= Chemotactic instability on a rectangular domain
{title2=$D(s)=\alpha s^2+(1+\alpha\kappa-\beta\kappa\eta)s+\kappa$}
For <diffusion>[cell diffusion], self-produced <chemotaxis>[chemotactic agent] and <logistic differential equation>[logistic proliferation], <Neumann boundary condition>[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 <Lyapunov stability>[stability] of the spatially uniform reaction system.
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