Alternative Density-Dependent Growth Laws

Why Compare Density-Dependent Growth Laws?

A growing cell population may slow as crowding, nutrient limitation, or other density-related effects increase. Several mathematical laws can represent this slowing, but they do not predict the same rate at every cell count. Comparing their per-capita growth rates helps reveal what each law assumes about density, while comparing trajectories shows how those assumptions matter over time.

The earlier material on logistic growth established the familiar model with a carrying capacity. Here, the focus is on how that model relates to a linear-crowding expression and how its predictions differ from saturating-inhibition and Gompertz growth. These laws are phenomenological descriptions: a good fit to cell counts does not, by itself, prove a particular biological mechanism.

Model Forms and Their Equilibria

Assume a single, well-mixed population whose count is represented by a continuous variable \(N(t)>0\). Each model below is autonomous: its parameters are constant over the time interval considered, and the rate depends on the current count rather than explicitly on time. These assumptions set aside random fluctuations, changing environments, and population mixtures.

The logistic model has a per-capita rate that declines linearly with count:

Here \(r>0\) is the low-density per-capita rate, in inverse time, and \(K>0\) is the positive equilibrium count, in cells. The rate is positive below \(K\), zero at \(K\), and negative above \(K\).

A linear-crowding expression writes the population rate as a linear increase minus a quadratic crowding term:

where \(c>0\) has units of inverse cells per unit time. This is not a distinct trajectory model from the logistic law: setting \(c=r/K\) makes the two equations identical. The different form can be useful when growth and crowding coefficients are the quantities being parameterized, but it does not create a new prediction.

One saturating-inhibition model lets the density-related reduction in per-capita growth approach a finite limit:

Here \(r>0\) is the rate as \(N\) approaches zero, \(q>0\) is the maximum density-related reduction, and \(H>0\) is a count scale. A finite positive equilibrium exists when \(q>r\), in which case

If \(q\le r\), the per-capita rate does not reach zero at any finite positive count, so this model has no positive finite equilibrium. When \(q>r\), its rate declines from \(r\) toward the negative limiting value \(r-q\) as \(N\) increases; unlike the logistic rate, it does not decline without bound.

The Gompertz law specifies a rate proportional to the logarithm of the ratio between equilibrium count and current count:

where \(a>0\) is an inverse-time parameter and \(K>0\) is the positive equilibrium. The rate is positive below \(K\) and negative above it. As \(N\) approaches zero from above, the Gompertz per-capita rate increases without bound; this is a mathematical feature to consider when using it at very low counts.

Worked Comparisons

Trajectories: A Derivation and a Numerical Comparison

For logistic growth, the trajectory can be obtained by separating variables when \(N\ne K\):

Let \(x=N/K\). Then \(dx/dt=rx(1-x)\), so separation and integration give \(\ln|x/(1-x)|=rt+C\) for \(x\ne1\). Applying \(x(0)=N_0/K\) and solving for \(N(t)\) yields

This formula applies to \(N_0>0\), including counts above \(K\). At \(N_0=K\), the solution is separately the constant trajectory \(N(t)=K\). The absolute value in the integrated logarithm is needed for initial counts above \(K\); omitting it would leave the real logarithm undefined in that case.

For Gompertz growth, set \(y=\ln(N/K)\). Since \(dy/dt=(1/N)dN/dt\), the model gives \(dy/dt=-ay\). Solving this linear equation and returning to \(N\) gives

The saturating-inhibition law also has a useful implicit trajectory. When \(q>r\), write \(K=rH/(q-r)\) and separate variables to obtain

The fraction decomposes as \(\frac{H/K}{N}+\frac{1+H/K}{K-N}\). Integrating between \(N_0\) and \(N(t)\), for a trajectory not at equilibrium, gives

This relation specifies the trajectory implicitly; it can be solved numerically for \(N(t)\). If \(N_0=K\), the trajectory is instead constant.

How to Choose and Diagnose a Growth Law

For each of these models with a finite positive equilibrium, the sign of \(g(N)\) indicates the direction of change on either side of that equilibrium. Below \(K\), these specified laws predict increasing counts; above \(K\), they predict decreasing counts. Their differences lie in the size and shape of the per-capita rate across density. Logistic growth becomes increasingly negative as \(N\) rises, saturating inhibition approaches a finite negative limit, and the Gompertz rate changes logarithmically and rises sharply as \(N\) becomes very small.

When diagnosing a model against serial cell counts, inspect more than whether the fitted equilibrium is plausible. Check whether predicted rates and trajectories track the observed changes at both low and high counts, whether estimated parameters imply the intended equilibrium, and whether residual discrepancies systematically vary with count or time. Predictions of negative counts, or extreme low-count rates that are biologically implausible, signal that the model or its domain of use needs attention. Sparse observations near a single density may not distinguish laws with similar local behavior.

The principal alternative to selecting one law is to fit several plausible forms and compare their predictive adequacy for the intended range of counts and times. This comparison should account for parameter uncertainty and avoid treating a better fit to the observed data as proof of a mechanism. The linear-crowding form should not be counted as an independent alternative to logistic growth when its parameters are mapped by \(c=r/K\). The saturating and Gompertz laws also have limitations: each imposes a particular shape on density dependence, and neither explicitly represents changing environments, multiple cell states, or stochasticity.

Common Pitfall