Symmetric competition model ($\dot{x}=\lambda x-\mu xy,\dot{y}=\lambda y-\mu xy$)

Two populations grow at the same per-capita rate and each loses members at a rate proportional to their product. On the invariant diagonal $x=y=z$, the logistic differential equation gives $z(t)=\lambda /[\mu +(\lambda -\mu )e^{-\lambda t}]$ for $z(0)=1$ and positive rates. The positive symmetric equilibrium $(\lambda /\mu ,\lambda /\mu )$ is stable along that diagonal but unstable transversely: $(x-y{)}^{\prime }=\lambda (x-y)$, while the Jacobian matrix has eigenvalues $-\lambda ,+\lambda$. Thus an exactly symmetric deterministic solution can relax toward coexistence while a small imbalance grows. A corresponding density-dependent Markov jump process has a controlled fluid limit on each fixed finite interval, rather than uniform control for all time.