Table Of ContentInterdisciplinary Applied Mathematics
(cid:2)olume 14
(cid:2)ditors
(cid:2)(cid:3)(cid:2)(cid:3) Antman (cid:4)(cid:3)(cid:5)(cid:3)Marsden
(cid:6)(cid:3) (cid:2)ir(cid:7)(cid:8)ich (cid:2)(cid:3) (cid:9)i(cid:10)(cid:10)ins
Mathematical Biology
(cid:6)(cid:3) (cid:11)lass(cid:12) (cid:3).D. Murray
Mechanics and Materials
(cid:13)(cid:3)(cid:14)(cid:3) (cid:15)(cid:7)hn
Systems and Control
(cid:2)(cid:3)(cid:2)(cid:3) (cid:2)astry(cid:12)P.S. (cid:4)rishnaprasad
Problemsinengineering,computationalscience,andthephysicalandbiolog-
ical sciences are using increasingly sophisticated mathematical techniques.
Thus, the bridge between the mathematical sciences and other disciplines
is heavily traveled. The correspondingly increased dialog between the dis-
ciplines has led to the establishment of the series: (cid:3)nterdisciplinary (cid:4)pplied
Mathematics.
The purpose of this series is to meet the current and future needs for the in-
teractionbetweenvariousscienceandtechnologyareasontheonehandand
mathematicsontheother.Thisisdone,(cid:5)rstly,byencouragingthewaysthat
mathematics may be applied in traditional areas, as well as point towards
new and innovative areas of applications(cid:6) and, secondly, by encouraging
other scienti(cid:5)c disciplines to engage in a dialog with mathematicians out-
lining their problems to both access new methods and suggest innovative
developments within mathematics itself.
The series will consist of monographs and high-level te(cid:7)ts from researchers
working on the interplay between mathematics and other (cid:5)elds of science
and technology.
Interdisciplinary Applied Mathematics
Springer Science+Business Media, LLC
Akira (cid:8)kubo, with Simon A. Levin
Di¤usion and
Ecological Problems:
Modern Perspectives
Second Edition
(cid:9)ith 114 Illustrations
SimonA.Levin
DepartmentofEcologyand
EvolutionaryBiology
Princeton(cid:10)niversity
Princeton,N(cid:3)08544
(cid:10)SA
slevin(cid:11)eno.princeton.edu
(cid:2)ditors
S.S.Antman (cid:3).E.Marsden
DepartmentofMathematics ControlandDynamicalSystems
and AppliedMathematics
InstituteforPhysicalScienceandTechnology CaliforniaInstituteofTechnology
(cid:10)niversityofMaryland Pasadena,CA91125
CollegePark,MD20742-4015 (cid:10)SA
(cid:10)SA
L.Sirovich S.(cid:9)iggins
DivisionofMailCode107-81 ControlandDynamicalSystems
Brown(cid:10)niversity MailCode107-81
Providence,(cid:12)I02912 CaliforniaInstituteofTechnology
(cid:10)SA Pasadena,CA91125
(cid:10)SA
MathematicsSub(cid:13)ectClassi(cid:5)cation(cid:14)2000(cid:15):92A17,35(cid:4)55
LibraryofCongressCataloging-in-PublicationData
Di¤usionandecologicalproblems:modernperspectives(cid:16)(cid:17)editedby(cid:18)Akira(cid:8)kubo,
SimonA.Levin.(cid:19)2nded.
p. cm.(cid:19)(cid:14)Interdisciplinaryappliedmathematics(cid:6)14(cid:15)
(cid:12)ev.ed.of:Di¤usionandecologicalproblems(cid:16)Akira(cid:8)kubo.1980.
Includesbibliographicalreferences.
ISBN 978-1-4419-3151-1 ISBN 978-1-4757-4978-6 (eBook)
DOI 10.1007/978-1-4757-4978-6
1.Ecology(cid:19)Mathematicalmodels. 2.Di¤usion(cid:19)Mathematicalmodels.
3.Biogeography(cid:19)Mathematicalmodels. I.(cid:8)kubo,Akira. II.Levin,SimonA.
III.(cid:8)kubo,Akira.Di¤usionandecologicalproblems. I(cid:2).Interdisciplinaryapplied
mathematics(cid:6)v.14.
QH541.15.M3(cid:8)38 2001
577(cid:2).01(cid:2)5118(cid:19)dc21 00-052258
Printedonacid-freepaper.
(cid:2)(cid:2)(cid:3)(cid:3)(cid:4)(cid:5)(cid:4)(cid:6)(cid:7)(cid:3) Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in (cid:2)(cid:3)(cid:3)(cid:4)
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenper-
missionofthepublisherSpringer Science+Business Media, LLC.
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ISBN978-1-4419-3151-1 SPIN10700945
(cid:5)edicated to J(cid:6)G(cid:6) Skellam
Preface to the Second Edition
The story of this edition is a testament to an almost legendary (cid:5)gure in
theoretical ecology and to the in(cid:20)uence his work and charisma has had on
the(cid:5)eld.Itisalsoastorythatcanonlybetoldbyatripbackintime,tothe
genesis of the First Edition and before.
Akira (cid:8)kubo and I were students together, but never knew it at the time.
HewasagraduatestudentatThe(cid:3)ohnsHopkins(cid:10)niversity,whereIwasan
undergraduate in mathematics. (cid:9)e both studied (cid:21)(cid:21)modern physics,(cid:22)(cid:22) taught
by Dino Franco (cid:12)asetti, and we decided years later that we musthave been
in the same class. Akira was then a chemical oceanographer, but ship time
andhisstomachdidnotagree.Soheturnedtotheory,andtherestishistory.
His impact has been phenomenal, and the First Edition of this book was
his most in(cid:20)uential work. Building on his famous work with dye-di¤usion
e(cid:7)periments, he turned his attention to organisms and created a unique
melding of ideas from physics and biology.
In the early 1970s, Lee Segel and I began to work on problems of plank-
tonic patchiness, following some pathways that were simultaneously being
e(cid:7)plored by Akira (cid:14)on di¤usive instabilities(cid:15). This brought Akira and me
together, and he presented me with a copy of his 1975 book, (cid:2)cology and
(cid:5)i(cid:7)usion, published in (cid:3)apanese by Tsuk(cid:13)i Shokan, Tokyo. I could under-
stand all of the Greek in the book, but none of the (cid:3)apanese. Still, I recog-
nized enough that was familiar to know that this was an important book,
but one whose in(cid:20)uence was likely to be a bit limited if it remained only in
(cid:3)apanese.IwasatthetimeEditoroftheBiomathematicsSeriesforSpringer-
(cid:2)erlag, and I encouraged Akira to produce an updated version of his work
for us, inlarge part so that I could read it. He readily accepted.
The ne(cid:7)t several years produced une(cid:7)pected bene(cid:5)ts for me. As Akira
developed his chapters and sent them to me for comments and editorial
suggestions, he began frequent pilgrimages from the Marine Sciences (cid:12)e-
search Center at Stony Brook to Cornell, where I was a faculty member.
This led to fruitful collaborations between us, while he became a de facto
member of our family, and an additional adviser for many of my students.
(cid:8)(cid:9)(cid:9)
viii PrefacetotheSecondEdition
As I read each chapter, I was amazed by the novel insights, the originality,
and the newvistas opened for me.
The success of (cid:5)i(cid:7)usion and (cid:2)cological (cid:8)roblems stimulated much re-
search on di¤usion and other spatial models in ecology over the past two
decades,anditbecameclearthananupdatedversion,greatlye(cid:7)panded,was
needed.Akirasettoworkonthisashislastpro(cid:13)ectbutnever(cid:5)nished.(cid:9)hen
he passed away in 1996, he left a mass of notes, mostly cryptic ones to him-
self, about changes he planned to make. His close friend, the late (cid:4)eiko
Parker,assembledthesefromamongthepapersheleft,andsentthemtome
with the simple statement that Akira asked her to do so and that I would
know what to do with them. Considering the state of the notes, this was a
daunting challenge, but one that I could not duck. Fortunately, Akira had
many friends. My (cid:13)ob was reduced primarilyto one of editor.
I approached Achi Dosan(cid:13)h, Mathematics Editor of Springer, who was
enthusiastic about the idea of a Second Edition of this popular book. I laid
out plans for the pro(cid:13)ect to friends and colleagues of Akira and found more
whowerewillingtopitchinthanIcoulduse.Theideawasthateachchapter
would be adopted by someone e(cid:7)pert in the area, who would amend what
Akira had written 20 years ago, guided by the notes he left, and then add
newmaterialwhereappropriatetotracethein(cid:20)uenceoftheoriginalbookin
later work. Eachchapter posed unique challenges, and the results I feel sure
would havegrati(cid:5)ed Akira.
It will be clear in the reading that some branches of the sub(cid:13)ect have
developedmuchmorethanothers(cid:6)thisofcourseistobee(cid:7)pected.Theearly
chaptersprovidedtheclassicalbackgroundforthesub(cid:13)ectandhavechanged
only in small doses. Some of the later chapters summarize e(cid:7)tremely active
areas of research and have e(cid:7)panded substantially. Furthermore, the cover-
age is somewhat idiosyncratic, as Akira would have wanted. For e(cid:7)ample,
noe¤ort hasbeenmadeto provide e(cid:7)tensivecoverage of thegreat advances
in metapopulation theory and interacting particle systems, though these are
introduced. Di¤usionremains theintegrative theme.
(cid:8)ne piece remains unchanged(cid:19)Akira(cid:22)s original preface. As Editor, I
resisted suggestions that it was too personal and not su(cid:23)ciently scienti(cid:5)c. A
reading of it will make clear that it is a gem, capturing a view of Akira(cid:22)s
personality and humanitythat could emerge in noother way.
This book is a gift from Akira(cid:22)s friends and colleagues to the memory of
our Sensei. Akira, we hope you like it. Itwas awork of love.
Princeton, New (cid:3)ersey Simon (cid:4)(cid:6) Levin
(cid:8)ctober 2001
Preface to the First Edition
This book is an e(cid:7)tended version of the (cid:3)apanese edition, (cid:2)cology and
(cid:5)i(cid:7)usion, published in 1975 by Tsuki(cid:13)i Shokan, Tokyo. All the chapters
are revised and up-to-date(cid:6) the last chapter is completely rewritten. These
changesre(cid:20)ectanincreasedawarenessoftheimportanceofspatialprocesses
such as di¤usion in population ecology. The number of references cited in
this book isalmost double that oftheoriginal (cid:3)apanese te(cid:7)t.
This book surveys a wide variety of mathematical models of di¤usion in
the ecological conte(cid:7)t. The introductory chapter provides a brief history of
di¤usionproblemsinecologytogetherwithadiscussionontheuseofmathe-
matical models. The material presented throughout this book is almost en-
tirely concerned with deterministic di¤erential equation models. Chapter 2
presents the basics of di¤usion ranging from a simple random walk model
to environmental turbulent di¤usion. Chapters 3 and 4 deal with the appli-
cation of physical di¤usion models to the dispersal of passive abiotic and
bioticpropertiesinecosystems.Adetaileddiscussionofmathematicalmodel-
ing for animal dispersal is presented in Chapter 5(cid:6) in Chapter 6 some e(cid:7)am-
ples of modeling animal di¤usion with or without biological processes are
given.
The tone of the book changes slightly in Chapter 7, which deals with the
dynamics of animal grouping such as insect swarms and (cid:5)sh schools. Here
two counteracting processes, advection and di¤usion, are recognized to be
importantin modeling.
Chapter 8 is concerned with animal movements in home ranges, and
Chapter 9 e(cid:7)amines the role of advection and di¤usion in the patchy distri-
bution of organisms. The book ends with Chapter 10, which presents an
e(cid:7)tensive discussion of mathematical models of dispersing populations with
intra- and inter-species interactions(cid:6) some topics of interest are traveling
wavesofdispersingpopulations,modelsfordensity-dependentdispersal,and
di¤usion-induced instability in interacting populations.
The book is written with the primary intent of providing scientists, par-
ticularly physicists but also biologists, with some background of the mathe-
matics and physics of di¤usion and how they can be applied to ecological
(cid:9)(cid:10)
(cid:7) PrefacetotheFirstEdition
A swarm of desert locusts, from the air. Swarm covering 1 km2 with locusts (cid:20)ying
mainly below 20 m. (cid:4)enya, 13 (cid:3)anuary 1953. Photography by H. (cid:3). Sayer (cid:14)from
(cid:12)ainey,1958(cid:15).
A patch of tracer dye in the sea, from the air. Patch covering 1 km2 with most
dyestayingupper5mofwater.Seao¤Argentinecoast,27(cid:3)uly1964.Photography
byB.(cid:4)atz.
PrefacetotheFirstEdition (cid:7)i
problems. The secondary intent is to provide a specialized te(cid:7)tbook for
graduatestudents,whoareinterestedinmathematicalecology.Thereaderis
assumedtohaveabasicknowledgeofprobabilityanddi¤erentialequations.
Although this book is a volume in the Biomathematics Series, the use of
mathematics is limited to the e(cid:7)tent that mathematical models contribute to
the interpretation of actual data or to gaining insight into the role of di¤u-
sion in modelecosystems.
A great deal of good luck was necessary for the completion of this book(cid:6)
infact,seven(cid:21)(cid:21)goodlucks(cid:22)(cid:22)combined.Forthe(cid:5)rstpartofthegoodluck,Dr.
(cid:3). G. Skellam, (cid:21)(cid:21)father of ecological di¤usion,(cid:22)(cid:22) to whom this book is dedi-
cated,encouragedaphysicaloceanographerwhowasgreatlyinspiredbythe
pioneer work of Skellam and who was madly determined to study rodent
dispersalonland.(cid:9)ithoutDr. Skellam(cid:22)sinterest inmyapproach,this book
would probablynot havebeenwritten.
For the second part of the good luck, Dr. H. C. Chiang, eminent ento-
mologist, has been so generous as to allow me to work with him on midge
swarming. I consider myself very lucky to have an association with Dr.
Chiang.Thetinymidgeshaveeducatedmeinthedangerofbuildingmathe-
maticalmodelsforbiologicalproblemsonsand.(cid:9)ithoutDr.Chiang(cid:22)sinter-
disciplinarymind,certainpertinentchapterswouldhavebeenmissinginthis
book.
For the third part of good luck, I work at the Marine Sciences (cid:12)esearch
CenterunderthedirectionofDr.(cid:3).(cid:12).Schubel,whoisamostbroadminded
leader. (cid:8)bviously, it is sometimes very di(cid:23)cult to (cid:13)ustify an oceanographer
who works with midges in a corn (cid:5)eld of Minnesota. Fortunately enough,
Dr. Schubel has a sense of scienti(cid:5)c humor and a perception of how knowl-
edgeinone(cid:5)eldcanaddtothatinanother.(cid:21)(cid:21)(cid:9)hoknows(cid:24)Manymillionsof
years from now, those darned insects might invade deep into the last re-
maining sanctuary on the earth (cid:14)i.e., the ocean(cid:15)(cid:25)(cid:22)(cid:22) (cid:9)ithout Dr. Schubel(cid:22)s
understanding, I would have had to resign from MS(cid:12)C to complete this
book.(cid:14)(cid:9)hatwouldhappenifIaskhimtogototheSerengetiNationalPark
to study wildebeest swarming, Inow wonder.(cid:15)
For the fourth part of good luck, I am indebted to Dr. S. A. Levin and
Mr. (cid:4). (cid:26)ano. Mr. (cid:26)ano was the science editor of Tsuki(cid:13)i Shokan who
kindly encouraged an amateur (cid:21)(cid:21)ecologist(cid:22)(cid:22) to write a book on ecological
modeling. The result was my (cid:3)apanese te(cid:7)t in 1975. (cid:10)nfortunately, the
Shokanwentbankruptimmediatelyafterward(cid:6)Ifeltbadaboutit,wondering
whether there was a causative relationship. Meanwhile, Dr. Levin, a dis-
tinguished mathematical ecologist, became aware of the book and made a
generous o¤er to publish an English translation of the book as a Lecture
NoteinBiomathematicsforwhichheservesasthemanagingeditor.Asdays
passed, the idea of translation evolved into a new book. Dr. Levin kindly
reviewed all my drafts and gave constructive comments and criticism.
Nonetheless, I may not have taken into account all the comments that he
