Density dependence
Density dependence means that demographic rates such as birth, death, survival or growth change when the population becomes more or less crowded.
What does “density” mean?
Population density is the number of individuals relative to available space or resources. In a simple non-spatial model, population size \(N\) is often used as a proxy for density.
For example, a population of 500 organisms may be extremely crowded in a small habitat but sparse in a very large one. So density is biologically more fundamental than raw abundance, even though many simple models use \(N\).
Density-independent growth as a starting point
Exponential growth assumes a constant per-capita growth rate:
\[\frac{1}{N}\frac{dN}{dt}=r.\]The model does not change the per-capita rate when the population becomes larger. This is therefore density-independent growth.
Negative density dependence
Negative density dependence occurs when population performance becomes worse as density increases.
Possible mechanisms include competition for food, space, light or nesting sites, increased disease transmission, waste accumulation, territorial conflict and stronger predation near dense populations.
The logistic model represents this idea by
\[\frac{1}{N}\frac{dN}{dt}=r\left(1-\frac{N}{K}\right).\]As \(N\) increases, the per-capita growth rate decreases.
How logistic density dependence works
The factor
\[1-\frac{N}{K}\]controls the strength of density limitation.
| Population level | Density effect | Interpretation |
|---|---|---|
| \(N\ll K\) | factor close to 1 | little density limitation |
| \(N=K/2\) | factor equals \(1/2\) | per-capita growth is reduced by half |
| \(N=K\) | factor equals 0 | net per-capita growth is zero |
| \(N\gt K\) | factor negative | population tends to decline |
Per-capita growth and density
Why negative density dependence can stabilise a population
Suppose population size falls below the carrying capacity. Per-capita growth becomes positive, so the population tends to increase. If population size rises above the carrying capacity, per-capita growth becomes negative, so the population tends to decline.
This feedback pushes the population toward \(K\).
Positive density dependence
Positive density dependence occurs when population performance improves as density increases, at least over some range.
This can happen when individuals benefit from finding mates, group defence, cooperative hunting, pollination, social organisation or collective modification of the environment.
At very low density, these benefits may be weak. Increasing density can therefore increase per-capita growth.
Connection with Allee effects
An Allee effect is a form of positive density dependence at low population density.
In a strong Allee effect, the per-capita growth rate can become negative below a threshold \(A\). One common model is
\[\frac{1}{N}\frac{dN}{dt}=r\left(1-\frac{N}{K}\right)\left(\frac{N}{A}-1\right).\]Below \(A\), the population tends to decline; between \(A\) and \(K\), it tends to grow.
Negative and positive density dependence together
Density dependence can act on different demographic processes
It is not necessary for density to affect every biological rate equally.
| Process | Possible density-dependent effect |
|---|---|
| birth | fertility may fall when food becomes limited |
| death | mortality may increase with crowding |
| survival | juvenile survival may decrease when competition is strong |
| development | growth to the next stage may slow at high density |
| dispersal | individuals may leave crowded areas more often |
Competition as a mechanism
Suppose each individual has access to fewer resources as population size increases. Then birth rates may decrease, death rates may increase, or both can happen.
The net per-capita growth rate can therefore be written conceptually as
\[\text{per-capita growth}=\text{birth contribution}-\text{death contribution},\]with one or both contributions depending on \(N\).
Density dependence is not the same as carrying capacity
Carrying capacity is one possible consequence of negative density dependence. Density dependence itself is the mechanism by which demographic rates change with crowding.
A model can contain density dependence without having a simple fixed carrying capacity, especially if environmental conditions change through time.
Density dependence versus density independence
| Density-dependent influence | Density-independent influence |
|---|---|
| competition for food | storm |
| crowding | temperature shock |
| mate limitation | fire |
| disease transmission that increases with contact | flood |
This distinction is useful, but real systems often contain both types of influence simultaneously.
Density dependence can be delayed
Population responses are not always immediate. For example, high density this year may reduce body condition and fertility next year.
Such delayed density dependence can produce oscillations or more complicated dynamics rather than simple convergence to an equilibrium.
Density dependence in stage-structured populations
Density may affect only particular stages. Seedling establishment can be strongly density-dependent while adult survival remains nearly unchanged.
In a projection matrix, this means some fertility, survival or transition entries depend on the current population state rather than remaining constant.
Density dependence in epidemiology
The idea also appears in infectious-disease models, but terminology must be used carefully. Infection rates may depend on how frequently individuals contact one another, and that relationship can change with population density.
In ecological epidemiology, one distinguishes forms such as density-dependent transmission and frequency-dependent transmission according to how contact scales with population size.
Why the exact mathematical form matters
Two models can both be described as density-dependent yet behave very differently.
Linear logistic feedback gives one pattern. An Allee-effect term can introduce a threshold. Saturating functions can make the effect weak at very high density. Time delays can create oscillations.
Estimating density dependence from data
One common idea is to examine whether per-capita population growth changes systematically with abundance or density.
If per-capita growth tends to decrease as density rises, that supports negative density dependence. If it rises at low density, that may indicate positive density dependence or an Allee effect.
However, environmental variation and observation error can obscure the relationship, so statistical analysis is usually required.
Why density dependence matters for harvesting
Harvesting changes population density, and density dependence determines how biological production responds.
Under logistic growth, reducing a very dense population can temporarily increase per-capita growth because competition is relaxed. But reducing a population too far can be dangerous if a strong Allee effect is present.
Deterministic and stochastic effects
A deterministic density-dependent model describes the average direction of change. Real populations also experience random births, deaths and environmental variation.
Near thresholds or extinction, stochastic fluctuations can therefore push a population into a different region of the density-dependent dynamics.