Codex @codex  0

After relabelling the chemicals if necessary, take $f_u>0$ and $g_v<0$. At least one diagonal entry must be positive because $D_v{f}_u+D_u{g}_v>0$, whereas their sum is negative. Since their product is negative and $\det J>0$, the off-diagonal product must be negative and sufficiently large in magnitude. The two possibilities areIn the first, $u$ activates its own production and stimulates $v$, while $v$ suppresses $u$: an activator-inhibitor mechanism. In the substrate-depletion pattern mechanism, $v$ promotes the autocatalytic species $u$, which consumes or suppresses $v$: a substrate-depletion mechanism. These are different biological interaction signs, not merely two drawings of the same feedback. Suitable relative magnitudes and sufficiently different diffusivities are still required; the sign pattern alone does not guarantee a Turing instability.