= Solution
For $q=\eta=1$, $J_*$ has <eigenvectors> $(1,1)$ and $(1,-1)$ with <eigenvalues> $-1$ and $-1/3$. The Neumann spatial modes are $\cos(j\pi x/L)$, $j=0,1,\ldots$. Combining each with those two reaction directions gives temporal rates
$$
-1-(j\pi/L)^2,\qquad-1/3-(j\pi/L)^2.
$$
Every linear perturbation therefore decays at least as fast as $e^{-t/3}$. The slowest mode is the spatially uniform antisymmetric change of the two <biological populations>. Thus
$$
\boxed{m-2/3=O(e^{-t/3}),\qquad n-2/3=O(e^{-t/3}).}
$$
Small nonlinear perturbations have the same bound, not just a formal mode calculation. In one spatial dimension take sufficiently small smooth Neumann data, or small data in a norm such as $H^1$ that controls quadratic products. The linear semigroup decays at rate $1/3$, and the nonlinear remainder is quadratic. Standard small-data variation-of-constants estimates first give decay $e^{-ct}$ for any fixed $c<1/3$. Choose $1/6<c<1/3$; then the remainder is $O(e^{-2ct})$, and its Duhamel convolution with $e^{-(t-s)/3}$ is $O(e^{-t/3})$ because $2c>1/3$. This sharpens the preliminary bound to the displayed rate. Special initial data with no slow-mode component may decay faster, but the general estimate cannot be improved uniformly.
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