The fundamental principle of linear stability analysis for the reaction-diffusion equation is that the stability of a stationary solution $\tilde{u}$ is determined by the sign of the linearization coefficient, $f^{\prime}(\bar{u})$. The system's deviation from the stationary state, $v$, follows the linear PDE $\frac{\partial v}{\partial t}=D \nabla^2 v+f^{\prime}(\tilde{u}) v$. Utilizing the logistic growth model $f(u)=0.5 u(1-u)$, two stability scenarios were demonstrated: the non-trivial stationary solution $\tilde{u}=1$ is stable because $f^{\prime}(1)=-0.5$ is negative, causing the initial deviation $v$ to decay exponentially; conversely, the trivial solution $\tilde{u}=0$ is unstable because $f^{\prime}(0)=+0.5$ is positive, causing $v$ to grow exponentially away from zero. In both cases, the diffusion term ensures the perturbation is smoothed and spread across the spatial domain.
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