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.