Tentative Title: The theory of coevolution The genetic theory of

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Tentative Title: The theory of coevolution
The genetic theory of coevolution
Author: Scott L. Nuismer
Coevolution between species drives diversification, promotes rapid and sustained evolutionary change,
and facilitates major evolutionary transitions. When combined with its clear relevance to applied
problems such as virulence evolution and the spread of invasive species, the recent explosion of studies
exploring the coevolutionary process is unsurprising. To a large extent, empirical research on
coevolution has been regularly and comprehensively synthesized by John N. Thompson in his wonderful
books, Interaction and coevolution (1982), The coevolutionary process (1994), and The geographic
mosaic of coevolution (2004). Although Thompson’s books include results of theoretical studies, the
sheer scope of his books precludes detailed treatment of coevolutionary theory. Consequently, the
mathematical models and key assumptions upon which our understanding of the coevolutionary
process rests remain scattered throughout the primary literature. The central goal of this book is to
present the key mathematical underpinnings of coevolution in a way that makes them accessible to a
broad range of biologists.
Although this book will synthesize coevolutionary theory, it is not intended to be an exhaustive review.
Instead, each chapter will develop mathematical models of specific coevolutionary processes from first
principles. These models will then be used to illustrate key assumptions, mathematical techniques, and
coevolutionary results that define and inform broad areas of coevolutionary research. Early chapters will
develop and analyze the most basic coevolutionary models upon which more advanced treatments are
based. Later chapters will build on these basic models to tackle more challenging questions using
sophisticated mathematical approaches that define modern coevolutionary research. By developing the
mathematical formalism from the ground up, I hope to make coevolutionary theory accessible to a wide
range of biologists. Furthermore, this structure will maximize the utility of the book as a template for
graduate seminars and courses focused on species interactions and coevolution. The availability of
online Mathematica notebooks developed to accompany each chapter will enable individual
investigation and provide a scaffold from which individuals can build more complex models tailored to
their specific interests and study systems.
Tentative Outline
Introduction
Chapter 1. A brief history of coevolutionary theory
-
Why the time has come to re-define coevolution. Its really about reciprocal selection. This is
what all the models study. The theory of coevolution is actually a theory of reciprocal
selection.
Chapter 3. The structure of coevolutionary models
Infection matrices. What are they? Pattern vs process. How could we identify them? Evolutionary and
experimental sampling effects
Interaction functions. What are they? How could we identoify the,m?
Section I. The origins of coevolutionary theory
What was the empirical motivation?
Section II. Fundamentals of coevolution
Chapter 1. Janzen and reciprocal evolutionary change
i. Reciprocal natural selection
ii. Evolutionary fundamentals
iii. Pre-requisites for coevolution
Section II. Coevolution in simple genetic systems
Chapter 2. Coevolution of major genes
i.
ii.
iii.
A brief introduction to infection genetics
A generic single locus framework
Infection genetics and coevolutionary dynamics
Chapter 3. Coevolution of polygenic traits
i.
ii.
iii.
iv.
The phenotypic interface of coevolution
A general framework for the analysis of coevolving quantitative traits
The coevolutionary dynamics of matching
The coevolutionary dynamics of escalation
Section III. Coevolution and Demography
Chapter 4. Coevolution in Lotka-Volterra systems
i.
ii.
iii.
Merging coevolution and demography
Separation of time scales approximations and adaptive dynamics
Coevolution of predator and prey
Chapter 5. Coevolutionary epidemiology
i.
ii.
Integrating coevolution into epidemiological models
The coevolution of virulence
Section IV. Coevolutionary genetics
Chapter 6. Dominance and segregation
i.
ii.
Diploid infection genetics and dominance
Coevolutionary dynamics, segregation, and deviations from Hardy-Weinberg
Chapter 7. Multiple loci, epistasis, and recombination
i.
ii.
Multi-locus infection genetics and epistasis
Coevolutionary dynamics, recombination, and linkage disequilibrium
Chapter 8. Modifier models and the coevolution of genome structure
i.
ii.
iii.
iv.
What are modifier models and why are they useful?
QLE approximations
The coevolution of ploidy levels
The red queen and coevolutionary models for the evolution of sex
Section V. Spatial structure and local adaptation
Chapter 9. Local adaptation, trait matching, and the geographic mosaic
i.
ii.
Overview of empirical patterns
The geographic mosaic theory
Chapter 10. The coevolutionary dynamics of local adaptation
i.
ii.
Quantifying local adaptation
Local adaptation in coevolving metapopulations
Chapter 11. The coevolutionary dynamics of trait matching
i.
ii.
Quantifying trait matching
Trait matching in coevolving metapopulations
Section VI. Multi-species interactions and community structure
Chapter 12. Coevolution in three species interactions
i.
ii.
iii.
Genetic correlations and interaction epistasis
Diffuse vs. pairwise coevolution
The dynamics of diffuse vs. pairwise coevolution
Chapter 13. Coevolution OF NETWORKS
Section VII. MacroCoevolution
Chapter 13. Coevolution and phylogeny
Section VIII. Three Myths of Coevolution
1. Local adaptation = winning
2. GFG genetics exists
3. Coevolution is a special case
Introduction
Coevolution: moving beyond the special case scenario upon which pop gen and q gen is built. Here I am
going to argue for a twfold revolution: 1) rethink Janzenian definition and focus instead on Thompsonian
selection mosaics which are, after all, what all coevolutionary models are made of 2) thrust coevolution
into the general center of P-Gen and Q-gen. For too long coevolution has been relegated to “ecology”
and PQ-gen has marched merrily along in ignorant bliss.
AS RICHARD GOMULKIEWICZ ONCE TOLD ME: IT IS THE POPULATION GENETICISTS WHO ARE THE
CONCIENSCE OF BIOLOGY. It is my hope that this book serves as a counter-point and acts in this way: as
the conscience of coevolutionary biology.
WE SHOULD MAINTAIN JANZEN’S DEFINITION TO COMBAT THE FORCES OF SLOVENLY SLOPPY
THINKING. AS MIKE LYNCH WROTE “THE DETAILS MATTER AND THEY MATTER A LOT” . JANZEN
BROUGHT RIGOUR TO COEVOLUTION; A RIZGOUR THAT HAS FADED.
COEVOLUTION HAS BECOME ISOLATED DUE TO THE SRINGENCY OF THE DEFINITION. HOWEVER, A
MORE ACCURATE VIEW OF MOST COEVOLUTIONARY THEORY IS AS A ***GENERALIZATION OF
STANDARD Q and P GEN*** which relaxes key assumptions. IT IS TIME FOR COEVOLUTION TO ZGO
MAINSTREAM AND NO LONGER DEFINED BY SOMEWHAT QUIRKY AND PARTICULAR METHODOLOGIES
AND QUESTION. THIS SHIT IS FUNDAMENTAL.
ALTHOUGH COEVOLUITON IS OFEN VIEW AS RARE OR AS A SPECIAL CASE, FROM A MATHEMATICAL
STANDPOINT, IT IS IRREFUTABLE THAT COEVOLUTIONARY THEORY IS ACTUALLY A GENERALIZATION OF
SINGLE SPECIES PGEN AND QGEN.
IN ADDITION TO NOT REFLECTING WHAT WE ATUALLY MODEL, JANZEN’s DEFINITION IS FAR TOO
DIFFICULT TO DOCUMENT IN THE NATURAL WORLD. WE ALMOST NEVER DOCUMENT EVOLUTION LET
ALONE COEVOLUITON. AN AUXILARY PURPOSE OF THIS BOOK IS THAT THE TIME HAS COME TO MOVE
COEVOLUTION INTO THE PURVIEW OF ORDINARY EVOLUTIONARY TECHNIQUES. THUS, AS WITH MUCH
OF EVOLUTIONARY BIOLOGY, WE SHOULD FOCUS ON THE PRE-REQUISITES. IS THERE
COEVOOLUTIONARY SELECTION FOR TRAITS WITH ADDITIVE GENETIC VARIANCE…
I HAVE A LOVE HATE RELATIONSHIP WITH JANZEN’S DEFINITION. ON THE PLUS SIDE, IT MADE
COEVOLUTION CONCRETE AND ALLOWED US TO FOCUS ON ONE SPECIFIC PHENOMENON IN A
RIGOROUS WAY AND BROUGHT THE FOCUS IN ON PAIRS OF INTERACTING TRAITS. AT THE SAME TIME,
HOWEVER, JANZEN’S DEFINITION IS SO IMPRACTICAL IT HAS ALIENATED MANY RESEARCHERS AND
PUSHED COEVOLUTION INTO THE FRINGE. WORSE YET, I WILL DEMONSTRATE AND ARGUE
THROUGHOUT THIS BOOK, THAT THE VAST BULK OF COEVOLUTIONARY MODELS INCLUDE THE PREREQUISOITES FOR JANZENIAN COEVOLUTION BUT DO NOT per se STUDY JANZENIAN COEVOLUTION.
Chapter 1. A brief history of coevolutionary theory (COMBINE WITH INTRODUCTION?)
Although Ehrlich and Raven popularized coevolution, the modern idea of coevolution — and the term —
was first formalized by Mode in 1950. Not really clear why Ehrlich and Raven is considered so important
when, at least to a theoretician, it was mode who nailed it. Thompson has provided an exhaustive
history of coevolution in his books. My goal here is somewhat different. Specifically, I hope to provide a
view of the historical development of coevolutionary modeling; one informed by the sequence of
influential models related to the topic developing from Mode’s first published model to now…
Empirical/Conceptual Motivation
1. Flor, Flax, and plant disease
Mode and Jayakar
2. Epidemiology
Anderson and May
Frank
3. Evolution of sex
Frank, Parker, Lively, Hamilton, NEE
4. Community ecology and assembly and stability
Roughgarden, Abrams, Dieckman
5. Spatial structure and local adaptation (or call this modern era where we know coevolution matters
for everything? Here we see coevolution being brought into the general fold of population genetics
and quantitative genetics. No more freaky special case modeling, just a relaxation of normal pop gen
and q gen assumptions)
GMTC papers, Morand, Gandon, Gavrilets
6. Population genetic and quantitative genetic approaches with analytical results
Seger 1988
Gavrilets
Mode (1958). “A mathematical model for the co-evolution of obligate parasites and their hosts”
EHRLICH AND RAVEN (1964)
Jayakar (1970)
Roughgarden and ecological coevolution leading the field astray… Down the path of any reciprocal
influence = coevolution…
Jaenike (1978)????
Slatkin and Maynard Smith (1979).
Hamilton (1980) (696 Cites)
LAWLOR, LR; SMITH, JM (1976). CO-EVOLUTION AND STABILITY OF COMPETING SPECIES. (210 Cites).
JANZEN (1980)
May and Anderson (1983). EPIDEMIOLOGY AND GENETICS IN THE COEVOLUTION OF PARASITES AND HOSTS (507
Cites)
Kiester, Lande, and Schemske (1984) MODELS OF COEVOLUTION AND SPECIATION IN PLANTS AND THEIR
POLLINATORS (92 Cites)
TAPER, ML; CASE, TJ (1985) QUANTITATIVE GENETIC MODELS FOR THE COEVOLUTION OF CHARACTER
DISPLACEMENT (139 Cites)
Seger (1988)
Nee, S (1989). ANTAGONISTIC CO-EVOLUTION AND THE EVOLUTION OF GENOTYPIC RANDOMIZATION
(55 Cites)
HOWARD, RS; LIVELY, CM (1994). PARASITISM, MUTATION ACCUMULATION AND THE MAINTENANCE OF
SEX. (198 Cites).
Morand (1996) (115 cites)
Ebert, Dieter; Hamilton, William D. (1996). TREE. Sex against virulence: The coevolution of parasitic
diseases (164 Cites).
DIECKMANN, U; MARROW, P; LAW, R (1995). EVOLUTIONARY CYCLING IN PREDATOR-PREY
INTERACTIONS - POPULATION-DYNAMICS AND THE RED QUEEN. (120 cites)
Dieckmann, U; Law, R (1996). The dynamical theory of coevolution: A derivation from stochastic
ecological processes. (329 Cites)
Gandon et al (1996) (284 Cites)
Abrams (1997)
Caldarelli, G; Higgs, PG; McKane, AJ. (1998). Modelling coevolution in multispecies communities (110
cites)
Gavrilets (1997 and 1998)
FRANK, SA (1993). COEVOLUTIONARY GENETICS OF PLANTS AND PATHOGENS (120 cites)
Frank (1993, 1994) and Frank Parker debates
Pathogens and sex in plants Article Author: Parker, Matthew (1994)
Modern times, the GMTC and the emergence of coevolutionary genetics, community ecology
Chapter 2. Fundamentals of Coevolution: Pre-requisites for Reciprocal Evolutionary Change
In 1980, Janzen defined coevolution as reciprocal evolutionary change in interacting species, launching
the modern era of coevolutionary research. Specifically, Janzen defined coevolution as
Janzen 1980: “an evolutionary change in a trait of the individuals in one population in response to a trait
of the individuals of the second population, followed by an evolutionary response by the second
population to the change in the first”
Janzen’s definition provided a simple yet precise definition of coevolution DISTINCTLY EVOLUTIONARY.
VERY DIFFERENT FROM Roughgarden types of MODELS WHICH USED THE TERM IN A DISTINCTLY
ECOLOGICAL FASHION. MORE IN LINE WITH MODE, AND JAYAKAR WHO DEVELOPED TRUE
COEVOLUTIONARY MODELS PRECEEDING JANZEN.
Formalizing Janzen’s verbal definition mathematically is straightforward and provides valuable
insights into the pre-requisites required for coevolution to occur. To take a concrete example, imagine
the interaction between the plant species XX and its long tongued pollinating fly XX. In this system, the
relative lengths of the flies tongue, x, and the plants corolla, y, are of central importance (REFS). If we
disregard the possibility of coevolution, evolutionary change in the length of these quantitative traits
can be predicted using the classical breeder’s equation:
βˆ†π‘₯Μ… = β„Žπ‘₯2 𝑆π‘₯
(1a)
βˆ†π‘¦Μ… = β„Žπ‘¦2 𝑆𝑦
(1b)
where β„Žπ‘₯2 and β„Žπ‘¦2 are the heritabilities of fly tongue and plant corolla respectively, and Sx and Sy are the
selection differentials acting on these traits. Obviously, as written in (1), coevolution does not occur and
the traits of the interacting species evolve independently in response to a constant and predictable
strength of selection. This is a classical simplifying assumption common to the majority of population
genetic and quantitative genetic theory (Figure 1a). If, however, we relax the assumption that selection
acting on the length of plant corolla is independent of fly tongue length and vice versa (Figure 1b), we
can easily extend the classical breeders equation in (1) to allow for coevolution as defined by Janzen:
βˆ†π‘₯Μ… = β„Žπ‘₯2 𝑆π‘₯ (πœ‘π‘¦ )
(2a)
βˆ†π‘¦Μ… = β„Žπ‘¦2 𝑆𝑦 (πœ‘π‘₯ )
(2b)
where the selection differentials Sx and Sy are now explicitly recognized to be functions of the
distribution of phenotypes φx and φy within the interacting species. Equations (2) demonstrate two
important pre-requisites for coevolution Sensu Janzen:
1. There must be additive genetic variation for the traits mediating an interspecific interaction
2. Selection acting on the traits mediating the interaction must depend upon the distribution of
phenotypes within the trait of the interacting species
The basic structure of coevolutionary models
The pre-requisites for coevolution imposed by Janzen’s definition suggest that all coevolutionary
models must have two fundamental ingredients: 1) a mechanism of inheritance and 2) a mechanism of
reciprocal selection. As a crude generalization, most models accomplish this using either quantitative or
population genetic approaches.
Models based within the framework of quantitative genetics assume that a trait (or suite of
traits) within one species interact functionally with a trait (or suite of traits) within another species.
Reciprocal selection is then produced because the outcome of an interaction between two individuals is
a function of their phenotypes. This interaction function must include a phenotype x phenotype
interaction for the model to be truly coevolutionary. Figure X illustrates several examples of interaction
functions which do, and do not, meet this criterion. In general, inheritance within such quantitative
genetic models of coevolution is implicit, with additive genetic variance or heritability assumed to be a
fixed model parameter (see Chapter 4 for details)
Models based within the framework of population genetics assume that a gene (or collection of
genes) within one species interact with a gene (or collection of genes) within the interacting species. In
such cases, reciprocal selection is generated because the outcome of encounters between individuals
depends on the interaction between their genotypes (e.g., infect/resist). Because genotypes are
discrete, the outcome of encounters is defined by an interaction matrix which must include genotype x
genotype interactions for the model to be truly coevolutionary. Figure X shows several interaction
matrices which do, and do not, meet this criterion. In contrast to the quantitative genetic framework,
population genetic models of coevolution explicitly model inheritance and thus allow the influence of
genetic details such as segregation and recombination to be studied (See chapters 4 and 5 for details).
In summary, all coevolutionary models include mechanisms of inheritance and reciprocal
selection. Although the details and subtleties of inheritance and the mechanistic basis of interaction
differ widely among models, all are united by a single essential feature: an interspecific interaction
between the phenotypes or genotypes of individuals which determines the fitness of the interacting
organisms.
Evaluating the pre-requisites for coevolution in natural populations
An obvious question facing anyone studying an interspecific interaction is whether or not the
species are coevolving. Although superficially simple, answering this question in the wild has proven to
be spectacularly challenging, with only a handful of compelling examples of coevolution (REFS).
Although the absence of compelling examples may at first seem to suggest coevolution is rare and
unimportant, the reality is that demonstrating coevolution in the strict sense outlined by Janzen is a
virtually insurmountable task. Imagine trying to document XXX over the course of a single dissertation.
The true difficulty of documenting coevolution is driven home by taking a sober look at the list of natural
systems in which even garden variety evolution has been demonstrated. Set aside, Darwin’s finches,
XXX, and XXX, and the list of natural systems where evolutionary change has been demonstrated in the
wild over any substantial period of time becomes small indeed. As a consequence, much research has
instead focused on the much more approachable task of evaluating the strength of selection and
quantifying additive genetic variance. In other words, much evolutionary research on natural
populations focuses on establishing that the pre-requisites for evolutionary met (additive genetic
variance and selection), rather than documenting evolutionary change per se. I argue that this is exactly
the shift we must make within coevolutionary biology in order to finally come to grips with the question
of how important coevolution is within natural populations.
The fundamental pre-requisites for coevolution are heritable variation for the relevant traits in
the interacting species and reciprocal selection. If we are willing to accept the fact that heritable
variation exists for virtually every quantitative trait studied to date (REFS), it becomes clear that the
great unknown is the prevalence and strength of reciprocal selection. If reciprocal selection is common,
it is likely that coevolution is a fundamentally important force; if it is rare, coevolution is nothing more
than an evolutionary side show. It is imperative that over the next ten years, reciprocal selection be
estimated in a sufficient number of systems for key generalizations to emerge about the strength of
reciprocal selection in the wild.
Although significantly more challenging than estimating the strength of selection in single
species systems, appropriate statistical and methodological approaches are available for estimating the
strength of reciprocal selection. Irrespective of approach, any empirical estimation of reciprocal
selection requires two key pieces of information: 1) Phenotypes/Genotypes of interacting individuals
and 2) Fitness of interacting individuals.
 Quantifying the strength of reciprocal selection in natural systems is the single most pressing
challenge within coevolutionary biology.
Although Janzen’s definition focuses explicitly on traits and thus lends itself to a quantitative genetics
formalization, it can be extended to a simpler population genetic framework without any loss of
precision. Specifically, if we now imagine that some subset of genes
βˆ†π‘₯Μ… = 𝐺π‘₯
Μ…π‘₯
1 πœ•π‘Š
Μ…π‘₯ πœ•π‘₯Μ…
π‘Š
βˆ†π‘ ≈ 𝑠(𝑝𝑦 )𝑝π‘₯ π‘žπ‘₯
Although written in quantitative genetics terms, (2) apply equally to population genetics where “gene”
can more or less be subsititued for trait…
equations (2) are too general to provide any real mathematical insight into coevolutionary dynamics,
they are invaluable for what they teach us about empirical investigation of coevolution and as a starting
point for the development of predictive models.
Chapter 3. Coevolution of major genes
One of the simplest possible models of coevolution arises when the outcome of encounters
between interacting species is mediated by a single haploid gene in each of the interacting species.
Although clearly an oversimplification, such single locus haploid models have provided valuable insights
into the process of coevolution and are perhaps not such terribly poor approximations to reality in some
systems. Take, for instance, the interaction between … L
Biological Motivation
Begin with an introduction about interactions between genes… Perhaps NOD? Vir? Or, a cartoon
example with cute little cartoon molecules? EXPLAIN HOW THE INTERACTION WOULD WORK, THEN SAY,
BUT HOW COULD WE MODEL THIS? HAPLOID
Translating Biology into Equations
We will take as our mathematical starting point the classical expression for evolutionary change
in a haploid diallelic locus originally derived by Wright (REF):
βˆ†π‘ =
Μ…
π‘π‘ž πœ•π‘€
Μ… πœ•π‘
𝑀
(1)
Clearly, what we need to know is the mean fitness of the two species as a function of the genotype
frequencies within the population. With this information in hand, coevolutionary dynamics can be
predicted using (1).
Equation (1) suggests our goal should be to calculate the mean fitness for each of the interacting
species. Since a population mean fitness is nothing but the average fitness of the population, our road
forward involves specification of the fitnesses of each of the possible genotypes. How can we derive
this? Let’s use the information we gathered from our cartoon example… Assuming that individuals of the
two species encounter one another at random, the fitness of each genotype is simply the sum of the
fitness consequences of possible encounters weighted by the probability of each type of encounter:
π‘Šπ΄ = 1 + 𝑠π‘₯ [πœ‹(𝐴, 𝐡)𝑝𝑦 + πœ‹(𝐴, 𝑏)π‘žπ‘¦ ]
(2a)
π‘Šπ‘Ž = 1 + 𝑠π‘₯ [πœ‹(π‘Ž, 𝐡)𝑝𝑦 + πœ‹(π‘Ž, 𝑏)π‘žπ‘¦ ]
(2b)
π‘Šπ΅ = 1 + 𝑠𝑦 [πœ‹(𝐴, 𝐡)𝑝π‘₯ + πœ‹(π‘Ž, 𝐡)π‘žπ‘₯ ]
(2c)
π‘Šπ‘ = 1 + 𝑠𝑦 [πœ‹(𝐴, 𝑏)𝑝π‘₯ + πœ‹(π‘Ž, 𝑏)π‘žπ‘₯ ]
(2d)
Can I just push this through in a completely general way? I don’t see why not…
Using (2), it is straightforward to calculate the population mean fitness for each species. Specifically, the
population mean fitness for species x is simply the expected fitness of genotypes segregating within the
population:
Μ…π‘₯ = π‘Šπ΄ 𝑝π‘₯ + π‘Šπ‘Ž π‘žπ‘₯ = 𝑠π‘₯ 𝑝π‘₯ [πœ‹(𝐴, 𝐡)𝑝𝑦 + πœ‹(𝐴, 𝑏)π‘žπ‘¦ ] + 𝑠π‘₯ π‘žπ‘₯ [πœ‹(π‘Ž, 𝐡)𝑝𝑦 + πœ‹(π‘Ž, 𝑏)π‘žπ‘¦ ]
π‘Š
(3a)
̅𝑦 = π‘Šπ΅ 𝑝𝑦 + π‘Šπ‘ π‘žπ‘¦ = 𝑠𝑦 𝑝𝑦 [πœ‹(𝐴, 𝐡)𝑝π‘₯ + πœ‹(𝐴, 𝑏)π‘žπ‘₯ ] + 𝑠π‘₯ π‘žπ‘₯ [πœ‹(π‘Ž, 𝐡)𝑝π‘₯ + πœ‹(π‘Ž, 𝑏)π‘žπ‘₯ ]
π‘Š
(3b)
Plugging (3) into (1) yields expressions for evolutionary change in our two interacting species:
A weak coevolution Approximation
INCLUDE A BOX ON APPROXIMATION
Application: Infection genetics and the GFG vs. MA debate
Table 1. Interaction matrix for a single haploid diallelic locus. Each possible interaction between
genotypes is specified by a probability of interaction πœ‹(𝑋, π‘Œ)
Species x genotype
A
a
Species y genotype
B
πœ‹(𝐴, 𝐡)
πœ‹(π‘Ž, 𝐡)
b
πœ‹(𝐴, 𝑏)
πœ‹(π‘Ž, 𝑏)
Chapter 4. Coevolution of polygenic traits
Biological Motivation
Begin with an introduction about interactions between quantitative traits… Give a colorful example…
Translating Biology into Equations
KEEP IT GENERAL HERE
A quantitative genetic approximation
We will take as our mathematical starting point the equation:
βˆ†π‘§Μ… = 𝐺
Μ…
1 πœ•π‘Š
Μ… πœ•π‘§Μ…
π‘Š
Rather than derive this well-worn equation from first principles, it is sufficient here to simply realize that
its key assumptions are that selection is relatively weak and additive genetic variance is fixed.
Mathematical Technique: DESCRIBING FREQUENCY DISTRIBUTIONS USING STATISTICAL MOMENTS (IN A
BOX)
Application: TRAIT MATCHING?
NOTE THAT TAYLOR EXPANDING ASSUMING WEAK COEVOLUTION LEADS TO STRAIGHT UP EVOLUTIOn:
SUGGESTS THAT IF COEVOLUTION IS GENERALLY WEAK, WE WON’T BE ABLE TO DETECT IT AND IT WILL
LOOK JUST LIKE ORDINARY OLD EVOLUTION (AND BEHAVE LIKE IT)
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