The constant philopater hypothesis: a new life history invariant for dispersal evolution

Abstract Surprising invariance relationships have emerged from the study of social interaction, whereby a cancelling‐out of multiple partial effects of genetic, ecological or demographic parameters means that they have no net impact upon the evolution of a social behaviour. Such invariants play a pivotal role in the study of social adaptation: on the one hand, they provide theoretical hypotheses that can be empirically tested; and, on the other hand, they provide benchmark frameworks against which new theoretical developments can be understood. Here we derive a novel invariant for dispersal evolution: the ‘constant philopater hypothesis’ (CPH). Specifically, we find that, irrespective of variation in maternal fecundity, all mothers are favoured to produce exactly the same number of philopatric offspring, with high‐fecundity mothers investing proportionally more, and low‐fecundity mothers investing proportionally less, into dispersing offspring. This result holds for female and male dispersal, under haploid, diploid and haplodiploid modes of inheritance, irrespective of the sex ratio, local resource availability and whether mother or offspring controls the latter's dispersal propensity. We explore the implications of this result for evolutionary conflict of interests – and the exchange and withholding of contextual information – both within and between families, and we show that the CPH is the fundamental invariant that underpins and explains a wider family of invariance relationships that emerge from the study of social evolution.


Appendix S2. Reproductive success
The probability that a focal juvenile wins a breeding site is k ta (x,z) = 1/(∑ b ∈ I F bt σ bta (1x bta )+∑ q∈T p q (∑ b∈I F bq σ bqa z bqa )(1-c)) , with a ∈{f,m}, I = {1, 2, …, n}, T = {1, 2, …, n p }, σ btf = 1-σ bt , and σ btm = σ bt ; and where: F is the fecundity of a focal mother; x the probability of dispersal of a focal offspring; and z the population average probability of dispersal of an offspring. The reproductive success of a rank-i mother in a type-t patch through her successful daughters that become rank-j mothers in a type-q patch is w itf → jqf = F it (1-σ it )((1-x itf )k tf (x,z)η tq +x itf (1c)∑ e∈T p e k ef (x,z)η eq )(1-ϕ), where ϕ is the fraction of genes a daughter inherits from her father. The reproductive success of a rank-i mother in a type-t patch through her successful sons that mate with rank-j mothers in type-q patches is w itf→jqm = F it (1-σ it )((1-x itf )k tf (x,z)η tq +x itf (1-c) ∑ e ∈ T p e k qf (x,z)η eq )µ, where µ is the fraction of genes a son receives from his mother. The reproductive success of a rank-i father in a type-t patch (i.e. a father that mates with a rank-i mother in a type-t patch) through his successful daughters that become rank-j mothers in type-q patches is w itm→jqf = F it σ it ((1-x itm )k tm (x,z)η tq +x itm (1-c)∑ e∈T p e k em (x,z)η eq )ϕ. The reproductive success of a rank-i father in a type-t patch through his successful sons that mate with rank-j mothers in type-q patches is w itm → jqm = F it σ it ((1-x itm )k tm (x,z)η tq +x itm (1-c))∑ e ∈ T p e k em (x,z)η eq )(1-µ). In the asexual reproduction model, there is no male component in the reproductive success expressions, in which case we drop the subscript 'f' from the reproductive success expressions of females, and we set ϕ = 0.

Appendix S3. Stable class frequencies and reproductive values
The expressions of the reproductive success of individuals define a transition matrix, which is given by The elements of the right-eigenvector of matrix A (corresponding to the leading eigenvalue) give the frequency of each class (Taylor & Frank 1996;Grafen 2006), which is u it = 1/(n.n p ), for all i ∈ I, t ∈ T. The elements of the left-eigenvector of matrix A (corresponding to the leading eigenvalue) give the reproductive values for individuals of each class (Fisher 1930;Taylor & Frank 1996;Grafen 2006). The class-reproductive values are c f = ϕ/(ϕ+µ) and c m = µ/(ϕ+µ).

Appendix S4. Fitness
The fitness of a focal individual is the sum of its different reproductive success components weighted by corresponding reproductive values, all divided by the mean reproductive value of the focal class. This is /v itm , for females and for males, respectively. The fitness of a random individual is given by the sum of fitness weighted by the frequency and reproductive value of each class. This is

Appendix S5. Selection gradient
The selection gradient is given by the slope of fitness on the breeding value of the focal recipient (i.e. the beneficiary of social behaviours): where: the slope of fitness on phenotypes give the marginal fitness effect of the behaviour; and the slope of the actor's (i.e. the enactor of a social behaviour) breeding value (i.e. the heritable component of an actor's phenotype), denoted by G, on the focal recipient's breeding value (i.e. the heritable component of a focal recipient's phenotype), denoted by g, gives the kin selection relatedness coefficients (Taylor & Frank 1996;Frank 1998;Rodrigues and Gardner 2013). Expanding the RHS of this equation, we find that the condition for the evolution of a slightly higher dispersal rate of a daughter is -r itf ω t υ t +(1c)r itf ∑ q∈T p q ω q υ q +ω t υ t h f ∑ j∈I U itf ρ ijtf > 0, whereas the condition for the evolution of a slightly higher dispersal rate of a son is -r itm ω t υ t +(1-c)r itm ∑ q∈T p q ω q υ q +ω t υ t h f ∑ j∈I U itm ρ ijtm > 0, where: ω tf = k tf (z,z) is the probability a single female wins a breeding site in a type-t patch; ω tm = k tm (z,z) is the probability that a single male wins a breeding site in a focal type-t patch; h tf = k tf (z,z)∑ i∈I F it (1σ it )(1-z itf ) is the probability a random juvenile female after dispersal is born in the focal type-t patch; h tm = k tm (z,z)∑ i∈I F it σ it (1-z itm ) is the probability that a random juvenile male after dispersal is born in the focal type-t patch; U jqf = (F jq (1-σ jq )(1-z jqf ))/(∑ b∈I F bq (1-σ bq )(1-z bqf )) is the frequency of the rank-j mother's daughters among the native daughters of a type-q patch; and U jqm = (F jq σ jq (1z jqm ))/(∑ b∈I F bq σ bq (1-z bqm )) is the frequency of the rank-j mother's sons among the native sons of a type-q patch. Under allomaternal control, we need to consider coefficients of relatedness between the allomother i and the offspring ϑ who are under the control of the allomother, in which case r it = r iϑt . To determine the selection gradient for the evolution of the sex allocation strategy, we follow the methodology outlined for the evolution of dispersal. However, we assume that the sex allocation strategy σ is an evolving trait rather than a parameter. Hamilton's rule for the evolution of the sex allocation strategy is given by equation (4) and (5).

Appendix S6. Relatedness
We assume a neutral population, and for each of the reproductive systems we define recursion equations for the coefficients of consanguinity in successive generation between juveniles, which we then solve for equilibrium. The coefficients of consanguinity allow us to derive the coefficients of relatedness between interacting individuals (Bulmer 1994, Rodrigues andGardner 2013). We focus on three coefficients of consanguinity: (1) the coefficient of consanguinity between ! 4! opposite-sex offspring (denoted by f); (2) the coefficient of consanguinity between female offspring (denoted by γ); and (3) the coefficient of consanguinity between male offspring (denoted by η). All the recursion equations have the form X t´ = ∑ q∈T π(q|t)(P SqX Y q + (1-P SqX )Z q ), where: π(q|t) is the probability that a type-t patch was a type-q patch in the previous generation; X is a coefficient of consanguinity (i.e. f, γ, or η); P Sqf = ∑ i∈I ((F iq (1-σ iq )/∑ j∈I F jq (1-σ jq ))(F iq σ iq /∑ j∈I F jq σ jq )), is the probability that two opposite-sex offspring sampled at random before dispersal are siblings; P Sqγ = ∑ i∈I (F iq (1-σ iq )/∑ j∈I F jq (1-σ jq )) 2 is the probability that two female offspring sampled at random before dispersal are siblings; P Sqη = ∑ i∈I (F iq σ iq /∑ j∈I F jq ρ jq ) 2 is the probability that two male offspring sampled at random before dispersal are siblings; Y q is the probability that two siblings share genes in common; and Z q is the probability that two non-siblings share genes in common.
The variables X, Y, and Z, depend on the type of reproduction, on the type of inheritance, and on the type of patch. In table 1 and 2 we define these variables for each case. The coefficients of consanguinity can then be used to define the coefficients of relatedness between interacting individuals. The relatedness between: (1) a mother and her daughters is r MD = p MD / p M ; (2) a mother and her sons is r MS = p MS / p M ; (3) a mother and a daughter the other mother is r MF = p MF / p M ; (4) a mother and a son of another mother is r MM = p MM / p M .