Source: BIOL 180 Lecture, University of Washington
Tags: population growth, exponential growth, lambda, per capita rate of increase, population density, density-dependent, density-independent, carrying capacity, population ecology
Difficulty: Introductory | Prerequisites: Basic algebra (exponents, natural logarithms)
Population ecology sits at the heart of biology: understanding how populations grow, shrink, and stabilise underpins everything from conservation planning to disease modelling. This lecture introduces two foundational pieces. First, exponential growth models, which describe how populations expand when resources are unlimited. Second, the factors that limit growth in the real world, split into density-dependent and density-independent categories. You should be comfortable with basic exponential and logarithmic functions before diving in.
Populations growing without limits follow exponential curves, described by two interchangeable equations (one using lambda, one using r and e). In the real world, growth is checked by density-dependent factors (competition, predation) and density-independent factors (weather, natural disasters).
Lambda (λ), the finite rate of increase
The ratio by which a population multiplies over one discrete time period. If λ = 1.5, the population grows by 50% each period. Think of it as the multiplication factor you apply each generation or time step.
r, the per capita (intrinsic) rate of increase
The instantaneous rate of population change per individual. Calculated as r = ln(λ). In simple terms, r tells you how fast the population is changing at any given moment, rather than over a fixed interval.
N₀ (N-zero), initial population size
The number of individuals in the population at the start of the observation period (time = 0).
Nₜ (N-sub-t), population size at time t
The number of individuals after t time periods have elapsed.
Population density
The number of individuals per unit area or volume. Calculated as (number of individuals) / (area or volume). Think of it as how crowded a population is in its habitat.
Density-dependent factors
Factors whose impact on birth or death rates changes as population density changes. Generally biotic in nature (competition for food, predation, disease spread). In simple terms, these hit harder the more crowded the population gets.
Density-independent factors
Factors that affect birth and death rates regardless of how many individuals are present. Generally abiotic: weather events, natural disasters, fires. Think of it as forces that do not care how crowded the population is.
Lambda (λ) measures population change over a discrete time period. It is defined as λ = e^r.
r, the per capita rate of increase, measures instantaneous change. It is defined as r = ln(λ).
When λ > 1, the population is growing. When λ < 1, it is declining. When λ = 1, population size is stable.
When r > 0, the population is growing. When r < 0, it is declining. When r = 0, no change.
Discrete-time model: Nₜ = N₀ · λᵗ
Use this when reproduction happens in distinct pulses (seasonal breeding, annual generations).
Continuous-time model: Nₜ = N₀ · e^(rt)
Use this when reproduction can happen at any time (bacteria dividing, humans reproducing year-round).
Both equations solve for any unknown variable if you know the other three: Nₜ, N₀, λ (or r), and t.
The discrete model suits organisms with non-overlapping generations or seasonal reproduction.
The continuous model suits organisms that reproduce continuously.
Both yield the same result at integer time steps, since λ = e^r.
Defined as the number of individuals per unit area (or volume, for aquatic organisms).
Formula: density = (number of individuals) / (area or volume).
Both abiotic factors (temperature, rainfall, habitat availability) and biotic factors (competition, predation) influence density.
Their effect on survival and reproduction scales with population density.
Generally driven by biotic interactions:
Intraspecific competition (food, nesting sites, mates)
Predation rates increasing as prey become easier to find
Disease spreading more readily in crowded populations
As density rises, birth rates tend to drop and death rates tend to rise, creating a natural brake on growth.
Their effect on birth and death rates does not change with population density.
Typically abiotic:
Severe weather (droughts, floods, cold snaps)
Catastrophic events (volcanic eruptions, wildfires, hurricanes)
A hurricane kills the same proportion of a population whether there are 100 or 100,000 individuals present.
These factors can cause sudden, sharp declines that the population had no way to "regulate" away.
Relationship between λ and r:
λ = e^r
r = ln(λ)
Discrete growth equation:
Nₜ = N₀ · λᵗ
Continuous growth equation:
Nₜ = N₀ · e^(rt)
Population density:
Density = (number of individuals) / (area or volume)
Exponential growth models are used in epidemiology to project early-stage disease outbreaks, when the pathogen has not yet exhausted its pool of susceptible hosts. Conservation biologists use the same equations to estimate how quickly an endangered species can recover once threats are removed.
Density-dependent regulation explains why fish farms must carefully manage stocking density: overcrowding increases disease transmission and competition for dissolved oxygen, reducing yield.
Confusing λ and r. Lambda is a multiplier applied over a discrete interval; r is an instantaneous rate. They are related (λ = e^r), but they are not the same quantity and cannot be swapped in equations.
Assuming exponential growth continues forever. The exponential model assumes unlimited resources. Real populations eventually face limiting factors that slow or reverse growth. The exponential model describes a special case, not the default.
Thinking density-independent means unimportant. Density-independent factors like hurricanes or droughts can devastate populations. "Independent of density" refers to how the factor operates, not to its severity.
Mixing up biotic and abiotic. Students sometimes label predation as density-independent. Predation is biotic and typically density-dependent: predators often take more prey when prey are concentrated.
⚠️ Be able to convert between λ and r in both directions. Exam questions frequently give you one and ask you to find the other.
⚠️ Know when to use Nₜ = N₀ · λᵗ versus Nₜ = N₀ · e^(rt). If the question specifies discrete generations, use λ. If it specifies continuous growth, use r and e.
⚠️ Expect a question asking you to classify a given factor as density-dependent or density-independent and explain why.
⚠️ Be prepared to calculate Nₜ given N₀, λ (or r), and t, or to solve for any one of those variables given the other three.
True or false: λ = 2.0 means the population doubles each time period. True.
Fill in the blank: r = ln(___). λ (lambda).
True or false: A volcanic eruption is an example of a density-dependent factor. False. It is density-independent (abiotic, affects the population regardless of its size).
Fill in the blank: Population density is calculated as the number of individuals divided by ___. Area (or volume).
True or false: If r = 0, the population size is stable. True.
Q: A population of 500 bacteria has a λ of 1.2 per hour. What is the population size after 3 hours?
A: Nₜ = N₀ · λᵗ = 500 · 1.2³ = 500 · 1.728 = 864 bacteria.
Q: If λ = 1.5, what is the per capita rate of increase (r)?
A: r = ln(λ) = ln(1.5) ≈ 0.405.
Q: A forest fire kills 40% of a deer population regardless of how many deer are present. Is this a density-dependent or density-independent factor? Explain.
A: Density-independent. The fire is an abiotic event whose impact does not scale with the number of deer in the area.
Q: Give two examples of density-dependent factors and explain why they intensify as population density increases.
A: Competition for food (more individuals means each gets a smaller share) and disease transmission (higher density increases contact rates between individuals, accelerating spread).
Q: A population of 1,000 organisms grows continuously at r = 0.03 per year. What is the population size after 10 years?
A: Nₜ = N₀ · e^(rt) = 1,000 · e^(0.03 · 10) = 1,000 · e^0.3 ≈ 1,000 · 1.3499 ≈ 1,350 organisms.
Exponential growth sets the stage for logistic growth models (coming next), which add carrying capacity (K) as the upper bound that density-dependent factors impose. Understanding λ and r here is essential for interpreting survivorship curves and life tables later in the ecology unit.
Density-dependent and density-independent factors also connect to community ecology: interspecific competition, predator-prey dynamics, and the concept of ecological niches all build on the idea that biotic interactions intensify with density.
Exponential growth, geometric growth, population growth rate, lambda, finite rate of increase, intrinsic rate of increase, per capita growth rate, r, ln lambda, e^r, Nt = N0 lambda^t, continuous growth model, discrete growth model, population density, density-dependent regulation, density-independent regulation, biotic factors, abiotic factors, competition, predation, disease, catastrophic events, population ecology, BIOL 180, intro biology, University of Washington