Science:Math Exam Resources/Courses/MATH102/December 2017/Question 13 (d)

MATH102 December 2017

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Question 13 (d)

A biochemical reaction in which a substance is both produced and consumed is described by the differential equation $${\frac {dc}{dt}}={\frac {4c^{3}}{1+c^{3}}}-rc,$$ where $c(t)$ denotes the concentration of the substance, $r>0$ is a constant and the units are omitted. (d) Draw a state-space (phase) diagram for the concentration $c$ in the case that $r<r_{\text{tan}}$. Label each positive steady state, and determine if they are stable or unstable. You do not need to calculate the steady states explicitly–just show their relative position on the diagram. Remark: you can solve this part without having calculated $r_{\text{tan}}$ in part (c)

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Hint

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Solution
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