1180:Lab5
Goals For This Week
We’ll learn how to solve ODEs with more than one state variable. As an example, we’ll look at a predator prey model. The fitzugh-nagumo model is desribed in section 5.8.
We’re finally ready to solve a system of differential equations with multiple parameters. This is the predator-prey system we talked about in class.
db dt dp dt
= (
= (
λ
−
−
δ ǫp
+
)
ηb b
) p
There are two state variables : b and p ; and four parameters : λ , ǫ , δ and η (that’s ‘lambda’,‘epsilon’,‘delta’ and ‘eta’). First, we shall write the right hand side of our equation. We’ll start with the skeleton-I’ll call the function predprey.
predprey<-function(t,y,p){
}
Then we should code up the right hand side of each individual equation.
predprey<-function(t,y,p){ db=(l-e*p)*b dp=(-d+n*b)*p
}
Now we return the result, but this is tricky as we have to combine db and dp and then designate it as a list.
1

predprey<-function(t,y,pars){ db=(l-e*p)*b dp=(-d+n*b)*p list(c(db,dp))
}
The problem with this code is that it uses all of these variables that we haven’t yet defined (l,e,...etc).
The values for these variables are stored in y and p,but we have to use with to ‘extract’ them.
This is our final function.
predprey<-function(t,y,pars){ with(as.list(c(y,pars)),{ db=(l-e*p)*b dp=(-d+n*b)*p list(c(db,dp))
})
}
Now we can run the function yini=c(b=100,p=2000) tlist=seq(0,20,.1) parameters<-c(l=1,d=1,n=.001,e=.001) result=ode(func=predprey,y=yini,t=tlist,parms=parameters) plot(result)
Over the last couple of weeks I’ve had you combine the output of many runs into a single graph using the lines command. In truth, plot is smarter than that.
yini1=c(b=100,p=2000) yini2=c(b=100,p=1000) yini3=c(b=500,p=500) tlist=seq(0,20,.1) parameters<-c(l=1,d=1,n=.001,e=.001) result1=ode(func=predprey,y=yini1,t=tlist,parms=parameters) result2=ode(func=predprey,y=yini2,t=tlist,parms=parameters) result3=ode(func=predprey,y=yini3,t=tlist,parms=parameters) plot(result1,result2,result3)
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This will only work when we are plotting a data frame (which result1,result2 and result3 are because they were made for us by ODE).
Save this Plot
Another way to visualize this data is with a phase plane. We’ll plot the prey on the horizontal and the predator on the vertical. The code below creates a phase plane picture with all three results.
plot(result1[,"b"],result1[,"p"],type=’l’,xlab=’Bacteria(Prey)’,ylab=’Amoeba(Predator)’) lines(result2[,"b"],result2[,"p"],lty=2,col=’red’) lines(result3[,"b"],result3[,"p"],lty=3,col=’green’)
If the plot only fills half the window close the plot window and try again.
Save this Plot
1.
Predator Prey System
•
Include time series and phase plane plot from above.
•
On the phase plane, does the population travel clockwise or anticlockwise.
•
Describe what is happening to the populations and why they cycle.
2.
Fitzugh-Nagumo Equation dv dt dw dt
=
=
− ǫ ( v v
( v
−
− a
γw )
)( v
−
1)
− w + I
•
Let I = 0, ǫ = .
01, a = .
3 and γ = 2 .
5. Solve with ode from t=0 to t=100. Use 3 sets of initial conditions.Let
w (0) = 0 and let v (0) = .
2, .
4 and .
6.
•
Plot the time series together.
•
Plot the phase portraits together.
•
With these parameters, this system has an activation threshold. Based, on your graphs, what is meant by that?
•
Now let v (0) = .
2 and change I to .01,.05 then .1. This represents a constant input current.
•
Plot the time series together
•
Plot the phase portraits of these results together.
•
What happens when you increase the input current?
3