Critical-domain-length conjecture for controllability of competitive systems

Consider the diffusive Lotka–Volterra strongly competitive system on an interval $(0,L)$ with boundary-constrained controls, parameters $a>0$ and $b>1$ satisfying $b>a$, initial data $y_1^0,y_2^0\in\mathcal{C}^0((0,L);[0,1])$, and equilibria $(0,1)$ and $(w_1^*,w_2^*)$. A barrier is a solution preventing controlled trajectories from reaching the relevant target. Critical-domain-length conjecture. For every $a>0$ and $b>1$ with $b>a$, there exists $L^*>\pi$ such that: for every $0<L<L^*$ and every initial data $y_1^0,y_2^0\in\mathcal{C}^0((0,L);[0,1])$, the system is asymptotically controllable to $(0,1)$; whereas for every $L>L^*$, a barrier exists, and consequently some initial data $y_1^0,y_2^0\in\mathcal{C}^0((0,L);[0,1])$ cannot be controlled to $(0,1)$ or $(w_1^*,w_2^*)$. The conjecture formalizes the observed transition from controllability on shorter intervals to barriers on sufficiently long intervals. The paper explains that numerical simulations support this behavior, but a rigorous proof is missing because an analytical construction of a barrier for a suitable $L^*$ is not known.