1180:Lab4
James Moore
January 29th, 2012
Goals For This Week
This week, we’ll learn how to solve ODEs with multiple parameters and focus on analyzing
a complex 1-D ODE.
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A model of T Cell growth in the lymph node
Let x be the number of T Cells in your lymph node.
x2
dx
= α − δx + γ
dt
(x + 1)2
(1)
They enter at a rate α, leave at a rate δ and grow at a rate γ. As this is a multiparameter model,
you’ll have to code the right hand side function like this.
f=function(t,x,p){
with(p,{list(g*x^2/(x+1)^2-d*x+a)})
}
In addition, when calling ODE, pass it arguments like this.
result=ode(func=f,t=tlist,y=x1/2,parms=list(g=5,d=1,a=0))
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Questions
For all questions, label the horizontal axis ‘time’ and the vertical axis ‘T Cells’. For these problems
let α = 0 and δ = 1.
1. Find the equilibria in terms of γ. You should find that there will either be 1 or 3 solutions?
For what values of γ are there three solutions? (Hint, when are the solutions real?)
2. Find the stability of the equilibrium at x = 0.
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3. Choose a value for γ such that there is only one equilibria and plot a solution (use initial
conditions x = 1 and run the simulation long enough so that you can see what happens to
the solution). What do you expect to happen if you use different initial conditions? Why?
4. Choose a value for γ such that there are three equilibria. Calculate what these equilibria are
and record them. Also, calculate the stability of each point. One equilibrium should be 0,
call the others x1 and x2 . Run ode with these parameters and 6 different initial conditions:
one for each equilibrium, one between 0 and x1 , one between x1 and x2 and one greater than
x2 . Plot all these results on the same axes. Explain what this plot looks like based on your
prior calculations? If you can’t, try running the simulation longer. What happens to the T
Cell population in each case.
Now let α = .01 and remake the same plot as the previous question. Finally, let α = .1 and make
the same plot. For each plot, write down the observed equilibria and stability.
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