trivial solution in NDSolve

I want to find b[x] and c[x] in the interval 0

b'[x] - d[x] = 0,
c[x] - e[x] = 0,
lambda* d'[x] + b[x] (theta - k c[x]/(s+c[x])) = 0,
e'[x]-c[x]b[x]/(s+c[x]) = 0,


which b'[x] is first derivative of b[x] respect to x and constants of equations are:

lambda=0.2;k=2;s=0.5;theta=1;


and boundary conditions are:

c[1]=1, e[0]=0, d[0]=0, d[1]=0


The code I used is:

lambda=0.2;k=2;s=0.5;theta=1;

sol=NDSolve[{D[b[x],x] -d[x]==0,D[c[x],x] -e[x]==0,lambda*D[d[x],x]
+b[x] (theta - k c[x]/(s+c[x]))==0,
D[e[x],x] -c[x]b[x]/(s+c[x])==0,
c[1]==1,e[0] == 0,d[0]==0,d[1]==0},{b,c,d,e}, {x, 0, 1}]

p1=Plot[b[x] /. sol,{x, 0, 1},PlotStyle ->Red]
p2=Plot[c[x] /. sol,{x, 0, 1}]


but it returns just trivial solution:

c[x]=1, b[x]=0


What can I do to get nontrivial solutions of these equations? Could anyone help me? And one more question, I need answers which are in the interval 0<c[x]<1 and 0<b[x]<1. How can I consider these in solving equations?

Shooting method:

lambda=1/5;
k=2;
s=1/2;
theta=1;

sols = Map[First[
NDSolve[{
D[b[x],x] -d[x]==0,D[c[x],x] -e[x]==0,
lambda D[d[x],x]+b[x] (theta - k c[x]/(s+c[x]))==0,
D[e[x],x] -c[x]b[x]/(s+c[x])==0,
c[1]==1,e[0] == 0,d[0]==0,d[1]==0},{b,c,d,e}, {x, 0, 0.5},
Method ->  "BoundaryValues" ->
{"Shooting",  "StartingInitialConditions" -> {c[0] == #}}]] &, Range[0,0.2, 0.1]];

Plot[Evaluate[c[x]/.sols],{x,0,1},PlotRange->All]
Plot[Evaluate[b[x]/.sols],{x,0,1},PlotRange->All]


Reference:

NDSolve's output ignores multiple valid solutions

• Could you please let me know what is Range[-5, 0, 1] in your code? And also I have another question: Is there any way to get answers for b[x] and c[x] which are positive in all interval 0<x<1? – sara nj Aug 8 '16 at 5:31
• Range[-5,0,1] is the list{-5,-4,-3,-2,-1,0} for starting values of b[0] – Young Aug 8 '16 at 5:35
• Does the first graph show 3 different answers for c? Does your code show all the answers available for c[x] and b[x]? – sara nj Aug 8 '16 at 5:36
• So changing the range to another range would change the solutions, right?Actually I know that answers are in the interval 0<b[x]<1 and 0<c[x]<1. So should I change the range regarding to these information? – sara nj Aug 8 '16 at 5:38
• It doesn't show all the answers ... changing the range could reveal more "solutions" – Young Aug 8 '16 at 5:40