Limiting-profile conjecture for non-cooperative Fisher–KPP traveling waves

Let \boldsymbol{p}_{\eta}_{\eta>0} and c_{\eta}_{\eta\geq0} satisfy

$$\left\{\begin{matrix}(\eta,\mathbf{p}_{\eta},c_{\eta})\in\mathscr{E}&\text{for all }\eta>0,\\ c_{0}=\lim\limits_{\eta\to0}c_{\eta}.\end{matrix}\right.$$

Assume $(H_{7})$. Stability is understood as local asymptotic stability with respect to $(E_{KPP}^{0})$. Limiting-profile conjecture. (i) If both $\alpha_{1}\mathbf{e}_{1}$ and $\alpha_{2}\mathbf{e}_{2}$ are stable and $c_{\alpha_{1}\mathbf{e}_{1}\to\alpha_{2}\mathbf{e}_{2}}\neq0$, then, after suitable translations, the traveling waves converge in $(\mathscr{C}_{loc}^{2}(\mathbb{R},\mathbb{R}^{2})\cap\mathscr{L}^{\infty}(\mathbb{R},\mathbb{R}^{2}))\times\mathbb{R}$ as $\eta\to0$ to a semi-extinct traveling wave $(p_{0}\mathbf{e}_{i},c_{0})$ of $(E_{KPP})_{0}$ connecting $0$ to $\alpha_{i}\mathbf{e}_{i}$, where $i=1$ if $c_{\alpha_{1}\mathbf{e}_{1}\to\alpha_{2}\mathbf{e}_{2}}>0$ and $i=2$ if $c_{\alpha_{1}\mathbf{e}_{1}\to\alpha_{2}\mathbf{e}_{2}}<0$. (ii) If there is a unique stable state $v_{s}\in\{\alpha_{1}\mathbf{e}_{1},\alpha_{2}\mathbf{e}_{2},\mathbf{v}_{m}\}$, exactly one of the following occurs: after suitable translations, the waves converge to a component-wise monotonic traveling wave of $(E_{KPP})_{0}$ connecting $0$ to $v_{s}$; or there are two translations whose separation tends to $+\infty$, with one translated limit a semi-extinct front $(p_{front}\mathbf{e}_{i},c_{0})$ connecting $0$ to $\alpha_{i}\mathbf{e}_{i}$ and the other a component-wise monotonic traveling wave $(\mathbf{p}_{back},c_{0})$ connecting $\alpha_{i}\mathbf{e}_{i}$ to $v_{s}$, for a unique $i\in\mathsf{I}_{u}$.