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Defineso $S=N(1-\theta )$ and $I=N\theta$. Part (a) already givesUsing the quotient rule,and hence the reduced Plant SI model with logistic total population is

For every $N(0)>0$, logistic growth gives $N(t)\to 1$. The asymptotic prevalence equation is thereforeIf $\beta <1$, its only nonnegative equilibrium is $\theta =0$, which attracts all prevalences. If $\beta >1$, $heta=0$ is unstable and the stable value is ${\theta }_{\ast }=1-1/\beta$. Combining these values with $N_{\ast }=1$ recovers exactly the equilibria and outcomes in part (b).

Insertinto the continuity equation. Its zeroth-order part is the homogeneous cosmological continuity equationSince $\bar{\rho }$ is spatially homogeneous, the terms of first order in $ϵ$ giveOn the other hand, the quotient rule and the background equation implySubstitution of the first-order equation yields the linearized cosmological continuity equationThe ratio $\delta \rho /\bar{\rho }$ is the density contrast.

Letbe the vertical absolute vorticity. Taking the vertical curl of the momentum equation removes both the tidal potential and the isothermal pressure force, because each is a gradient field. The two-dimensional vorticity equation isThe surface-density equation isThe quotient rule now givesSince $\zeta =W/\Sigma$ is the vortensity,