The amount of vegetation eaten in a day by a grazing animal is a function of the amount V of food available (measured as biomass in units such as pounds per acre). This relationship is called the functional response. If there is little vegetation available, the daily intake will be small, sincethe animal will have difficulty finding and eating the food. As the food biomass increases, so does the daily intake. Clearly, though, there is a limit to the amount the animal will eat, regardless of the amount of food available. This maximum amount eaten is the satiation level.In addition to the kangaroos, the major grazing mammals of Australia include merino sheep and rabbits. For sheep the functional response isS = 2.8 - 2.8e -a.aivand for rabbits it isH= 0.2 - 0,2e -0.00sVHere S and H are the daily intake (measured in pounds), and Vis the vegetation biomass (measured in pounds per acre).(a) Find the satiation level for sheep and that for rabbits. (Round your answers to three decimal places.)sheeprabbitsIb(b) One concern in the management of rangelands is whether the various species of grazing animals are forced to compete for food. It is thought that competition will not be a problem if the vegetation biomass level provides at least 85% of the satiation level for each species.What biomass level guarantees that competition between sheep and rabbits will not be a problem? (Round your answer to two decimal places.)Ib/acreNeed Help? Read It

The amount of vegetation eaten in a day by a grazing animal is a function of the amount V of food available (measured as biomass in units such as pounds per acre). This relationship is called the functional response. If there is little vegetation available, the daily intake vill be small, since the animal will have difficulty finding and eating the food. As the food biomass incre ases, so does the daily intake. Clearly, though, there is a limit to the amount the animal will eat, regardless of the amount of food available. This maximum amount eaten is the satiation level. In addition to the kangaroos, the major grazing mammals of Australia include merino sheep and rabbits. For sheep the functional response is $ 2.8 - 2.0e 0.0r, and for rabbits it is H- 0.2 - 0.2e 0.00 Here S and H are the daily intake (measured in pounds), and Vis the vegetation biomass (measured in pounds per acre). (a) Find the satiation level for sheep and that for rabbits. (Round your answers to three decimal places.) sheep Ib Ib rabbits (b) One concern in the management of rangelands is whether the various species of grazing animals are forced to compete for food. It is thought that competition will not be a problem if the vegetation biomass level provides at least 85% of the satiation level for each species. What biomass level quarantees that competition between sheep and rabbits will not be a problem? (Round your answer to tvo decimal places.) Ib/acre Need Help? Read

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section2.4: Average Rate Of Change Of A Function

Problem 4.2E: bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.

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The amount of vegetation eaten in a day by a grazing animal is a function of the amount V of food available (measured as biomass in units such as pounds per acre). This relationship is called the functional response. If there is little vegetation available, the daily intake will be small, since
the animal will have difficulty finding and eating the food. As the food biomass increases, so does the daily intake. Clearly, though, there is a limit to the amount the animal will eat, regardless of the amount of food available. This maximum amount eaten is the satiation level.
In addition to the kangaroos, the major grazing mammals of Australia include merino sheep and rabbits. For sheep the functional response is
S = 2.8 - 2.8e -a.aiv
and for rabbits it is
H= 0.2 - 0,2e -0.00sV
Here S and H are the daily intake (measured in pounds), and Vis the vegetation biomass (measured in pounds per acre).
(a) Find the satiation level for sheep and that for rabbits. (Round your answers to three decimal places.)
sheep
rabbits
Ib
(b) One concern in the management of rangelands is whether the various species of grazing animals are forced to compete for food. It is thought that competition will not be a problem if the vegetation biomass level provides at least 85% of the satiation level for each species.
What biomass level guarantees that competition between sheep and rabbits will not be a problem? (Round your answer to two decimal places.)
Ib/acre
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Expert Solution

Step 1

Given relations are

$S = 2.8 - 2.8 e^{- 0.01 v} H = 0.2 - 0.2 e^{- 0.006 v}$

where $S$ and $H$ are daily intake of sheep and rabbits respectively measured in pound and $v$ is the vegetation biomass measured in pound per acre.

To find station level for sheep and rabbit.

To find station level for sheep, let us differentiate $S$ w.r.t $v$ . It results

$\frac{d S}{d v} = 2.8 e^{- 0.01 v} \frac{d^{2} S}{d v^{2}} = - 0.028 e^{- 0.01 v}$

Clearly $\frac{d S}{d v} \to 0$ as $v \to \infty$ and as $v \to \infty$ , $\frac{d^{2} S}{d v^{2}} \to - \infty < 0$ . Therefore, satiation level for the sheep will go for $v \to \infty$ and in that case $S_{\infty} = 2.8$ . So, satiation level for the sheep is 2.8 lb.

Similarly, for the rabbits, differentiating $H$ w.r.t $v$ , it results

$\frac{d H}{d v} = 0.0012 e^{- 0.006 v} \frac{d^{2} S}{d v^{2}} = - 0.0000072 e^{- 0.006 v}$

Clearly $\frac{d S}{d v} \to 0$ as $v \to \infty$ and as $v \to \infty$ , $\frac{d^{2} S}{d v^{2}} \to - \infty < 0$ . Therefore, satiation level for the rabbit will go for $v \to \infty$ and in that case $H_{\infty} = 0.2$ . So, satiation level for the sheep is 0.2 lb.

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