Separation of Variables, System Progress, and Phase Diagrams
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(1) Solve by separation of variables and solution to the ODE (see attached file for formula).
(2) Solve the equation (see attached file for formula).
(3) For the given graph, re-sketch a graph for x01 and indicate the system progress from x01 as t --> infinity. Identify any critical points as stable, unstable, and semi-stable (see attached file for formula).
(4) Draw and label the phase diagram. Indicate the system progress (using arrows and explain. Identify any critical points at stable, unstable, or semi-stable: (see attached file for formula).
(5) Draw and label the phase diagram. Indicate the system progress (using arrows and explain. Identify any critical points at stable, unstable, or semi-stable: (see attached file for formula).
(6) A harvesting model is represented (by the formula in the attached file):
See attached file for more information.
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Solution Summary
The solution discusses the separation of variables, system progress and phase diagrams.
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1.
The equation is:
(1.1)
Then:
(1.2)
Applying initial condition:
(1.3)
Therefore:
(1.4)
The solutions look like:
2.
The equation is:
(2.1)
This is a separable equations:
(2.2)
At t=0 we have therefore:
(2.3)
As we see the solution approaches
(2.4)
Note that at this limit
(2.5)
That is, is a stationary point of the equation.
At that point:
(2.6)
The first derivative at the equilibrium point is negative, hence the equilibrium point is stable.
3.
The phase diagram is:
There are two equilibrium points and corresponding to the situation
• Point If then and decreases - moving away from
• Point If then and decreases - moving towards and away from
• Point If then and increases - moving away from
We see that is a semi-stable point. If we start slightly to its ...
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