Chemotactic instability on a rectangular domain ($D(s)=\alpha s^2+(1+\alpha \kappa -\beta \kappa \eta )s+\kappa$)

For cell diffusion, self-produced chemotactic agent and logistic proliferation, Neumann cosine modes reduce linear stability of the uniform state to a two-by-two matrix indexed by $s={\pi }^2(n^2/L_1^2+m^2/L_2^2)$. Its trace is negative and its determinant is $D(s)=\alpha s^2+(1+\alpha \kappa -\beta \kappa \eta )s+\kappa$. Chemotaxis can destabilize a mode precisely when $D(s)<0$, despite stability of the spatially uniform reaction system.

When $\alpha =\kappa =\eta =1$, the instability threshold of a spatial mode with squared wave number $s$ is $\beta =s+2+1/s$. On a rectangle $L_1=1$, $L_2=1/2$, modes varying in both coordinates have $s\ge 5{\pi }^2$. Since the threshold increases for $s>1$, the first such mode is $(n,m)=(1,1)$, at ${\beta }_{crit}=5{\pi }^2+2+(5{\pi }^2{)}^{-1}$. At equality it is neutral; strict instability begins above this value.