65 55
95 60
  1. Graph the data points and determine which Holling-type function fits the data best.
  2. Using the meanings of the parameters a and n , determine values for those parameters by examining a graph of the data. Recall that n measures what prey value results in the half-maximum of the predator value.
  3. Plot the resulting Holling-type I, II, and III functions on top of the data. Was the result from part a. correct?
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Practice Key Terms 8

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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