5.1 Population Growth Models

Population growth models give ecologists a way to describe and predict how population sizes change over time. From simple exponential models to logistic models that factor in environmental limits, these tools are foundational for conservation planning, pest management, and understanding how species interact with their environments.

Exponential vs Logistic Growth

The two core models you need to know differ in one key assumption: whether resources are limited.

Exponential growth assumes unlimited resources. With nothing slowing reproduction down, the population grows faster and faster, producing a characteristic J-shaped curve. Think of bacteria dividing in a fresh petri dish with plenty of nutrients.

Logistic growth adds realism by incorporating carrying capacity (K), the maximum population size an environment can sustain. Growth starts fast but slows as the population approaches K, producing an S-shaped curve. A deer population in a forest with finite food and space follows this pattern more closely.

Real populations rarely follow either model perfectly. Predator-prey interactions, disease outbreaks, and seasonal changes all create deviations. Still, these two models are the starting framework for nearly everything else in population ecology.

Mathematical Representations

The exponential growth equation:

$\frac{dN}{dt} = rN$

• r = intrinsic rate of increase (per capita growth rate under ideal conditions)
• N = current population size

The logistic growth equation:

$\frac{dN}{dt} = rN\left(\frac{K - N}{K}\right)$

• K = carrying capacity

Notice that the logistic equation is just the exponential equation multiplied by $\left(\frac{K - N}{K}\right)$. That fraction acts as a brake. When N is small relative to K, the fraction is close to 1 and growth looks nearly exponential. As N approaches K, the fraction shrinks toward 0 and growth stalls. If N ever exceeds K, the fraction goes negative and the population declines.

Population Growth Curves

Growth Phases

When you plot population size over time, most populations move through recognizable phases:

1. Lag phase — Slow initial growth. The population is small and still establishing itself. A newly introduced species in a new habitat often shows this.
2. Exponential (log) phase — Rapid increase. Resources are abundant and competition is low. Algal blooms in nutrient-rich water are a classic example.
3. Deceleration phase — Growth rate slows as resources become scarcer and competition, predation, or disease intensifies.
4. Stationary (plateau) phase — Population size levels off near carrying capacity. An established bird population in a stable ecosystem fluctuates around K.
5. Decline phase — Can occur if the population overshoots K and depletes resources, or if a major environmental stressor hits.

Not every population passes through all five phases. Some crash before reaching a plateau; others oscillate around K indefinitely.

Curve Analysis

When you look at a population growth curve, here's what to pay attention to:

• The steepness of the exponential phase reflects the intrinsic growth rate (r). A steeper rise means a higher r.
• The height of the plateau indicates the carrying capacity (K).
• Fluctuations around the plateau often signal density-dependent factors at work, such as predator-prey cycles (e.g., lynx and snowshoe hare populations cycling together).
• Comparing growth curves across species or locations can reveal differences in resource availability, competitive ability, or environmental quality.

Calculating Growth Rates

Basic Growth Rate Calculations

Instantaneous growth rate (r) can be rearranged from the exponential model:

$r = \frac{1}{N} \cdot \frac{dN}{dt}$

This tells you the per capita rate of population change at a specific moment.

Finite rate of increase (λ) converts r into a multiplicative factor:

$\lambda = e^r$

If $\lambda = 1.05$, the population grows by 5% per time step. If $\lambda < 1$, the population is shrinking.

Per capita growth rate over a discrete interval:

$\text{per capita growth rate} = \frac{N_2 - N_1}{N_1 \cdot t}$

where $N_1$ and $N_2$ are population sizes at two time points and $t$ is the time between them.

Useful Derived Metrics

Doubling time — how long it takes a population to double in size:

$t_d = \frac{\ln(2)}{r}$

A population with $r = 0.1$ per year has a doubling time of about 6.93 years.

Generation time (T) — the average time between a parent's reproduction and its offspring's reproduction:

$T = \frac{\ln(R_0)}{r}$

where $R_0$ is the net reproductive rate (the average number of offspring a female produces over her lifetime). These metrics are especially useful for comparing species: organisms with short generation times and high r values (like insects) can grow explosively, while those with long generation times (like elephants) grow much more slowly.

Predicting Population Sizes

Exponential and Logistic Predictions

To predict future population size under exponential growth:

$N_t = N_0 \cdot e^{rt}$

• $N_0$ = initial population size
• $r$ = intrinsic growth rate
• $t$ = time elapsed

For logistic growth, the prediction equation is:

$N_t = \frac{K}{1 + \left(\frac{K - N_0}{N_0}\right) \cdot e^{-rt}}$

For populations with non-overlapping generations (annual plants, some insects), a discrete-time geometric model works better:

$N_{t+1} = N_t \cdot \lambda$

This simply multiplies the current population by the finite rate of increase each generation.

Beyond the Basic Models

The exponential and logistic equations treat all individuals as identical. Real populations have age structure, size classes, and variable survival rates. More advanced approaches account for this:

• Matrix population models divide the population into age or stage classes and use matrices to project growth. These are common for managing forest tree populations or species with distinct juvenile and adult stages.
• Stochastic models add random variation to growth parameters, producing a range of possible outcomes rather than a single prediction. These are valuable for wildlife populations affected by unpredictable events like droughts or storms.

The accuracy of any prediction depends on how well the model's assumptions match reality and how reliable the input data is. For high-stakes decisions (endangered species recovery, invasive species control), ecologists often compare results from multiple models to gauge uncertainty and improve confidence in their forecasts.