What Population Counts Can Tell Us
A population that grows by a roughly constant proportion per unit time may be compatible with exponential growth. If growth slows as the population becomes larger, density dependence is a possible explanation: competition, limited resources, or other mechanisms may reduce birth rates or increase death rates. Counts alone do not reveal which mechanism is operating, but changes in per-capita growth and variability provide useful diagnostics.
We describe population size by a continuous-time birth-death process \((X_t)_{t\ge0}\) on the state space \(S=\{0,1,2,\ldots\}\), defined on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), with filtration \((\mathcal{F}_t)_{t\ge0}\). At state \(n\), a birth changes the population to \(n+1\) at rate \(\lambda_n\), and a death changes it to \(n-1\) at rate \(\mu_n\), where \(\mu_0=0\). Rates are nonnegative and finite at each state. The instantaneous drift is the birth rate minus the death rate; the instantaneous variance rate depends on both rates, not just their difference.
For observations at times separated by \(h>0\), the change is \(\Delta=X_{t+h}-X_t\). The finite-interval quantity \(\Delta/(hX_t)\), when \(X_t>0\), is a realized per-capita change per unit time. It is not automatically an estimate of the infinitesimal drift: over a finite interval, multiple births and deaths may occur, and the population may move through several states. The calculations below distinguish this descriptive quantity from local model rates.
Local Drift and Variability
Write \(N_h\) for the number of jumps in \((t,t+h]\). To justify extracting local moments from one-jump probabilities, assume that, conditional on \(X_t=n\), the probabilities of one birth and one death are respectively \(\lambda_n h+o(h)\) and \(\mu_n h+o(h)\), and that \(\mathbb{P}(N_h\ge2\mid X_t=n)=o(h)\). Also assume the moment-tail condition \(\mathbb{E}[\Delta^2\mathbf{1}_{\{N_h\ge2\}}\mid X_t=n]=o(h)\). This last condition controls the size as well as the probability of increments from multiple-jump paths; the probability condition alone is not enough for moment calculations.
Proof. Put \(A=\{N_h\ge2\}\). On the one-birth event, \(\Delta=1\); on the one-death event, \(\Delta=-1\); and on the no-jump event, \(\Delta=0\). The contributions to the first moment from \(A\) have absolute value at most \(\mathbb{E}[|\Delta|\mathbf{1}_A\mid X_t=n]\). By Cauchy-Schwarz, this is at most \(\{\mathbb{E}[\Delta^2\mathbf{1}_A\mid X_t=n]\mathbb{P}(A\mid X_t=n)\}^{1/2}=o(h)\), since both factors inside the product are \(o(h)\). The contribution to the second moment from \(A\) is \(o(h)\) by the stated moment-tail condition. Thus the one-birth and one-death probabilities give \(\mathbb{E}[\Delta\mid X_t=n]=\lambda_n h-\mu_n h+o(h)\) and \(\mathbb{E}[\Delta^2\mid X_t=n]=\lambda_n h+\mu_n h+o(h)\). Finally, \(\operatorname{Var}(\Delta\mid X_t=n)=\mathbb{E}[\Delta^2\mid X_t=n]-\mathbb{E}[\Delta\mid X_t=n]^2\); the squared mean is \(O(h^2)\), so divided by \(h\) it tends to zero. This proves both limits. \(\square\)
The proposition gives two different signals. The drift \(b_n=\lambda_n-\mu_n\) is the local expected change per unit time. The variance rate \(a_n=\lambda_n+\mu_n\) measures local fluctuation intensity. Births and deaths can nearly cancel in the drift while both contribute positively to variability. For \(n>0\), per-capita drift is \(b_n/n\), while the local variance rate of the relative change \(\Delta/n\) is \(a_n/n^2\).
Proof. In the exponential model, \(b_n=\lambda n-\mu n=n(\lambda-\mu)\); division by \(n>0\) gives \(\lambda-\mu\). In the density-dependent model, \(b_n=\beta n(1-n/K)-\delta n=n\{\beta(1-n/K)-\delta\}\), so division by positive \(n\) gives the stated expression. For two positive states \(n_1<n_2\), the difference between the latter per-capita drifts is \(-\beta(n_2-n_1)/K\), which is negative when \(\beta>0\). If \(\beta=0\), it is constant instead. \(\square\)
Worked Diagnostics
Worked Example: A Trajectory with Slowing Per-Capita Change
Suppose one population is observed at equally spaced times, with counts \(10,20,30,40\). Compute the realized per-capita change per interval and describe what the pattern suggests.
The successive absolute changes are \(20-10=10\), \(30-20=10\), and \(40-30=10\). Dividing each by the starting count for that interval gives \(10/10=1\), \(10/20=1/2\), and \(10/30=1/3\). Because the intervals have equal length, dividing all three values by the common interval length would not change their ordering. The observed absolute change is constant while the realized per-capita change falls.
Check: The counts increase by \(10\) at each step, since \(20-10=10\), \(30-20=10\), and \(40-30=10\). The per-capita values are \(1\), \(1/2\), and \(1/3\), respectively. This pattern is consistent with slowing per-capita growth, but one trajectory does not establish density dependence: random fluctuations, observation error, or changing conditions can also produce it.
Worked Example: Comparing Drift and Variability Rates
Compare an exponential model with \(\lambda=3/2\), \(\mu=1\), at population size \(n=4\), and a density-dependent model with \(\beta=2\), \(K=10\), and \(\delta=1/2\), at sizes \(n=2\) and \(n=8\). Compute the local drift, variance rate, per-capita drift, and relative variance rate.
For the exponential model at \(n=4\), the birth rate is \((3/2)4=6\) and the death rate is \(1\cdot4=4\). Thus the drift is \(6-4=2\), the variance rate is \(6+4=10\), the per-capita drift is \(2/4=1/2\), and the relative variance rate is \(10/4^2=10/16=5/8\).
For the density-dependent model at \(n=2\), the birth rate is \(2\cdot2(1-2/10)=4(4/5)=16/5\), and the death rate is \((1/2)2=1\). Therefore the drift is \(16/5-1=11/5\), the variance rate is \(16/5+1=21/5\), the per-capita drift is \((11/5)/2=11/10\), and the relative variance rate is \((21/5)/2^2=21/20\).
At \(n=8\), the birth rate is \(2\cdot8(1-8/10)=16(1/5)=16/5\), and the death rate is \((1/2)8=4\). The drift is \(16/5-4=-4/5\); the variance rate is \(16/5+4=36/5\); the per-capita drift is \((-4/5)/8=-1/10\); and the relative variance rate is \((36/5)/8^2=36/320=9/80\).
Check: In the exponential model, the per-capita drift is \(3/2-1=1/2\), agreeing with \(2/4\). In the density-dependent model the formula gives \(2(1-2/10)-1/2=2(4/5)-1/2=8/5-1/2=11/10\) at \(n=2\), and \(2(1-8/10)-1/2=2(1/5)-1/2=2/5-1/2=-1/10\) at \(n=8\). The calculations show that drift and variability need not move in the same direction: at \(n=8\), the drift is negative but the variance rate remains positive.
Worked Example: Replicate Changes at Two Population Sizes
Suppose six replicate intervals of length \(h=1\) start at \(n=2\) and produce changes \(-1,0,1,0,1,-1\). Six other intervals, also of length \(1\), start at \(n=8\) and produce changes \(-2,-1,0,1,2,0\). Compute the empirical mean and central second moment of the changes, and compare relative variability.
For the first group, the sum of changes is \(-1+0+1+0+1-1=0\), so the empirical mean is \(0/6=0\). The sum of squared changes is \(1+0+1+0+1+1=4\), so the empirical central second moment, using divisor \(6\), is \(4/6=2/3\). For the second group, the sum is \(-2-1+0+1+2+0=0\), giving mean \(0/6=0\). The sum of squared changes is \(4+1+0+1+4+0=10\), giving central second moment \(10/6=5/3\).
Using these empirical second moments as descriptive measures, the relative quantities are \((2/3)/2^2=(2/3)/4=1/6\) for the first group and \((5/3)/8^2=(5/3)/64=5/192\) for the second. The larger group has a larger raw second moment, \(5/3>2/3\), but a smaller second moment relative to squared population size, since \(1/6=32/192>5/192\).
Check: Both empirical means are zero, so the empirical central second moments equal the corresponding average squared changes: \(4/6=2/3\) and \(10/6=5/3\). The relative calculations use the starting sizes squared, \(2^2=4\) and \(8^2=64\), and give \(1/6\) and \(5/192\). These are finite-interval descriptive summaries, not exact instantaneous variance rates.
Limits of the Diagnostic
Observed counts are noisy realizations, not the model's drift function. A single sequence can show an increasing, decreasing, or irregular per-capita change even when the underlying rates follow one of these simple models. Replicate intervals starting at comparable population sizes help separate average change from variability, but intervals must be interpreted in light of their duration: longer intervals allow more jumps and more movement between states. The local moment result requires control of multiple-jump contributions; a small probability of multiple jumps by itself does not justify discarding their contribution to a moment.
Model assumptions also matter. A finite-capacity birth rate is only one form of density dependence; different biological mechanisms imply different functions of \(n\). Environmental changes can alter rates over time, while unrecorded immigration or departures can change counts without matching the specified birth-death mechanism. Measurement error can obscure small changes or create apparent fluctuations. And a trajectory that appears to level off does not, by itself, identify a capacity \(K\) or establish the particular rate formula used here.
Common mistake. For the density-dependent example, it is incorrect to calculate per-capita drift as just the birth rate divided by \(n\), thereby omitting deaths. At \(n=8\), that incorrect calculation gives \((16/5)/8=2/5\), a positive value. The correct calculation includes the death rate: \((16/5-4)/8=(-4/5)/8=-1/10\). Births alone do not determine net growth.
A practical diagnosis therefore compares several features: the direction of per-capita drift across population sizes, the variability of counts relative to their size, and whether the observed pattern persists across replicates and time periods. Agreement with a model's qualitative prediction is evidence of compatibility, not proof that the model is uniquely correct.
Check Your Understanding
Use the rate formulas and distinguish local model quantities from finite-interval observations.
- In the exponential model with birth rate \(\lambda n\) and death rate \(\mu n\), what are the per-capita drift and local variance rate at state \(n>0\)?
- For the density-dependent model, calculate the per-capita drift at \(n=5\) when \(\beta=2\), \(K=10\), and \(\delta=1/2\).
- Why does the probability of two or more jumps being \(o(h)\) not, by itself, control their contribution to the second moment?
- If two groups have different raw increment variances, which population-size normalization corresponds to the local variance rate of relative change?
- Give one reason why a single observed decline in per-capita change does not establish density dependence.