Food Webs and Graphs

Margaret (Midge) Cozzens1    1 Rutgers University, Piscataway, NJ, USA

2.1 Introduction

The study of food webs has occurred over the last 50 years, generally by ecologists working in natural habitats with specific relatively narrow interests in mind. At the outset, a few mathematicians became interested in the graph-theoretic properties of food webs and their corresponding competition graphs; however, linkages between ecologists’, mathematicians’, and conservationists’ interests and results were few and far between. This chapter introduces food webs and various corresponding graphs and parameters to those interested in important research areas that link mathematics and ecology. A basic background on food webs and graphs is provided, with exercises to further illustrate the concepts. Each section has exercises which reinforce the newly introduced concepts. These exercises, sometimes open-ended, together with additional references to preliminary work may be used as springboards to numerous research questions presented in the chapter. Research questions accessible to biology and mathematics students of all levels, with references to previous work, are provided. We should note that all specific research questions are, at least in part, open questions, thus providing ample opportunities for student involvement with actual research. The citations accompanying each research question give more background and starting points for exploration.

The main goals for this chapter are to provide the necessary background that will allow the reader to:

• Recognize various relationships between organisms, and look for patterns in food webs.

• Use graphs and directed graphs (digraphs) to model complex trophic relationships.

• Determine trophic levels and status within a food web, and the significance of these levels in calculating the relative importance of each species (vertices) and each relationship (arcs) in a food web.

• Use a food web to create the corresponding competition (predator or niche) overlap graph, and projection graphs to determine the dimensions of a community’s habitat.

• Determine the competition number of a graph and its significance for a community’s ecological health.

• Inform conservation policy decisions by determining what happens to the whole food web and habitat if a species becomes extinct (nodes are removed) or prey relationships change (arcs).

2.2 Modeling Predator-Prey Relationships with Food Webs

Have you ever played the game Jenga? It’s a game where towers are built from interwoven wooden blocks, and each player tries to remove a single block without the tower falling. The player who crashes the tower of blocks loses the game. Food webs are towers of organisms. Each organism depends for food on one or many other organisms in an ecosystem. The exceptions are the primary producers—the organisms at the foundation of the ecosystems that produce their energy from sunlight through photosynthesis or from chemicals through chemosynthesis. Factors that limit the success of primary producers are generally sunlight, water, or nutrient availability. These are physical factors that control a food web from the “bottom up.” On the other hand, certain biological factors can also control a food web from the “top down.” For example, certain predators, such as sharks, lions, wolves, or humans, can suppress or enhance the abundance of other organisms. They can suppress them directly by eating their prey or indirectly by eating something that would eat something else. Understanding the difference between direct and indirect interactions within ecosystems is critical to building food webs. For example, suppose your favorite food is a hamburger. The meat came from a cow, but a cow is not a primary producer—it can’t photosynthesize! But a cow eats grass, and grass is a primary producer. So, you eat cows, which eat grass. This is a simple food web with three players. If you were to remove the grass, you wouldn’t have a cow to eat. So, the availability and growth of grass indirectly influences whether or not you can eat a hamburger. On the other hand, if cows were removed from the food web, then the direct link to your hamburger would be gone, even if grass persisted.

Primary producers, also called basal species, are always at the bottom of the food web. Above the primary producers are various types of organisms that exclusively eat plants. These are considered to be herbivores, or grazers. Animals that eat herbivores, or each other, are carnivores, or predators. Animals that eat both plants and other animals are omnivores. An animal at the very top of the food web is called a top predator.

Through the various interactions in a food web, energy gets transferred from one organism to another. Food webs, through both direct and indirect interactions, describe the flow of energy through an ecosystem. By tracking the energy flow, you can derive where the energy from your last meal came from, and how many species contributed to your meal. Understanding food webs can also help to predict how important any given species is, and how ecosystems change with the addition of a new species or removal of a current species.

Food webs are complex! In this chapter, we explore the complexity of food webs in mathematical terms using a physical model, called a directed graph (digraph), to map the interactions between organisms. A digraph represents the species in an ecosystem as points or vertices (singular = vertex) and puts arrows for arcs from some vertices to others, depending on the energy transfer, that is, from a prey species to a predator of that prey.

The species that occupy an area and interact either directly or indirectly form a community. The mixture and characteristics of these species define the biological structure of the community. These include parameters such as feeding patterns, abundance, population density, dominance, and diversity. Acquisition of food is a fundamental process of nature, providing both energy and nutrients. The interactions of species as they attempt to acquire food determine much of the structure of a community. We use food webs to represent these feeding relationships within a community.