Science:Math Exam Resources/Courses/MATH215/December 2011/Question 07 (a)

MATH215 December 2011

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Question 07 (a)
Consider the system of equations ${\begin{aligned}x'&=x(2-x-y)\\y'&=y(1-x)\end{aligned}}$ (a) Find and plot the critical (or fixed) points. Find and plot both nullclines, that is the set of points where x' = 0 and where y' = 0.

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Hint
To be a critical point, both $x'$ and $y'$ must be 0. You may need to consider multiple cases.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. For the nullclines, we consider $\ x'=x(2-x-y)=0$: $\ x'=0$, if $\ x=0$ or $\ y=2-x$. Now we consider $\ y'=y(1-x)=0$: $\ y'=0$, if $\ y=0$ or $\ x=1$. For the critical points, $\ x'=y'=0$ must hold. Hence, the critical points are $\ (x_{0},y_{0})=(0,0),(2,0),(1,1)$. Plotted in the phase space: