Notice that and 2-678-67] (usar][a]) 07] ([307][2]) = = [1.1 -0.1] [R₁ [1.1 -0.1] [1.1 -0.1 [Ro A5 9-36 A-9 420 0.3 0.7 [1.1 -0.17 0.3 0.7 Co We can generalize this to find a formula for = = A² = [1.1 -0.1] [R₂ [1.1 -0.1] [1.3362 1.0085 3 ¹] ² [2] = 4³[2]. [Ro R₁ [2] = Cn -0.3362] -0.0085 Co 0.3 0.7 [1.4942 -0.4942] 1.4827 -0.4827 = An 2 [Rn [Cn] Selected powers of matrix A are listed below, rounded to 4 decimal places. Find the population vectors after 5 years, 10 years, 20 years, and 50 years. Round to whole numbers of animals (that is, round the value of Rn to two decimal places and round the value of Cn to an integer). A¹0 = , Ro representing the populations in year n: A50 1.1 -0.1] [Ro [1.1 -0.1] = [1.4463 -0.4463] 1.3389 -0.3389 [1.5000 -0.5000] 1.5000 -0.5000 What appears to be the long-term progonosis for the rabbit and cougar populations with the initial population vector [19] ² ? If the population stabilizes, what is the ratio of rabbits to cougars?
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.
![Notice that
and
[2₂] - [0.3
=
=
=
R3
[2]-6.3
[1.1 -0.1] [R₁ [1.1
0.3 0.7
A5
[1.1 -0.1 [Ro
0.3
0.7 Co
420
1.1 -0.1] [Ro
Ta]-[307] (6307][2])
[1.1 -0.1] [R₂]
0.7
[1.1 -0.17
0.3 0.7
=
3
We can generalize this to find a formula for
=
= A²
= [0.3
=
[1.3362 -0.3362]
1.0085 -0.0085
[1.1 -0.1]
R₁
[2] =
Cn
[1.4942 -0.4942]
1.4827 -0.4827
[Ro
Co
A³
0.3 0.7
2
[Rn
[Cn]
= An
Selected powers of matrix A are listed below, rounded to 4 decimal places. Find the population
vectors after 5 years, 10 years, 20 years, and 50 years. Round to whole numbers of animals
(that is, round the value of Rn to two decimal places and round the value of Cn to an integer).
A¹0 =
07] ([307][2])
,
Ro
representing the populations in year n:
[1.1 -0.1]
A50 =
[1.4463 -0.4463]
1.3389 -0.3389
[1.5000 -0.5000]
1.5000 -0.5000
What appears to be the long-term progonosis for the rabbit and cougar populations with the
initial population vector
[19] ² ? If the population stabilizes, what is the ratio of rabbits to
cougars?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6195f099-697f-489e-94ae-c77f1a063c08%2F37b0b70c-75a0-48e9-a77e-69dba75b2d76%2Fj6bw2m8_processed.png&w=3840&q=75)
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