Conjecture on uniform convergence under weak competition and fear

Let $(u,v)$ solve the reaction diffusion system referenced as

, with fear function $k(x)$ satisfying the assumptions in

. Suppose

$$\widetilde{k}<\frac{1}{a_{2}}\left(\frac{b_{1}b_{2}}{c_{1}}-c_{2}\right),$$

and the parameters satisfy the restrictions in

$$and Theorem~$$

. Uniform-convergence conjecture. The solution $(u,v)$ converges uniformly to the spatially homogeneous state $(u^*,v^*)$ as $t\to\infty$. This conjecture is motivated by numerical simulations and follows a preceding lemma for suitably chosen fear functions, but its general validity is not established in the supplied text.