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Springer Optimization and Its Applications 88 Natali Hritonenko Yuri Yatsenko Mathematical Modeling in Economics, Ecology and the Environment Second Edition Springer Optimization and Its Applications VOLUME 88 Managing Editor Panos M. Pardalos (University of Florida) Editor–Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (Lehigh University) Y. Ye (Stanford University) Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the past several decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences. TheseriesSpringerOptimizationandItsApplicationspublishesundergraduate and graduate textbooks, monographs and state-of-the-art expository work that focusonalgorithmsforsolvingoptimizationproblemsandalsostudyapplications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization,optimalcontrol,discreteoptimization,multiobjectiveprogramming, description of software packages, approximation techniques and heuristic approaches. For furthervolumes: http://www.springer.com/series/7393 Natali Hritonenko (cid:129) Yuri Yatsenko Mathematical Modeling in Economics, Ecology and the Environment NataliHritonenko YuriYatsenko DepartmentofMathematics SchoolofBusiness PrairieViewA&MUniversity HoustonBaptistUniversity PrairieView,TX,USA Houston,TX,USA ISSN1931-6828 ISSN1931-6836(electronic) ISBN978-1-4614-9310-5 ISBN978-1-4614-9311-2(eBook) DOI10.1007/978-1-4614-9311-2 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013953869 MathematicsSubjectClassification(2010):44–01,49–01,49N90,91B76,93A,97M40 ©SpringerScience+BusinessMediaNewYork1999,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Mathematicalmodelingistheartofdescribingreal-worldphenomenaintheterms ofmathematicalconcepts.Theincreasingimportanceofmathematicalmodelingin solvingeconomic–environmentalproblemsisarelevanttrendinmodernresearch. Relations between human society and the environment are interdisciplinary and include technological, scientific, economic, biological, demographic, social, and political aspects. This textbook presents various mathematical models used in economics, ecology, and environmental sciences and discusses connections among them. Topics include economic growth and technological development, population dynamics and human impact on the environment, resource extraction andscarcity,airandwatercontamination,rationalmanagementofeconomyandthe environment, climate change, and global dynamics. The authors focus on deter- ministic models and their investigation techniques, including discrete and contin- uousmodels,differentialandintegralequations,optimizationandoptimalcontrol, andsteady-stateandbifurcationanalyses. This expository textbook offers an attractive collection of a wide range of modelsrangingfromtheclassicCobb–Douglasproductionfunction,Solowmodels of economic growth, Lotka–Volterra and McKendrick–McCamy population models, Hotelling and Dasgupta–Heal models of exhaustible resource, and Forrester and Meadows models of world dynamics to modern models of technological change and environmental protection that have so far appeared only in scientific journals. The authors demonstrate that the same models can be used to describedifferenteconomicandenvironmentalprocessesandsimilarinvestigation methodsareapplicabletoanalyzevariousmodels. Themaingoalsofthistextbookare: (cid:129) To expose modern practice of applied mathematical modeling in economics, populationbiology,andenvironmentalsciences (cid:129) To describe relations among various economic, population, and environmental models (cid:129) To demonstrate how integrated mathematical models are built from simple components v vi Preface (cid:129) To explain investigation techniques for considered models and to provide an interpretationoftheobtainedresults This textbook is intended for graduate and upper-division undergraduate students,faculty,academics,andindustrypractitionersineconomicsandenvironmen- tal sciences as well as for a wide mathematical audience. It also presents a self- containedintroductionforresearcherscomingintothefieldforthefirsttime. Textbook Features SincethepublicationofthefirsteditionofthisbookbyKluwerAcademicin1999, the authors have been regularly contacted by universities from across the world aboutusingitasatextbook.ThefirsteditionwasrepublishedinChinain2006by Science Press as Volume 23 of their “Series of Mathematical Masterpieces Abroad” and translated entirely into Chinese by the Renmin University of China Pressin2011. Thesecondeditionisentirelyrevisedandupdatedcomparedtothefirstedition. Obsoletematerialhasbeenreplacedwithnewandmorerelevantmodels,references have been essentially updated, and exercises have been added to the end ofevery chapter. Solutions to all end-of-chapter exercises and other supplement materials are not included due to space constraints but will be available to instructors who adoptthistextbookintotheircourses(contacttheauthorsatnahritonenko@pvamu. edu or the publisher). The present edition has been classroom tested. The authors havesuccessfullyuseditsdraftinteachingundergraduateandgraduatecoursesin mathematical modeling at several universities in the United States, Europe, andAsia. Most of the material is modular to allow for various course configurations, emphasizing certain economic, biological, or environmental applications. The authors strive to give an instructor substantial flexibility in designing a syllabus andusingtheirpreferredmathematicaltools.Themajorityofchaptersarerelatively independentandcanbecoveredinfullorpartiallyandinanarbitraryorder.Some exceptionsarethefollowing: (cid:129) Chapter2andSect.3.1arerecommendedforabetterunderstandingofChaps.4, 5,and10–12. (cid:129) Chapter5continuesChap.4andismathematicallymoreadvanced. (cid:129) Chapter7canbecoveredafterChap.6andismoreadvanced. (cid:129) Chapter9canbecoveredafterChap.8andismoreadvanced. To better understand a specific modeling problem, students need an integrated understanding of mathematical modeling. To address this need, the textbook explores a variety of diverse mathematical models from different applied areas andprovideselementsoftheiranalysis,ratherthanmerelyfocusingonacomplete analysisof few problems. It includes theorems with occasional proofs where they Preface vii arereasonableandeffective.Specialattentionisgiventothestep-by-stepconstruc- tion of models, choice of control variables, analysis of arising mathematical problems and their interaction, qualitative behavior of model trajectories, and appliedinterpretation. Thesetofconsideredeconomic–environmentalmodelsisrepresentativeenough todemonstratehownewproblemsandprocessesunderstudydeterminethechoice ofmathematicaltools.ThemodelsofChaps.2,3,6,10,and11usemostlyordinary differential equations, whereas Chaps. 7–9 consider partial differential equations andChaps.4and5useintegralequations. This textbook explains how complex models are constructed from common simple modules that describe elementary economic and environmental processes. The economic models of Chaps. 2 and 3 are used as blocks in the models of Chaps. 4, 5, 10, 11, and 12. The models of resource extraction of Chap. 10 are usedasblocksinthemodelsofChaps.11and12.Themodelswithenvironmental controlofChap.11andtheworldmodelsofChap.12considerpopulationmodels of Chap. 4 and environmental pollution models of Chaps. 8 and 9 in an aggregatedform. Recommended Courses Thepresenteditionisdesignedtoserveasatextbookforone-and/ortwo-semester courses in mathematical modeling. The entire content of this textbook covers a two-semestergraduatecourseinAppliedMathematicalModelingorMathematical ModelsandMethods.Inone-semestercourses, somechapters andsections canbe omitted without affecting the logical development of the material. Depending on thechapterschosen,thistextbookcanfitbothundergraduateandgraduatecourses. Specifically, it can be used for several undergraduate courses in Departments of Mathematics, Environmental Sciences, Environmental Research, Ecosystem Sci- ence and Management, Management Sciences, Science and Technology, and so on. The table below lists some of the courses for which this textbook is recommended. Samplecourse Level Content AppliedMathematicalModeling Undergraduate Chaps.1–4,6,10,12 MathematicalMethodsinEconomicsandEnvironment Undergraduate Chaps.1–4,8,10,11 MathematicalModelinginEconomics Undergraduate Chaps.1–5,10,12 ModelsofBiologicalSystems Undergraduate Chaps.1,6,7,10,12 EnvironmentalModels Undergraduate Chaps.1,2,6,8–12 AppliedOptimizationModels Undergraduate Chaps.1–7,10,11 MathematicalModels(formathematicsmajors) Undergraduate Chaps.1,2,4,5,6–10 MathematicalModeling(fornon-mathematicsmajors) Undergraduate Chaps.1–3,6,10,12 AppliedMathematicalModeling Graduate Chaps.1–4,6,8,10–12 MathematicalModelsandMethods(twosemesters) Graduate Chaps.1–12 viii Preface The presentation level requires mathematical knowledge of basic university mathematicscourses,includingCalculusand,ideally,DifferentialEquations.The authorsavoidusingadvancedmathematicalconceptsorprovidethemasAppendi- ces,e.g.,inChaps.2and5. Review of Textbook Content Chapter1exploresthestepsofappliedmathematicalmodelingandprovidesabrief overviewofitsconcepts,notations,andtools.Theremainingchaptersaredivided intothreeparts. Part I “Mathematical Models in Economics” (Chaps. 2–5) 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 sophisti- catedmodelsofintegratedsystems.Chapters2and3analyzeaggregatenonlinear 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 appendixtoChap.2containsareview ofextremumconditions(maximumprinci- ple)fortheoptimalcontrolproblemsstudiedinthisandsubsequentchapters. Part II “Models in Ecology and Environment” (Chaps. 6–9) explores various mathematicalmodelsusedinpopulationandenvironmentalproblems.Itcoverstwo large topics: models of biological communities and their rational exploitation (Chaps. 6 and 7) and models of pollution propagation in the atmosphere and water reservoirs (Chaps. 8 and 9). Some basic models of Chaps. 2–10 are only briefly discussed because they can be found in more specialized textbooks. How- ever, more complex models constructed from these components are explained in detail. PartIII“ModelsofEconomic–EnvironmentalSystems”isdevotedtointegrated models of economic and environmental dynamics. Chapter 10 describes various modelsofnonrenewableresourceextraction,includingthewell-knownHotelling’s rule and Dasgupta–Heal modelof 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 accu- mulation and abatement and adaptation to environmental damage. Chapter 12 offers a brief glance at the history and mathematical structure of famous models ofglobalchange,fromtheClubofRomemodelstothemodernintegratedassess- mentmodels,theirspecifics,achievements,andlimitations. PrairieView,TX,USA NataliHritonenko Houston,TX,USA YuriYatsenko Acknowledgements Theauthorswouldliketoexpresstheirspecialthankstotheirdearcolleaguesand collaboratorsProfs.DaurenAdilbekov,Noe¨lBonneuil,SeilkhanBoranbaev,Raouf Boucekkine,ThierryBrećhet,BenitoChen,RenanGoetz,ElinaGrigorieva,David Greenhalgh, Aliakbar Haghighi, Boyan Jovanovic, Nobuyuki Kato, Andre de Korvin, Sergey Lyashko, Janos Turi, Angels Xabadia, and George Zaccour for longdiscussionsandfriendlyadvice. TheauthorsarealsogratefultoProfs.RezaAhangar,ShairAhmad,LindaAllen, SebastianAnita,TedBarton,AlainBensoussan,GeorgeBitros,SteveBleiler,John Boland, Alberto Bucci, Michael Caputo, Felix Chernousko, Constantin Conduneanu, Allen Donald, Saber Elaydi, Gustav Feichtinger, Jerzy Filar, Rafail Gabasov,AnahitGalstyan,MaryAnnHorn,RichardB.Howarth,MimmoIannelli, PeterKort,YuriLedyaev,UrszulaLedzewicz,MariaLeite,SuzanneLenhart,Omar Licandro, Klaus Puhlman, Roy Radner, Gerald Rambally, Jose´ Ramoń Ruiz- Tamarit,S.TrivikramaRao,CatherineRoberts,SureshSethi,KatherineSchubert, OlliTahvonen,CuongLeVan,VladimirVeliov,JayWalton,andDavidZilberman fortheirkindassistance,usefulremarks,andconversationsthattheymayhavevery wellforgotten. This textbook benefited from the support of NSF, NATO, International Soros Foundation, Isaac Newton Institute for Mathematical Sciences (University of Cambridge,UK),MathematicalScienceResearchInstitute(UniversityofBerkeley, CA),InstituteforMathematicsanditsApplications(UniversityofMinnesota,MN), American Institute of Mathematics (Palo Alto, CA), and Prairie View A&M University. Finally, our deep appreciation goes to our students Marcia Brown, Frankson Collins, Idrissa Diarra, Esmaeli Djavidi, Michelle Jackson, Khavansky Johnson, SantosPedraza,SheriStewart,KassoumTraore,andOlgaYatsenko,whocarefully studiedtheoriginalmanuscript,askingnumerousquestionsandmakingsuggestions thatimprovedourwork. ix