You are here

Mathematical Modeling in Economics, Ecology and the Environment

Natali Hritonenko and Yuri Yatsenko
Publication Date: 
Number of Pages: 
Springer Optimization and Its Applications 88
[Reviewed by
Mary Flagg
, on

The authors explain the content of the book nicely in the Preface:

Chapter 1 explores the steps of applied mathematical modeling and provides a brief overview of the concepts, notations and tools. The remaining chapters are divided into three parts:


Part I “Mathematical Models in Economics” (Chapters 2–3) is devoted to the mathematical modeling of economic systems. This area of modeling is well established, with its own terminology, classification and investigation methods. The considered models are used later in Part III as components in more sophisticated models of integrated systems. Chapter 2 and 3 analyze aggregate nonlinear economic-mathematical models based on production functions. Chapters 4 and 5 concentrate on the models of economic and technological development under improving technology, described by integral or partial differential equations. Part I focuses on the qualitative analysis and optimization in considered models. An appendix to Chapter 2 contains a review of extremum conditions (maximum principle) for the optimal control problems studied in this and subsequent chapters.


Part II “Models in Ecology and Environment” (Chapters 6–9) explores various mathematical models used in population and environmental problems. It covers two large topics: models of biological communities and their rational exploitation (Chapters 6 and 7) and models of pollution propagation in the atmosphere and water reservoirs (Chapters 8 and 9). Some basic models of Chapters 2-10 are only briefly discussed because they can be found in more specialized textbooks. However, more complex models constructed from these components are explained in detail.


Part III “Models of Economic-Environmental Systems” is devoted to integrated models of economic and environmental dynamics. Chapter 10 describes various models of nonrenewable resource extraction, including the well-known Hotelling’s rule and Dasgupta-Heal model of economic growth with an exhaustible resource. Chapter 11 focuses on economics of climate change and explores aggregate optimization models of economic-environmental interactions such as pollution accumulation and abatement and adaptation to environmental damage. Chapter 12 offers a brief glance at the history and mathematical structure of famous models of global change, from the Club of Rome models to the modern integrated assessment models, their specifics, achievements and limitations.

This book contains a wealth of information on many of the models used in describing economic and environmental changes. The first chapter describes the steps to creating a model, and in subsequent chapters the authors are careful to point out how the more complicated models are built from the simple ones by following the steps laid out in Chapter 1. Each chapter starts with a simple model, usually one that can be analytically solved. The authors proceed to make the model more realistic by allowing other quantities to vary. The more complicated models are analyzed for equilibrium conditions and stability analysis. Some numerical results are presented to help students understand the qualitative nature of the solutions. Each model is explained nicely with the appropriate assumptions, variables and relationships.

The distinguishing feature of the book is Part III, integrating the economics with the environmental models in language that is accessible to a mathematically educated person without experience in the field. Although the last few chapters speak in generalities instead of complicated specifics, they are very valuable for understanding the larger picture of the complexity of global problems.

The second edition has been revised to include new data and remove obsolete information. Each chapter comes with exercises at the end. The exercises in the first few chapters ask the students to organize and categorize the information on different models, as well as practice some of the basic mathematical techniques. Other exercises ask students to compute analytical solutions in the cases where they exist and show that certain functions satisfy the equations of a particular model. They also ask students to continue the qualitative analysis begun in the chapter. Additionally, each chapter comes with a reference list including books suggested for further student reading.

Having discussed the best features of the book, I must mention its limitations as well. This is advertised as a textbook for a graduate level or upper-level undergraduate course in mathematical modeling. It is well suited as a graduate text, assuming students will use the references provided to fill in the details. As an undergraduate textbook, it is likely to overwhelm students. Perhaps if the professor teaching the class has knowledge and experience in this area, and uses his or her experience to describe situations to apply these models and provides contexts for the definitions, this book would be a good choice. But for a professor without a great deal of experience in the field teaching an undergraduate modeling course, it would not be an appropriate choice. Similarly, if the intended audience did not have any experience in economics, it may not be the best choice for a textbook.

The book also does not leave much room for discovery. The list of models in the first half of the book will keep the students discovering some of the simpler models for themselves. That may not be possible in a textbook of this type, but it is something to consider when looking for a textbook.

Overall this is a very well written and very thorough introduction to mathematical modeling in economics and the environment. It is a valuable “first source” on basic modeling that is accessible to a graduate student level or professional audience. It would be challenging for an upper level undergraduate with some background in mathematics and economics.

Mary Flagg is an Assistant Professor of Mathematics in the Department of Mathematics, Computer Science and Cooperative Engineering at the University of St. Thomas in Houston, Texas. She has a Ph. D. in Mathematics with a specialty in algebra from the University of Houston and a M. S. in Chemical Engineering from the California Institute of Technology. She chose to review this book as she looks for ways to update the undergraduate curriculum in mathematics at the University of St. Thomas (a small liberal arts school) by adding problem solving opportunities in applied mathematics. 


1. Introduction: Principles and Tools of Mathematical Modeling

Part I. Models of Controlled Economic Systems

2. Aggregate Models of Economic Dynamics
3. Modeling of Technological Change
4. Models with Heterogeneous Capital
5. Optimization of Economic Renovation

Part II. Models in Ecology and Environment

6. Mathematical Models of Biological Populations
7. Modeling and Control of Biological Communities
8. Models of Air Pollution Propagation
9. Models of Water Pollution Propagation

Part III. Models of Economic-Environmental Systems

10. Modeling of Non-Renewable Resources
11. Modeling of Environmental Protection
12. Models of Global Dynamics: from the Club of Rome to Integrated Assessment