The evolution of bequeathal in stable habitats

Abstract Adults sometimes disperse, while philopatric offspring inherit the natal site, a pattern known as bequeathal. Despite a decades‐old empirical literature, little theoretical work has explored when natural selection may favor bequeathal. We present a simple mathematical model of the evolution of bequeathal in a stable environment, under both global and local dispersal. We find that natural selection favors bequeathal when adults are competitively advantaged over juveniles, baseline mortality is high, the environment is unsaturated, and when juveniles experience high dispersal mortality. However, frequently bequeathal may not evolve, because the fitness cost for the adult is too large relative to inclusive fitness benefits. Additionally, there are many situations for which bequeathal is an ESS, yet cannot invade the population. As bequeathal in real populations appears to be facultative, yet‐to‐be‐modeled factors like timing of birth in the breeding season may strongly influence the patterns seen in natural populations.

relatively well studied (Clobert, Baguette, Benton, & Bullock, 2012;Ronce, 2007), developing a greater understanding of bequeathal can teach us about the other side of the same behavioral coin and adds a new dimension to our understanding of breeding dispersal (Harts, Jaatinen, & Kokko, 2016;Johst & Brandl, 1999;Paradis, Baillie, Sutherland, & Gregory, 1998). Studying the exceptions to the norm in evolutionary ecology is often illuminating and can provide fresh insights into well-studied biological processes. As dispersal is a fundamental driver affecting the ecology, evolution, and population persistence of organisms (Bowler & Benton, 2005), understanding the conditions which favor particular types of dispersal is of much importance.
Empirical studies of bequeathal are rare (Berteaux & Boutin, 2000). However, reported rates of bequeathal are as high as 68% in red squirrels (Boon, Reale, & Boutin, 2008) and 30% in kangaroo rats (Jones, 1986). All known examples of bequeathal occur in commodity-dependent species that require valuable resources such as a dens, burrows, middens, or resource caches for survival and reproduction (Lambin, 1997). When resources critical to survival and reproduction are substantial and difficult to secure, parents may boost offspring fitness by bequeathing the natal site. Such offspring may stand a better chance of defending the natal territory than dispersing and acquiring a new one. Parents, on the other hand, are often in a better position to detect vacant territories or challenge existing ownership because of their enhanced experience and competitive skills. However, these benefits must be balanced against potentially high conflict between parents and offspring, especially in viscous populations (Kuijper & Johnstone, 2012). This conflict of interest is inherent to a variety of social systems involving resource inheritance, such as cooperative breeding (Koenig, Pitelka, Carmen, Mumme, & Stanback, 1992), and primatively eusocial and eusocial societies (Myles, 1988).
Other factors, including parental age and condition, offspring size and competitive ability, territory quality, and population density, are thought to affect bequeathal (Lambin, 1997;. However, with so many inputs, interpreting and synthesizing results from multiple studies is challenging. Some studies have found no relationship between parental age/condition and bequeathal , while others have found an increase in bequeathal with age (Descamps, Boutin, Berteaux, & Gaillard, 2007). It seems clear that density matters, but how and when is unclear; bequeathal has been found to increase with local density in kangaroo rats (Jones, 1986) but to decrease with density in Columbian ground squirrels and red squirrels (Boutin, Tooze, & Price, 1993;Harris & Murie, 1984;. Similarly, inconsistent patterns of density-dependent dispersal have been observed across vertebrate taxa (Matthysen, 2005).
Part of the problem in interpreting the current evidence is the lack of a general theoretical framework for understanding bequeathal dynamics (Berteaux & Boutin, 2000). The problem of bequeathal lies at the intersection of parent-offspring conflict and dispersal, both long and large literatures (Anderson, 1989;Clobert, Danchin, Dhondt, & Nichols, 2001;Clobert et al., 2012;Godfray, 1995;Hamilton & May, 1977;Trivers, 1974). But very little work has directly addressed bequeathal. Price (1992) used dynamic programming to investigate optimal bequeathal for a single female, finding that timing of breeding was an important determinant of its adaptive value. But as the model did not include any population, just an individual female, it is difficult to interpret. Bequeathal, as a special form of dispersal, is inherently game theoretic, generating powerful frequency dependence. A game-theoretic model by Kokko and Lundberg (2001) comes closest to our target; in that, it examines dispersal from and competition for territorial breeding sites, combined with conflict between an adult and a single offspring. However, their model examined residency in seasonal habitats with different productivity and survivorship, and it failed to find any bequeathal-like pattern among the evolutionarily stable strategies.
As a first step to building a theoretical framework for bequeathal, we present a simple bequeathal model. Our model considers parentoffspring conflict, competition for territories, local and global dispersal, and survival rates of adults and juveniles with overlapping generations.
Like Kokko and Lundberg (2001), we consider production of a single offspring to avoid complications arising from sibling competition. This assumption is unrealistic in many cases, but allows for understanding of other factors before advancing to more complicated models. Unlike Kokko and Lundberg (2001) but like Hamilton and May (1977), we study a stable, uniform habitat, in order to eliminate many well-studied causes of dispersal in spatially and temporally variable environments. This is also unrealistic, but again allows for understanding the basic evolutionary logic of bequeathal, before studying it in stochastic environments, in which dispersal may be favored for other reasons.
A great deal of work remains to be done, extending these first models to consider facultative responses and additional strategies such as reproductive queuing. Still, even the simple models we analyze here are capable of producing a number of surprising dynamics. Therefore, they are worth understanding in themselves before productive work can begin on extending them.
The major result of our analysis is that bequeathal is favored by the comparative advantage adults have in competing for sites. This advantage arises because there is more competition to acquire a new site than to retain an existing site. Since adults are better competitors, comparative advantage favors sending the better warrior to the most difficult battle. However, inclusive fitness considerations tend to work against bequeathal. Under clonal reproduction, the adult and juvenile will agree that the best warrior serves in the harshest battle. But since adults and juveniles are imperfectly related, they disagree, under some range of costs and benefits. Any factor that reduces the adult's costs will therefore help bequeathal evolve. Such factors include adults having high mortality risk and low residual reproductive value, such as at the end of life. Conversely, any factor that reduces juvenile benefits will work against bequeathal. For example, if juveniles are fragile, having high baseline mortality, then it makes little sense to bequeath territory to them. We outline the mathematical argument that leads to these conclusions, ending the paper with a discussion of unmodeled factors that may also strongly influence the facultative use of bequeathal in natural populations.

| MODEL DEFINITION
We use a mix of methods-including formal analysis, numerical sensitivity analysis, and individual-based simulation-to construct and understand our models of bequeathal. We begin by defining the global and local dispersal models analytically. Table 1 summarizes the symbols used in the models, each of which is explained in the following sections.

| Population and life cycle
Imagine a population of organisms with overlapping generations, living at N spatially separated sites. Only one adult can survive and reproduce at each site, and each adult produces one same-sex (female) juvenile offspring each breeding season. The life cycle proceeds in the following sequence: (a) birth of offspring, (b) dispersal of either the offspring or parent, (c) competition for site occupancy, and (d) probability of survival to the next breeding season.
Juveniles reproductively mature in one breeding season. At the end of each breeding season, adults and juveniles may die, prior to reproduction in the next season. Let s A be the probability a resident adult (A) survives to the next breeding season. Let s J be the corresponding juvenile survival probability. When s A = s J = 1, all sites will remain occupied.
The environment will be saturated. When either survival probability is less than one, some open sites may exist. Thus, these models allow us to examine the effects of saturation and open environments, as emergent properties of vital parameters, rather than exogenous assumptions.

| Heritable strategies
Assume reproduction is sexual and haploid. Also, assume two pure heritable strategies, Bequeath (B) and Stay (S). Both strategies are expressed in adults. A bequeathing adult always disperses after reproduction, arriving at an "away" site. This leaves its offspring behind to compete to retain the natal "home" site. A staying adult always evicts its offspring, forcing it to compete for an "away" site, while the adult remains behind to compete to retain the "home" site.
We have also analyzed an infinite alleles model that allows continuously varying strategies between pure Bequeath and pure Stay, using a heritable probability of bequeathing. The continuous strategy space produces the same results, in this case, owing to a lack of geometric mean fitness effects (bet hedging), stable internal equilibria, and evolutionary branching. Therefore, we stick to the discrete strategy case in this paper, for ease of understanding. The individual-based simulation code we include in the Supplemental can be toggled to continuous strategy space for comparison.

| Dispersal
We have analyzed two extreme dispersal models, a global model and a local model. In the global model, all sites are equidistant; consequently, dispersal from any site has an equal probability of arriving at any other site. In the local dispersal model, sites are arranged in a ring, and individuals can disperse only to one of two neighboring sites, at random. Real dispersal patterns are probably intermediate between these two extremes.
We assume that dispersal is costly, carrying a chance of dispersalrelated mortality. These costs may be due to increased predation risk during dispersal, energetic costs, or limited knowledge of resource availability in new sites. Let d A be the probability that an adult survives dispersal and arrives at a new site. Let d J be the probability that a juvenile survives dispersal. Typically, d A > d J , and so we focus on that condition, considering whether it is necessary or not for bequeathal to be an ESS.

| Competition
Individuals must compete to retain or colonize sites. All individuals who disperse into or remain in a site compete for it. We assume a lottery-type competitive model, in which all individuals arriving or residing at a site simultaneously compete for it. Adults have an advantage over juveniles in competition, and we express this advantage as a relative advantage C A > 1. The probability that an adult retains or occupies a site with n A other adult competitors and n J juvenile competitors is as follows: After competition, a single individual survives to occupy each site.

| Expected fitness
Using the assumptions above, we can write expected inclusive fitness expressions for B and S. We fully develop the global dispersal model first, before specifying how the local model differs. The global model can be derived for any population frequency of Bequeath, p, while the local model cannot. However, both models can be analyzed for the ESS conditions of both B and S. .

| Global dispersal and fitness
Let p be the proportion of the population with strategy B. Let R be the proportion of sites with a resident adult, at the start of each breeding season. The goal is to compute the probability n A adults and n J juveniles immigrate to a particular site. Under the assumption that dispersal events are independent of one another, the probability that n A adults and n J juveniles arrive at a particular site will be multinomial with three categories (adult, juvenile, and none) and N − 1 trials. As the number of sites N grows large, the distribution approaches a bivariate Poisson, just like a binomial distribution with low probability approaches univariate Poisson as the number of trials becomes large. Therefore, in the limit N → ∞: The juvenile stays at the site, competing with n A adult immigrants and n J juvenile immigrants. The juvenile survives the season with probability s J .
The other component of fitness is the probability of acquiring the away site to which the adult disperses. This is as follows: If the bequeathing adult survives dispersal, it competes with a resident R of the time, in addition to another n A adult immigrants and n J juvenile immigrants. Since the number of sites is very large, the distribution of immigrants here is the same as before, not conditional on the focal immigrant, because dispersal events are independent in the Poisson process. If the number of sites were small, or dispersal were local, this would not be true, as we explain later.
Finally, we devalue fitness from the offspring, due to imperfect inheritance. This gives us inclusive fitness: where ρ is the coefficient of relatedness between the adult and juvenile. For a typical example, this would be ρ = 0.5. But for a maternally inherited trait, it might be ρ = 1.
The fitness expression for the Stay strategy is constructed similarly: where: Note that, while the expressions W(B) and W(S) are presented as inclusive fitness expressions, they are just expected growth rates.
No weak selection approximation or other assumptions typical of other inclusive fitness models have been made.

| Local dispersal and fitness
The local dispersal model is analogous. However, the probability Pr (n A , n J ) under local dispersal cannot be approximated by a Poisson distribution, even at N → ∞, because at most two sites (neighbors) contribute dispersers to any focal site. Additionally, the disperser pool is no longer independent of a focal disperser arriving at an away site.
Furthermore, it is not easy to specify the distribution of immigrants for any population frequency of Bequeath, p, because local dispersal generates spatial correlations in genotypes-the population residency rate R will not tell us the relevant residency probability at every locale.
It is possible, however, to completely define the model for invading B and invading S, that is, for p ≈ 0 and p ≈ 1. This allows us to conduct standard ESS analysis, even though we will not be able to find the location of any internal equilibria. This turns out to be sufficient for this model. But we have also verified all of these inferences using individual-based simulation, which is available through the link in Supplemental Materials.
Constraining p ∊ {0, 1}, the distribution of immigrants is now defined by a simple binomial process, as each neighboring site contributes an immigrant half of the time (it can go in either direction), discounted by the probabilities of residency R and dispersal survival d A and d J . In other words, each immigrant is a coin flip from a biased coin with probability of arrival of π = R(pd Whether there are one or two "coins" to flip depends upon our focus. When focusing on a home site, there are two neighbors who may contribute immigrants. But when focusing on an away site, the focal disperser counts as one of the neighbors, and so there is only one "coin" to flip. With these facts in mind, we can define inclusive fitness much as before. The expressions add little insight, and so we include them only in the Appendix. The Mathematica notebook in the Supplemental contains all of these expressions and computes fitness differences from them.

| MODEL RE SULTS
There are two antagonistic forces that strongly influence when Bequeath can be an ESS. The first is the comparative advantage that adults have in competition. This advantage favors Bequeath. The second force, opposed to the first, is the conflict of interest between parent and offspring that arises from sexual reproduction. Baseline survival, dispersal survival, and dispersal pattern (local or global) all interact with these two forces.
Even a model as simple as this one is very complex. Therefore, we explain these two antagonistic forces first, without reference to dispersal pattern or baseline and dispersal survival rates. We consider how local and global dispersal differ, through their effects on comparative advantage and conflict of interest. Then, we vary adult and juvenile survival rates to show how they interact with adult comparative advantage and parent-offspring conflict of interest.

| Bequeathal is favored by comparative advantage
Assume for the moment that s A = s J = 1 and that d A = d J = 1 so that there is no baseline nor dispersal mortality. As can be seen by substituting these values in Equation 4, these assumptions imply that all sites are always occupied (R = 1), a saturated environment. Figure 1 illustrates the nature of invasion and stability under these conditions. Each of the four diagrams in Figure 1 illustrates movement from and into a focal "home" site for a rare invader, as well as movement from and to an "away" site the invader attempts to claim. This is a cartoonish representation of the full model, but will serve to explain the basic forces in the model, before moving on to nuances.
Consider only the top two diagrams for now, (a)  Now consider the amount of competition at home and away. To retain the home site, the adult S individual competes with, on average, one other adult. Any competitive advantage of adults has no effect here, because all immigrants are adult, when B is common. In contrast, to acquire the away site, the lone juvenile disperser competes with a juvenile resident and, on average, one adult immigrant.
Therefore, there is one additional competitor at the away site, and the juvenile must contend with its disadvantage against an adult (assuming C A > 1). So, Stay sends its juvenile to an away site at which it must compete against, on average, one additional juvenile. Also, any competitive advantage of adults hurts Stay, because as C A increases, the chance of acquiring the away site decreases. For very large C A , the only way for a S juvenile to acquire an away site is for no adults to immigrate.
The situation is nearly reversed when Bequeath invades, as shown in Figure 1b. Now a B juvenile remains home and competes with, on average, one other juvenile. Immigrants are all juvenile now, because Stay is common. Competitive advantage of adults (C A > 1) is again irrelevant for the invader retaining the home site. But at the away site, the dispersing B adult does better as C A increases, since its competitive advantage reduces the impact of any immigrant juveniles. If C A = 1, the dispersing adult acquires the away site one-third of the time, on average. But for very large C A , it will acquire the away site one-half of the time.
Considering Figure 1a,b the principle reason that Bequeath can be an adaptation is that it uses the comparative advantage of adults by allocating the better warrior, the adult, to the worse battlefield, the away site. In contrast, Stay allocates the worse warrior, the juvenile, to the worse battlefield. In the mathematical Appendix (Equations A1, A5a,b), we show that, as long as no other forces are in play (C A > 1 and ρ = 1), Bequeath is always an ESS and Stay is never an ESS.
The same principle applies to the local dispersal model, illustrated by Figure 1c,d. However, the excess competition at away sites, compared to the home site, is smaller than in the global dispersal model.
This fact has no impact on the long-run dynamics, as long as ρ = 1.
B is still favored by comparative advantage and uniquely an ESS. So, we postpone discussion of local dispersal until the next section.

| Sexual reproduction and conflict of interest
The principle of comparative advantage will not uniquely determine the evolutionary result, unless the juvenile and adult have no conflict of interest. When ρ = 1, there is no conflict of interest, and selection favors allocating the adult to the more dangerous away site. The adult and juvenile always agree. But for ρ < 1, there is a conflict of interest, with bequeathal representing a costly action by the adult.
As ρ gets smaller, selection favors adults choosing the easier battle, which is always the home site. However, large C A can compensate, allowing B to continue to be stable, even when ρ is so small that B can no longer invade the population.
To appreciate how conflict of interest and comparative advan- To understand these results, consider Stay to be a "selfish" strategy while Bequeath is "cooperative." A Bequeath adult disperses at a personal cost, because there is more competition at the away site, leaving the easier home site for the juvenile to defend. When ρ = 1, the interests of the adult and juvenile are completely aligned, and so the adult favors the strategy that results in the greatest joint success (family growth). But when ρ < 1, the adult and juvenile will disagree.  Now the cost of dispersal is greatly increased. If ρ = 1, this has no effect, because the adult will still agree to disperse, since both the adult and juvenile must pay the same dispersal cost (25%).
But as long as ρ < 1, the cost quickly becomes too great for the adult, favoring Stay. The region in which B can be an ESS is greatly reduced.
Combining 25% baseline and dispersal mortality, in panel (d), demonstrates a strong interaction between these two forms of mortality. To further understand the effects of the mortality parameters, we proceed in the next sections by fixing ρ = 0.5, representing sexual reproduction, and allowing adult and juvenile survival rates to vary independently. Note that Stay can be both an ESS and nonviable, as sometimes happens in models with both ecological and evolutionary dynamics. Also, note that the conditions for viability refer to expectations. Many parameter combinations will lead to extirpation with high probability, even when they strictly satisfy the conditions above. Populations near the purple region are highly endangered.

| Baseline mortality
Perhaps counterintuitively, Bequeath does best when adult survival, s A , is low while juvenile survival, s J , is high. When adult survival is low, the residual reproductive value of an adult is also low.
This effectively reduces the cost to the adult of bequeathing the home site. Since the adult will likely die anyway, better for it to provide a benefit to the offspring. However, unless s J is also sufficiently large, the juvenile will not live to enjoy any bequeathed benefit. As    Regardless, the existence of these mixed equilibria sheds light on the general conditions that favor both B and S, and therefore aids in understanding dispersal strategy more generally. Specifically, we are struck by how hard we had to search to find mixed equilibria in these models. Unless dispersal and mortality are tuned in precise ways, selection will not favor a mix of B and S.

| D ISCUSS I ON
We have developed and analyzed two very simple models of bequeathal. In the first, dispersal is global and random. In the second, dispersal is local and random. In both models, a single adult breeder occupies a site and produces a single juvenile offspring. Genes in the adult determine whether it evicts the juvenile, forcing it to disperse, or rather bequeaths the site to the juvenile, dispersing itself.
Both adults and juveniles must compete with nonrelated individuals to retain or acquire breeding sites, and adults are advantaged in such competition. Adults and juveniles experience mortality during dispersal and at the end of each breeding season. Depending upon survival parameters, the habitat may or may not be saturated, but it is always uniform and static, with respect to the number and productivity of breeding sites.
Based on our results, bequeathal is most likely to be adaptive under the following conditions.

In unsaturated habitat. An unsaturated environment, with vacant
breeding sites, reduces the competition a bequeathing adult faces.

When adults easily defeat juveniles in contests for breeding sites.
Our models make no distinction between experience-related and sizerelated competitive advantages.

3.
When adults are superior to juveniles in dispersal survival. Our models do not address whether superior survival is due to greater knowledge of the habitat or greater experience avoiding predation or even greater body size.

When adults have less residual reproductive value than their off-
spring. This can be true, for example, when an adult is less likely to p Rt survive to breed a second time than a juvenile is to survive to adulthood.
These conditions do not seem too restrictive, and indeed, all of them have been suggested in the empirical literature as conditions that may favor bequeathal. As described in Section 3, these conditions are interactive and can sometimes counteract one another.
Our analysis also finds many situations in which bequeathal does not evolve, even when these conditions are satisfied (for empirical examples that fail to detect bequeathal, see Lambin, 1997;Selonen & Wistbacka, 2017). The major reason is that bequeathal is a cooperative behavior that may impose substantial fitness costs on the adult. As a result, often even when bequeathal is adaptive-can be maintained by natural selection-it may not be able to invade the population. For most of the parameter space in our models, bequeathal is most challenged when it is rare. This positive frequency dependence creates large regions in which both bequeathal and juvenile dispersal are evolutionarily stable, making it hard to know what to predict.
Prediction is made more challenging once we remember that models of this sort are rarely valuable for their direct quantitative predictions. As the first formal models of bequeathal, these had to be simple to be productive. Despite their simplicity, they exhibit complex dynamics that demonstrate the basic trade-offs inherent in bequeathal, trade-offs that are likely to operate in more complex models as well as in real populations.

| Facultative response
The strategies we have modeled so far are inflexible. Bequeathal in nature, like other modes of resource inheritance, is more likely part of a portfolio of dispersal strategies that individuals deploy facultatively, as conditions change (Myles, 1988). Models without explicit plasticity can sometimes be usefully interpreted as guides to plastic response. There are also risks that plasticity will generate novel feedback. In that case, attempting to interpret evolutionary dynamics as behavioral dynamics may frustrate and confuse. Still, it is useful to consider facultative interpretations of our results, as it helps to integrate our models with the existing literature, as well as guide future theorizing.
We have assumed that adult competitive ability, C A , is constant across individuals. If instead adults vary in competitive ability and have some knowledge of it, then dispersal strategy may be contingent. We found that bequeathal is favored and easier to maintain when C A is large, suggesting that larger and more aggressive individuals might do better pursuing bequeathal.
There is also the possibility that individuals who already occupy a site have a prior residency advantage over immigrant intruders (Kokko, López-Sepulcre, & Morrell, 2006;Maynard Smith & Parker, 1976). This could apply to both nonbequeathing adults and juveniles who inherit breeding sites. If such an advantage were only to apply to adults, then the conditions favoring bequeathal would be reduced.
An animal using bequeathal facultatively should be more likely to bequeath in unsaturated habitat than in a saturated one Harris & Murie, 1984;.
Unsaturated habitat favors bequeathal, because it reduces competition at an away site. Thinking ecologically, stochastic disturbance that creates new unoccupied habitat, or rather removes a large portion of the population, may encourage bequeathal.
Provided adults enjoy higher dispersal survival than do juveniles, facultative bequeathal following disturbance or an increase in baseline mortality may allow a population to rescue itself. This is because habitat saturation would be higher under bequeathal than under juvenile dispersal. Such a mechanism can work in our models. If it can also function in natural populations, even rare bequeathal following disturbance may be ecologically important, because it will allow populations to persist in otherwise challenging habitats.
Bequeathal may also be a facultative strategy at end of life (Descamps et al., 2007). We found that, when adults experience higher baseline mortality than do juveniles, selection tends to favor bequeathal. This is because an adult with low survival expectation has low residual reproductive value. In more complex life histories, where, for example, the survival probability changes with age, it might be possible that young adults will be selected to evict offspring, while older adults are selected to bequeath.
Another aspect of life history that may lead to facultative bequeathal is timing of birth (Price, 1992). When females give birth late in the season, juveniles may not have sufficient time to grow to a size that would allow them to successfully disperse and compete for a breeding site. In contrast, an offspring born early in the season may have an advantage, competing against an average juvenile. If so, bequeathal may be favored late in the breeding season, even when it cannot be favored early in the season. Evidence consistent with this has been found in plateau pikas (Zhang et al., 2017).
Finally, we have treated habitat saturation as a uniform factor. In reality, local saturation matters more than global saturation. Adults who know their range and are aware of open sites may do better bequeathing, even though the same individuals might do better to evict offspring, if the local environment were more saturated. Along similar lines, the models could be expanded to include a flexible search strategy during dispersal (McCarthy, 1999), such that dispersers are more likely to colonize empty sites and avoid those that are occupied.

| Future directions
Conspicuously absent strategies in our models are site sharing and floating. In the wider literature, for example, Brown and Brown (1984), and in other models of breeding dispersal, such as Kokko and Lundberg (2001), adults may share sites with offspring. While sharing a site, offspring either postpone reproduction or reproduce at a reduced rate, while adults suffer some cost of sharing. A sharing strategy could be introduced into our models. Instead of bequeathing or evicting the offspring, the adult could allow the juvenile to remain at the natal site, a strategy seen in red squirrels (Berteaux & Boutin, 2000) as well as bushy-tailed woodrats (Moses & Millar, 1994). Parameters would be needed to specify juvenile and adult reproductive rates at a shared site, and unless juvenile reproductive rate is zero, some additional aspects of dispersal strategy would be needed to address conflict between offspring of both residents.
In this way, the models could begin to integrate with the reproductive skew and reproductive queuing literatures (Koenig et al., 1992;Keller & Reeve, 1994;Clutton-Brock, 1998;Kokko & Johnstone, 1999;Johnstone, 2000;Cant & English, 2006.) Similarly, our models could be expanded to include the possibility of floating, or waiting in interstitial habitat for breeding sites to become available (Penteriani, Ferrer, & Delgado, 2011 increase in habitat saturation. To explore these ideas, we envision an expanded strategy space in which adults both evict a certain number of offspring (from zero to all) and determine whether the adult itself disperses (bequeaths). The bequeathal strategy studied in this case would correspond to adult dispersal and eviction of all-but-one offspring from the natal site. However, many other dispersal patterns would be possible within this strategy space, including total eviction with adult residency and all-but-one eviction with adult residency.
A feature of bequeathal in many species is that a durable resource-often a den, burrow, or cache-is bequeathed together with the territory. Our models ignored the construction and persistence of such resources. Presumably, there is some cost of building a den, and if adults are better able to afford these costs, then our models may underestimate bequeathal's adaptiveness.
As a first sketch of a model with dynamic site resources, suppose that each site is also characterized by the presence or absence of a den. When a site has a resident, a den can be maintained. In the absence of a resident, a den has a probability of decaying. A den can be constructed at a site at a fitness cost k A for adults and k J for juveniles, where k A < k J . We think this model could be analytically specified under global dispersal, generating a three-dimensional dynamical system in which the frequency of bequeathal, the residency rate, and the proportion of sites with dens would all evolve together.
Our models have ignored males, treating them as ambient and causally inert. Provided that males are carriers of the bequeathal allele and that there is no shortage of males, this assumption may be harmless. However, suppose instead that males also depend upon the same sites for survival. Then, different dispersal strategies may be favored, depending upon both an individual's sex and the sex of its offspring. As observed instances of bequeathal appear to be sexbiased toward both females (Fisher et al., 2017)  This requires researchers to track the relatedness of juveniles and adults in a population, location of adults and juveniles after breeding, and availability of potential territories in space across several breeding seasons to adequately identify and test the predictions of our model.

ACK N OWLED G M ENTS
This paper is dedicated to the memory of Parry M. R. Clarke, our dear friend and colleague who did most of the theoretical work presented here. We thank members of the UC Davis Animal Behavior and Human Behavioral Ecology programs for their attention and advice on early versions of this work. We also thank two anonymous reviewers for their thoughtful and constructive feedback.

CO N FLI C T O F I NTE R E S T
None declared.

APPENDIX I N CLUS I V E FITN E SS I N TH E LO C A L D I S PE R SA L M O D EL
Limiting our definitions to p ϵ {0, 1}: Inclusive fitness is defined identically to the global dispersal model, using the probabilities above: W(B) = ρ Pr (home|B) + Pr (away |B), W(S) = Pr (home|S) + ρ Pr (away|S).

B EQ U E ATH A L I S A N E SS U N D E R A S E X UA L R EPRO D U C TI O N
Let d A = d J = s A = s J = ρ = 1. Under these conditions, the environment will remain saturated, and so R = 1. Under these conditions, the average number of immigrants to each site is 1, and all dispersers are adults. Since the environment remains saturated, the average persite success of a common strategy must be 1 (the carrying capacity).
Therefore, we only need to compute the S invader fitness and compare it to 1 to prove whether B is an ESS.

G LO BA L D I S PER SA L
The probability distribution of adults arriving to a site simplifies to a straight Poisson probability: The probability that a mutant S adult retains a home site is now: And the probability the dispersing juvenile S acquires the away site is as follows: These expressions, and their sum, are not so easy to evaluate for any C A > 1. But we can inspect the limits and still deduce that B is an ESS for any C A > 1. First, consider when C A = 1. Then: which sum to 1. So when C A = 1, there are of course no differences between B and S strategies, so they have the same fitness.
(10) Pr (n A ) = exp ( − 1) n A ! (11) Pr (n A ) 1 1 + n A  (1) lim which sums to less than 1. Since the effect of increasing C A on Pr (away|S) is to reduce it, B is an ESS for any C A > 1.
A similar argument proves that B can always invade a population of S, under the same conditions.

LO C A L D I S PER SA L
The probability that a mutant S adult retains a home site is as follows: And the probability the S juvenile acquires an away site is: Under asexual reproduction, mutant fitness is just the sum of these two expressions. This sum is never greater than 1-resident fitness-provided C A > 1. Therefore, B is an ESS.
A similar argument shows that B can always invade S, under the same conditions.

GLOBAL DISPERSAL
When ρ < 1, B is not an ESS for any C A > 1. But B is an ESS for C A → ∞.
To demonstrate this result, assume again d A = d J = s A = s J = 1. As a result, again R = 1. However, now assume 0 < ρ < 1. Resident fitness will not be 1 now, but instead some fraction of 1, as offspring fitness is discounted by ρ. So we must calculate both resident and invader fitness.
A resident B juvenile retains home site with probability: And a resident B adult acquires an away site with probability: And resident B inclusive fitness is given by W(B) = ρ Pr (home|B) + Pr (away|B). Invader fitness is as in the previous section, but with inclusive fitness W(S) = Pr (home|S) + ρ Pr (away|S).
Therefore, B is never an ESS, when C A = 1 and ρ < 1.
A similar argument demonstrates that S is always an ESS under the same conditions. Now consider when C A → ∞. Again, taking limits: And now W(B) > W(S) for any ρ > 0.
Therefore, B is an ESS, once C A is sufficiently large. We cannot prove analytically how large C A must be to cross the threshold required to make B an ESS. But we can be sure such a threshold exists, as the effect of C A on the probabilities of winning sites is monotonic.

LO C A L D I S PER SA L
In the case of local dispersal, an exact condition can be derived. B is an ESS, provided: Unfortunately, nothing can be gained by inspecting this inequality directly, aside from noting that greater relatedness favors bequeathal. This inequality defines the dashed blue boundary in  > 14 + C A (25 + C A (1 − 4C A )) 3(4 + C A (7 + C A )) .