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# Initial Conditions, Sources, Sinks, Nodes and Bifurcation

Our problem is to determine which values of D result in extinction and which result in survival. This can be done by studying equation (*), treating D as a bifurcation parameter see Section 2.6

a. Using technology to study solutions to an equation (*) for parameter values of 08, 0.7, 0,4, and 0.5. For each choice of D, use several diflerent initial conditions x0. What are your observations?

b. Find the equilibrium solutions ... as a function of the parameter D and determine whether they are sinks, sources, or nodes. (Assume that the only meaningful equilibrium solutions are those for which 0 < x < 10.) Construct the bifurcation diagram for equation (9. (Sec Section 2.6.) At what value of I) does a bifurcation occur? Explain the significance of this bifurcation with regard to the fate of the H coil.

c. Use technology to test the validity of your bifurcation analysis in part (b) by examining the olutions of equation (*) again, choosing various values of D very close to the bifurcation value you found in part (b). Are your observations as expect?

d. Assuming that the growth vessel is kept at volume of 20 liters, at what speed should the chemostat pump be run in order to maintain a steady-state E. coil population of 8 microgram/liter? A population of 4.5 microgram/liter?

I am really having trouble with this course on Differential Equations. The book that we are using is A Modern Introduction to Differential Equations by Henry Ricardo.

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dx/dt = [(0.81(10-x) / (3 + (10 - x))) - D]x
or, dx/dt = x[0.81(10-x)/(13-x)] - Dx

here x(t) denotes the concentration of E.coli at time t. D is the pump rate divided by the volume. This makes sense that when D will be zero the concentration of E.Coli. will completely depend on time and that is what this equation shows.

To begin, we notice that x = 0 is always a solution. ...

#### Solution Summary

Differential Equations, Initial Conditions, Sources, Sinks, Nodes and are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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