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Sagot :
Answer:
[tex](0,0)[/tex] [tex](4000,0)[/tex] and [tex](500,79)[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required
Determine the equilibrium solutions
We have:
[tex]\frac{dR}{dt} = 0.09R(1 - 0.00025R) - 0.001RW[/tex]
[tex]\frac{dW}{dt} = -0.02W + 0.00004RW[/tex]
To solve this, we first equate [tex]\frac{dR}{dt}[/tex] and [tex]\frac{dW}{dt}[/tex] to 0.
So, we have:
[tex]0.09R(1 - 0.00025R) - 0.001RW = 0[/tex]
[tex]-0.02W + 0.00004RW = 0[/tex]
Factor out R in [tex]0.09R(1 - 0.00025R) - 0.001RW = 0[/tex]
[tex]R(0.09(1 - 0.00025R) - 0.001W) = 0[/tex]
Split
[tex]R = 0[/tex] or [tex]0.09(1 - 0.00025R) - 0.001W = 0[/tex]
[tex]R = 0[/tex] or [tex]0.09 - 2.25 * 10^{-5}R - 0.001W = 0[/tex]
Factor out W in [tex]-0.02W + 0.00004RW = 0[/tex]
[tex]W(-0.02 + 0.00004R) = 0[/tex]
Split
[tex]W = 0[/tex] or [tex]-0.02 + 0.00004R = 0[/tex]
Solve for R
[tex]-0.02 + 0.00004R = 0[/tex]
[tex]0.00004R = 0.02[/tex]
Make R the subject
[tex]R = \frac{0.02}{0.00004}[/tex]
[tex]R = 500[/tex]
When [tex]R = 500[/tex], we have:
[tex]0.09 - 2.25 * 10^{-5}R - 0.001W = 0[/tex]
[tex]0.09 -2.25 * 10^{-5} * 500 - 0.001W = 0[/tex]
[tex]0.09 -0.01125 - 0.001W = 0[/tex]
[tex]0.07875 - 0.001W = 0[/tex]
Collect like terms
[tex]- 0.001W = -0.07875[/tex]
Solve for W
[tex]W = \frac{-0.07875}{ - 0.001}[/tex]
[tex]W = 78.75[/tex]
[tex]W \approx 79[/tex]
[tex](R,W) \to (500,79)[/tex]
When [tex]W = 0[/tex], we have:
[tex]0.09 - 2.25 * 10^{-5}R - 0.001W = 0[/tex]
[tex]0.09 - 2.25 * 10^{-5}R - 0.001*0 = 0[/tex]
[tex]0.09 - 2.25 * 10^{-5}R = 0[/tex]
Collect like terms
[tex]- 2.25 * 10^{-5}R = -0.09[/tex]
Solve for R
[tex]R = \frac{-0.09}{- 2.25 * 10^{-5}}[/tex]
[tex]R = 4000[/tex]
So, we have:
[tex](R,W) \to (4000,0)[/tex]
When [tex]R =0[/tex], we have:
[tex]-0.02W + 0.00004RW = 0[/tex]
[tex]-0.02W + 0.00004W*0 = 0[/tex]
[tex]-0.02W + 0 = 0[/tex]
[tex]-0.02W = 0[/tex]
[tex]W=0[/tex]
So, we have:
[tex](R,W) \to (0,0)[/tex]
Hence, the points of equilibrium are:
[tex](0,0)[/tex] [tex](4000,0)[/tex] and [tex](500,79)[/tex]

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