Ecology Density Dependence and Single-Species Population Dynamics
Tancredi Caruso
  • LAST MODIFIED: 27 July 2016
  • DOI: 10.1093/obo/9780199830060-0155


Natural populations possess an intrinsic tendency to exponential growth, which can, for example, be easily realized under laboratory conditions. The tendency to exponential growth is so pervasive that it can be considered a fundamental law of biology. However, exponential growth is observed in natural populations very rarely: several processes such as resource and space limitation, competition, disease, predation, and other factors control and limit population size. In terms of population dynamics, density dependence means that the velocity at which population grows (i.e., growth rate) varies with time as a function of population density N. This is a nonlinear process that in the simplest models causes population to settle at some typical value such as the so-called carrying capacity K. Still, real populations show a remarkable tendency to fluctuations, and these fluctuations may be periodic or aperiodic and depend, to various degrees, on variation in demographic properties, environmental conditions, resources, and also interactions with other organisms. This article focuses on single-species population dynamics, with a particular emphasis on models that do not explicitly consider interactions between different species. There are a phenomenal number of observational and experimental studies of single-species population dynamics with density dependence. In these studies, empirical and theoretical ecologists have tried for several decades to unravel the main mechanisms that underpin the demography of natural populations. One of the main goals is often to develop phenomenological models to project the temporal trajectories of populations, especially for species of strategic interest for food production and ecosystem management.

General Overviews

The most basic model of density dependence in single-species population is the continuous logistic-growth equation. One version of the model is first introduced in Verhulst 1838, although the model is often referred to as the Verhulst-Pearl model: Pearl and Reed 1920 is the first to make the model available to the broad scientific community. The most important and complex forms of logistic models can be derived from the basic logistic-growth equation. Excellent introductions to the main logistic-growth models are chapter 2 in Gotelli 2008 and chapter 6 in Case 2000. There, the authors help the reader with very effective illustrations and a straightforward mathematics, which is, for example, used to clarify the link between exponential and logistic growth. For populations with overlapping generations and continuous growth, the logistic-growth equation is easily derived from the exponential-growth equation. The derivation is based on modifying the exponential-growth model by introducing a term that Krebs 1994 and Gotelli 2008 interpret as the unused portion of the carrying capacity: this term depends on the ratio between population density N and carrying capacity K. If N is small relative to K, the population is, at least approximately, growing exponentially. This rapid growth can, for example, happen because in a small population, competition for resources might not be limiting. However, N rapidly approaches K, and the growth rate progressively decreases. When N is about K, the growth rate tends to zero and the population has reached the population size sustainable at the given environmental conditions (carrying capacity). If N happens to overshoot K, the growth rate becomes negative and N returns to K after a certain time. K is a stable equilibrium, and perturbations that move the population away from K, either positively or negatively, are always absorbed: sooner or later population returns to its carrying capacity. In chapter 5 of Pastor 2008 and chapter 23 of Robinson 2004, full mathematical details are given for the basic logistic-growth equation. Mathematicians study this type of density-dependent dynamics as good examples of first-order nonlinear differential equations. However stable these models are, real populations typically fluctuate in time. These fluctuations, too, can be due to the nonlinearity of density-dependence feedback, which can sometimes produce unexpected patterns. Density dependence is a nonlinear phenomenon, and the impressive phenomena that can emerge from nonlinearity are particularly clear in discrete models for populations with non-overlapping generations, such as those of many species of arthropods with a short-lived adult stage. These models have great historical importance, as they contributed to unveiling the intriguing universe of chaotic dynamics: May 1976 still represents an excellent introduction to the topic, and see also Gotelli 2008.

  • Case, Ted J. 2000. An illustrated guide to theoretical ecology. New York: Oxford Univ. Press.

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    One of the best and most comprehensive introduction to theoretical ecology; makes use of an impressive number of elegant illustrations and numerous examples drawn from many classic study cases. The level of mathematics is generally high yet manageable for the non-mathematically inclined ecologist.

  • Gotelli, Nicholas J. 2008. A primer of ecology. 4th ed. Sunderland, MA: Sinauer.

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    A real primer for those who have acceptable levels of basic mathematics but minimal calculus and no background in mathematical ecology. The book can be used by undergraduates at their first experience of ecological modeling while remaining a valuable compendium and effective summary for the scientist.

  • Krebs, Charles J. 1994. Ecology: The experimental analysis of distribution and abundance. 4th ed. New York: HarperCollins.

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    A classic textbook of ecology with many examples, including applications of the simple density-dependence single-species model.

  • May, Robert M., ed. 1976. Theoretical ecology. Dorking, UK: Blackwell.

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    A classic book edited by one of the founders of theoretical ecology and featuring contributions by ecologists and biologists who have made the history of the discipline. Chapter 2 focuses on density dependence and represents a phenomenal introduction to the ethos of pure theoretical ecology.

  • Pastor, John. 2008. Mathematical ecology. Hong Kong: Wiley-Blackwell.

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    A text for students of ecology who have a somewhat good basis of calculus and matrix algebra. A delightful feature of the book is the rich set of references to the historical background of ecological models.

  • Pearl, Raymond, and Lowell J. Reed. 1920. On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences of the United States of America 6:275–288.

    DOI: 10.1073/pnas.6.6.275E-mail Citation »

    The paper that has made the logistic-growth equation available to the modern scientific community.

  • Robinson, James C. 2004. Ordinary differential equations. Cambridge, UK: Cambridge Univ. Press.

    DOI: 10.1017/CBO9780511801204E-mail Citation »

    A textbook of mathematics for students who have a strong basis in calculus and matrix algebra: it is one of the best books for grasping in a relatively simple way the deepest mathematical aspects of the most simplistic ecological models.

  • Verhulst, Pierre F. 1838. Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique 10:113–117.

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    Population models with density dependence set off in this paper from the early 20th century.

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