Best choice of conditions to solve a system of non linear differential equations

Question

I would ask you to take a look at the following problem:

1. I have this system of ODE: $$ \begin{cases} \dot{x}(t)= x(t)[r_1(t) -b_1(t)x(t- \tau(t)) - c_1(t)y^m(t)] & t\neq t_k\\ \dot{y}(t)= y(t)[- r_2(t)- b_2(t) y(t- \sigma(t)) +c_2(t) x(t) y^{m-1}(t)] & \\ \Delta x(t_k)= x(t_k^+)- x(t_k)=d_k x(t_k) & t=t_k\\ \Delta y(t_k)= y(t_k^+)- y(t_k) = f_k y(t_k) & \end{cases}. $$
2. $x(t)$ and $y(t)$ are discontinuous function at points $t_k$.
3. All these functions: $r_1(t), \, r_2(t), \, b_1(t), \, b_2(t), \, c_1(t),\mbox{ and } c_2(t)$ are positive and periodic with the same period $\omega$.
4. $\tau(t)$ and $\sigma(t)$ are continuously differentiable positive periodic functions with the same period $\omega$ such that $\tau'(t) <1 \mbox{ and } \sigma'(t)<1$.
5. $m\in (0,1)$
6. $-1 < d_k \leq 0, \, -1 < f_k \leq 0$ are constants for $k=1,2, \cdots,$ such that there exists a positive integer $q$ $t_{k+q}=t_k + \omega$.

I am trying to make the right choice for these parameters to solve this system with Mathematica.