The prey-refuge conjecture for preventing blow-up in a three-species food chain

Consider the three-species food-chain model given by equations (1.3)–(1.1) on a bounded domain with Neumann boundary conditions. Let the parameters satisfy $c<\frac{w_3}{D_3}$, and let $(u_0,v_0,r_0)$ be an initial condition such that the solution $r$ of equation (1.3) blows up in finite time, meaning

$$\lim_{t\to T^*<\infty}\int_{\Omega}r(\mathbf{x},t)\,d\mathbf{x}=\infty.$$

For the modified model with a prey refuge, let $\Omega_1\subset\Omega$ denote a refuge patch. Prey-refuge conjecture. There exists a patch $\Omega_1$ such that every single refuge patch of measure at least $|\Omega_1|$, with the same parameter set and initial condition, yields globally existing solutions for the modified model. In particular, the corresponding solution $r$ does not blow up in finite time.

This conjecture asks whether a sufficiently large prey refuge can prevent blow-up that occurs without a refuge, and whether the required threshold depends only on the refuge measure rather than its geometry. The source presents this as an open question motivated by the change of sign in the coefficient of $r^2$ between refuge and non-refuge regions.