## 3 Results
### Example  
As an example, let us consider the problem in the unit disk \(D\) such that \(x_{1}^{2}+x_{2}^{2}<1\), bounded by the circle \(\Gamma\):\[\varepsilon^{4}\Delta u-u_{t}=u(u-a(\mathbf{x}))(u-1)+uv,\ \ \varepsilon^{2} \Delta v-v_{t}=v-b(\mathbf{x})u,\ \ \mathbf{x}\in D,\ t\in\mathbb{R}^{+};\] \[\frac{\partial u}{\partial u_{\Gamma}}(\mathbf{x}_{\Gamma},t)= \frac{\partial v}{\partial u_{\Gamma}}(\mathbf{x}_{\Gamma},t)=0,\ \ \mathbf{x}_{\Gamma}\in\Gamma,\ t\in\mathbb{R}^{+}; \tag{42}\] \[u(\mathbf{x},0)=u^{0}(\mathbf{x}),\ \ v(\mathbf{x})=v^{0}( \mathbf{x}),\ \mathbf{x}\in\overline{D}.\]  
In the case of positive \(a(\mathbf{x})\) and \(b(\mathbf{x})\), this problem belongs to the \"PN\" type of quasi-monotonicity. Let \(\gamma\) be an ellipse \(4x_{1}^{2}+9x_{2}^{2}=1\). Hence, \(D^{(+)}=\{(x_{1},x_{2}):4x_{1}^{2}+9x_{2}^{2}<1\}\) and \(D^{(-)}=D\backslash\overline{D}^{(+)}\):  
\[a(\mathbf{x}):= \begin{cases}1,&\mathbf{x}\in\overline{D}^{(-)}\\ 0.2,&\mathbf{x}\in D^{(+)}\end{cases} ;\ \ b(\mathbf{x}):= \begin{cases}0.5,& \mathbf{x}\in\overline{D}^{(-)}\\ 0.2,&\mathbf{x}\in D^{(+)}\end{cases} ,\]  
In the domain \(D^{(-)}\), we have \(f^{(-)}(u,v)=u(u-1)^{2}+uv\). Assuming \(v\geq 0\), the equation \(f^{(-)}=0\) has a unique root \(\varphi^{(-)}\equiv 0\), and therefore  
\[h^{(-)}(v)=v,\ \ \psi^{(-)}\equiv 0,\ \ \Phi^{(-)}(\tilde{v}^{(-)}):=\sqrt{2 \int\limits_{0}^{\tilde{\sigma}^{(-)}}vdv}=\tilde{v}^{(-)},\ \ \
u^{(-)}(v)\equiv 1,\]  
\[\Psi^{(-)}(\tilde{u}^{(-)},q)=\sqrt{2\int\limits_{0}^{\tilde{\sigma}^{(-)}}(u (u-1)^{2}+uq)du}=\tilde{u}^{(-)}\sqrt{\frac{1}{2}\left(\tilde{u}^{(-)}\right) ^{2}-\frac{4}{3}\tilde{u}^{(-)}+1+q}. \tag{43}\]  
In the domain \(D^{(+)}\), we have \(f^{(+)}(u,v):=u(u-0.2)(u-1)+uv\). The equation \(f^{(+)}=0\) has three roots:  
\[\varphi^{(1)}\equiv 0,\ \ \varphi^{(2)}(v)=0.6-\sqrt{0.16-v},\ \ \varphi^{(+)}(v)=0.6+\sqrt{0.16-v}.\]  
Only the function \(\varphi^{(+)}(v)\) satisfies Proposition 1, and therefore  
\[h^{(+)}(v)=v-0.2(0.6+\sqrt{0.16-v}),\ \ \psi^{(+)}\equiv 0.145,\]  
\[\Phi^{(+)}(\tilde{v}^{(+)})=\sqrt{2\int\limits_{\psi^{(+)}}^{ \tilde{v}^{(+)}}(v-0.2(0.6+\sqrt{0.16-v}))dv}=\sqrt{\left(v^{2}-0.24v+\frac{0. 8}{3}(0.16-v)^{3/2}\right)\Big{|}_{\varphi^{(+)}}^{\bar{\varphi}^{(+)}}},\] \[\
u^{(+)}(v)=1-\frac{0.1}{\sqrt{0.16-v}};\ \ \
u^{(+)}(v)>0,\ \text{for}\ 0 \leq v<0.15;\]  
\[\Psi^{(+)}(\tilde{u}^{(+)},q)=\sqrt{2\int\limits_{\varphi^{(+)} (q)}^{\tilde{u}^{(+)}}(u(u-0.2)(u-1)+uq)du}=\sqrt{\left(0.5u^{4}-0.8u^{3}+(0.2 +q)u^{2}\right)\big{|}_{\varphi^{(+)}(q)}^{\bar{\varphi}^{(+)}}=}\] \[= |\tilde{u}^{(+)}-\varphi^{(+)}(q)|\sqrt{0.25(\tilde{u}^{(+)})^{2} +(0.5\sqrt{0.16-q}-0.1)\tilde{u}^{(+)}+0.1\sqrt{0.16-q}+0.01-0.25q}. \tag{44}\]  
The functions \(\Phi^{(\mp)}(\tilde{v}^{(\mp)})\) exist for each \(q\in[\psi^{(-)},\psi^{(+)}]\), and the functions \(\Psi^{(\mp)}(\tilde{u}^{(\mp)},q)\) exist for \(q\in[0,0.12]\). The equation \(H_{0}^{v}(q)=0\) has the unique solution \(q_{0}\cong 0.08\), and the equation \(H_{0}^{u}(p,q_{0})=0\) has the unique solution \(p_{0}\cong 0.181\). Aside from that, for the expressions mentioned in Remark 2, we have  
\[h^{(-)}(q_{0})-h^{(+)}(q_{0})=0.2+\sqrt{0.16-q_{0}}>0,\] \[f^{(-)}(p_{0},v_{0})-f^{(+)}(p_{0},v_{0})=-0.8p_{0}(p_{0}-1)>0.\]  
Thus, Propositions 1-5 are valid for Problem (42). The corresponding stationary problem therefore has the following zeroth-order asymptotic approximation to its solution:  
\[U_{0}(\mathbf{x},\varepsilon)= \begin{cases}\hat{u}^{(-)}(\sigma),& \sigma\leq 0,\quad\quad\quad\quad 4x_{1}^{2}+9x_{2}^{2}\geq 1;\\ \varphi^{(+)}(\hat{v}^{(+)}(\tau))+\hat{u}^{(+)}(\sigma),&\sigma\geq 0,\ \tau\geq 0, \ 4x_{1}^{2}+9x_{2}^{2}\leq 1;\end{cases}  \tag{45}\] \[V_{0}(\mathbf{x},\varepsilon) \begin{cases}\vartheta^{(-)}(\tau),& \tau\leq 0,\ 4x_{1}^{2}+9x_{2}^{2}\geq 1;\\ \vartheta^{(+)}(\tau),&\tau\geq 0,\ 4x_{1}^{2}+9x_{2}^{2}\leq\ 1.\end{cases}