Data fitting from coupled differential equations

• To: mathgroup at smc.vnet.net
• Subject: [mg93291] Data fitting from coupled differential equations
• From: margeorias at gmail.com
• Date: Mon, 3 Nov 2008 05:25:11 -0500 (EST)

Hi,

I'm working on a predator-prey model for 2-species:
n1'[t] == a1 n1[t] - b1 n1[t]^2 + c1 n1[t] n2[t]
n2'[t] == a2 n2[t] - b2 n2[t]^2 + c2 n1[t] n2[t]
n1[0] == n10,
n2[0] == n20

I have real data for (t,n1[t],n2[t]), i.e. data={t,n1[t],n2[t]} and my
objective is to estimate the coefficients a1,a2,b1,b2,c1,c2.

To do this, I generate synthetic data by solving the above equations
using {n10,n20,a1,a2,b1,b2,c1,c2}={0.01,0.001,2,3,1,1,1,2} and sample
the solutions,i.e.
data = Table[{t, n1[t], n2[t]} /. sol, {t, 0, 20, 0.25}];

I then set-up the function below:

sse[a1_?NumericQ,a2_?NumericQ,b1_?NumericQ,b2_?NumericQ,c1_?
NumericQ,c2_?NumericQ]:=
Block[{sol,n1,n2},
sol=NDSolve[{
n1'[t]==a1 n1[t]-b1 n1[t]-c1 n1[t] n2[t],
n2'[t]==a2 n2[t]-b2 n2[t]-c2 n1[t] n2[t],
n1[0]==0.01,n2[0]==0.001},
{n1,n2},{t,0,20}][[1]];
Plus@@Apply[((n1[#1]-#2)^2+(n2[#1]-#3)^2)&,data,{1}]/.sol]

and use FindMinimum to retrieve the coefficients:
FindMinimum[sse[a1,a2,b1,b2,c1,c2],{{a1,1.},{a2,1.},{b1,1.},{b2,1.},
{c1,1.},{c2,1.}},AccuracyGoal->20,PrecisionGoal->18,WorkingPrecision-
>40]

My problem however is that the resulting coefficients are not the same
as the original one, ie:
{a1->-9.49992942438124754467310,a2->-9.50000018840483650972573,b1-
>-9.50007117410767898402213,b2->-9.50000041008883222559689,c1-
>-9.50000036535088621114653,c2->-9.50000030036113407572174}

I can improve the results by making starting points very close to the
actual value of the coefficients i.e. {a1,2}. Well this somehow
defeats the purpose and in reality, I don't know whats the actual
value of the coefficients - they range from -Infinity to Infinity.How
do I go about obtaining the correct coefficients? Is there something
wrong with the function above?

Another thing, wouldn't NMinimize be appropriate for this as well
since I want to find the global minimum of sse? I have tried it but it
gave the wrong values for the coefficients as well :(

I'm trying to make FindFit work on the coupled differential equations
but no luck yet. Any guidance on this?

Appreciate any feedback you have.

Thanks again.

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