I have met a system of non-linear equations as follows, $$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$ $$\frac{\mathbb{d}z_k}{\mathbb{d}t}=(1-\alpha)y_k\sum_s{s^az_s}+(\alpha y_k-\beta)z_k,$$ $$\frac{\mathbb{d}x_k}{\mathbb{d}t}=\beta z_k,$$ where $0\leq t\leq\infty$, $k\in\mathbb{N}$ and $1\leq k\leq n$, and $a<0$.

I think this system has no analytic solutions as $t\rightarrow\infty$. Is there any method to approximate its solutions? How to estimate the impact of $\alpha$ on solutions, i.e. $\sum x_k(\infty)$ increases as $\alpha$ increases or decreases?