Delayed differential equation with convolution

Question

So I have a differential equation :

$\left\{\begin{split} & c'(t) = \alpha q(t) c(t)^{2/3} - \beta(1 - q(t))c(t)\\ &\chi'(t) = \kappa c(t) q(t) \\ & q(t) = \int_{-\infty}^t \Big( \frac{\chi(u)}{\chi(u) + \chi_c}e^{-(t-u)/t0}\Big)du\end{split} \right.$

I wrote the following DDE :

alpha = 0.1
beta = 0.1
kappa = 0.1
t0 = 0.5
chic = 1/2


sol = NDSolve[{c'[t] == alpha*c[t]^(2/3)*q[t] - beta*(1 - q[t])*c[t], 
   chi'[t] == kappa*c[t]*q[t], 
   q[t] == Integrate[
     chi[u]/((chi[u] + chic))*Exp[-(t - u)/t0], {u, -5, t}], 
   q[0] == 1/2, chi[t /; t <=  0] == 1/2*(Tanh[-t*10]), 
   c[t /; t <= 0] == 20}, {c, chi}, {t, 0, 14}]

But I'm getting those errors :

Item 2 requested in "Delayed time 1 = 2 computed at 3 = 4 did not evaluate to a real number." out of range; 1 items available.

General::stop: Further output of StringForm::sfr will be suppressed during this calculation.

NDSolve::rdelay: Delayed time u = 2 computed at 3 = 4 did not evaluate to a real number.

I don't understand what it means. Could you explain the erros plz ? And how to fix it ?

Thx

Answer 1

If you write the third integral equation as a differential equation, it will work:

sol = NDSolve[{
      c'[t] == alpha * c[t]^(2 / 3) * q[t] - beta * (1 - q[t]) * c[t], 
      chi'[t] == kappa * c[t] * q[t], 
      q'[t] + q[t] / t0 == chi[t] / ((chi[t] + chic)),
      q[0] == 1/2, 
      chi[t /; t <= 0] == 1 / 2 * (Tanh[- t * 10]),
      c[t /; t <= 0] == 20
      }, 
      {c, chi}, {t, 0, 14}
   ]