Let $\Psi(a) = \frac{a}{2}$ if $a>0$ and $0$ if $a\le 0$.

Now we consider the following coupled system of nonlinear ODEs: $$\begin{aligned}&\frac{1}{2}\sigma_1^2 u_1''(x) + \mu_1 u_1'(x) + \Psi(-u_1(x))u'_1(x) - (r+q_1)u_1(x) + q_1 u_2(x) =0, \\ &\frac{1}{2}\sigma_2^2 u_2''(x) + \mu_2 u_2'(x) + \Psi(-u_2(x))u'_2(x) - (r+q_2)u_2(x) + q_2 u_1(x) =0, \end{aligned}$$ with initial value $$u_1(0) = u_2(0) = -p <0,$$ where $\sigma_1,\sigma_2,\mu_1,\mu_2,q_1, q_2$ and $r$ are positive constants. My question is: can we show that such a system has a solution $(u_1, u_2)$ on the domain $[0,\infty)$ satisfying $$\lim_{x\to \infty} u_1(x) = \lim_{x\to \infty} u_2(x) =0?$$

Any suggestion will be greatly appreciated.