MATH102 December 2014
• QA 1 • QA 2 • QA 3 • QA 4 • QA 5 • QA 6 • QA 7 • QA 8 • QB 1 • QB 2 • QB 3 • QB 4 • QB 5 • QB 6 • QB 7 • QC 1 • QC 2(a) • QC 2(b) • QC 2(c) • QC 2(d) • QC 3 •
Question C 02 (b)
The graphs to the right show two functions,
where , and . Suppose that is the concentration of a substance involved in a chemical reaction and satisfies the equation
Determine the stability of each steady state. Explain how you arrived at your conclusions either in words or using a diagram.
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A steady state is stable if states that are initially close enough to that steady state will get closer to it with time. A steady state is unstable, if states that are initially very close to it eventually move away from that steady state. For example, if we start above(below) a steady state, and the derivative is positive(negative), then the state will increase(decrease) and eventually move away from the steady state so it is an unstable steady state.
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To determine the stability of each steady state, we just need to find the increasing and decreasing regions. As represents the concentration of a substance, we only need to consider the positive half of the real axis. The three steady states divide it into three intervals
If we choose a initial condition in the interval , as is negative, would decrease and approach the steady state as time goes on. So is stable. Similarly, we find is unstable and is stable.
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