Skip to main content

Über dieses Buch

Updated to textbook form by popular demand, this second edition discusses diverse mathematical models used in economics, ecology, and the environmental sciences with emphasis on control and optimization. It is intended for graduate and upper-undergraduate course use, however, applied mathematicians, industry practitioners, and a vast number of interdisciplinary academics will find the presentation highly useful.
Core topics of this text are:
· Economic growth and technological development
· Population dynamics and human impact on the environment
· Resource extraction and scarcity
· Air and water contamination
· Rational management of the economy and environment
· Climate change and global dynamics
The step-by-step approach taken is problem-based and easy to follow. The authors aptly demonstrate that the same models may be used to describe different economic and environmental processes and that similar investigation techniques are applicable to analyze various models. Instructors will appreciate the substantial flexibility that this text allows while designing their own syllabus. Chapters are essentially self-contained and may be covered in full, in part, and in any order.

Appropriate one- and two-semester courses include, but are not limited to, Applied Mathematical Modeling, Mathematical Methods in Economics and Environment, Models of Biological Systems, Applied Optimization Models, and Environmental Models. Prerequisites for the courses are Calculus and, preferably, Differential Equations.



Chapter 1. Introduction: Principles and Tools of Mathematical Modeling

This chapter provides a brief overview of the goals, general principles, and specific tools of mathematical modeling, specifically for economic and environmental systems. Section 1.1 analyzes the role and structure of the modeling process in scientific research and decision-making practice. Section 1.2 explores and compares different types of mathematical models (deterministic and stochastic, continuous and discrete, linear and nonlinear). Section 1.3 outlines selected concepts and results of calculus and the theory of differential and integral equations used in this textbook.

Natali Hritonenko, Yuri Yatsenko

Mathematical Models in Economics


Chapter 2. Aggregate Models of Economic Dynamics

This chapter explores aggregate optimization models of the neoclassic economic growth theory, which are based on the concept of production functions. The models are described by ordinary differential equations and involve static and dynamic optimization. Section 2.1 analyzes production functions with several inputs, their fundamental characteristics, and major types (Cobb–Douglas, CES, Leontief, and linear). Special attention is given to two-factor production functions and their use in the neoclassic models of economic growth. Sections 2.2 and 2.3 describe and analyze the well-known Solow–Swan and Solow–Ramsey models. Section 2.4 contains maximum principles used to analyze dynamic optimization problems in this and other chapters.

Natali Hritonenko, Yuri Yatsenko

Chapter 3. Modeling of Technological Change

Economic data for the last two centuries has demonstrated the presence of a self-sustaining mechanism of cumulative productivity growth, known as

technical progress


technological change

. In modern times, the technical progress affects not only the efficiency of the economy, but also the natural environment and entire lifestyle of human society. Section 3.1 provides a comprehensive review of major directions and trends in the modeling of technical progress, including: autonomous, induced, exogenous and endogenous, embodied and disembodied technological change, and technological change as a separate sector of the economy. Section 3.2 analyzes the classic Solow–Swan, Shell, and Ramsey models of economic dynamics with exogenous technological change. Section 3.3 explores modern one- and two-sector models with endogenous technological change, physical and human capital, and knowledge accumulation. Substitution, diffusion, and evolution models of technological innovations are briefly discussed in Sect. 3.4.

Natali Hritonenko, Yuri Yatsenko

Chapter 4. Models with Heterogeneous Capital

This chapter explores economic growth models with heterogeneous capital and labor described by the integral or partial differential equations. Such models are imperative in explaining economic development under embodied technological change. Section 4.1 describes the well-known macroeconomic growth models with vintage capital of R. Solow and L. Johansen and analyzes links among them. Optimization vintage capital models at a firm level are portrayed in Sect. 4.2. Section 4.3 considers models with investment into different vintages of capital. The last section discusses two fundamental replacement problems of Operations Research: the serial replacement of a single machine and the parallel replacement of several machines.

Natali Hritonenko, Yuri Yatsenko

Chapter 5. Optimization of Economic Renovation

This chapter analyzes optimization problems in the economic models with heterogeneous capital and labor of

Chap. 4

. Such models are important in explaining economic modernization under improving technology. Section 5.1 provides a qualitative analysis of the continuous-time optimization problem of one-machine replacement from Sect.


using standard tools of nonlinear optimization. Section 5.2 explores the optimal modernization of vintage capital in a profit-maximizing firm under environmental constraints. Section 5.3 investigates an optimization problem with nonlinear utility in the Ramsey vintage capital model of Sect.


. A balanced growth regime is established and analyzed under exponential technology and labor. It possesses new properties compared to the linear utility case. Section 5.4 contains a mathematical appendix that derives extremum conditions for vintage capital models using variation techniques and Lagrange multipliers.

Natali Hritonenko, Yuri Yatsenko

Models in Ecology and Environment


Chapter 6. Mathematical Models of Biological Populations

The study of development and interaction of biological species is an important direction of modern research. As one of the central parts of this study, mathematical modeling assists in understanding the behavior of populations and provides reliable forecasts and recommendations for sustainable policies and management. A variety of mathematical models of biological populations and their investigation techniques have been developed, but practice requires new models that consider, for instance, aftereffect and joint influence of different exogenous and endogenous factors. This chapter explores well-known population models that have become a foundation to contemporary models widely used in practice. Section 6.1 presents population models based on ordinary differential equations and basic elements of their analysis. Section 6.2 explores different types of interaction among species and offers a detailed analysis of predator–prey models. Section 6.3 discusses partial differential and integral models of population dynamics.

Natali Hritonenko, Yuri Yatsenko

Chapter 7. Modeling of Heterogeneous and Controlled Populations

Age-structured population models, considered in Chap. 6, have become a traditional tool in biological modeling and are widely used in other disciplines. They possess well-developed investigation techniques. However, the size of individuals is a more important parameter than their age for some species. Section 7.1 considers two size-structured models that describe a population with natural reproduction and a fully managed population and explores links between size- and age-structured models. Nonlinear models of heterogeneous populations with intraspecies competition and their investigation techniques (steady-state and bifurcation analyses) are discussed in Sect. 7.2. An endogenous control is introduced into the model to address management problems in farming, fishery, forestry, and other applications. Controlled age- and size-dependent models are considered in Sect. 7.3.

Natali Hritonenko, Yuri Yatsenko

Chapter 8. Models of Air Pollution Propagation

The modeling of environmental contamination is a complex subject that requires considering heterogeneous natural and human factors distributed in space and time. Corresponding models involve partial differential equations or their discrete analogues. This chapter presents and analyzes models of pollution propagation in the atmosphere. Section 8.1 reviews basic definitions and properties of pollution propagation in air and water. Starting with simple models, Sect. 8.2 derives partial differential equations for air pollution transport and diffusion with pollution sources. The next two sections are devoted to two air pollution control problems: the location of new plants and control of pollution intensity of existing plants. The last section discusses the structure and features of more complicated atmospheric pollution models. Models of water pollution are considered in Chap. 9.

Natali Hritonenko, Yuri Yatsenko

Chapter 9. Models of Water Pollution Propagation

The modeling of pollution dissemination in water is based on a mathematical description of hydrodynamic, hydraulic, and physical–chemical processes that control pollution transfer in water reservoirs. Section 9.1 describes the structure and classification of such models. Section 9.2 discusses a general three-dimensional model of water pollution, which includes transport and diffusion of pollutants in dissolved state, suspension, and ground deposits, adsorption and sedimentation of pollutants, subjected to water dynamics and boundary conditions of water reservoirs, and pollution sources. Section 9.3 describes a two-dimensional horizontal model of water pollution dissemination. Section 9.4 presents a simple one-dimensional model and its analytic solutions for one point source of pollutant. Section 9.5 explores compartmental models of water pollutions and related problems of water pollution control.

Natali Hritonenko, Yuri Yatsenko

Models of Economic-Environmental Systems


Chapter 10. Modeling of Nonrenewable Resources

This chapter is devoted to an important economic–environmental problem: the modeling of the optimal extraction and utilization of nonrenewable (exhaustible) resources. Nonrenewable resources are natural resources that cannot be replaced as quickly as they are being consumed. Examples of such resources include fossil fuels (petroleum, coal, and natural gas) and mineral resources (iron, gold, and other). The models of Section 10.1 consider the resource extraction process in isolation from other economic activities. Section 10.2 investigates the economic growth model with a two-factor Cobb–Douglas technology that uses physical capital and exhaustible resource to produce aggregate product.

Natali Hritonenko, Yuri Yatsenko

Chapter 11. Modeling of Environmental Protection

This chapter is devoted to analytic modeling of the relations between the economy, society, and the environment, with a focus on combating climate change. Section 11.1 describes simple models of these relationships and discusses mitigation and adaptation as two major human responses to environmental damage. Sections 11.2 and 11.3 investigate economic models with optimal investments into mitigation of environmental pollutions. Section 11.4 analyzes optimal investments into mitigation and adaptation against environmental damage. These economic–environmental models are formulated as social planner problems with mitigation and adaptation investments as separate variables. The steady-state analysis of optimal investments leads to essential implications for associated long-term environmental policies.

Natali Hritonenko, Yuri Yatsenko

Chapter 12. Models of Global Dynamics: From Club of Rome to Integrated Assessment

Human activities can cause negative global changes in the environment, such as global warming, excessive pollution of the environment, extinction of some species, degradation of entire geographic areas, and others. Section 12.1 discusses the modeling of global changes as a scientific problem of great complexity and importance. Section 12.2 describes the first global models—models of world dynamics developed by J. W. Forrester, D. H. Meadows, and M. D. Mesarovic in the 1970s. They represented first systematic attempts to analyze the global dynamics and produced outcomes that attracted significant public attention. These models evolved in the 1990s into a more quantitative approach known as integrated assessment models, which are explored in Sect. 12.3.

Natali Hritonenko, Yuri Yatsenko


Weitere Informationen

Premium Partner

in-adhesivesMKVSNeuer Inhalt

BranchenIndex Online

Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.



Technisches Interface Design - Beispiele aus der Praxis

Eine gute Theorie besticht nur darin, dass am Ende einer Entwicklung sinnvolle und nutzergerechte Produkte herauskommen. Das Forschungs- und Lehrgebiet Technisches Design legt einen starken Wert auf die direkte Anwendung der am Institut generierten wissenschaftlichen Erkenntnisse. Die grundlegenden und trendunabhängigen Erkenntnisse sind Grundlage und werden in der Produktentwicklung angewendet. Nutzen Sie die Erkenntnisse aus den hier ausführlich dargestellten Praxisbespielen jetzt auch für Ihr Unternehmen.
Jetzt gratis downloaden!


Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.