Bounded and Unbounded Population Growth

Why Boundedness Depends on the Growth Law

A cell count can rise rapidly without establishing that growth will continue without limit. A model with a constant net growth rate predicts exponential increase whenever proliferation exceeds loss. A density-regulated model instead reduces per-capita growth as the population grows, which can produce a finite limiting population. The distinction matters when extrapolating cell-count trajectories: the same short-term increase may be consistent with very different long-term behavior.

Here, \(N(t)\) denotes the number of cells at time \(t\), measured in cells. We consider deterministic, continuous-time models with \(N(0)=N_0>0\). A trajectory is bounded if there is a finite constant \(M\) such that \(N(t)\leq M\) for every \(t\geq 0\). It is unbounded if no such constant exists. These definitions concern the model's mathematical prediction, not a claim that a real culture can contain infinitely many cells.

Constant Net Growth: A Linear ODE With Exponential Trajectories

In the constant-net-growth model, \(g\) is proliferation minus loss per cell per unit time, assumed constant and independent of population size. The model also assumes a well-mixed population and no external cell input or removal beyond the processes represented in \(g\). Separating variables and integrating gives \(\ln(N(t)/N_0)=gt\), so the trajectory is \(N(t)=N_0e^{gt}\). Although the differential equation is linear in \(N\), its solution is exponential.

The sign of \(g\) determines boundedness. If \(g>0\), then \(e^{gt}\) grows without limit as \(t\) increases, so \(N(t)\) is unbounded. If \(g=0\), the population remains at \(N_0\). If \(g<0\), the population decreases toward zero and is bounded above by \(N_0\). This last case describes declining cell abundance, not extinction at a finite time: the continuous exponential trajectory remains positive for every finite \(t\).

Density Regulation: A Finite Bound From Negative Feedback

The logistic model makes per-capita growth depend on population size: \(N^{-1}dN/dt=r(1-N/K)\). For \(r>0\) and \(K>0\), per-capita growth is positive when \(0<N<K\), zero at \(N=K\), and negative when \(N>K\). These assumptions represent a stable density-dependent limitation: crowding or resource limitation increasingly offsets growth as the population approaches capacity. The model has no immigration, discrete division events, or explicit delay in that feedback.

Separating variables and integrating the logistic equation yields the trajectory below. The constant is set by \(N(0)=N_0\), giving the stated solution.

The solution makes the bound precise. If \(N_0<K\), then \(K/N_0-1>0\); the denominator is at least 1, so \(0<N(t)\leq K\). If \(N_0>K\), then \(-1<K/N_0-1<0\). Since \(0<e^{-rt}\leq1\), the denominator lies between \(K/N_0\) and 1, so \(K\leq N(t)\leq N_0\). In both cases the denominator remains positive, ruling out a singularity and establishing a finite upper bound. As \(t\) tends to infinity, \(e^{-rt}\) tends to zero and \(N(t)\) tends to \(K\).

Reading Boundedness From Birth and Death Parameters

A useful density-regulated form separates low-density proliferation and loss from a crowding penalty: \(\frac{dN}{dt}=(b_0-d_0)N-cN^2\). Here \(b_0\) and \(d_0\) are nonnegative low-density rates in time\(^{-1}\), and \(c>0\) is a density-feedback coefficient in cells\(^{-1}\) time\(^{-1}\). The quadratic term means that the per-capita net rate, \(b_0-d_0-cN\), declines as \(N\) rises.

Write \(r=b_0-d_0\). When \(r>0\), the equation is logistic with \(K=r/c\), so a positive initial population remains bounded. When \(r=0\), the equation becomes \(dN/dt=-cN^2\), whose solution is \(N(t)=N_0/(1+cN_0t)\); it declines and is bounded by \(N_0\). When \(r<0\), both \(rN\) and \(-cN^2\) are negative for \(N>0\), so the population decreases and is likewise bounded. Thus \(c>0\) prevents unbounded positive growth in all three cases, although only \(r>0\) gives a positive equilibrium capacity.

Diagnostics, Alternatives, and Limits

A finite sequence of increasing counts cannot by itself establish that growth is unbounded: a density-regulated population can increase for a long time while still approaching a finite limit. Conversely, a short apparent plateau does not prove that the limit is a stable carrying capacity. Measurement variation, changing proliferation or death rates, depletion of a resource, and changes in the culture environment can all produce patterns that a constant-parameter model misses.

Inspect whether the assumed mechanism is consistent with the setting. For constant net growth, a plot of log count against time should be approximately linear if the model is appropriate, because \(\ln N(t)=\ln N_0+gt\). For logistic growth, estimates of per-capita growth should decline with density and approach zero near the proposed \(K\). Systematic curvature in the log-count plot challenges constant \(g\); continued positive per-capita growth near the fitted capacity challenges the logistic interpretation. These diagnostics can reveal mismatch but do not, by themselves, identify its cause.

The logistic model is a compact alternative to constant net growth when density dependence is plausible, but its simplicity is also a limitation. It treats feedback as immediate and smooth, and its fitted \(K\) is conditional on the modeled conditions. A population can exceed an estimated capacity temporarily if assumptions change or if the model is not appropriate; the theorem guarantees boundedness only under the stated logistic model and parameter conditions. More detailed mechanisms may be needed when resources, compartments, or time-varying rates matter.

Common Mistake

A common error is to reason that “the ODE is linear, so cell count grows linearly in time.” The correction is to distinguish the equation's dependence on \(N\) from the shape of its solution: \(dN/dt=gN\) is linear in \(N\), but solving it gives \(N(t)=N_0e^{gt}\). If \(g>0\), that solution is unbounded. Boundedness follows only when the rate or another modeled mechanism prevents indefinite positive growth; it is not implied by calling an equation linear.