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The cubic saturation population model is the ordinary differential equationHere $n\ge 0$ represents a population density, $\alpha$ is the low-density per-capita growth rate, and $K$ is the positive saturation density. The per-capita inhibition is quadratic in population density. This makes a different density-dependence assumption from the logistic equation.

Let $f(n)=\alpha n-\beta n^3$. The nonnegative equilibria are $0$ and $K$. Since $f^{\prime }(0)=\alpha >0$ and $f^{\prime }(K)=-2\alpha <0$, the zero equilibrium is unstable to a positive perturbation and $K$ is asymptotically stable. Positive initial data remain positive and approach $K$.

The substitution $m=n^2$ gives $\dot{m}=2\alpha m-2\beta m^2$, a logistic equation. Consequently, for $n(0)=n_0>0$,This explicit solution shows the initial exponential population growth when $n_0\ll K$, followed by saturation. The model describes a chosen phenomenological inhibition law; a biological application must justify why the inhibition scales with the square of population density.

Population growth is change in population abundance through births, deaths and movement. For a stated region, the accounting identity iswhere the terms are birth, death, immigration and emigration counts per unit time. A growing local population need not have more births than deaths if immigration supplies the difference.

In a closed population with constant per-capita birth and death rates $b,d$, the identity becomes $\dot{N}=(b-d)N=rN$, giving exponential growth $N(t)=N_0{e}^{rt}$. For $r>0$ the doubling time is $\log 2/r$; a negative $r$ gives exponential decline. These are assumptions about rates, not a universal prediction for every biological population.

The logistic growth equation instead uses $\dot{N}=rN(1-N/K)$ with carrying capacity $K$. Its per-capita growth decreases with abundance, representing one form of negative density dependence. Resource limitation, age structure and environmental variation can require other mechanisms. In particular, two populations with equal present abundance but different reproductive age distributions need not have equal future population growth.

Thomas Robert Malthus made the comparison between population growth and the growth of food supply explicit in his 1798 An Essay on the Principle of Population by Thomas Robert Malthus. In Chapters I–II, he assumed that unchecked numbers could follow a geometric progression, taking doubling every 25 years as his rule, while food supply gained a fixed amount in each such period, an arithmetic progression. These were different hypotheses about two quantities, rather than a calculation from a single universal growth law.^[ref]

With equal initial scales, his comparison can be written as $N_k=N_0{2}^k$ and $F_k=F_0(1+k)$ after $k$ periods of 25 years. Then $N_k/F_k=(N_0/F_0)2^k/(1+k)$ grows without bound. The corresponding continuous exponential growth rate for numbers would be $\log (2)/25$ per year. The mathematics describes the consequence of the assumed rates; it does not establish that actual birth, death, human migration and food-production rates stay fixed. Demography can test population assumptions against counts, age composition and measured rates.