Mathematical Ecology

by

Kevin S. McCann

• LAST REVIEWED: 09 September 2021
• LAST MODIFIED: 23 May 2012
• DOI: 10.1093/obo/9780199830060-0004

Introduction

Most people do not think of mathematics when they think of ecology. Nonetheless, mathematical ecology has a long and storied history, with mathematics playing a major role in the development of a simplifying framework for studying nature’s vastly complex ecological systems. Ecologists are often interested in how populations, communities, and ecosystems change in space and time; fortunately, there exists a branch of mathematics developed specifically to deal with dynamics, known as dynamical systems theory. Math continues to play a significant, and arguably growing, role in the development of ecology today. In this article, major papers and books that introduce the vast area of research that is mathematical ecology are identified.

Historical Background

The first major use of mathematics in ecology occurred in the early 1900s. From this early work, the single species logistic growth equation and models for species interactions emerged. The seminal models of Alfred Lotka (Lotka 1925) and Vito Volterra (Volterra 1926) produced basic ecological models for competition and predation—now referred to as the Lotka-Volterra models—that have formed the cornerstone for much of ecology today. Hastings 1997 conceptually introduces these models and similar models in an introductory book on population biology, while Kingsland 1995 is an engaging and thorough account of the history of population ecology, with special emphasis on the role of mathematics. Finally, a wonderful introductory book focused on the basic mathematic tools behind evolutionary and ecological models is Otto and Day 2007. In the middle of the 20th century, Robert MacArthur and colleagues bolstered the use of mathematics by using basic mathematical models to develop community and behavioral ecology (e.g., MacArthur and Pianka 1966). MacArthur used his conceptual abilities to champion the role of competition in determining the structure of ecological communities. This work led to island biogeography theory (MacArthur and Wilson 1967), a classic theory that accounts for patterns in diversity on islands of different area and distance from mainland ecosystems. Shortly after MacArthur’s seminal contributions, a cornerstone for theoretical ecology, a number of physicists entered ecology in the 1970s, and their use of mathematical tools drove a dramatic increase in dynamical systems theory in ecology. Robert May, a trained physicist, was one of the first to recognize that chaotic dynamics emerged from even simple models (May 1973), while he and others extended the basic two-species models to mathematical systems of many interacting species (May 1973). Today, the strong role of mathematical theory remains. Levin, et al. 1997 points out that increases in computational ability and the need to understand the influence of scale have led to a number of modern-day mathematical challenges that await resolution by ecologists.

• Hastings, Alan. 1997. Population biology: Concepts and models. New York: Springer.

  This is an introductory book that covers common models and concepts in ecology, and acts as an introductory resource for those wishing to begin understanding the use of mathematical models in ecology.

• Kingsland, Sharon E. 1995. Modeling nature: Episodes in the history of population ecology. 2d ed. Chicago: Univ. of Chicago Press.

  A thorough and well-told story of the history of population ecology, with special emphasis on the role of mathematics. This book is very accessible and an excellent entry point into the role, and history, of mathematics and theory in ecology up to the 1980s.

• Levin, Simon A., Bryan Grenfell, Alan Hastings, and Alan S. Perelson. 1997. Mathematical and computational challenges in population biology and ecosystems science. Science 275:334–343.

  DOI: 10.1126/science.275.5298.334

  This article discusses some of the major recent challenges for mathematical ecology and acts as a good reference point for understanding where mathematical ecology is presently active. Simon Levin is one of the major players in this area of ecology.

• Lotka, Alfred J. 1925. Elements of physical biology. Baltimore: Williams & Wilkins.

  This seminal contribution employs a mathematical framework for single species growth and interacting species, as well as the major ecosystem cycles. This important book set the stage for mathematical ecology.

• MacArthur, Robert H., and Eric R. Pianka. 1966. On the optimal use of a patchy environment. American Naturalist 100:603–610.

  DOI: 10.1086/282454

  MacArthur and Pianka developed the framework for optimal foraging theory (OFT), which states that organisms forage in such a way as to maximize their net energy intake per unit time. This seminal paper is a large part of many aspects of theoretical ecology today as well as evolutionary ecology.

• MacArthur, Robert H., and Edward O. Wilson. 1967. The theory of island biogeography. Princeton, NJ: Princeton Univ. Press.

  This is a major contribution to one of the more successful theories in ecology. This book details the simple assumptions behind the theory of the island biogeography model and its predictions. It remains central to understanding the geographic distribution of species diversity.

• May, Robert M. 1973. Stability and complexity in model ecosystems. Princeton, NJ: Princeton Univ. Press.

  This famous book added mathematics to the notion that diversity, in and of itself, drives ecosystem stability. This book serves as a good reference for this active area in ecology, arguing that diversity, if anything, tends to drive instability. May argues that inherent biological structure must lie behind the sustainability of complex ecosystems.

• Otto, Sarah P., and Troy Day. 2007. A biologist’s guide to mathematical modeling in ecology and evolution. Princeton, NJ: Princeton Univ. Press.

  This contribution walks students of biology through the mathematics behind common biological models. This book differs from other books in that the authors focus on giving the reader a guided tour of the basic mathematical tools behind models. Written for students of all backgrounds and quite accessible.

• Volterra, Vito. 1926. Fluctuations in the abundance of a species considered mathematically. Nature 118:558–560.

  DOI: 10.1038/118558a0

  The seminal paper by Vito Volterra, in which he derived the classic Lotka-Volterra equations for predator-prey dynamics. The paper, and comments by Lotka that accompanied the publication, offers an interesting look into the early work of these researchers who apparently derived similar models separately.