Mathematical Ecology of Populations and Ecosystems

John Pastor

• Publisher: Wiley-Blackwell
• Year: 2011
• Print ISBN: 9781405177955, 9781405188111
• eBook ISBN: 9781444358452
• Edition: 1
• Topic: Biological Sciences
• Subtopic: Biology
• Index: Biological Sciences

MATHEMATICAL ECOLOGY

Population ecologists study how births and deaths affect the dynamics of populations and communities, while ecosystem ecologists study how species control the flux of energy and materials through food webs and ecosystems. Although all these processes occur simultaneously in nature, the mathematical frameworks bridging the two disciplines have developed independently. Consequently, this independent development of theory has impeded the cross-fertilization of population and ecosystem ecology. Using recent developments from dynamical systems theory, this advanced undergraduate/graduate level textbook shows how to bridge the two disciplines seamlessly. The book shows how bifurcations between the solutions of models can help understand regime shifts in natural populations and ecosystems once thresholds in rates of births, deaths, consumption, competition, nutrient inputs, and decay are crossed.

Mathematical Ecology is essential reading for students of ecology who have had a first course in calculus and linear algebra or students in mathematics wishing to learn how dynamical systems theory can be applied to ecological problems.

Part 1

Preliminaries

1

What is mathematical ecology and why should we do it?

Let’s begin by looking again at the photograph in the Prologue and imagine yourself walking through this forest. What do you see? Jot down a few things (this is your first exercise). They need not be profound – in fact, it is best not to try to make them profound. After all, Darwin constructed the most profound theory in biology by asking ordinary questions about barnacles, birds, and tortoises, amongst many other things.

Perhaps you see big trees and little trees and think that big trees are older than little trees. You also might notice that there are more little trees than big trees, and so not every little tree grows up to be a big tree – most die young. But the little trees must come from somewhere, namely seeds produced and shed by the bigger trees. These are the core ideas of population ecology.

Or perhaps you might notice that there are some dead needles and leaves on the ground and some standing dead trees which will eventually fall to the soil, the result of the deaths of those young trees and plant parts. You also note that the live trees have roots in the soil formed partly from those dead leaves and logs and surmise that the trees obtain some nutrients from them. These are the core ideas of ecosystem ecology.

These two views of the forest look very different, but they both contain biological objects that interact with each other through hypothesized processes. When we model a biological object such as a population, we begin by offering an analogy between it and a mathematical object. Mathematically we will term these analogs state variables. The processes usually represent a transfer of something (live individuals, seeds, nutrients) from one biological object to another. Processes will be modeled by mathematical operations, such as addition, multiplication, subtraction, or powers. One or more operations and the objects they operate on will be encapsulated into an equation, specifically an equation which relates how one state variable partly determines the state of itself and perhaps another at some point in the future. These equations will contain, besides mathematical operations and state variables, some parameters, whose values remain fixed while the state variables change. Each state variable will be described by one equation. The time-dependent behavior of the state variables and the magnitudes of the state variables at equilibrium are called the time-varying and equilibrium solutions of the model, respectively. We then use the rigor of mathematics to work through the logic of our thinking to gain some insight into the biological objects and processes.

Therefore, mathematical ecology does not deal directly with natural objects. Instead, it deals with the mathematical objects and operations we offer as analogs of nature and natural processes. These mathematical models do not contain all information about nature that we may know, but only what we think are the most pertinent for the problem at hand. In mathematical modeling, we have abstracted nature into simpler form so that we have some chance of understanding it. Mathematical ecology helps us understand the logic of our thinking about nature to help us avoid making plausible arguments that may not be true or only true under certain restrictions. It helps us avoid wishful thinking about how we would like nature to be in favor of rigorous thinking about how nature might actually work.

What equations should we choose to use to model the dynamical relations amongst the state variables? Of course, there are an infinite number of equations we can choose, but we prefer equations that are simple to understand, are derived from simple “first principles,” have parameters and operations that correspond to some real biological process and are therefore potentially measurable, and produce surprising results that lead to new observations. These four properties of these equations are components of mathematical beauty. They are important criteria by which we judge the utility of an equation or model because they help clarify our thinking. They often force our thinking into new directions.

This is all well and good, but why should we play this game? Why not just state hypotheses as clearly as we can and do the experiments to test them? One reason is that we are often not sure of either the internal logic of our ideas and hypotheses or their consequences. For example, state variables often affect and are affected by another state variable. This mutual interaction between state variables is termed feedback. Feedbacks are common in ecological systems – in fact, they are characteristic of all interesting ecological systems. Systems with internal feedbacks are almost impossible to completely understand in an intuitive way. Without a clear understanding of how the feedback works, it is also very difficult to do an experiment which manipulates the feedback. It is easy to understand a chain of events where X influences Y and Y influences Z, but what if Z also affects X? What then happens to Y? By writing a system of equations, one for each of the state variables and using the rules of mathematics, we can examine the logical structure of feedbacks and their consequences.

Examining the properties of a system of equations allows us to pose further questions and determine how their answers might follow logically from their structure and properties. For example, the population ecologist might wonder how the proportion of individuals of a given age class changes over time, whether the proportional distribution over all age classes ever settles down to a stable distribution, and what that distribution is. The ecosystem ecologist might note that the world surrounding the forest contributes material to it (in rainwater, for example) and the forest contributes material back to the surrounding world (in the water leaching out of the soil). He or she might wonder what difference it makes how and where the material enters and leaves the ecosystem. Both ecologists might also wonder what happens if we harvest some of a population or ecosystem: does the population or ecosystem recover to its earlier state? How will it recover? Can we harvest so much that the population or ecosystem will never recover? And what exactly is meant by “recover”?

Examining these equations also allows us to uncover hidden assumptions about our ideas and ask what happens when we relax those assumptions. For example, we have assumed that each equation in our model applies equally well to every species that is reasonably similar to the one we are studying. Well, do they? What difference does it make if they aren’t similar to each other? How different do things have to be to make a difference in the system’s behavior? How do different species affect each other? How does including additional tropic levels or other components affect the behavior of the models?

Finally, mathematical modeling allows us to rigorously connect the two different views of population and ecosystem ecologists. For example, the ecosystem ecologist notices that the forest floor contains layers corresponding to different ages of leaf litter from many years in the past. One year’s leaf litter is transferred into older decay classes with each passing year. If the leaves are decomposing, something is being lost from each age class of litter. The ecosystem ecologist pauses and notices that these ideas bear a great deal of resemblance to the age class model of the population ecologist. Can we take the equations for the dynamics of the live populations and extend them belowground into the leaf litter? This shows the real power of mathematical abstraction. Once you recognize a structural correspondence between two different systems, then the same equations and same mathematical techniques could apply to both. If it turns out that this is the case, then the ecologist has discovered some underlying principle of organization in nature, a principle which he or she did not expect when first observing a particular forest (or prairie or lake) and jotting down what first caught his or her eye.

And that is what mathematical ecology is about.

The nature of theoretical problems and their relation to experiment

In the process of abstracting nature into a mathematical model, we run into a number of theoretical problems. These are distinct from the sorts of problems experimenters have to deal with. Most ecologists are familiar with experimental questions such as measuring the response of an individual, population, or ecosystem to manipulations, or determining the proper number of samples required to detect a difference between mean values of measurements. In contrast to these experimental problems, mathematical models of ecological systems address a variety of theoretical questions regarding the logical consistency and consequences of ideas (Caswell 1988). While measuring devices are the tools of the experimental ecologist, equations are the tools of the mathematical ecologist. Equations are used to examine the following theoretical pro...

Table of contents

• Cover
• Dedication
• Title
• copyright
• Prologue
• Preface
• Acknowledgments
• Part 1: Preliminaries
• Part 2: Populations
• Part 3: Ecosystems
• Part 4: Populations and ecosystems in space and time
• References
• Index