Population Ecology
Exponential, logistic, and stochastic growth models
2026-04-09
3-part lecture (with breaks)
Part 1 Exponential growth
Part 2 Logistic growth — density dependent
Part 3 Logistic growth with stochasticity
Part 1: Why study population growth?
- Conservation of threatened species
- Sustainable use of natural resources
- Informing management of invasive species, fisheries, and epidemics
Part 1: Exponential population growth
Density independent growth \[\frac{dN}{dt}=rN\]
Exponential growth
What is required for a population to grow?
How many births and how many deaths?
\[N_{t+1} = N_t + B - D + I - E\]
- \(B\) = Births
- \(D\) = Deaths
- \(I\) = Immigration
- \(E\) = Emigration
If we assume immigration and emigration are equal, the change in population size simplifies to:
\[\Delta N = B - D\]
Exponential growth
\[\Delta N = B - D\]
- More births than deaths: the population grows
- More deaths than births: the population declines
Exponential growth
Change in population (\(dN\)) over a very small interval of time (\(dt\)):
\[\frac{dN}{dt}=B-D\]
Births and deaths are expressed as per-capita rates:
\[B = bN\] \(b\) = instantaneous birth rate
[births / (individual · time)]
\[D = dN\]
\(d\) = instantaneous death rate
[deaths / (individual · time)]
Exponential growth
Substituting gives change in population over time:
\[\frac{dN}{dt}=(b-d)N\]
\[\frac{dN}{dt}=(0.55-0.50)N\]
\[\frac{dN}{dt}=(0.55-0.50)\times100\]
\[\frac{dN}{dt}=0.05\times100\]
Exponential growth
Letting \(r = b - d\), the intrinsic rate of increase, gives the continuous exponential growth equation:
\[\frac{dN}{dt}=rN\]
Exponential growth
\(N\) = population size
\(r\) = intrinsic rate of increase
\(r\) determines whether a population grows or declines:
- \(r = 0\) no change
- \(r > 0\) population grows
- \(r < 0\) population declines
The differential equation describes the growth rate, not the population size.
Exponential growth
The size of an exponentially growing population at time \(t\):
\[N_t = N_0e^{rt}\]
The discrete version gives population size per time-step:
\[N_{t+1} = N_t + r_dN_t\]
\[N_{t+1} = 100 + 0.05 \times 100 = 105\]
\(N_t\) = population size at time \(t\)
\(r_d\) = discrete growth rate
Exponential growth
Theoretical populations with different values of \(r\)
- \(r = 0\) no change
- \(r > 0\) growth
- \(r < 0\) decline
Taking the natural log of population size linearises exponential growth.
Exponential growth
Log of exponential growth
Growth rates across species
| Virus |
300.0 |
3.3 minutes |
| Bacterium |
58.7 |
17 minutes |
| Protozoan |
1.59 |
10.5 hours |
| Hydra |
0.34 |
2 days |
| Flour beetle |
0.101 |
6.9 days |
| Brown rat |
0.0148 |
46.8 days |
| Domestic cow |
0.001 |
1.9 years |
| Mangrove |
0.00055 |
3.5 years |
| Southern beech |
0.000075 |
25.3 years |
Growth rate vs population size
Growth rate over population size increases proportionally — not absolutely.
Exponential growth — summary
Exponentially growing populations have a doubling time determined by \(r\) — not fixed at one year
\(r\) is constant, so growth is unbounded
Surely no species can grow exponentially forever?
- Correct! Enter density dependence — Part 2.
Assumptions of the exponential model
Take 5 minutes to discuss the assumptions of the exponential growth model
( ͡° ͜ʖ ͡°)
Part 2: Logistic population growth
Density dependent growth \[\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)\]
Logistic growth
Populations do not grow forever — they reach a carrying capacity (\(K\)).
\(K\) = the maximum population size supported by available resources (food, shelter, space).
\[\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)\]
\(N\) = population size \(r\) = intrinsic rate of increase \(K\) = carrying capacity
Logistic growth
The term \(\left(1 - \frac{N}{K}\right)\) acts as a density penalty:
- Crowded populations incur a large penalty; sparse populations incur a small one.
\[\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)\]
When \(N = K\): \(\quad\frac{N}{K} = 1\), so \(\left(1 - \frac{N}{K}\right) = 0\), and \(\frac{dN}{dt} = 0\)
The population stabilises at \(K\)
Logistic growth
When \(N\) is small relative to \(K\), the penalty is small and growth is nearly exponential.
\[\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)\]
As \(N\) grows, a larger fraction of capacity is used and the penalty increases:
- \(K = 100\), \(N = 7\): unused capacity \(= 1 - (7/100) = 0.93\) → growing at 93% of exponential rate
- \(K = 100\), \(N = 98\): unused capacity \(= 1 - (98/100) = 0.02\) → growing at 2% of exponential rate
The maximum growth rate occurs at \(N = K/2\)
Logistic growth
Thanos should have studied population ecology
If a species is at carrying capacity, halving the population maximises its growth rate.
But species differ in their \(r\) and \(K\) — the effect of halving is not equal across all species.
Logistic growth
Paramecium growing to capacity.
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Logistic growth — worked example
\[N_{t+1} = N_t + r_dN_t\left(1- \frac{N_t}{K} \right)\]
\[N_{t+1} = 100 + 0.05 \times 100 \left(1- \frac{100}{200} \right)\]
Compare with the exponential model: \(N_{t+1} = 105\)
The density penalty has already reduced growth by 2.5 individuals.
Logistic growth — rate vs size
Unlike exponential growth, the logistic growth rate peaks at \(K/2\) and returns to zero at \(K\).
Logistic vs exponential growth rates
Logistic — rate vs population size
Exponential — rate vs population size
Assumptions of the logistic model
Take 5 minutes to discuss the assumptions of the logistic growth model
(ᵔᴥᵔ)
Part 3: Introducing stochasticity
Stochasticity
Everything so far has been entirely deterministic — but the real world is not.
\[N_{t+1} = N_t + r_dN_t\left(1- \frac{N_t}{K} \right)\]
Environmental stochasticity
- Populations go through good and bad years
- Represented by adding variability to \(r_d\)
Demographic stochasticity
- By chance, births or deaths may cluster in time
- Represented by making birth and death probabilities explicit in \(r\)
Variability can also be incorporated into \(K\).
We will focus on environmental stochasticity.
Stochasticity
Even with a positive average growth rate, stochasticity can drive extinction.
Recap
We have covered:
Discrete and continuous forms of exponential growth
- also known as density independent growth
Discrete and continuous forms of logistic growth
- also known as density dependent growth
Stochastic logistic growth
Model assumptions
Key points
Exponential growth: population grows in proportion to its size, indefinitely
Logistic growth: a density penalty slows growth as \(N\) approaches \(K\)
- Growth rate is greatest at \(N = K/2\)
Stochasticity: can drive extinction even when average growth is positive; deterministic models cannot capture this
Assumptions matter — think critically about these for your assignment
Things to be aware of
Continuous vs discrete equations use different notation and can give different results for large \(r\)
Many variants of population growth models exist, e.g.: \[N_{t+1} = \lambda N_t\left(1- \frac{N_t}{K} \right)\] Concepts transfer, but parameter meanings and outputs may differ.
This is just the beginning — models extend to age-structured populations, competition, predation, and more.
Tips for the assignment
Attend labs and office hours
Focus on the two key graphs: population size vs time, and growth rate vs population size
Use the Shiny app to explore parameter effects — but do not submit screenshots as figures
Check your outputs for logical sense: if numbers look extreme, trace back your inputs and equations
