= Solution
The fastest temporal growth occurs for the lowest spatial <eigenvalue>, with $p=q=r=1$. Its growth rate is
$$
s_{111}=\gamma-\frac{3\mu\pi^2}{L^2}.
$$
For a generic initial condition its coefficient is nonzero, so this term determines the long-time behavior. In fact, every nonzero, nonnegative initial density has $a_{111}>0$, since the product of the three first sine <eigenfunctions> is strictly positive in the interior. Consequently
$$
\boxed{L_{\mathrm{crit}}=\pi\sqrt{\frac{3\mu}{\gamma}},\qquad \text{exponential growth precisely when }L>L_{\mathrm{crit}}.}
$$
For $L<L_{\mathrm{crit}}$ every term decays. At equality the first term is stationary and all higher terms decay: equality does not produce exponential growth. This is a <reaction-diffusion equation> threshold arising from competition between production and loss through the <Dirichlet boundary condition>.
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