After working through this module, you should be able to:
Suppose a population begins at 20 individuals, grows at 8% per year, and lives in an environment that can support about 500 individuals.
The unconstrained model is
\[ \frac{dP}{dt}=rP. \]
It assumes that the same proportional growth rate continues no matter how large the population becomes. To represent limited food, space, or other resources, introduce a carrying capacity \(K\):
\[ \frac{dP}{dt}=rP\left(1-\frac{P}{K}\right). \]
| Quantity | Meaning | Example units |
|---|---|---|
| \(P\) | current population | individuals |
| \(P_0\) | initial population | individuals |
| \(r\) | unconstrained proportional growth rate | per year |
| \(K\) | carrying capacity | individuals |
| \(t\) | time | years |
| \(\Delta t\) | length of one simulation step | years |
The new factor
\[ 1-\frac{P}{K} \]
measures the fraction of the environment’s capacity that remains available.
| Current population | Constraint term | What the model does |
|---|---|---|
| \(P\) is small compared with \(K\) | close to 1 | behaves almost like unconstrained growth |
| \(P=K/2\) | \(1/2\) | growth remains positive but is reduced |
| \(P=K\) | 0 | population remains constant |
| \(P>K\) | negative | population decreases toward capacity |
flowchart LR
R[Growth rate r] --> G[Growth flow]
P[Population P] --> G
K[Carrying capacity K] --> C[Constraint 1 - P/K]
P --> C
C --> G
G --> Pflowchart LR R[Growth rate r] --> G[Growth flow] P[Population P] --> G K[Carrying capacity K] --> C[Constraint 1 - P/K] P --> C C --> G G --> P
Over a time step of length \(\Delta t\), approximate the change with
\[ \Delta P \approx rP_{old}\left(1-\frac{P_{old}}{K}\right)\Delta t. \]
Therefore,
\[ P_{new}=P_{old}+rP_{old}\left(1-\frac{P_{old}}{K}\right)\Delta t. \]
This is still the same reusable simulation pattern:
\[ \boxed{\text{new amount}=\text{old amount}+\text{rate of change}\times\text{time passed}} \]
Use \(P_{old}=20\), \(r=0.08\) per year, \(K=500\), and \(\Delta t=1\) year.
\[ \text{constraint}=1-\frac{20}{500}=0.96 \]
\[ \text{growth flow}=(0.08)(20)(0.96)=1.536 \]
\[ P_{new}=20+(1.536)(1)=21.536. \]
start_time <- 0 # years
end_time <- 100 # years
delta_t <- 1 # years per step
growth_rate <- 0.08 # per year
carrying_capacity <- 500 # individuals
initial_population <- 20 # individualstime <- seq(from = start_time, to = end_time, by = delta_t)population <- numeric(length(time))
population[1] <- initial_populationfor (i in 2:length(time)) {
population_old <- population[i - 1]
constraint <- 1 - population_old / carrying_capacity
growth <- growth_rate * population_old * constraint
change <- growth * delta_t
population[i] <- population_old + change
}The only change from Module 2.2 is the constraint term inside the growth flow.
results <- data.frame(time, population)
results$constraint <- 1 - results$population / carrying_capacity
results$growth <- growth_rate * results$population * results$constraint
head(results) time population constraint growth
1 0 20.00000 0.9600000 1.536000
2 1 21.53600 0.9569280 1.648672
3 2 23.18467 0.9536307 1.768769
4 3 24.95344 0.9500931 1.896647
5 4 26.85009 0.9462998 2.032659
6 5 28.88275 0.9422345 2.177146Check that the first update agrees with the hand calculation.
plot(
results$time,
results$population,
type = "l",
lwd = 3,
col = "#214f73",
ylim = c(0, carrying_capacity * 1.1),
xlab = "Time (years)",
ylab = "Population (individuals)",
main = "Constrained population growth"
)
abline(
h = carrying_capacity,
lty = 2,
lwd = 2,
col = "#b65335"
)
legend(
"bottomright",
legend = c("Population", "Carrying capacity"),
col = c("#214f73", "#b65335"),
lty = c(1, 2),
lwd = c(3, 2),
bty = "n"
)
The resulting S-shaped curve is called logistic growth. Growth is initially slow because the population is small, becomes faster as the population increases, and then slows as the population approaches carrying capacity.
target_population <- 0.90 * carrying_capacity
target_positions <- which(results$population >= target_population)
first_target <- target_positions[1]
results[c(first_target - 1, first_target), ] time population constraint growth
68 67 447.6636 0.10467276 3.748655
69 68 451.4123 0.09717545 3.509295Inspecting the crossing row and the preceding row confirms that we found the first simulated time at or above the target.
The growth column records the modeled flow at each time. Find its largest value:
fastest_position <- which(results$growth == max(results$growth))[1]
results[fastest_position, ] time population constraint growth
42 41 253.4035 0.4931931 9.998147For this model, growth is fastest when the population is near \(K/2\). Before that point, adding more individuals increases the total growth flow. After that point, resource limitation has the stronger effect.
Use the same initial population and growth rate, but remove the constraint:
unconstrained_population <- numeric(length(time))
unconstrained_population[1] <- initial_population
for (i in 2:length(time)) {
population_old <- unconstrained_population[i - 1]
growth <- growth_rate * population_old
change <- growth * delta_t
unconstrained_population[i] <- population_old + change
}plot(
time,
population,
type = "l",
lwd = 3,
col = "#214f73",
ylim = c(0, 1000),
xlab = "Time (years)",
ylab = "Population (individuals)",
main = "Constrained and unconstrained growth"
)
lines(time, unconstrained_population, lwd = 3, lty = 2, col = "#b65335")
abline(h = carrying_capacity, lty = 3)
legend(
"topleft",
legend = c("Constrained", "Unconstrained", "Capacity"),
col = c("#214f73", "#b65335", "black"),
lty = c(1, 2, 3),
lwd = c(3, 3, 1),
bty = "n"
)
The unconstrained model eventually exceeds any fixed bound. The constrained model instead approaches an equilibrium near \(K\).
Carrying capacity is not a wall that the population can never cross. It is an equilibrium toward which the model moves. Test an initial population above capacity:
above_population <- numeric(length(time))
above_population[1] <- 650
for (i in 2:length(time)) {
population_old <- above_population[i - 1]
constraint <- 1 - population_old / carrying_capacity
growth <- growth_rate * population_old * constraint
change <- growth * delta_t
above_population[i] <- population_old + change
}
plot(
time,
above_population,
type = "l",
lwd = 3,
col = "#6b4c8a",
ylim = c(0, 700),
xlab = "Time (years)",
ylab = "Population (individuals)",
main = "Starting above carrying capacity"
)
abline(h = carrying_capacity, lty = 2, col = "#b65335")
When \(P>K\), the constraint is negative, so the modeled change is negative. The population moves downward toward carrying capacity.
The logistic differential equation has the exact solution
\[ P(t)=\frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}. \]
results$exact <- carrying_capacity /
(1 + ((carrying_capacity - initial_population) / initial_population) *
exp(-growth_rate * results$time))
results$error <- results$population - results$exact
tail(results) time population constraint growth exact error
96 95 494.3547 0.011290587 0.4465244 494.0659 0.2888496
97 96 494.8012 0.010397538 0.4115772 494.5171 0.2841384
98 97 495.2128 0.009574384 0.3793086 494.9344 0.2784404
99 98 495.5921 0.008815767 0.3495220 495.3202 0.2719300
100 99 495.9416 0.008116723 0.3220337 495.6769 0.2647617
101 100 496.2637 0.007472656 0.2966726 496.0066 0.2570725As in Module 2.2, a smaller time step generally brings the Euler simulation closer to the analytical solution. That verifies the numerical implementation; it does not prove that the model’s assumptions fit a real population.
The constrained-growth model assumes:
Real populations may overshoot a changing capacity, experience delays, migrate, or fluctuate because of seasons and random events. The logistic model is useful because it captures one important feedback—not because it captures everything.
For each experiment, make a prediction before changing the code.