Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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I only need question 4, Thank you.
In the presence of enough food and lacking predators and competitors, a population of rabbits
will increase by a fixed percentage each spring. For this set of exercises, we will say that the
rabbit population increases by 10% each year. Thus, if the initial population is R0, then the
population the following spring will be R1 = (1.1)R0 rabbits. In two years, the population will be
R2 = (1.1)R1 = (1.1)2R0. The number of rabbits after n years will be
Rn = (1.1)nR0,
and clearly the rabbit population grows without bound. We will assume that the rabbit population
is measured in hundreds of rabbits (so that R = 1 represents 100 rabbits).
Suppose now that we introduce a small number of cougars into the environment to keep the rabbit
population under control.
Let’s suppose that the amount of rabbits eaten each year is proportional to the cougar population.
Then the change in the rabbit population is governed by the equation
Rn+1 = (1.1)Rn − (0.1)Cn,
where Rn and Cn represent the rabbit and cougar populations in year n, respectively, rabbits measured in hundreds and cougars measured in singles. The constant 0.1 is the kill rate.
![If there are no cougars, what does the equation Rn+1 = (1.1)Rn - (0.1)Cn say about the
change in the population of rabbits? Does that make sense?
The rate of growth of the cougar population will depend on both the number of cougars and
the number of rabbits. We will assume that the growth of the cougar population is governed
by the equation
Cn+1 = (0.3) Rn+ (0.7)Cn.
If there are no rabbits, what does the equation Cn+1 = (0.3)R₁ + (0.7)Cn say about the
change in the population of cougars? Does that make sense?
We can combine these two equations into a system:
Let A =
1
-0.1
[0.300.7]
[Rn+1
1.1
[+] = [0.3 -0.7] [²].
Cn+1
Is matrix A invertible? How do you know?
Suppose that the initial populations of rabbits is 1000 and the initial population of cougars
is 2. Then we can represent the intial population by
Find the population vector after a year. Round to whole numbers of animals (that is, round
the value of R₁ to two decimal places and round the value of C₁ to an integer).](https://content.bartleby.com/qna-images/question/6195f099-697f-489e-94ae-c77f1a063c08/f68d2408-a0c0-403c-acbf-1211387a9c8d/zn11zl_processed.png)
Transcribed Image Text:If there are no cougars, what does the equation Rn+1 = (1.1)Rn - (0.1)Cn say about the
change in the population of rabbits? Does that make sense?
The rate of growth of the cougar population will depend on both the number of cougars and
the number of rabbits. We will assume that the growth of the cougar population is governed
by the equation
Cn+1 = (0.3) Rn+ (0.7)Cn.
If there are no rabbits, what does the equation Cn+1 = (0.3)R₁ + (0.7)Cn say about the
change in the population of cougars? Does that make sense?
We can combine these two equations into a system:
Let A =
1
-0.1
[0.300.7]
[Rn+1
1.1
[+] = [0.3 -0.7] [²].
Cn+1
Is matrix A invertible? How do you know?
Suppose that the initial populations of rabbits is 1000 and the initial population of cougars
is 2. Then we can represent the intial population by
Find the population vector after a year. Round to whole numbers of animals (that is, round
the value of R₁ to two decimal places and round the value of C₁ to an integer).
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