91Ó°ÊÓ

The total number \(P\) of prey taken by a predator depends on the availability of prey. \(C\). \(S\). Holling proposed a function of the form \(P=c n /(1+d n)\) to model the number of prey taken in certain situations. \({ }^{25}\) Here \(n\) is the density of prey available, and \(c\) and \(d\) are constants that depend on the organisms involved as well as on other environmental features. Holling took data gathered earlier by T. Burnett on the number of sawfly cocoons found by a small wasp parasite at given host density. In one such experiment conducted, Holling found the relationship $$ P=\frac{21.96 n}{1+2.41 n} $$ where \(P\) is the number of cocoons parasitized and \(n\) is the density of cocoons available (measured as number per square inch). a. Draw a graph of \(P\) versus \(n\). Include values of \(n\) up to 2 cocoons per square inch. b. What density of cocoons will ensure that the wasp will find and parasitize 6 of them? c. There is a limit to the number of cocoons that the wasp is able to parasitize no matter how readily available the prey may be. What is this upper limit?

Step by step solution

01

Plot the Function

To draw the graph of \(P = \frac{21.96n}{1 + 2.41n}\), calculate \(P\) for values of \(n\) ranging from 0 to 2. Use a suitable interval (for example, every 0.1) to get points to plot. Calculate a few sample values: \(n=0\), \(P=0\); \(n=1\), \(P \approx 9.11\); \(n=2\), \(P \approx 12.86\). Connect these points on a graph to visualize the relationship.

02

Solve for Specific Prey Number

To find the density \(n\) that results in \(P = 6\), use the given relationship: \(6 = \frac{21.96n}{1 + 2.41n}\). Multiply both sides by \(1 + 2.41n\) and solve for \(n\): \(6 + 14.46n = 21.96n\). Simplify to find \(n = \frac{6}{7.5} \approx 0.8\). Thus, approximately 0.8 cocoons per square inch will ensure 6 are parasitized.

03

Determine the Upper Limit of Prey Count

Determine the upper limit by examining the behavior of the function as \(n\) approaches infinity. The term \(\frac{21.96n}{1 + 2.41n}\) approaches \(\frac{21.96}{2.41} \approx 9.11\) as \(n\) becomes very large. This indicates a maximum number of 9 cocoons that the wasp can parasitize.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Holling's Function

Holling's function is a mathematical model that describes the relationship between the number of prey consumed by a predator and the density of that prey in the environment. This concept is pivotal in understanding predator-prey dynamics in ecological systems.
The general form of the function is given by:

• \(P = \frac{c n}{1 + d n}\)

Here:

• \(P\): the number of prey consumed
• \(n\): the prey density
• \(c\) and \(d\): constants specific to the predator-prey relationship

This function suggests that as prey become more abundant, the rate of consumption by predators increases but eventually reaches a limit due to saturation. This saturation happens because predators can only consume a finite amount of prey over a given time, regardless of availability.

Cocoon Density

Cocoon density is the measure of how many cocoons are present per unit area, such as per square inch. In the case of the Holling's function specific to the experiment conducted by Burnett and analyzed by Holling, the cocoon density serves as \(n\) in the equation:
\[P = \frac{21.96 n}{1 + 2.41 n}\]
Cocoon density is a crucial factor because it directly influences the number of prey (cocoons) that are parasitized by the wasp.
When trying to determine how many cocoons will be parasitized based on density, you need to understand how changes in density affect the outcome predicted by the function.

This variable allows us to predict practical outcomes in natural settings, such as knowing what densities are too low for effective parasitization and which are optimal or saturating.

Graphing Functions

Graphing functions is a visual way to understand how the independent variable (in this case, cocoon density \(n\)) affects the dependent variable \(P\) (number of cocoons parasitized).
To graph the given function \(P = \frac{21.96 n}{1 + 2.41 n}\), you start by calculating \(P\) for a series of \(n\) values, such as 0, 0.5, 1, 1.5, and 2 cocoons per square inch.

This process involves:

• Calculating specific \(P\) values for chosen \(n\)
• Plotting these points on a coordinate plane, with \(n\) as the x-axis and \(P\) as the y-axis
• Connecting the points to form a smooth curve

The shape of this graph typically shows a curve that rises steeply at first and then levels off as it reaches the saturation point. This visual representation makes it easier to identify trends, such as where the maximum parasitization occurs, without using calculations alone.

Maximum Parasitization Limit

In ecological models, the maximum parasitization limit is the theoretical upper limit on how many prey a predator can consume, regardless of prey abundance. For the function examined by Holling:

• \[P = \frac{21.96 n}{1 + 2.41 n}\]

As \(n\) becomes very large (approaches infinity), the denominator \(1 + 2.41 n\) has less impact on the equation, resulting in the function approaching a maximum value:

• The function approaches \(\frac{21.96}{2.41} \approx 9.11\)

This means that, in this model, no matter how abundant the prey, the wasp can only parasitize around 9 cocoons per square inch. This limit arises from the biological constraints the parasite faces, such as time and energy needed to continue parasitizing more prey. This understanding is crucial in ecological management to avoid overestimating the impact predators can have on prey populations.