A forest has a population of cougars and a population of squirrels. Let a represent the number of cougars (in hundreds) above some level, denoted with 0. So x = 3 corresponds NOT to an absence of cougars, but to a population that is 300 below the designated level of cougars. Similarly, let y represent the number of squirrels (in hundreds) above a level designated by zero. The following system models the two populations over time: x' = 0.5x + y y' - x 2.5y Solve the system using the initial conditions (0) = 0 and y(0) = 1. t t x(t) = X e (t+1) e y(t) = t e te
A forest has a population of cougars and a population of squirrels. Let a represent the number of cougars (in hundreds) above some level, denoted with 0. So x = 3 corresponds NOT to an absence of cougars, but to a population that is 300 below the designated level of cougars. Similarly, let y represent the number of squirrels (in hundreds) above a level designated by zero. The following system models the two populations over time: x' = 0.5x + y y' - x 2.5y Solve the system using the initial conditions (0) = 0 and y(0) = 1. t t x(t) = X e (t+1) e y(t) = t e te
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
A forest has a population of cougars and a population of squirrels. Let xx represent the number of cougars (in hundreds) above some level, denoted with 0. So x=−3x=-3 corresponds NOT to an absence of cougars, but to a population that is 300 below the designated level of cougars. Similarly, let yy represent the number of squirrels (in hundreds) above a level designated by zero. The following system models the two populations over time:
x'=−0.5x+yx′=-0.5x+y
y'=−x−2.5yy′=-x-2.5y
Solve the system using the initial conditions x(0)=0x(0)=0 and y(0)=1y(0)=1.
x(t)x(t) = e−32t−(t+1)e−32tIncorrect
y(t)y(t) = e−32t−te−32tCorrect
Choose the graph that best represents the solution curve.
![### Population Dynamics Model for Cougars and Squirrels
In this example, we study the population dynamics of cougars and squirrels in a forest using a system of differential equations.
Let \( x \) represent the number of cougars (in hundreds) above some base level, denoted with 0. Hence, \( x = -3 \) does not indicate the absence of cougars, but rather that the population is 300 below the designated base level. Similarly, let \( y \) represent the number of squirrels (in hundreds) above a base level designated by zero. The system of equations modeling the interaction of the two populations over time is given by:
\[
\begin{cases}
x' = -0.5x + y \\
y' = -x - 2.5y
\end{cases}
\]
#### Initial Conditions
We solve the system using the initial conditions: \( x(0) = 0 \) and \( y(0) = 1 \).
#### Solution Analysis
The solutions to this system are given by:
\[
\begin{aligned}
x(t) &= \frac{e^{-\frac{3}{2}t}}{2} - (t + 1)e^{-\frac{3}{2}t} \\
y(t) &= \frac{e^{-\frac{3}{2}t}}{2} - te^{-\frac{3}{2}t}
\end{aligned}
\]
After verifying the solutions, we see that the correct solution set that satisfies the given initial conditions is:
\[
\boxed{y(t) = \frac{e^{-\frac{3}{2}t}}{2} - te^{-\frac{3}{2}t}}
\]
The incorrect solution attempt for \( x(t) \) is:
\[
\boxed{x(t) = \frac{e^{-\frac{3}{2}t}}{2} - (t + 1)e^{-\frac{3}{2}t}}
\]
This example demonstrates a typical application of differential equations in modeling ecological systems, showing how closely interacting species can influence each other's populations over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22e84192-4b16-4dc2-817c-afb771201e2b%2Fb5abd85c-abdc-4cd5-a098-f8d7ccf9638c%2Fqw5ylzr_processed.png&w=3840&q=75)
Transcribed Image Text:### Population Dynamics Model for Cougars and Squirrels
In this example, we study the population dynamics of cougars and squirrels in a forest using a system of differential equations.
Let \( x \) represent the number of cougars (in hundreds) above some base level, denoted with 0. Hence, \( x = -3 \) does not indicate the absence of cougars, but rather that the population is 300 below the designated base level. Similarly, let \( y \) represent the number of squirrels (in hundreds) above a base level designated by zero. The system of equations modeling the interaction of the two populations over time is given by:
\[
\begin{cases}
x' = -0.5x + y \\
y' = -x - 2.5y
\end{cases}
\]
#### Initial Conditions
We solve the system using the initial conditions: \( x(0) = 0 \) and \( y(0) = 1 \).
#### Solution Analysis
The solutions to this system are given by:
\[
\begin{aligned}
x(t) &= \frac{e^{-\frac{3}{2}t}}{2} - (t + 1)e^{-\frac{3}{2}t} \\
y(t) &= \frac{e^{-\frac{3}{2}t}}{2} - te^{-\frac{3}{2}t}
\end{aligned}
\]
After verifying the solutions, we see that the correct solution set that satisfies the given initial conditions is:
\[
\boxed{y(t) = \frac{e^{-\frac{3}{2}t}}{2} - te^{-\frac{3}{2}t}}
\]
The incorrect solution attempt for \( x(t) \) is:
\[
\boxed{x(t) = \frac{e^{-\frac{3}{2}t}}{2} - (t + 1)e^{-\frac{3}{2}t}}
\]
This example demonstrates a typical application of differential equations in modeling ecological systems, showing how closely interacting species can influence each other's populations over time.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
