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., Competing Interests: 3.2The 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 advantage interact, in Figure 2 we map regions of stability for B and S for combinations of ρ and C A. Focus for now on only the upper left, panel (a), the enlarged plot with labeled regions. The horizontal axis is the magnitude of C A, expressed as the base‐2 logarithm, a “fold” value. If you folded a piece of paper in half 10 times, then its thickness would be 210 layers, a 10‐fold increase in thickness. Likewise, you can read the value log 2 C A = 10 as a 10‐fold increase in adult competitive ability, relative to a juvenile. The vertical axis is ρ, from complete conflict at the bottom to complete agreement at the top. The colored regions represent different combinations of ρ and C A for which B and S are not evolutionarily stable. In the orange regions, S is not an ESS. In the blue regions, B is not an ESS. In the white region, both B and S are evolutionarily stable. The red and blue curves show the boundaries for the different dispersal models, with global dispersal represented by the solid curves and local by the dashed. Figure 2Effects of relatedness, ρ, and adult competitive ability, C A, on stability of Bequeath and Stay. In each panel, horizontal axis is the logarithm of C A and vertical axis is ρ. The shaded regions indicate combinations of C A and ρ for which either Bequeath (orange) or Stay (blue) is the only ESS. In the white region, both Bequeath and Stay are ESSs. Boundaries for the global dispersal model are solid. Boundaries for the local dispersal model are dashed. Each panel shows regions for different combinations of dispersal survival and baseline survival. Top row: s A = s J = 1. Bottom row: s A = s J = 0.75. Left column: d A = d J = 1. Right column: d A = d J = 0.75In Figure 2a, there is no dispersal mortality nor baseline mortality. At the top, the results correspond to the inferences in the previous section: The comparative advantage of adults renders Bequeath an ESS (and Stay not an ESS) for all C A > 1 (log 2 C A > 0). But as ρ decreases, the orange regions become increasingly restricted to large C A values. By the time ρ reaches 0.5, corresponding to sexual reproduction, either only S is an ESS (blue regions) or both B and S are ESSs. At the limit ρ = 0, B is never an ESS, although if C A is large enough, even tiny amounts of relatedness are sufficient for B to be an ESS. We prove this result in the Appendix. 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. Provided C A is large enough, B can remain stable. But for small C A, B may not be an ESS. The reason B can be stable even when it cannot invade is because of positive frequency dependence. When B is rare, the adult is dispersing into a site with a resident S adult, in addition to any juvenile immigrants from other sites. For an adult, competing against another adult for the away site is much harder than defending the home site from invading juveniles. But as B increases in frequency, more and more away sites are occupied by juveniles left behind by B adults. It is simultaneously true that more adults enter the dispersal pool, and so adults invade the away site. But this effect happens at both the home and away site and so does not affect the relative cost of adult dispersal. This means that Bequeath does better the more common it becomes, because the away site becomes easier to win, reducing the costliness of adult dispersal. The boundaries for global and local dispersal, shown by the solid and dashed curves, sometimes differ greatly. The major effect of local dispersal is to make it harder for either strategy to invade the population. Local dispersal makes the white region larger, and so more combinations of parameters lead to both B and S being evolutionarily stable. To understand why, it is helpful to refer again to Figure 1. Under local dispersal, at most two individuals can immigrate into any site. Therefore, while the average number of immigrants remains the same as in the global model, the distribution is different. First, the probability of zero immigrants at the home site is reduced under local dispersal. Under global dispersal, the probability of zero immigrants is exp (−1) ≈ 0.37, while under local dispersal, it is only 0.25 (the chance of two coin flips coming up tails). This makes the effective amount of competition greater under local dispersal. Second, the focal disperser now counts for one of the immigrants at the away site. So a rare strategy disperser now competes against, on average, one resident and one‐half immigrant, instead of one resident and one immigrant, as under global dispersal. Indeed, the probability of no additional immigrants at the away site has increased to 0.5 under local dispersal, in contrast to 0.37 under global dispersal. This reduced competition at the away site and increased competition at the home site help Bequeath, by reducing the effective cost of adult dispersal. It is still true that average competition at the away site is greater than average competition at home. But a smaller difference under local dispersal means that B can be stable for smaller values of ρ than it can under global dispersal. Simultaneously, Stay becomes stable under local dispersal for larger values of ρ. The sword of local dispersal cuts both ways: A smaller cost for a dispersing adult is also a smaller benefit for a resident juvenile. This means that Bequeath gains less under local dispersal than it does under global, resulting in both dashed boundaries in Figure 2a receding and increasing the range of conditions for which both B and S are ESSs. The other plots in Figure 2 show the interaction of ρ and C A under different values of dispersal and baseline survival. In Figure 2c, baseline survival for both adults, s A, and juveniles, s J, is reduced by 25%. This creates open habitat, effectively reducing competition at the away site. Under global dispersal, the steady‐state residency becomes R ≈ 0.56. Under local dispersal, R ≈ 0.61. Competition at the home site is also reduced, as fewer other sites have residents to produce immigrants. But this reduction in the disperser pool applies equally to home and away sites. In aggregate, lowered baseline survival benefits Bequeath, by reducing the relative intensity of competition at the away site. This results in an increased orange region, a reduction in the region in which Stay can be an ESS. In Figure 2b, we instead reduce dispersal survival by 25%, setting d A = d J = 0.75. Dispersal mortality has the opposite effect, to aid Stay over Bequeath. Unlike a reduction in baseline survival, a reduction in dispersal survival does not necessarily result in open habitat. Here, the environment remains saturated at R = 1. Since residents always survive, as long as any individual arrives at a site, the site will remain occupied, eventually filling the environment. 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.