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# Mathematical Modeling and Simulation

## Introduction

Mathematical modeling and simulation are important research and monitoring tools used to understand biological communities and their relationships to the environment. Mathematical models are collections of variables, equations, and starting values that form a cohesive representation of a process or behavior. Because interactions among the members of biological communities and components of the abiotic environment are extremely complex, mathematical models are useful for understanding how ecosystems function and for making predictions about managing ecosystems.

There are numerous types of mathematical models used by ecologists and environmental scientists. Some models isolate the key factors that drive elements of a system. Other mathematical models are comprehensive simulations that include as many components and interactions as possible. Mathematical models also cover different spatial and temporal scales: from the smallest tidepool ecosystem to the entire planet; from a single day to millions of years.

## Historical Background and Scientific Foundations

Mathematical models and simulations are used scientifically as a tool for improving the understanding of the ecology of a region and managerially as a tool for making decisions regarding resource and environmental issues. Depending on the scope and sensitivity of the system, developing mathematical models of an environment can be an extremely interdisciplinary undertaking. It might require input from ecologists, biologists, chemists, physicists, engineers, and computer scientists, as well as social scientists, economists, and politicians. Results from mathematical models might be shared among scientists, policy makers, and various government officials, often from different localities and jurisdictions.

Mathematical models usually have three basic parts. These are the variables and their definitions, the equations into which the variables are incorporated, and starting values for the variables. When mathematical models are applied to ecological situations, more information is required. For example, an ecological model requires the user to assign a meaning to the variables, to know the units of the variables, and to bind the ranges of values over which the variables are realistic. It might also be useful to know the way that the values of the variables are measured, the ecological context of the variables, the reproducibility of the values of the variables, among other information.

There are two basic approaches to building a mathematical model of an ecosystem. The first is called the compartmental system approach or a conceptual model. This type of model is often depicted using box and flow illustrations. The boxes represent a pool or a compartment of something found in an ecosystem. For example, a box might represent the total weight of carbon in the phytoplankton in a certain region of the ocean. The box is then impacted by several flows, visually represented by arrows into and out of the box. An arrow into the phytoplankton carbon box might represent the rate of conversion of atmospheric carbon dioxide into carbohydrate by photosynthesis. An arrow out of the box might represent the loss of carbon from the community of phytoplankton because of predation.

Conceptual models most often put emphasis on the gross dynamics of a whole ecosystem. They tend to be rather general models that can be applied to many different systems. For example, the phytoplankton model discussed earlier may be altered for use in either the Pacific Ocean or the Atlantic Ocean. However, conceptual models tend to be imprecise. The rates of photosynthesis in the Atlantic and Pacific Oceans are likely very different. This is because the climate conditions, the nutrients that drive photosynthesis, and the species of phytoplankton all differ from place to place. These factors are largely omitted from conceptual or compartmental system models.

In contrast, the qualitative model, which is also referred to as the experimental components approach, control model, or stressor model, incorporates as many interactions and components of a system as possible. Instead of using a single compartment to represent the phytoplankton carbon, a qualitative model would represent the various populations of phytoplankton individually. It would take experimental results of photosynthetic rates for the different species and use this information to come up with a rate of carbon dioxide assimilation for the entire community. Qualitative models tend to be more precise than compartmental models, but they are also very specific. It would be difficult to use a qualitative model of phytoplankton growth designed to represent Pacific Ocean phytoplankton when modeling the phytoplankton in the Atlantic Ocean. Qualitative models can be extremely useful for understanding how ecosystems will respond to environmental stresses.

Depending on the needs of the users and the ecological questions addressed, several types of models may be developed to represent a single ecosystem. Ecological models can also be embedded within one another. One part of the system may be represented by a qualitative model, while another part is represented by a conceptual model. Many ecological models have a geographical component. Geographical information systems (GIS) and geospatial analyses are often used in conjunction with ecological simulations.

Mathematical models and simulations are used to better understand a multitude of ecological issues. For example, simulations are commonly used to model biogeochemical cycling within aquatic systems. They are used to understand the way that toxic materials move through ecosystems. These toxic materials include pesticides, heavy metals, and radionuclides. Ecological models of ground water flow and plume diffusion in air are often incorporated into these ecotoxicology models. In terrestrial systems, scientists have developed agricultural models and forestry models. One of the most common types of mathematical models involves population dynamics. Population models are used to predict and understand the age structure of fish, the age structure of trees, and rates of primary production and photosynthesis. Predator-prey models (the Lotka-Volterra equations) are commonly used to predict the population dynamics of predator and prey when external environmental impacts are minimal.

## WORDS TO KNOW

ABIOTIC: A term used to describe the portion of an ecosystem that is not living, such as water or soil.

CONCEPTUAL MODEL: the collection of all populations of organisms in an area.

ECOSYSTEM: all of the physical properties of a location.

POPULATION MODEL: A mathematical model that can project the demographic consequences to a species as a result of certain changes to its environment.

PREDATOR-PREY MODEL: A mathematical model used to analyze interaction between two species in an ecosystem.

## Impacts and Issues

Using mathematical models appropriately involves some challenges. Users need to be aware of the assumptions in any model of an ecological system. For example, boundary conditions, time steps, valid ranges of variables, appropriateness of equations chosen to represent actual behavior or measurements, and the stability of the underlying mathematics can all impact whether or not the model actually answers the questions it sets out to answer. Mathematical modelers spend a significant amount of time testing simulations to assess how well the model corresponds to expected patterns from the real system and how the model changes when perturbations to the system occur.

Perhaps the most well publicized and most controversial types of ecological simulations are climate models. Because the global climate is such a complex system, it is very difficult to incorporate all of the factors that influence it. As a result, climate simulations tend to be conceptual models. Conceptual models are fundamentally imprecise, and depending on starting values and assumptions, a wide range of predictions can be made. This led to controversy over the validity of the concept of climate change resulting from anthropogenic factors. Through years of research and study, climate change models have became more refined and the predictions made by the models more cohesive. The Intergovernmental Panel on Climate Change (IPCC) has followed and assessed much of the climate change modeling over the last decade. The models that IPCC scientists have studied are broad-scale and identify the drivers and stressors that affect global climate change.

## BIBLIOGRAPHY

Books

Grant, William, and Todd Swannack. Ecological Modeling. Malden, MA: Blackwell Publishing, 2007.

Hadlock, Charles R. Mathematical Modeling in the Environment. Boca Raton, FL: CRC Press, 2007.

Odum, Eugene, and Gary W. Barrett. Fundamentals of Ecology, 5th ed. St. Paul, MN: Brooks Cole, 2004.

Pastorok, Robert A., Steven M. Bartell, Scott Ferson, and Lev R. Ginzberg, eds. Ecological Modeling in Risk Assessment: Chemical Effects on Populations, Ecosytems, and Landscapes. Boca Raton, FL: CRC Press, 2002.

Web Sites

Intergovernmental Panel on Climate Change. “Home Page.” http://www.ipcc.ch/ (accessed March 4, 2008).

International Society for Ecological Modeling. “Home Page.” October 25, 2007. http://www.isemna.org/ (accessed March 2, 2008).

National Park Service. “Developing Conceptual Models of Relevant Ecosystem Components.” February 20, 2008. http://science.nature.nps.gov/im/monitor/ConceptualModels.cfm (accessed March 2, 2008).

Smithsonian Environmental Research Center. “Ecological ModelingLab.” http://www.serc.si.edu/labs/ecological_modeling/index.jsp (accessed March 2, 2008).

Juli Berwald