Table Of ContentInterdisciplinary Applied Mathematics 49
Frithjof Lutscher
Integrodiff erence
Equations in
Spatial Ecology
Interdisciplinary Applied Mathematics
Volume 49
Editors
AnthonyBloch,UniversityofMichigan,AnnArbor,MI,USA
CharlesL.Epstein,UniversityofPennsylvania,Philadelphia,PA,USA
AlainGoriely,UniversityofOxford,Oxford,UK
LeslieGreengard,NewYorkUniversity,NewYork,USA
Advisors
L.Glass,McGillUniversity,Montreal,QC,Canada
R.Kohn,NewYorkUniversity,NewYork,USA
P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA
AndrewFowler,UniversityofOxford,Oxford,UK
C.Peskin,NewYorkUniversity,NewYork,USA
S.S.Sastry,UniversityofCalifornia,Berkeley,CA,USA
J.Sneyd,UniversityofAuckland,Auckland,NewZealand
RickDurrett,DukeUniversity,Durham,NC,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/1390
Frithjof Lutscher
Integrodifference Equations
in Spatial Ecology
123
FrithjofLutscher
MathematicsandStatistics
UniversityofOttawa
Ottawa,ON,Canada
ISSN0939-6047 ISSN2196-9973 (electronic)
InterdisciplinaryAppliedMathematics
ISBN978-3-030-29293-5 ISBN978-3-030-29294-2 (eBook)
https://doi.org/10.1007/978-3-030-29294-2
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Tomyteachers.
Foreword
Integrodifference equations are relative newcomers to the world of mathematical
ecology. First emerging in the 1970s and 1980s, these discrete-time, continuous-
space models with generalized growth and dispersal components have increased
in popularity, and the number of variations has grown over time. Much of this
increaseinpopularityhasbeendrivenbytherealization,onthebiologicalside,that
suchmodelscanbemorecloselyalignedwithrealecologicalsystemsthancanthe
more traditional reaction–diffusion models. Growth is often seasonal, so discrete-
time dynamics makes sense, and dispersal is often complex and nondiffusive, so
generalized dispersal kernels give added realism. From a broader perspective, the
move toward integrodifference equations reflects a desire to forge more intimate
connections between theory and biological observation than has been previously
found. This desire, shared by ecologists and mathematicians alike, reflects the
healthystateofthefieldoftheoreticalecology.
Whileoriginallydesignedtomodelthespace-timedynamicsofasinglespecies
with discrete, nonoverlapping generations, integrodifference equations have now
been extended to cover a range of additional features, such as age-structure,
interactions between different species, stochasticity, heterogeneous environments,
and complex, multi-staged dispersal strategies, to name a few. The importance of
getting the details of ecology exactly right has grown. Ecologists are no longer
satisfiedwithcrudecaricaturesofthesystemstheystudy.Thereisamovetoensure
thatmodelswillnowprovidefaithfulreflectionsofdetailedandspecificquantitative
hypotheses about the biological processes. Integrodifference equations have a role
toplaybyinjectingmuch-neededrealismintospace-timemodelsinecology.
Although typically providing more ecological realism than reaction–diffusion
models, integrodifference models nonetheless retain many qualitative features
found in reaction–diffusion models. The critical patch-size problem, describing
the minimum area needed for population persistence, exists for both models, as
do the traveling wave and spreading speeds problems, describing the wave of
advanceofinvasivepopulations.Patternformationviadispersal-driveninstabilities
can also be found in both models, as can many other features. In other words,
integrodifferenceequationshavemuchincommonwithparabolicpartialdifferential
vii
viii Foreword
equations.Bothcouplegrowthanddispersalinasimilarway,andthisgivesriseto
deeper mathematical properties that are shared and govern fundamental properties
of solutions. For example, under reasonable assumptions, each equation has a so-
called comparison principle, which states that the operator is order preserving:
initially ordered solutions remain ordered for all time. In turn, this feature allows
for the construction of super- and sub-solutions, which can be very useful in
determiningqualitativebehaviorssuchaspopulationpersistenceortravelingwaves.
Thus,thestudyofintegrodifferenceequationshasmanymathematicalfeaturesthat
willresonatewithresearchersversedinthetheoryofreaction–diffusionequations.
Despitethesesimilarities,integrodifferencemodelsalsoexhibitdifferenceswith
reaction–diffusion models. Many of the differences arise from the very general
natures of the growth and the dispersal models found in the integrodifference
framework.Thegrowthterm,essentiallyadiscrete-timemap,canexhibitahump-
shaped feature, ecologically describing the process of overcompensation, which
may lead to chaos, even for a single species. This property of chaotic dynamics
cannot be found for single-species differential equation models. In fact, it is well
knownthatthreeinteractingspeciesarerequiredbeforechaoscanariseinasystem
of differential equations. The dispersal kernel, expressed as a probability density
function, can take any number of exotic forms, including heavy tails. These can
lead to unusual outcomes, for example, giving rise to an invasion processes that
continually accelerate with time. What a wonderful mathematical playground in
whichtoundertakequantitativeanalysis!
The timing of this book is perfect. While the number of papers developing and
analyzing integrodifference equations is growing quickly, it is still just possible to
compile a reference list that includes every significant publication written in the
area. In this volume, Frithjof Lutscher has attempted the monumental task, and in
doing so has created a comprehensive survey of the field. The book is, however,
muchmorethanasurveyofintegrodifferenceequations.Therearesignificantnew
resultsinthebook,particularlyinlatersections.Thesenewresultsmeshseamlessly
withexistingknowledge,fillingingapsandextendinganalyses.
Taken in its entirety, the book provides a synthesis, a woven tapestry of ideas,
based on the mathematical structures underlying integrodifference equations, and
on the biological insights that these mathematical structures yield. Imprinted on
the book is the perspective of an applied mathematician, embracing structure and
strivingforclarityofthought.Assuch,thebookisajoytoread.
Opportunitiesofthistype,towriteabookthatcarvesoutadevelopingfield,are
rareandvaluable.WhenIreadthebook,IamgladthatFrithjofLutscherundertook
thewritingbecausehehasdoneatremendousjobofit.Thiswillbetheperfecttext
for both graduate students who are new to the field and for seasoned researchers
whoneedtolookupspecificresults.Thistextistimelyandwilldefinethefieldof
integrodifferenceequationsformanyyearstocome.
PenderIsland,BC,Canada MarkLewis
June2019
Preface
Ecosystemsaremarvelousassemblagesofindividualsthatgrow,reproduce,interact
with one another, move about in space, and eventually die. Ecosystems also
provide essential services to humans, from oxygen production and recycling of
organic matter to food provision and pollination of agricultural crops. At the
same time, ecosystems are in peril from human activity, such as overexploitation,
landscape fragmentation, and various effects of global change. Spatial ecology
aims to understand the role that individual movement, population interaction, and
landscapecharacteristicsplayingeneratingthepatternsofspeciesdistributionsthat
we observe in space and time, in particular questions of population persistence,
populationspread,andstability.
Dynamic mathematical models are powerful tools for understanding natural
phenomena in the physical sciences and increasingly also in the life sciences.
Mathematical methods, from the rigor of model formulation to the depth of
analysis and the power of computation, are indispensable when dealing with the
wide range of spatial and temporal scales that are inherent in many of the most
fundamental questions of spatial ecology. Dynamical systems are also fascinating
objects to study in their own right. The interaction between dynamical systems
models and their applications in ecology and other fields is a constant source of
mutual challenges and inspiration. The vision of this book is to introduce the
reader to a class of models known as integrodifference equations and show how
the fascinating, interdisciplinary circle of observation, modeling, analysis, and
interpretation enhances ecological understanding and mathematical theory at the
sametime.
Integrodifferenceequationsaremodelsforthetemporalevolutionofthedensity
of one or more populations from one generation to the next. These models are
tailor-made to adequately represent the dynamics of organisms with a particular
life cycle, where population growth and individual dispersal occur separated in
time but synchronized in the population. Ecological examples contain many plant
and insect species, particularly in temperate climates. Mathematically speaking,
integrodifferenceequationsdefinediscretedynamicalsystems(recursions)onsome
appropriatefunctionspace.Theirmathematicalanalysisandecologicalapplication
ix
x Preface
have seen great progress in the past three decades. This book provides the first
comprehensive introduction to the subject and serves as a reference guide to the,
bynow,sizableliteratureonallaspectsoftheseequations.
Thebookisdividedintothreeparts:
IBasicsandFoundations. Chapters 1–8 contain the most important aspects of
modeling with and analysis of integrodifference equations, using the simplest
possible scenarios. The focus is on extinction, persistence, stability, and spread
of a single species in simple landscapes. Basic ideas for numerical approaches
areprovided.
IIApplicationsandApproximations. Chapters 9–12 consider various aspects
of real-world applications. The simplifying assumptions from the first part
are modified where necessary to more adequately describe realistic situations.
Variousmethodsofmodelsimplificationarepresented.
IIIExtensionsandChallenges. Chapters 13–17 discuss substantial and signif-
icant extensions to the simpler models in order to tackle more challenging
biological questions, such as the dynamics of stage-structured populations,
interactingpopulations,andspatialandtemporalvariationintheenvironment.
Throughout this book, questions of population dynamics and their application
to real-world systems motivate all models and mathematical analysis, and all
theoreticalandcomputationalresultsarediscussedinrelationtoecologicaltheory
and implications. The greatest progress is made where ecology and mathematics
cometogethertoinspireeachothertowarddeeperunderstandingineachdiscipline
andtheirinterplay.
