F. Density Dependence

1. Density-dependent or -independent?

In the models used up to now population dynamics were density-independent, as all vital rates of the population and its growth rate were dependent only on the environment, not on density of the population. Density independence is a condition implied in the time-invariance assumption. If, however, any of the vital rates aij depend also on density of the entire population or of one of its stages (represented by n(t)), then density dependent processes are taking place. In that case, the matrix A ={aij} becomes a variable, which varies annually as the density of the population changes: At = f(n(t')), where t'=t or t+1. Therefore, transition models can be applied only as simulations of how population density and population growth rate change in time. For each step of the projection, the affected vital rates are recalculated as a function of the relevant density component.

2. Forms of density dependence

There are many kinds of density-dependent processes, which can be differentiated according to effect, cause, response, mechanism and severity. The negative effects of density dependence are usually better known than the positive ones. Negative density-dependent effects occur if a vital rate decreases as density increases, positive effects occur if both increase.

Competition among the individuals in a population is a very common mechanism of density dependence, caused by limited availability of a necessary commodity, usually a resource, or space (sites). The population responds with a drop in one or more vital rates, usually growth and survivorship, but also fecundity and recruitment. At low density, individuals do not interfere with each other, so that population growth is density-independent. As density increases in time, however, resources may become less available, each individual acquires less of the resource, and negative density dependence manifests itself. It becomes stronger if the population continues growing, at the lower growth rate.

Predation can have a similar negative density-dependent effect, as when at low density prey is left alone, but at high density it gets decimated down to a minimum (GUD, giving-up-density) or becomes extinct. The net result both in the case of competition and predation is that the growth of the population is unaffected at low density, but depressed at high density. In many cases this may lead to regulation of the population around a maximum density. In logistic models this is called the carrying capacity K of the environment (for the particular organism that is considered). Other negative density-dependent phenomena are infectuous desease and parasitism, which may cause outbreaks at high host population density.

Positive density-dependent effects never regulate population density, as is inherent in all positive feedback mechanisms, which magnify their effects. These processes are unstable, as they tend to change to density independence or negative interactions. As will be discussed later, negative density dependence can also be unstable, when it leads to the death of all individuals or to zero recruitment.

A known example of positive density dependence is the Allee-effect (Allee 1931), when population growth rate is very low at low density, due to the low chance to find mates (or to be pollinated, in plants). Such populations, if newly established, grow very slowly in density, but eventually experience accelerated growth rates. At these higher densities, negative density dependence may start as some resource becomes scarce, while the encounter rate of mates is at its maximum. Other examples of positive density dependence are large animal aggregations (bird colonies, herds, fish shoals), where there is "safety in numbers". This phenomenon contrasts with negative density-dependent predation, as in the positive case the "enemy" is swamped and cannot consume more. Many plants "evade" seed predation by producing large numbers of seeds at once, saturating seed-eaters.

In predator populations, cooperative groups manage to gather sufficient amounts of food, while small groups would not. Pollination of flowers also has positive density-dependent components, besides the Allee-effect (as wind-dispersed pollen from few plants may just miss all available stigmata). Insects may be attracted to dense patches of a flowering plant species and pollinate them efficiently, but ignore sparse stands.

Other examples of positive density-dependent effects are the numbers of young bird hatchlings in a nest, or the position of a tree in a forest stand, where individuals benefit from physical protection by others against extreme weather conditions.

3. Processes affected

In the previous examples of density dependence, different vital rates are affected. Predation typically affects survivorship, while resource competition may lead to lower growth rates, survivorship and/or birth rate (fertility or fecundity, recruitment). Likewise, the positive examples affect fertility (Allee-effect, dense flowering), survivorship (animal aggregation), and growth rates and survivorship (cooperation).

Iindividuals in different life-cycle stages within a population differ in size and/or shape, and thus in their ability to acquire resources and in their response to environmental conditions. These lead to stage-specific differences in survivorship, growth and fertility.  Therefore, density-dependent processes are likely to affect different transitions (vital rates) between stages. In the previous example of bird hatchlings that kept each other warm, later they may suppress weaker siblings, even kick them out of the nest. In the case of the Canadian song sparrow (Arcese and Smith 1988), fecundity, but not survivorship, of the females was affected by nest density due to decreased food supply per breeding bird. In the example of seed production in plants, fecundity (seeds/plant) and fertility (seedlings/plant) stay high at high seed density, but seedling survivorship may be lower than at low density (Silvertown 1982). One consequence of the variation in effects on defferent rates and stages of a structured population, is that the density-dependent phenomena not only change population growth rates, but stage distributions and their fluctuations as well.

4. Detection and description of density dependence

In order to be certain whether a process is in fact density-dependent, its effects need to be quantified. Thus, the studied (partial) vital rates (young per female, or seeds per plant, survivorship of larvae, etc.) show a decrease with increasing density (or an increase, in the positive case). However, it is sometimes impossible to compare situations where different quantities and values are involved, of different organisms in different environments.

In order to quantify the strength or severity of the density dependence, a commonly used method is k-value analysis. It is based on the "killing power" (also calculated for life tables), where k = log(initial density/final density), which is compared with log(initial density). Initial and final density can be denoted, respectively, as B (before) and A (after), so that k = log(B/A) = log(B) - log(A). In many cases the density-dependent process is not a single effect with a distinction between before and after, but B and A take the meaning of "without" and "with" exposure to the process, or compared to the lowest density. It should be noted that there are problems using this relationship as a statistical test. It is strictly speaking not allowed, since x and y are not independently sampled. Nevertheless, the shape of the k-value curve is very informative of the severity of the density-dependent phenomenon. Because they are independent of the actual value of the vital rates concerned, k-values enable comparisons between different processes, effects and populations. Examples of k-value analysis of density dependent effects on survivorship, fecundity and groth in various organisms are discussed by Begon et al. (1990), and can be viewed here.

The shapes of the k vs B curve is described by its slope b, which can vary from 0 to infinity. At b = 0, there is no difference between before and after that depends on density. A may still be less than B, but this is density-independent. If 0 < b < 1, the vital rate is somewhat depressed at higher density, representing a weak negative density-dependent effect. If b = 1, the vital rate is suppressed to such an extent that the stage to which it leads stays the same (A = constant) (Proof). In this case, the same number of individuals survive, no matter with how many they started. This is exact compensation, where due to competition or predation, the numbers of births and deaths are equal. Thus, 0 < b < 1 means under-compersation, where there are still more births than deaths but less than without the density-dependent process. Over-compensation (with b approaching infinity) occurs when survivorship, growth or fertility become more and more suppressed by crowding and depletion of resources, so that ultimately there is no more resource to support any individuals and all die, or no new individuals are recruited. This leads to either extinction of the population or large fluctuations in its density, in contrast to exact compensation, when the population is in a state of equilibrium.

5. The role of resources and other organisms

Density-dependent processes also differ in the environmental controls or factors, as opposed to the population responses discussed before. In many populations of all kinds of organisms, sites for establishment are limited in number, or a consumable resource such as prey or soil nutrients has a finite rate of supply. In these cases, limited availability is a condition for having density-dependent phenomenon of inter-specific competition.

Populations may also be controlled by exploitation (predation and herbivory), depending on how the consumers behave (Begon et al. 1990, Chapter 9). For instance predator saturation, leading to increasing chance to escape for prey at high density, occurs either by the inability of the consumer to handle more prey (a so-called functional response), or other constraints of their population growth rate (which is a numerical response). On the other hand, many consumers prefer high-density food sources, as mentioned before, having negative density-dependent effects on the consumed population.


Allee, W. C. 1931. Animal aggregations. A study in general sociology. Univ. of Chicago Press, Chicago, USA.

Arcese, P. and J. N. M. Smith. 1988. Effects of population density and supplemental food on reproduction in song sparrows. Journal of Animal Ecology 57: 119-136.

Begon, M., J. L. Harper and C. R. Townsend. 1990. Ecology : individuals, populations, and communities. Cambridge, Blackwell Scientific Publ.

Silvertown, J. W. 1982. Introduction to plant population ecology . Longman, London.