(M1.1 C) A steady state of a differential model is a set of values that make all of the rate equations zero, simultaneously. For example, in a typical SIR model (one in which recovereds stay recovered and don't become susceptible again), if there are no infecteds (I = 0) to begin with, S', I', and R' are all equal to zero and S, I, and R never change. In other words, if I = 0, then S and R can be whatever they want and you will have a steady state. Consider the bunnies and foxes model 1 B. 10 1 BF 25000 1 B' BF - 200 1 F 250 = B(0) = Bo F(0) = Fo. Find all ordered pairs (B, F) that produce steady states, that is to say that both B' and F' are zero simultaneously (at the same time).
(M1.1 C) A steady state of a differential model is a set of values that make all of the rate equations zero, simultaneously. For example, in a typical SIR model (one in which recovereds stay recovered and don't become susceptible again), if there are no infecteds (I = 0) to begin with, S', I', and R' are all equal to zero and S, I, and R never change. In other words, if I = 0, then S and R can be whatever they want and you will have a steady state. Consider the bunnies and foxes model 1 B. 10 1 BF 25000 1 B' BF - 200 1 F 250 = B(0) = Bo F(0) = Fo. Find all ordered pairs (B, F) that produce steady states, that is to say that both B' and F' are zero simultaneously (at the same time).
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Steady State of a Differential Model**
A steady state of a differential model is a set of values that make all of the rate equations zero, simultaneously. For example, in a typical SIR model (one in which recovereds stay recovered and don't become susceptible again), if there are no infecteds (\(I = 0\)) to begin with, \(S'\), \(I'\), and \(R'\) are all equal to zero and \(S\), \(I\), and \(R\) never change. In other words, if \(I = 0\), then \(S\) and \(R\) can be whatever they want and you will have a steady state. Consider the bunnies and foxes model:
\[
B' = \frac{1}{10}B - \frac{1}{200}BF
\]
\[
F' = \frac{1}{25000}BF - \frac{1}{250}F
\]
\[
B(0) = B_0
\]
\[
F(0) = F_0
\]
Find all ordered pairs \((B, F)\) that produce steady states, that is to say that both \(B'\) and \(F'\) are zero simultaneously (at the same time).
**Explanation:**
- \(B'\) and \(F'\) represent the change in population of bunnies (\(B\)) and foxes (\(F\)) over time.
- The terms \(\frac{1}{10}B\) and \(\frac{1}{200}BF\) in the equation for \(B'\) describe bunny population growth and reduction due to fox predation.
- The terms \(\frac{1}{25000}BF\) and \(\frac{1}{250}F\) in the equation for \(F'\) describe fox population growth due to bunny availability and natural fox death rate.
The objective is to find values of \(B\) and \(F\) where the populations become stable, i.e., do not change over time, represented by \(B' = 0\) and \(F' = 0\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb2fa90c-1147-4575-bb50-301a6f95fb8e%2F55e1ea35-869c-44e0-b207-0034cd4c8da9%2Fi3ynfi_processed.png&w=3840&q=75)
Transcribed Image Text:**Steady State of a Differential Model**
A steady state of a differential model is a set of values that make all of the rate equations zero, simultaneously. For example, in a typical SIR model (one in which recovereds stay recovered and don't become susceptible again), if there are no infecteds (\(I = 0\)) to begin with, \(S'\), \(I'\), and \(R'\) are all equal to zero and \(S\), \(I\), and \(R\) never change. In other words, if \(I = 0\), then \(S\) and \(R\) can be whatever they want and you will have a steady state. Consider the bunnies and foxes model:
\[
B' = \frac{1}{10}B - \frac{1}{200}BF
\]
\[
F' = \frac{1}{25000}BF - \frac{1}{250}F
\]
\[
B(0) = B_0
\]
\[
F(0) = F_0
\]
Find all ordered pairs \((B, F)\) that produce steady states, that is to say that both \(B'\) and \(F'\) are zero simultaneously (at the same time).
**Explanation:**
- \(B'\) and \(F'\) represent the change in population of bunnies (\(B\)) and foxes (\(F\)) over time.
- The terms \(\frac{1}{10}B\) and \(\frac{1}{200}BF\) in the equation for \(B'\) describe bunny population growth and reduction due to fox predation.
- The terms \(\frac{1}{25000}BF\) and \(\frac{1}{250}F\) in the equation for \(F'\) describe fox population growth due to bunny availability and natural fox death rate.
The objective is to find values of \(B\) and \(F\) where the populations become stable, i.e., do not change over time, represented by \(B' = 0\) and \(F' = 0\).
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