= Symmetric competition model
{title2=$\dot x=\lambda x-\mu xy,\quad\dot y=\lambda y-\mu xy$}
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: $(x-y)'=\lambda(x-y)$, 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.
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