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Supporting information to: “The constant philopater hypothesis: a new life
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history invariant for dispersal evolution”
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Antonio M. M. Rodrigues1,2,*, Andy Gardner3
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1. Department of Zoology, University of Cambridge, Downing Street, Cambridge
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CB2 3EJ, United Kingdom.
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2. Wolfson College, Barton Road, Cambridge CB3 9BB, United Kingdom.
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3. School of Biology, University of St Andrews, St Andrews KY16 9TH, United
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Kingdom.
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*
Corresponding author, email: ammr3@cam.ac.uk
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Contents:
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Appendix A. Ecological dynamics
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Appendix B. Reproductive success
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Appendix C. Stable class frequencies and reproductive values
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Appendix D. Fitness
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Appendix E. Selection gradient
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Appendix F. Relatedness
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Appendix G. Convergence stability
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Appendix H. Table 1 & 2
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References
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Appendix A. Ecological dynamics
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Here, we follow the ecological dynamic of patches resource-availability outlined in
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the main text. This can be described by a transition matrix, which is given by
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𝜂11
𝐏=( ⋯
𝜂1𝑛p
⋯
⋱
⋯
𝜂𝑛p 1
⋯ ),
𝜂𝑛p 𝑛p
(A1)
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where ηij is the probability that a type-i patch becomes a type-j patch the next season.
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At ecological equilibrium, there is a stable frequency of the different types of patches
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in the population, which is given by the elements of the right-eigenvector of matrix P.
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We denote the frequency of type-i patches at equilibrium by pi. If we define a random
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variable Tt, denoting the state of a focal patch in season t, then the coefficient of
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correlation between two successive seasons, denoted by τ, is defined as τ ≡
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cov(Tt,Tt+1)√(var(Tt)var(Tt+1)), where -1 ≤ τ ≤ 1 (Rodrigues and Gardner 2012). An
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environment is temporally stable when τ = 1, temporally unpredictable when τ = 0,
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and seasonal when τ = -1.
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Appendix B. Reproductive success
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The probability that a focal juvenile wins a breeding site is kta(x,z) = 1/(∑b∈IFbtσbta(1-
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xbta)+∑q∈Tpq(∑b∈IFbqσbqazbqa)(1-c)) , with a {f,m}, I = {1, 2, …, n}, T = {1, 2, …,
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np}, σbtf = 1-σbt, and σbtm = σbt. The reproductive success of a rank-i mother in a type-t
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patch through her successful daughters that become rank-j mothers in a type-q patch
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is witf→jqf = Fit(1-σit)((1-xitf)ktf(x,z)ηtq+xitf(1-c)∑e∈Tpekef(x,z)ηeq)(1-ϕ), where ϕ is the
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fraction of genes a daughter inherits from her father. The reproductive success of a
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rank-i mother in a type-t patch through her successful sons that mate with rank-j
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mothers in type-q patches is witf→jqm = Fit(1-σit)((1-xitf)ktf(x,z)ηtq+xitf(1-c)
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∑e∈Tpekqf(x,z)ηeq)μ, where μ is the fraction of genes a son receives from his mother.
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The reproductive success of a rank-i father in a type-t patch (i.e. a father that mates
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with a rank-i mother in a type-t patch) through his successful daughters that become
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rank-j mothers in type-q patches is witm→jqf = Fitσit((1-xitm)ktm(x,z)ηtq+xitm(1-
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c)∑e∈Tpekem(x,z)ηeq)ϕ. The reproductive success of a rank-i father in a type-t patch
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through his successful sons that mate with rank-j mothers in type-q patches is witm→jqm
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= Fitσit((1-xitm)ktm(x,z)ηtq +xitm(1-c))∑e∈Tpekem(x,z)ηeq)(1-μ). In the asexual
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reproduction model, there is no male component in the reproductive success
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expressions, in which case we drop the subscript ‘f’ from the reproductive success
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expressions of females, and we set ϕ = 0.
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Appendix C. Stable class frequencies and reproductive values
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The expressions of the reproductive success of individuals define a transition matrix,
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which is given by
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(𝑤itf→jqf )
𝑛.𝑛p ×𝑛.𝑛p
𝐀=(
(𝑤itf→jqm )
𝑛.𝑛p ×𝑛.𝑛p
(𝑤itm→jqf )
𝑛.𝑛p ×𝑛.𝑛p
(𝑤itm→jqm )
).
(C1)
𝑛.𝑛p ×𝑛.𝑛p
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The elements of the right-eigenvector of matrix A (corresponding to the leading
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eigenvalue) give the frequency of each class (Taylor & Frank 1996; Grafen 2006),
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which is uit = 1/(n.np), for all i  I, t  T. The elements of the left-eigenvector of
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matrix A (corresponding to the leading eigenvalue) give the reproductive values for
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individuals of each class (Fisher 1930; Taylor & Frank 1996; Grafen 2006). The
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class-reproductive values are cf = ϕ/(ϕ+μ) and cm = μ/(ϕ+μ). Under asexual-
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reproduction, cf = 1.
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Appendix D. Fitness
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The fitness of a focal individual is the sum of its different reproductive success
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components weighted by corresponding reproductive values, all divided by the mean
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reproductive value of the focal class. This is Witf = (∑j∈I∑q∈Twitf→jqfvjqf+
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∑j∈I∑q∈Twitf→jqmvjqm)/vitf, and Witm = (∑j∈I∑q∈Twitm→jgfvjgf+∑j∈I∑q∈Twitm→jqmvjqm)/vitm, for
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females and for males, respectively. The fitness of a random individual is given by the
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sum of fitness weighted by the frequency and reproductive value of each class. This is
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w = ∑j∈I∑q∈T uiqfviqfWiqf+∑j∈I∑q∈TuiqmviqmWiqm.
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Appendix E. Selection gradient
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The selection gradient is given by the slope of fitness on the breeding value of the
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focal actor: dw/dgit =∑j∈I∑q∈Tujqfviqf(∂Wjqf/∂xit)(dGjqf/dgit)+∑j∈I
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∑q∈Tujqmvjqm(∂Wjqm/∂xit)(dGjqm/dgit), where: the slope of fitness on phenotypes give the
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marginal fitness effect of the behaviour; and the slope of the recipient’s breeding
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value, denoted by G, on the actor’s breeding value, denoted by g, gives the kin
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selection relatedness coefficients (Taylor & Frank 1996; Frank 1998; Rodrigues and
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Gardner 2013). Expanding the RHS of this equation, we find that the condition for the
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evolution of a slightly higher dispersal rate of a daughter is –ritfωtυt+(1-
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c)ritf∑q∈Tpqωqυq+ωtυthf∑j∈IUitfρijtf > 0, whereas the condition for the evolution of a
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slightly higher dispersal rate of a son is –ritmωtυt+(1-c)ritm∑q∈Tpqωqυq
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+ωtυthf∑j∈IUitmρijtm > 0, where: ωtf = ktf(z,z) is the probability a single female wins a
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breeding site in a type-t patch; ωtm = ktm(z,z) is the probability that a single male wins
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a breeding site in a focal type-t patch; htf = ktf(z,z)∑i∈IFit(1-σit)(1-zitf) is the probability
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a random juvenile female after dispersal is born in the focal type-t patch; htm =
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ktm(z,z)∑i∈IFitσit(1-zitm) is the probability that a random juvenile male after dispersal is
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born in the focal type-t patch; Ujqf = (Fjq(1-σjq)(1-zjqf))/(∑b∈IFbq(1-σbq)(1-zbqf)) is the
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frequency of the rank-j mother’s daughters among the native daughters of a type-q
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patch; and Ujqm = (Fjqσjq(1-zjqm))/(∑b∈IFbqσbq(1-zbqm)) is the frequency of the rank-j
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mother’s sons among the native sons of a type-q patch. Under allomaternal control,
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we need to consider coefficients of relatedness between the allomother i and the
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offspring ϑ who are under the control of the allomother, in which case rit = riϑt. To
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determine the selection gradient for the evolution of the sex allocation strategy, we
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follow the methodology outlined for the evolution of dispersal. However, we assume
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that the sex allocation strategy σ is an evolving trait rather than a parameter.
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Hamilton’s rule for the evolution of the sex allocation strategy is given by equation
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(4) and (5).
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Appendix F. Relatedness
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We assume a neutral population, and for each of the reproductive systems we define
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recursion equations for the coefficients of consanguinity in successive generation
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between juveniles, which we then solve for equilibrium. The coefficients of
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consanguinity allow us to derive the coefficients of relatedness between interacting
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individuals (Bulmer 1994, Rodrigues and Gardner 2013). We focus on three
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coefficients of consanguinity: (1) the coefficient of consanguinity between opposite-
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set offspring (denoted by f); (2) the coefficient of consanguinity between female
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offspring (denoted by γ); and (3) the coefficient of consanguinity between male
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offspring (denoted by η). All the recursion equations have the form Xt´ =
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∑q∈Tπ(q|t)(PSqXYq+ (1-PSqX)Zq), where: π(q|t) is the probability that a type-t patch was
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a type-q patch in the previous generation; X is a coefficient of consanguinity (i.e. f, γ,
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or η); PSqf = ∑i∈I((Fiq(1-σiq)/∑j∈IFjq(1-σjq))(Fiqσiq/∑j∈IFjqσjq)), is the probability that two
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opposite-sex offspring sampled at random before dispersal are siblings; PSqγ =
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∑i∈I(Fiq(1-σiq)/∑j∈IFjq(1-σjq))2 is the probability that two female offspring sampled at
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random before dispersal are siblings; PSqη = ∑i∈I(Fiqσiq/∑j∈IFjqρjq)2 is the probability
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that two male offspring sampled at random before dispersal are siblings; Yq is the
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probability that two siblings share genes in common; and Zq is the probability that two
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non-siblings share genes in common. The variables X, Y, and Z, depend on the type of
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reproduction, on the type of inheritance, and on the type of patch. In table 1 and 2 we
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define these variables for each case. The coefficients of consanguinity can then be
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used to define the coefficients of relatedness between interacting individuals. The
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relatedness between: (1) a mother and her daughters is rMD = pMD / pM; (2) a mother
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and her sons is rMS = pMS / pM; (3) a mother and a daughter the other mother is rMF =
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pMF / pM; (4) a mother and a son of another mother is rMM = pMM / pM.
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Appendix G. Convergence stability
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To determine if a pair of optimal dispersal strategies is convergence stable (CS;
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Christiansen 1991; Eshel 1996; Taylor 1996) we define the matrix:
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𝜕
𝜕𝑊
𝜕𝑧nnp
(𝜕𝑔
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nnp
|
)
𝑥nnp =𝑧nnp
⋮
𝜕
(
𝜕𝑧nnp
𝜕
𝜕𝑊
𝜕𝑧11
(𝜕𝑔
nnp
⋱
𝜕𝑊
(𝜕𝑔 |
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⋮
𝑥11 =𝑧11
)
⋮
|
𝑥nnp =𝑧nnp
) |
.
⋮
𝜕
𝜕𝑧11
|
𝜕𝑊
(𝜕𝑔 |
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𝑥11 =𝑧11
)
)
∗ ,…,𝑧
∗
𝑧11 =𝑧11
nnp =𝑧nnp
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The set of optimal strategies (𝑧11 *,… , 𝑧𝑛𝑛𝑝 *) are convergence stable if the
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eigenvalues of matrix (F1) have negative real parts (Otto & Day 2007).
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(F1)
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Appendix H. Table 1 & 2
Table 1. Recursion equations and coefficients of consanguinity under maternal control
Coefficients of consanguinity
X´ = ∑q∈Tπ(q|t) (PSqXY+ (1-PSqX)Z)
X
Y
Z
pM
pMD
pMF
pMS
pMM
Asexual
γ
1
hhf
1
1
hhf
-
-
Sexual haploidy
f
½+½ hfhmf
¼hfhff+½hfhmf
1
½pM+½hfhmf
½hfhff+
½pM+½hfhmf
½hfhff+
+¼hmhmf
Sexual diploidy
Sexual haplodiploidy
f
f
γ
η
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½(½+½ hfhmf)
¼hfhff+½hfhmf
+½hfhmf
+¼hmhmf
½(½+½ hfhmf)
½hfhfγ
+½hfhmf
+½hfhm f
¼(½+½ hfhmf)+
¼hfhfγ+½hfhmf
½hfhmf+¼
+¼hmhmη
½+½hfhmf
hfhfγ
½hfhmf
½+½ hfhmf
½pM+½hfhmf
½hfhff+
½hfhmf
½pM+½hfhmf
½hfhmf
½+½ hfhmf
½pM+½hfhmf
½hfhfγ+
½hfhff+
½hfhmf
½+½hfhmf
hfhfγ
½hfhmf
Note: The coefficients of consanguinity are determined by solving the recursion equations for equilibrium: γ´ = γ under asexual reproduction; f´
= f under sexual reproduction with both haploid and diploid inheritance; and γ´ = γ, f´ = f, and η´ = η, under sexual reproduction with
haplodiploid inheritance. These coefficients are patch-type specific. We can then determine the coefficients of consanguinity between: a mother
and herself (pM); a mother and a daughter or a son (pMD or pMS); between a mother another mother’s daughter or son (pMF or pMM).
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Table 2. Coefficient of consanguinity under offspring control
Coefficients of consanguinity
pOF
pOFS
pOFF
pOM
pOMS
pOMM
asexual
1
1
hhf
-
-
-
Sexual haploidy
1
½pOF+½hfhmf
¼hfhff+½hfhm f
1
½ pOM+½hfhmf
¼hfhff+½hfhmf
+¼hmhmf
Sexual diploidy
Sexual haplodiploidy
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½+½hfhmf
½+½hfhmf
½(½+½hfhmf)
¼hfhff+½hfhmf
+½hfhmf
+¼hmhmf
¼(½+½hfhmf)
¼hfhfγ+½hfhmf
+½hfhm+¼
+¼hmhmη
+¼hmhmf
½+½hfhmf
1
½(½+½hfhmf)
¼hfhff+½hfhmf
+½hfhmf
+¼hmhmf
½+½hfhmf
hfhfη
Note: Having derived the coefficients of consanguinity γ, f, and, η we can derive the coefficients of consanguinity between: a female offspring
and herself (pOF); a female offspring and her sisters (pOFS); a female offspring and another mother’s daughters (pOFF); a male offspring and
himself (pOM); a male offspring and his brothers (pOMS); a male offspring and another mother’s sons (pOMM).
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