# A system of nonlinear ODEs

I am working with Mathematica to plot a system of Nonlinear ODEs, I did a program but it doesn't work. I don't know exactly where is the problem? If someone kindly can help with a remark or a suggestion. The program is next

b1 = 0.5;
b2 = 0.51;
b3 = 0.01;
b4 = 0.1;
c1 = 0.05;
c2 = 0.1;
delta1 = 0.0001;
delta2 = 0.0001;
delta3 = 0.00001;
alphaa = 0.7;
alphap = 0.7;
alphapc = 0.7;
alphacp = 0.7;
epsilon = 0.00001;
gamma = 1;
psi = 0.01;
xi[x_] = x^2 /(1 + x^2);
F[x_, y_] = alphaa*xi[y/(x + epsilon)];
G[x_, y_] = alphap*xi[y/(x + epsilon)];
H1[x_, y_] = alphapc*xi[y/(x + epsilon)];
H2[x_, y_] = alphacp*xi[x/(y + epsilon)];
H[x_, y_] = H1[x, y] - H2[x, y];
eqns = {a'[t] ==
gamma*q[t] - (b1 + b2 + delta1)*a[t] + b3 *c[t] + b4 *p[t] -
F[a[t], c[t]]*a[t]*c[t] - G[a[t], p[t]]*a[t]*p[t],
p'[t] ==
b2*a[t] - (b4 + c1 + delta2)*p[t] - H[c[t], p[t]]*p[t]*c[t] +
G[a[t], p[t]]*a[t]*p[t],
c'[t] ==
b1*a[t] + c1*p[t] - (b3 + c2 + delta3)*c[t] -
H[c[t], p[t]]*p[t]*c[t] + F[a[t], c[t]]*a[t]*c[t] - phi*c[t],
q'[t] == -gamma*q[t],
b'[t] == phi*c[t],
a[0] == 0, p[0] == 0, c[0] == 0, q[0] == 1, b[0] == 0
};
sol = DSolve[eqns, {a, p, c, d, b}, t]
"eqns/.sol/.{t\[Rule]RandomReal[]}"
Plot[{a[t] /. sol}, {t, 0, 100}]


Thank you in advance

There are two main problems in your code:

1. A typo in your differential equations: phi -> psi
2. Using the exact DSolve for such a complicated system instead of the numerical solver NDSolve.
eqns = {a'[t] ==
gamma*q[t] - (b1 + b2 + delta1)*a[t] + b3*c[t] + b4*p[t] -
F[a[t], c[t]]*a[t]*c[t] - G[a[t], p[t]]*a[t]*p[t],
p'[t] ==
b2*a[t] - (b4 + c1 + delta2)*p[t] - H[c[t], p[t]]*p[t]*c[t] +
G[a[t], p[t]]*a[t]*p[t],
c'[t] ==
b1*a[t] + c1*p[t] - (b3 + c2 + delta3)*c[t] -
H[c[t], p[t]]*p[t]*c[t] + F[a[t], c[t]]*a[t]*c[t] - psi*c[t],
q'[t] == -gamma*q[t], b'[t] == psi*c[t], a[0] == 0, p[0] == 0,
c[0] == 0, q[0] == 1, b[0] == 0};
sol = NDSolve[eqns, {a, p, c, d, b}, {t, 0, 10}];
Plot[{a[t] /. sol}, {t, 0, 10}]


• Thank you very much for your answer. I didn't see that the problem can be that. Many kinds regards Commented Jul 16, 2021 at 10:43
• @KamalKhalil Deleting a question when it has an answer is really bad practice here. For the future always post a minimal working similar example and never post confidential data/codes. Commented Jul 16, 2021 at 11:10
• @HansOlo Okay I will not do that. Thank you very much! Commented Jul 16, 2021 at 11:22