https://orcid.org/0000-0002-9875-4227
Figures
Abstract
Many social interactions happen indirectly via modifications of the environment, e.g. through the secretion of functional compounds or the depletion of renewable resources. Here, we derive the selection gradient on a quantitative trait affecting dynamical environmental variables that feed back on reproduction and survival in a finite patch-structured population subject to isolation by distance. Our analysis shows that the selection gradient depends on how a focal individual influences the fitness of all future individuals in the population through modifications of the environmental variables they experience, weighted by the neutral relatedness between recipients and the focal. The evolutionarily relevant trait-driven environmental modifications are formalized as the extended phenotypic effects of an individual, quantifying how a trait change in an individual in the present affects the environmental variables in all patches at all future times. When the trait affects reproduction and survival through a payoff function, the selection gradient can be expressed in terms of extended phenotypic effects weighted by scaled relatedness. We show how to compute extended phenotypic effects, relatedness, and scaled relatedness using Fourier analysis, which allow us to investigate a broad class of environmentally mediated social interactions in a tractable way. We use our approach to study the evolution of a trait controlling the costly production of some lasting commons (e.g. a common-pool resource or a toxic compound) that can diffuse in space and persist in time. We show that indiscriminate posthumous spite readily evolves in this scenario. More generally, whether selection favours environmentally mediated altruism or spite is determined by the spatial correlation between an individual’s lineage and the commons originating from its patch. The sign of this correlation depends on interactions between dispersal patterns and the commons’ renewal dynamics. More broadly, we suggest that selection can favour a wide range of social behaviours when these have carry-over effects in space and time.
Author summary
Organisms continually modify their environment by extracting resources, releasing toxins, or engineering habitats. These environmental modifications can have significant fitness consequences on other organisms, including those living far away in space and the distant future. To better understand the evolutionary relevance of such environmentally mediated social interactions, we quantify natural selection on a trait that influences biotic or abiotic environmental variables in a population composed of finite patches experiencing isolation by distance, as is typical of taxa facing limited dispersal. Using Fourier analysis, we show how selection on a trait due to its environmental effects can be understood in terms of the way that a focal individual influences the fitness of all future individuals at all spatial positions via a modification to the environment these individuals encounter, whose importance depends on the relatedness between these and the focal. Simply put, natural selection tends to favour traits whose environmental effects help (resp. harm) individuals that are more (resp. less) related than average, including those distantly in space and time. In addition to allowing for a full quantification of directional selection, this inclusive fitness perspective helps garner intuition for the evolution of environmentally mediated social interactions in general and of spite in particular.
Citation: Mullon C, Peña J, Lehmann L (2024) The evolution of environmentally mediated social interactions and posthumous spite under isolation by distance. PLoS Comput Biol 20(5): e1012071. https://doi.org/10.1371/journal.pcbi.1012071
Editor: Christian Hilbe, Max Planck Institute for Evolutionary Biology: Max-Planck-Institut fur Evolutionsbiologie, GERMANY
Received: September 22, 2023; Accepted: April 11, 2024; Published: May 30, 2024
Copyright: © 2024 Mullon et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting information files.
Funding: CM thanks the Swiss National Science Foundation for funding, including salary (PCEFP_3181243 to CM); JP acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d’Avenir) program, grant ANR-17-EURE-0010, and from the Max Planck Institute for Evolutionary Anthropology. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Organisms continually interact with one another in ways that significantly impact their survival and reproduction. Such social interactions are incredibly diverse. Still, they can usefully be classified as to whether they occur directly between individuals, such as grooming or fighting over resources, or as to whether they are indirectly mediated by environmental modifications, such as through the depletion or enrichment of resources, or the release or detoxification of pollutants [1]. Direct social interactions thus take place among contemporaries who are physically close to one another, while environmentally mediated interactions can extend much further in space and time. When environmental modifications have long-lasting and long-ranging effects, indirect social interactions may occur between individuals whose lifetimes show little or no overlap. This may lead to forms of trans-generational helping (e.g. when the underconsumption of a slowly renewable resource ensures healthy stock maintenance for future generations) or harming (e.g. when overconsumption leads to stock collapse and poor harvest in the future).
An extreme form of trans-generational social behaviour is posthumous altruism, which is a behaviour that results in a net reduction of the survival and/or reproduction of an actor to benefit only recipients living beyond the actor’s death. Several models have explored the biological scenarios that favour the evolution of posthumous altruism [2, 3]. But perhaps more striking is posthumous spite, which is a behaviour that results in a net reduction of the survival and/or reproduction of an actor to harm recipients that live only after the actor’s death. The conditions that may favour posthumous spite remain unclear. Understanding this, and more generally the evolution of environmentally mediated social behaviour, requires describing how selection shapes traits that underlie both direct and indirect social interactions.
The theory devoted to the evolution of quantitative traits influencing direct social interactions is well established (see, e.g. [4–9]). One of the main contributions of this body of work has been to highlight the importance of limited dispersal for determining how selection shapes social traits in populations that are subdivided into finite groups and spatially structured [4, 5]. Under limited dispersal, stochastic demographic effects resulting from finite group (or patch or deme) size generate genetic associations, whereby individuals expressing the same traits may be more or less likely to interact directly with one another than with individuals expressing alternative traits. The importance of such associations is enshrined in the fact that the selection gradient on a quantitative trait can be expressed as a marginal (or gradient) form of Hamilton’s rule [4, 5]. This gradient, which captures the first-order effects of selection, is sufficient to determine the trait values towards which a population converges under mutation-limited evolution ([7], i.e. to characterise convergence stability [10]).
The marginal form of Hamilton’s rule is computationally attractive because all the necessary information about interactions is summarized in pairwise relatedness coefficients evaluated under neutrality (i.e. in the absence of selection or trait variation). This remarkable simplification makes the selection gradient tractable under realistic demographic assumptions, particularly in populations exhibiting isolation by distance (e.g. lattice models [5, 11, 12]). This has opened the door to understanding the evolution of multiple types of direct social interactions in such populations (e.g. helping and harming [13–20]; sex ratio [21]; and dispersal [11, 22–24]).
In contrast, modelling the evolution of social interactions mediated by abiotic or biotic environmental variables is significantly more challenging in spatially structured populations. This is because computing the selection gradient in this case also requires computing the joint distribution of pairwise relatedness and environmental variables in the population (under neutrality [5, 25]). Generally, this distribution is the stationary solution to a high-dimensional stochastic dynamical system that is difficult to analyse or, in some cases, even to characterise. The challenge is apparent in models that allow for trait-driven changes in local demography. Even in the island model of dispersal, where the spatial structure is only implicit [26], there is typically no analytical solution to the distribution of demographic states within groups. As a result, evaluating the selection gradient on traits with effects on local demography and/or ecology is computationally challenging (e.g. [25, 27–30]). This “curse of dimensionality” becomes even more acute under isolation by distance, as the size of the state space on which relevant environmental (or demographic) variables fluctuate blows up exponentially, with the selection gradient now requiring the distribution of states among as well as within groups (e.g. eq. 22 in [25]).
To circumvent this challenge, two approximations have been suggested. One is the pair approximation developed for lattice-structured populations, where typically at most one individual lives in sites connected by stepping-stone dispersal [31–38]. Pair approximation is based on moment equations of the demographic state distribution that ignore third and higher-order moments. Under this approximation, the selection gradient can be written in the form of Hamilton’s marginal rule, thus allowing for a sharp understanding of some of the effects of demography on the evolution of social behaviour ([19, 20] see also [39]). However, pair approximation is not straightforward when considering arbitrarily complex dispersal patterns (e.g. [40]), patches with more than one individual, or trait-driven environmental state variables.
Another approximation relies on considering the dynamics of environmental state variables to be deterministic with a stable fixed point, so that there are no environmental stochastic fluctuations in the absence of genetic variation [41]. The selection gradient can then be readily expressed as a marginal Hamilton’s rule with inter-temporal fitness effects arising through trait-driven environmental modifications at different temporal distances. In addition to being simpler to compute than the original problem, this decomposition allows the delineation between a component of selection resulting from direct social interactions and a component arising indirectly through changes in the environmental dynamics. So far, this approach has been applied only to the island model—hence, in the absence of isolation by distance [41, 42]. For populations exhibiting isolation by distance, there exist general, marginal Hamilton’s rule like, formulas. These express the selection gradient in terms of inter-temporal and spatial effects of trait expression on the fitness of all possible recipients [2, 43]. However, how environmental modifications mediate the long-lasting and long-ranging fitness effects due to trait expression remains implicit in these general formulas. To better understand selection resulting from indirect social interactions via environmental feedback, fitness effects need to be unpacked in terms of trait-driven environmental modifications at different temporal and spatial distances.
Here, we do exactly that. We fully characterise the selection gradient on a trait that impacts the deterministic dynamics of environmental state variables that can be abiotic or biotic, and which feed back on the survival and reproduction of the evolving species in a general model of isolation by distance. Using Fourier analysis, we express this gradient in terms of extended phenotypic effects and relatedness coefficients scaled to local competition, both of which provide biological insights about the nature of selection and are straightforward to compute for a wide range of classical evolutionary models (e.g. the Wright-Fisher model and the Cannings model). We use our results to investigate the evolution of environmentally mediated helping and harming through space and time, such as via the production of a lasting common-pool resource or the release of a durable toxic compound. Our analyses indicate that indiscriminate spite where individuals suffer a cost to harm others living in the future, even beyond the actor’s death, readily evolves by natural selection. This result is in contrast with previous findings suggesting that indiscriminate spite rarely evolves when it occurs through direct interactions. More broadly, our model and results suggest that selection can favour a wide range of social behaviours when these are mediated in space and time through environmental feedbacks.
2 Models
2.1 Spatial structure, life cycle, traits and environmental variables
We consider a population of homogeneous individuals subdivided among D homogeneous patches (or demes or groups), each carrying N adult individuals (see Table 1 for a list of the key symbols used). The population is censused at discrete demographic time steps between which the following events occur in cyclic order: (a) reproduction and adult survival; (b) dispersal among patches; and (c) density-dependent regulation within patches such that each patch contains N adult individuals at the beginning of the next demographic time step.
Patches are arranged homogeneously in d dimensions, with Dj patches in dimension j ∈ {1, …, d}. For example, under a lattice structure in a one-dimensional habitat, D = D1 patches are arranged on a circle, while in a two-dimensional habitat, D = D1 × D2 patches are arranged on a torus. We denote by the set of all patches. The fact that patches are arranged homogeneously means that, at a technical level, we can endow the set
with an abelian group structure (see e.g. [44] for the use of such group structure in evolutionary biology), which will allow us to leverage techniques from Fourier analysis on finite abelian groups (Box 1).
Box 1. Fourier analysis on finite abelian groups
We assume that the set of patches is endowed with an abelian group structure, which allows us to consider more general spatial structures than just lattice models (e.g. hierarchical structures). The group
is defined as the direct product,
(I.A)
where
is the additive group of integers modulo Dj. The group
then consists of the set of all vectors i = (i1, …, id) with
together with addition (where addition between two vectors is component-wise). On such a group, the Fourier transform
of function f at
is given by
(I.B)
where the “character” function
(I.C)
with
, is defined for all
and
. Here, we followed the convention of population genetics (e.g. [5, 46, 47]) and defined the Fourier transform in terms of the character χk(h) (instead of its conjugate, given in Eq I.E, which is more standard in mathematics and engineering). As such, the Fourier transform of f gives the characteristic function of f when f is a probability distribution. For instance, the Fourier transform
is the characteristic function of the dispersal distribution mk. The original function is recovered by using
(I.D)
where
is the inverse transform of
at
, which is defined in terms of the conjugate of χk(h):
(I.E)
(e.g. [82]). Another property that we use in our analysis is the orthogonality relation between characters, i.e., the identity
(I.F)
(see [82], p. 169).
Each patch is characterized by a quantitative state variable representing a biotic or abiotic environmental factor, which we refer to as an environmental state variable (e.g., density of a common-pool resource or pollutant, habitat quality). Meanwhile, each individual in the population is characterised by a genetically determined quantitative trait (e.g. consumption of a resource, release of a pollutant, investment into habitat maintenance) that influences the environment and the individual’s survival and reproduction. We are interested in the evolution of this trait under the following three assumptions.
- Trait and environmentally mediated reproduction and survival. By expressing the evolving trait, individuals can directly affect the survival and/or reproduction of any other individual in the population. For example, individuals may engage in costly fights for resources in other patches and return to their own patch to share these resources with patch neighbours. The effects of trait expression on others are assumed to be: (a) spatially invariant, i.e. the marginal effect of an individual from patch
on the survival and/or reproduction of an individual in patch
only depends on the “distance” j − i between the two patches (where j − i is calculated from the abelian group operation, see Box 1); and (b) spatially symmetric, i.e. the marginal effect of an individual from patch i on an individual in patch j is equal to the effect from j to i. We refer to these two characteristics (a) and (b) together as spatial homogeneity. The survival and reproduction of an individual may also depend on the environmental state variable of each patch, also in a spatially homogeneous way (i.e. the marginal effect of the environmental state variable of a patch i on the survival and reproduction of an individual residing in patch j only depends on the distance j − i, and is equal to the effect from j to i).
- Dispersal. Each individual either stays in its natal patch or disperses to another patch. Dispersal occurs with non-zero probability so that patches are never completely isolated. We assume that dispersal is spatially homogeneous, i.e. the probability of dispersal from one patch i to another j depends only on the distance j − i = k between the two patches (spatial invariance), and is equal to the probability of dispersing the distance i − j = −k (spatial symmetry). We can thus write mk = m−k for the probability that an individual disperses to a patch at a distance k from its natal patch (with
).
- Trait and environmentally mediated environmental dynamics. Through trait expression, individuals can affect environmental state variables from one demographic time step to the next. For example, the environmental variable may be a common-pool resource that individuals absorb locally, or a pollutant produced by individuals which then diffuses in the environment. Such trait effects on the environment are also spatially homogeneous (i.e. the marginal effect of an individual from patch i on the environmental state variable of patch j only depends on the distance j − i and is equal to the effect from j to i). These trait-driven environmental modifications can thus lead to inter-temporal, environmentally mediated social interactions.
2.2 The focal individual, its fitness, and environmental dynamics
The spatial homogeneity underlying all processes described above means that the patch indexed as can be taken as a representative patch, and that any individual in this patch can be taken as a representative individual from the population. We refer to this patch and to this individual as, respectively, the focal patch and the focal individual. In general, we refer to patch
at time (or “generation”) t ≥ 0 prior to the focal generation as patch “k, t”. Thus, the focal patch corresponds to patch 0, 0. In the following, we introduce some notation to describe trait and environmental variation in the population (see Fig 1C for a summary diagram of our model). We denote by z• the realized value of the trait of the focal individual, and by zk,t the realized average trait of individuals other than the focal individual living in patch k, t [e.g. for a one-dimensional habitat, z1,1 is the average trait expressed in the adjacent patch (patch 1) one time point before the focal generation]. Hence, z0,0 denotes the average phenotype among the patch neighbours of the focal individual in the focal generation (thus excluding the focal from the average). We use
to denote the vector collecting all such realized phenotypes in
in lexicographic order, finishing with position D−1 = (D1 − 1, D2 − 1, …, Dd − 1). Finally, we use nk,t to denote the environmental state variable in patch k, t; we collect the environmental state variables across all patches in the vector
.
Schematic description of our model for a one-dimensional lattice habitat (see sections 2.1–2.2 for details). Each patch k ∈ {…, D − 1, 0, 1, …} at time t ∈ {0, 1, …} in the past is characterized by an environmental state variable nk,t (represented here by a cloud, e.g. water level, concentration of a pollutant, density of a resource), and the average trait value zk,t expressed by the individuals it carries (e.g. water absorption rate, detoxifying capacity, handling time; individuals represented here as palms). The environmental state n0,0 of the focal patch k = 0 at time t = 0 depends on all environmental states and traits of the previous generation according to the environmental map g (blue dashed arrows, Eq 2). In turn, the fitness of a focal individual with trait z• (in yellow) depends on all environmental states and traits expressed in its own generation according to the fitness function w (orange arrows, Eq 1). The two functions g and w thus characterise how evolutionary and environmental dynamics interact with one another through dual inheritance of traits and environmental state variables.
The fitness of the focal individual is determined by the function such that
(1)
is the expected number of successful offspring (i.e. offspring that establish as adults, including the surviving self) produced over one demographic time by the focal individual with trait z•, when the trait average among other individuals at the different spatial positions is z0,0, and environmental state variables are n0,0. These state variables are obtained from the solution to the system of equations
(2)
where
is a transition map determining the dynamics of the environmental state variables nk,t of all patches, which is a circular permutation of n0,t with nk,t as first element [e.g. for a one-dimensional lattice where d = 1, n0,t = (n0,t, n1,t, …, nD−1,t), n1,t = (n1,t, n2,t, …, n0,t), n2,t = (n2,t, n3,t, …, n0,t, n1,t), and so on]. The map g depends on (i) the traits in the whole population expressed at the previous generation via zk,t+1 (recall that t goes back in time), which is a circular permutation of the elements of z0,t+1 with zk,t+1 as first element [e.g. for a one dimensional lattice, z0,t+1 = (z0,t+1, z1,t+1, …, zD−1,t+1), z1,t+1 = (z1,t+1, z2,t+1, …, z0,t+1), z2,t+1 = (z2,t+1, z3,t+1, …, z0,t+1, z1,t+1), and so on]; and (ii) the environmental state variables of all patches at the previous generation via nk,t+1. Due to the recursive nature of Eq (2), the environmental state variables in the focal generation, n0,0, depend on the whole history of traits zH = (z0,1, z0,2, …) expressed in the population prior to the focal generation. As a result, the fitness of a focal individual may also depend on the traits expressed by all other previous individuals across space and time. To make this dependence explicit, we write the fitness of the focal individual as w(z•, z0,0, n0,0(zH)).
We make the additional assumption that in a monomorphic population where all individuals express the same resident phenotype z, the deterministic environmental dynamics described by the map g have a unique hyperbolically stable equilibrium point, identical in each patch, and satisfying
(3)
where z = (z, …, z) and
are vectors of dimension D whose entries are all equal to trait value z and environmental state variable
, respectively. This is sometimes called the spatially homogeneous or flat solution in multi-patch ecological systems (p. 235 in [45]). The existence of such a solution entails that, in the absence of genetic variation, all patches converge to the same environmental equilibrium
, which may depend on the resident trait z. One useful property of a monomorphic population at such an equilibrium is that fitness must be equal to one, i.e.
holds. This is because the total population size is constant, and consequently, each individual exactly replaces itself on average.
The fitness function (1) and the recursion (2) assume that fitness and the environmental dynamics can be written as functions of trait averages within patches. This said, this assumption does not limit us to only considering situations where effects within patches are additive. Indeed, because we are interested in convergence stability and thus in the first-order effects of selection (i.e. the first-order effects of trait variation), Eqs (1) and (2) are sufficient to model biological scenarios with arbitrary complicated non-additive traits effects among individuals within and/or between patches, for instance through complementarity or antagonism. Just a little care may be required when defining these expressions from an individual-based model (p. 95 in [5]).
2.3 Evolutionary dynamics
We assume that the quantitative trait evolves through rare mutations of small phenotypic effects, such that the evolutionary dynamics proceeds as a trait substitution sequence on the state space (i.e. the process of “long-term evolution” for finite populations described in [7]). We are interested in characterising convergence stable trait values, which are local attractors of the trait substitution sequence. To do so, we base ourselves on the first-order effects of selection on the fixation probability of a mutant that arises as a single copy in a population monomorphic for a resident trait value [7, 11, 12]. Technical details about our derivations can be found in S1 Text and accompanying boxes. Our main findings are summarized below.
3 Results
3.1 Recipient-centered perspective: Intra- and inter-temporal fitness effects
A trait value in the interior of
is convergence stable when
(4)
holds, where the function
(5)
referred to as the selection gradient, can be written as the sum of two terms, given by
(6a)
and
(6b)
(see Appendix A in S1 Text for a derivation). Here, all quantities are evaluated in a monomorphic population with all individuals expressing the resident trait value z, and the environmental state variable in all patches are at the environmental equilibrium
(Eq 3). A trait value z* satisfying condition (4) constitutes a candidate end-point of evolution. More specifically, z* is a mode of the stationary phenotypic distribution under the trait substitution sequence (see eq. A-5 in S1 Text and e.g. [7] for details). It is also possible for a mode of this distribution to lie on the boundary of a closed trait space, e.g. of
so that a mode sits at zmin and/or zmax. For a mode to be at zmin, s(zmin) < 0 must hold, in which case zmin is convergence stable. Similarly, if s(zmax) > 0 holds, then zmax is convergence stable and a mode of the stationary phenotypic distribution.
Both sw(z) and se(z) depend on (i) marginal fitness effects, i.e. on derivatives of focal fitness (that we interpret below), and (ii) the relatedness Rk,t between the focal individual and another randomly sampled individual from patch k, t. Such relatedness coefficient is defined as
(7)
where μ is the mutation rate at the evolving locus; Qk,t is the stationary probability that an allele sampled in the focal individual is identical by descent with a homologous allele sampled in another individual chosen at random from patch k, t under neutrality (i.e. in a population monomorphic for z); and
is the average probability of identity by descent between two homologous alleles sampled in two individuals living t generations apart. The probability of identity by descent Qk,t, and thus the relatedness coefficient Rk,t, may depend on the resident phenotype z, but we leave this dependence implicit for readability.
Relatedness Rk,t quantifies the extent to which an individual that is sampled from patch k, t is more (when Rk,t > 0) or less (when Rk,t < 0) likely than a randomly sampled individual to carry an allele identical by descent to one carried by the focal individual at a homologous locus. To illustrate this notion, consider a Wright-Fisher process (where there is no adult survival and individuals are semelparous), which is the reference model for probabilities of identity by descent under isolation by distance (e.g. [5, 46, 47]). For this model, the relatedness coefficients Rk,t for t = 1, 2, 3, … can be shown to be given by
(8)
where
and
is the Fourier transform (or characteristic function) of the mk probabilities (see Box 1 for definitions of Fourier transforms, character functions χk(h), and their inverses
; Appendix B in S1 Text for an example of the characteristic function of a dispersal distribution; and [48] for a derivation of Eq 8). The relatedness coefficient between two individuals in the same generation, Rk,0, is given by Eq (8) with t = 2 (i.e. Rk,0 = Rk,2 holds for all k). In a panmictic or randomly mixing population (where mk = 1/D for all k), relatedness between any two individuals is zero (i.e. Rk,t = 0 for all k and all t; as the Fourier transform reduces to
if h = 0 and 0 otherwise using property I.F in Box 1). However, as soon as dispersal is non-uniform (i.e. if mk ≠ mj for some k ≠ j), relatedness varies among individuals from different patches according to spatial and temporal distance. In particular, when dispersal is limited so that individuals have a tendency to remain in their natal patch, relatedness between the focal individual and individuals in the same patch from the same generation increases (R0,0 > 0). Because the average relatedness is zero (i.e.
holds from Eq 7), the focal individual must also be negatively related to individuals residing in at least one other patch (i.e. Rk,0 < 0 must hold for some k ≠ 0). Which patches those are depends on patterns of dispersal. Under short-range dispersal, the focal individual tends to be positively related to individuals in patches nearby and negatively related to individuals further away (Figs 2C and 3C). Under long-range dispersal, relatedness can be negative between individuals living in patches at intermediate distances (Figs 2D and 3D).
Panels A-B: Dispersal distribution mk in a lattice-structured population in a one-dimensional habitat (with D1 = 51). An offspring leaves its natal patch with probability 1 − m0 = m = 0.8 and disperses to a patch at a Manhattan distance that follows a truncated binomial distribution (eq. A-7 in Appendix B.1 in S1 Text) with mean in panel A, leading to short-range dispersal, and
in panel B, leading to long-range dispersal. The distance dispersed along each habitat dimension is uniformly distributed across all dimensions and directions (Appendix B.1 in S1 Text for details). Panels C-D: Relatedness Rk,t for the dispersal distributions shown in panels A and B, respectively (using Eq 8 with patch size N = 20 and no adult survival
). Panel C highlights how relatedness decays in time and space, becoming negative away from the focal deme when dispersal is short-range. In contrast, in panel D, where dispersal is long-range, relatedness is negative at intermediate and large distances, thus leading to a multimodal distribution of relatedness values. Panels E-F: Scaled relatedness κk,t for the dispersal distributions shown in panels A and B, respectively, under a Wright-Fisher model with fecundity effects (using Eq 21 with patch size N = 20). The trend of scaled relatedness is similar as that for relatedness. See S1 Data for how to generate these figures using Mathematica.
Panels A-B: Dispersal distribution mk in a two-dimensional habitat (with D1 = D2 = 13). An offspring leaves its natal patch with probability 1 − m0 = m = 0.8 and disperses to a patch at a Manhattan distance that follows a truncated binomial distribution with mean in panel A, leading to short-range dispersal, and
in panel B, leading to long-range dispersal (see Appendix B.2 in S1 Text for details). Panels C-D: Relatedness Rk,0 from the dispersal distributions shown in panels A and B, respectively (using Eq 8 with patch size N = 20 and no adult survival
). Panels E-F: Scaled relatedness, κk,10 in panel E and κk,1 in panel F, from the dispersal distributions shown in panels A and B, respectively, for a Wright-Fisher model (using Eq 21 with patch size N = 20). See S1 Data for how to generate these figures using Mathematica.
In Eq (6), the derivative ∂w/∂z• is the effect of a trait change in the focal individual on its own fitness. Likewise, ∂w/∂zk,t is the effect of a trait change in the whole set of individuals living in patch k, t on the fitness of the focal individual. Such effect is weighted by relatedness Rk,t in the expressions given by Eq (6). As such, sw(z) (Eq 6a) is the net effect of all intra-temporal effects on fitness, while se(z) (Eq 6b) is the net effect of all inter-temporal effects (i.e. all effects within and between demographic periods, respectively). More broadly, Eq (6) consists in the sum of effects on the fitness of a focal individual stemming from all individuals that currently live (Eq 6a) or have lived (Eq 6b) in the population. Here, the focal individual is the recipient of phenotypic effects, present and past (recipient-centered perspective). How past phenotypic effects are mediated by environmental dynamics is left implicit in Eq (6), contained in Eq (6b) through n0,0(zH). In the next section, we expose such environmental effects by unpacking Eq (6b) and shifting from a recipient-centered to an actor-centered perspective.
3.2 Actor-centered perspective: Environmentally mediated extended phenotypic effects
To understand natural selection on social traits, it is often helpful to see the focal individual as the actor, rather than the recipient, of phenotypic effects [4, 5, 49]. To shift to this perspective, we can leverage the space-time homogeneity of our model to see that ∂w/∂zk,t in Eq (6) is equivalent to the total effect of the focal individual on the fitness of the individuals in a patch at distance k at t steps in the future, and that relatedness Rk,t (Eq 8) quantifies the extent to which an individual sampled in a patch at distance k at t steps in the future is more (or less) likely than a randomly sampled individual to carry an allele identical by descent to one in the focal individual at a homologous locus [2]. These considerations readily lead to an actor-centered perspective for selection on intra-temporal effects, sw(z) (Eq 6a).
For selection on inter-temporal effects, se(z), we further need to unpack the phenotypic effects through the environmental dynamics in Eq (6b). To do so, we now let the time index t ≥ 0 denote time forward so that nk,t is the value of the environmental state variable in patch k at t time steps in the future of the focal generation (t = 0), and likewise let zk,t denote the collection of population phenotypes at time t in the future. Environmental dynamics forward in time are characterised by rewriting Eq (2) as
(9)
where
is equal to zk,0 except that the component z0,0 in this vector is replaced with
, i.e. the average phenotype in the focal patch including the focal individual (e.g. for a one-dimensional lattice,
). Eq (9) brings upfront the potential complexity of characterising the environmental consequences of a trait change in the focal individual. This is because the trait of the focal individual, z•, influences the environmental state variable of potentially any patch k over one generation, nk,1, which can in turn have knock-on effects in the future on nk,2, nk,3, and so on throughout space in an interactive way. To characterise such effects, we write
(10)
for the extended phenotypic effect of the focal individual on the environmental state variable in patch k at t generations in the future. Selection on inter-temporal effects se(z) (Eq 6b) can then be written in terms of these extended phenotypic effects as
(11)
(see Appendix C.1 in S1 Text for a derivation). Eq (11) indicates that se(z) consists in the total effect of the focal individual on the fitness of individuals in each patch k, t in the future, via a change in the environmental state of possibly all patches j − k, t, where fitness is weighted by their relatedness Rk,t to the focal individual. From the point of view of the focal individual, relatedness Rk,t can then be thought of as the “genetic value” of an individual randomly sampled in patch k, t in units of fitness. More specifically, Rk,t can be interpreted as the number of units of its own fitness that the focal individual is willing to exchange with an individual from patch k, t against one unit of theirs without changing the mutant’s probability of fixation at z*. Selection thus favours the focal individual sacrificing some of its own fitness to increase fitness in patch k, t when Rk,t > 0, and to decrease fitness when Rk,t < 0. How such sacrifice impacts the environment encountered by recipients is quantified by the extended phenotypic effect ej−k,t in Eq (11).
The remaining challenge is how to compute ek,t, given the complex repercussions that a perturbation has in time and space (i.e. how to quantify a perturbation in the coupled dynamical system defined by Eq 9). We show in Appendix C.2 in S1 Text that this can be achieved through Fourier analysis using the following building blocks. First, we let ψk be the focal individual’s effect on the environmental state variable of patch k over one generation. Owing to our space-time homogeneity assumptions, this effect can be calculated as
(12)
In practice, the rightmost expression is often more useful than the expression between equal signs in Eq (12) (as the rightmost expression only requires characterising the environmental map
of the focal patch, e.g. Eq 26 below). Similarly, we let
(13)
be the effect of the environmental state variable of one patch on the environmental state variable of another patch at distance k over one generation. With the above notation, and writing Ψ(h) and
for the Fourier transforms of ψk and ck respectively, the extended phenotypic effect can be efficiently computed as the inverse Fourier transform of
(see Box 1 for the definition of inverse Fourier transforms), namely
(14)
The form of
indicates that ek,t can be considered a perturbation in the dynamics of an environmental state variable that ripples into the future. This perturbation has its origin in a focal individual whose trait affects the environmental state variables of possibly multiple patches over one generation (captured by Ψ(h) in Eq 14). This one-generational change then propagates through space over t − 1 generations owing to the environmental dynamics, finally impacting the environment of individuals t generations downstream of the focal individual (captured by
in Eq 14). In Box 2, we generalize Eqs (11)–(14) to multi-dimensional environmental dynamics, that is, when multiple environmental state variables can be affected by the evolving trait and whose dynamics can interact with one another (e.g., metacommunity dynamics).
Box 2. Multi-dimensional environment
We generalize sw(z) and se(z) to the case where there are ne > 1 environmental state variables. We denote by the vector of such variables in patch k, t (where
is the value of the ith environmental variable). The fitness of the focal individual is now given by
(II.A)
where
is the value at t = 0 of
, whose elements are solutions of
(II.B)
where gi is the transition map for environmental variable i. We assume that in a monomorphic population z, there is a hyperbolically stable fixed point to environmental dynamics,
(II.C)
where
is a vector of dimension D whose entries are all given by
. With fitness given by Eq (II.A), selection on intra-temporal effects sw(z) remains unchanged, given by Eq (6a) with
substituted for w(z•, z0,0, n0,0(zH)). For selection on inter-temporal effects se(z), carrying out mutadis mutandis the same calculations as we have for the one-dimensional case, yields
(II.D)
where ei,k,t is the extended phenotypic effect on environmental variable i in patch k, t. This is computed as the inverse transform
(II.E)
where
is the i-th element of the vector
, which is obtained from
(II.F)
Here, the community matrix C(h) has its ij-th element given by
, where
(II.G)
is the effect of environmental variable j in the focal patch on environmental variable i in patch k, t. The vector Ψ(h) has i-th element given by Ψi(h), which is the Fourier transform of
(II.H)
Under the infinite island model of dispersal, where Rk,t = 0, ei,k,t = 0, and ci,0←j,k = 0 for all
except k = 0, Eq (II.D) reduces to eqs. 15–16 of [41].
Eq (11) together with Eqs (12)–(14) constitutes a basis to quantify and understand selection on traits that have inter-temporal effects through the environment under isolation by distance. These equations formalise the intuition behind inclusive fitness arguments for environmentally mediated social interactions, saying that natural selection tends to favor traits whose environmental effects benefit the fitness of relatives (i.e. individuals more likely to carry genes that are identical by descent). Here, the fundamental currency is individual fitness, and its exchange rate among individuals is given by relatedness. However, in many cases it is not fitness that is directly impacted by traits or the environment, but rather some intermediate payoff, such as energy, the amount of prey caught, or the size of a breeding territory. In turn, this payoff influences survival and reproduction, which together determine fitness. We next explore such scenarios.
3.3 Payoff-mediated fitness: Scaled relatedness or the genetic value of others in payoff units
3.3.1 Payoff and fitness.
Following much of evolutionary game theory (see e.g. [8] for a textbook treatment), we now consider cases where fitness depends on a payoff function summarizing social interactions (e.g. a case where the energy that an individual accrues depends on its foraging behaviour, the foraging behaviour of others, and how resources are distributed in the environment). We let this payoff function be , such that π(z•, z0,0, n0,0) is the payoff to the focal individual with phenotype z• when the collection of average phenotypes among all other individuals is z0,0 and the collection of environmental state variables across all patches is n0,0. We assume that the fitness of this focal individual can, in turn, be written as a function
of the payoff to self, the average payoff to a patch neighbour, and the average payoff to an individual from each patch other than the focal, i.e. as
(15)
where
(16)
is a (D + 1)-dimensional vector collecting the payoff π• to the focal individual, the average payoff π0 to a patch neighbour, and the average payoff πk to an individual from each patch k ≠ 0. As an argument to π0 in Eq (16), we used
to denote a vector that is equal to z0,0 except for its first entry, which is given by
, i.e. by the average trait among the neighbours of a neighbour of the focal individual. This captures the notion that the focal individual can influence the payoff of its neighbours.
Eq (15) allows individual fitness to depend on the payoff of all the individuals of its generation in an arbitrary way. This said, in most applications the survival and fecundity of an individual depend only on its own payoff. In this case, fitness may be written as
(17)
where
and
are survival and fecundity functions, respectively, and quantities with R as a superscript are defined so that
is the average survival in the focal patch, otherwise
for k ≠ 0; and
is the average fecundity in the focal patch, otherwise
for k ≠ 0. Suppose we set survival
to zero in Eq (17). In that case, we obtain the fitness function of the classical Wright-Fisher process (e.g. eq. 3 in [50], in the absence of environmental effects and for a circular stepping-stone model). More generally, if
is a positive constant and payoffs only influence fecundity
, Eq (17) implements a form of “death–birth” updating protocol (i.e. where individuals sampled at random to die are replaced by the offspring of selected individuals according to payoff, e.g. [36, 51]). Conversely, a “birth-death” updating is obtained by setting
to a positive constant and letting payoff influence survival
only. Eq (17) will constitute a useful platform to explore more specific examples later, even though many of our results hold for the more general relationship between payoff and fitness given by Eq (15).
3.3.2 Selection under payoff-mediated fitness.
If fitness is of the form of Eq (15), the selection gradient can be written as
(18)
where
(19)
and
(20)
(see Appendix D in S1 Text for a derivation). To understand Eq (18), it is first useful to interpret λk (Eq 20) as a coefficient of fitness interdependence through payoffs. Specifically, λk measures the effect on the fitness of the focal individual of a change in the payoff of an individual at a distance k, relative to the effect of the payoff of the focal individual on its own fitness. When positive, λk can thus be interpreted as the strength of competition, as it indicates how much an increase in the payoffs of an individual at distance k reduces the fitness of the focal individual. With this in mind, the coefficient κk,t (Eq 19) can be thought of as a measure of relatedness scaled to competition (or scaled relatedness for short, e.g. [7, 52, 53]; with Eq 19 extending to isolation by distance the formalization of this concept found in [54]). To see this, suppose that the evolving trait helps current patch neighbours (i.e. such that ∂π(z•, z0,0, n0,0)/∂zk,0 > 0) and consider the numerator of κ0,0 on the first line of Eq (19). This numerator consists in the relatedness R0,0 of a focal individual towards current patch neighbours, discounted by an inclusive fitness effect through increased competition, which is due to the increase in neighbours’ payoff caused by the focal individual helping them. The inclusive fitness effect consists of three relatedness weighted fitness effects: (i) −λ0/(N − 1), which captures the increase in competition experienced by the focal individual itself (where 1/(N − 1) is the frequency of the focal individual among the neighbours of an individual it helps); (ii) −λ0(N − 2)R0,0/(N − 1), which captures the increase in competition experienced by the relatives in the focal patch that are distinct from the individuals the focal individual is helping directly; and finally (iii) the total effect
, which captures the increase in competition experienced by relatives in other patches. The other two lines of Eq (19) are interpreted similarly: the relatedness towards neighbours from patch k, t discounted by an inclusive fitness effect on all recipients experiencing a change in fitness stemming from the focal changing the payoff of individuals in patch k, t (the denominator in Eq (19) standardizes these effects relative to the inclusive competitive effects induced by the focal individual changing its own payoff). More broadly, κk,t can be interpreted as the number of units of its own payoff that the focal individual is willing to exchange with a unit of payoff accruing to a randomly sampled individual from patch k, t without changing the mutant’s probability of fixation at a singular value z*. The scaled-relatedness coefficient κk,t can thus be seen as the genetic value of other individuals in patch k, t relative to that of the focal individual in units of payoff. For (k, t) ≠ (0, 0), κk,t then measures the net social discount rate of temporally delayed and spatially extended fitness effects [43].
From the considerations above, Eq (18) can be read as an inclusive fitness effect at the payoff level. That is, selection depends on how the focal individual influences its own payoff and the payoff of all other individuals across patches, now and in the future, weighted by their scaled relatedness κk,t. For recipients in the future (t ≥ 1), payoff effects are mediated by how the focal individual perturbs the environment in each patch (via ek−j,t in Eq 18), and in turn by how such environmental perturbation influences payoffs (via ∂π/∂nj,0 in Eq 18).
Box 3. Scaled-relatedness coefficients
With fitness given by Eq (17), we show in Appendix E in S1 Text that
(III.A)
where as usual, all functions are evaluated at the resident trait value z, a prime denotes a derivative, and
is the inverse Fourier transform of a function X(h) at
(Eq I.D of Box 1). The functions F and Gt are defined as
(III.B)
For fecundity effects under a Wright-Fisher process (
), the above reduces to F(h) = 0 and
, which yields Eq (21) of the main text when
(i.e. the payoff is directly fecundity).
Using Eq (III.A), we also show in Appendix F in S1 Text that the summary statistic K, for selection on environmentally mediated social interactions under local interactions (Eq 24), is given by
(III.C)
where
(III.D)
For fecundity effects under a Wright-Fisher process (
) the summary statistic K simplifies to
(III.E)
while for survival effects (
), it simplifies to
(III.F)
In Appendix E in S1 Text, we use Fourier analysis to compute scaled relatedness κk,t for the fitness model given by Eq (17) and an arbitrary dispersal distribution mk. Our results are summarized in Box 3. For example, we obtain that under a Wright-Fisher process,
(21)
holds, where
(22)
is the probability that, under neutrality, a gene descending from the focal individual is in patch k at t > 0 steps in the future (which depends on the characteristic function
of the dispersal distribution, Table 1). The collection of these probabilities,
, can thus be seen as the distribution of a standard random walk on the set of patches with step distribution characterized by the dispersal probabilities mk. When such a random walk leads to a probability pk,t that is greater than under a uniform distribution (i.e. when pk,t > 1/D holds), Eq (21) indicates that scaled relatedness κk,t is positive, so that selection favours environmental transformations that increase payoffs in patch k, t. Conversely, selection favours environmental transformations that decrease payoffs in patches where pk,t is smaller than under a uniform distribution (i.e. when pk,t < 1/D holds). Which patches are those depends on the dispersal distribution (compare Fig 2E with Fig 2F for short and long-range dispersal in a 1D lattice, and Fig 3E with Fig 3F for short and long-range dispersal in a 2D lattice).
The selection gradient (18) recovers previous results of social evolution theory, which provides a robustness check of our calculations. To see these connections, assume that there are no ecologically meditated interactions, that is, that ∂π(z•, z0,0, n0,0)/∂n0,0 = 0 holds, that −C(z) = ∂π(z•, z0,0, n0,0)/∂z• < 0 is a net payoff cost to self, and that Bk(z) = ∂π(z•, z0,0, n0,0)/∂zk,0 is a payoff benefit to individuals at distance k, which is typical of models under the heading of the evolution of “cooperation” or “altruism”. Further, suppose that individuals interact only with other individuals at a single distance k so that the selection gradient is proportional to −C(z) + κk,0Bk(z). Then, Eq (18) entails that selection favours such a helping behaviour if κk,0 > C(z)/Bk(z) holds, that is, if the scaled-relatedness coefficient is greater than the cost-to-benefit ratio. For a Wright-Fisher process, κk,0 reduces to −1/(DN − 1) ≤ 0 (Eq 21), which is always non-positive. Thus, helping cannot spread regardless of population structure because κk,0 > C(z)/Bk(z) cannot be satisfied as long as Bk(z) > 0. This result was first derived for a lattice-structured population for D → ∞ and k = 0 in [55], and for finite D and any k under a circular one-dimensional habitat in [5] (chapter 8, and generalized to other abelian group structures in [44] and [43]). More generally, the scaled-relatedness coefficient given in Box 3 recovers established conditions for the spread of helping and harming behaviour in lattice-structured populations under different biological assumptions, such as for survival effects or for fecundity effects with overlapping generations (e.g. [15, 36, 51, 56]; see [43] for the explicit connections to this literature). Finally, in the presence of ecologically mediated interactions, so that ∂π(z•, z0,0, n0,0)/∂n0,0 ≠ 0, Eq (18) recovers eq. (A.21) in [2] which holds for a Wright-Fisher process (to see this correspondence, set sk,t = N(∂π(z•, z0,0, n0,0)/∂n0,0)ek,t in Eq 18).
3.3.3 Local interactions.
In Eq (18), payoffs can depend on the traits expressed in and the environmental variables of all patches. In many instances, payoff effects can reasonably be assumed to be local, i.e. the payoff of an individual depends only on the traits and the environment of its patch. In this case, the payoff to the focal individual can be written as π(z•, z0,0, n0,0) = π(z•, z0,0, n0,0), and the selection gradient Eq (18) reduces to
(23)
where
(24)
summarizes selection on environmentally mediated social interactions (see Appendix F in S1 Text for a derivation and Eq III.C in Box 3 for more details). When K = 0, selection is blind to the effects of the trait on the environment, even if the environment affects payoffs (i.e. even if ∂π(z•, z0,0, n0,0)/∂n0,0 ≠ 0). When K > 0, selection favours trait values that improve the environment (i.e. the payoff in the future increases). Conversely, when K < 0, selection favours trait values that deteriorate the environment (i.e. the payoff in the future decreases). Which of these outcomes unfolds depends on the interaction between extended phenotypic effects ek,t and scaled-relatedness coefficients κk,t, with K > 0 when ek,t and κk,t tend to be of the same sign (i.e. when
holds), and K > 0 when they tend to be of opposite sign (i.e. when
holds). In the next section, we explore through an example how this interaction depends on dispersal and how environmental state variables of different patches influence one another.
3.4 Inter-temporal helping and harming through a lasting commons
To gain more specific insights into how isolation by distance influences the way selection shapes environmentally mediated interactions, consider a scenario where the environmental variable is some lasting commons (e.g. a common-pool resource or a toxic compound) that can move in space, and whose production depends on an evolving trait that is individually costly to express. We assume that the commons is a “good” when the environmental variable takes positive values () and a “bad” when it takes negative values (
). We also assume that the trait leads to the former when z > 0 and to the latter when z < 0, where the trait space is assumed to be
. The trait can thus be broadly thought of as environmentally mediated helping (increasing survival and reproduction to recipients) when z > 0, and as environmentally mediated harming (decreasing survival and reproduction to recipients) when z < 0.
We assume that fitness takes the form of Eq (17) with payoff given by
(25)
where B > 0 and C > 0 are parameters that respectively tune the effects of the environmental variable in the focal patch n0,0 and of the modifying trait z• of the focal individual on the payoff of the focal individual. These effects also depend on the shape parameters αB and αC, which we assume are positive integers, with αB odd (e.g. αB = 1) and αC even (e.g. αC = 2). Thus, the local commons increases (resp. decreases) payoffs when n0,0 > 0 (resp. n0,0 < 0) holds, but any trait expression, that is, any z• away from 0, is individually costly and reduces individual payoff, and thus fitness. We also assume that costs increase more steeply than benefits, i.e. that αC > αB holds.
Meanwhile, how the trait modifies the commons is determined by the environmental map g (Eq 2), which we assume is given by
(26) Eq (26) states that the commons changes from one demographic time point to the next due to three processes. First, the commons is modified or “produced” in a patch according to a function
of the average trait expressed in that patch, i.e.
in the focal patch and zk,0 otherwise (with k ≠ 0). We assume that P(0) = 0 holds, and that P is monotonically increasing, i.e. P′(z) > 0 for all
, where P′(z) denotes the derivative of P with respect to z. Second, each unit of commons “diffuses” or moves with probability dk to a patch at a distance k from its source patch, where the probability distribution defined by dk can be thought of as the environmental equivalent of the dispersal probability distribution mk. We let
denote the Fourier transform of this distribution for future use (i.e.
is to dk what
is to mk, Table 1). Third, a unit of commons decays with rate 0 < ϵ ≤ 1 from one time step to the next. Substituting Eq (26) into Eq (3) indicates that in a monomorphic population for z, the dynamics of the commons stabilises to
(27)
which is positive when z > 0, negative when z < 0, and whose absolute value increases as the rate of decay ϵ decreases, as expected. This equilibrium is unique because g is linear in P, and stable because ϵ > 0 holds.
With fitness of the form of Eq (17) and payoff Eq (25) only depending on local interactions (as in section 3.3.3), we can use Eqs (23) and (24) to characterise selection. With κk,t as given in Box 3, all that remains to be computed are the extended phenotypic effects, ek,t. To do so, we start by substituting Eq (26) into Eqs (12) and (13), to obtain ψk = P′(z)dk/N, and ck = (1 − ϵ)dk, which in turn yield and
. Substituting these expressions into Eq (14) then gives
, which leads to
(28)
for the extended phenotypic effect ek,t on patch k, t, where
(29)
Equation Eq (28) can be understood as follows. By marginally changing its trait value, a focal individual produces P′(z)/N additional units of commons. Each such unit decays with time according to (1 − ϵ)t−1, and ends up in patch k, t according to qk,t (Eq 29), which can be interpreted as the probability that a non-decaying unit of the commons modified in the focal patch is located in patch k t generations in the future. In fact, the collection
yields the distribution of a standard random walk on the set of patches with step distribution characterised by dk. Extended phenotypic effects thus depend critically on the way the commons moves in space as captured by dk (see Fig 4 for examples of ek,t in a 1D model).
Panel A: When the commons moves locally, extended phenotypic effects ek,t decay in time and space away from the focal deme (from Eq 28 with D1 = 31, movement probability d = 0.6 and expected movement distance , see Appendix G.5 in S1 Text for details on how movement is modelled; production function P(z) = Nz, i.e. each unit of z contributes to one unit of resource; decay rate ϵ = 0.2; other parameters: N = 20, m = 0.3,
). Panel B: In contrast, when the resource moves at greater distances, extended phenotypic effects ek,t are greatest further away from the focal deme (from Eq 28 with movement parameters d = 0.98 and
; production function P(z) = Nz; decay rate ϵ = 0.5; other parameters: same as Fig 2A). See S1 Data for how to generate these figures using Mathematica.
In turn, how selection depends on extended phenotypic effects is found by substituting Eqs (25) and (28) into Eq (23). From this, we obtain
(30)
where
(31)
can be thought of as the total expected genetic value in units of payoff of all the individuals in the future that are affected by a unit of the commons in the focal patch. Since the term multiplying Ω in Eq (30) is positive (recall that αB is odd and hence αB − 1 is even), the selection gradient increases with Ω, and the greater Ω is, the greater the z favoured by selection. In fact, the selection gradient reduces to s(0) ∝ Ω at z = 0 (under our assumptions about parameters and the commons production function P). This shows that in a population where the trait is initially absent (so that individuals have no effect on the commons), selection favours environmental modifications leading to a common good (z > 0) when Ω > 0, or to a common bad (z < 0) when Ω < 0. Put differently, selection favours environmentally mediated inter-temporal helping when, in the eyes of the focal individual, the recipient of such help on average has positive genetic value in units of payoff, and conversely, inter-temporal harming when it has negative genetic value.
Further insights can be generated if we assume the simple functional form P(z) = P0z. In this case, we obtain
(32)
The singular value z* (32) is convergence stable under our assumption that the cost of trait expression increases faster than the benefits (i.e. αC > αB). Therefore, z* is the mode of the stationary phenotypic distribution of the trait substitution sequence (we compute this distribution for the Wright-Fisher model in eq. A-136 in S1 Text). From Eq (32) it is clear that the absolute value of z* increases with the benefit-to-cost ratio B/C, with αB/αC, and with the environmental effect of the trait P0. However, whether z* is positive or negative (and thus whether helping or harming evolves), ultimately depends on the sign of Ω, i.e. whether the expected genetic value Ω of a modification to the commons is positive or negative in payoff units.
The impact of species dispersal and commons movement on Ω can be understood most easily by assuming that payoff influences fecundity under a Wright-Fisher process (i.e. and
in Box 3). In this case, Ω can be expressed as (see Appendix G.1 in S1 Text)
(33)
where cov(pt, qt) is the covariance between the distributions of the random walks of genes
(Eq 22) and commons
(Eq 29). This covariance is positive when there is a positive association between gene lineages and the commons these lineages modify. In other words, Ω is positive and helping is favoured when an environmental modification owing to the expression of a gene is most likely to be experienced by individuals living in the future and carrying identical-by-descent copies of that gene. Conversely, Ω tends to be negative and harming is favoured when this environmental modification is less likely to be experienced by future carriers.
While Eq (33) offers intuition on the biological conditions leading to positive or negative values of Ω, this quantity is more readily computed by noting that Ω = ϵKN/P′(z) holds, where K is defined in Eq (24), and by substituting eqs. (A-127) and (A-128) in S1 Text into Eq (III.E) in Box 3. Doing so, we obtain
(34) Fig 5A and 5B give the sign of Ω under a binomial model for the distance of both the dispersal of the focal species and the movement of the commons (see Appendix G.5 in S1 Text for details). These figures show that such model of dispersal allows for both positive and negative values of Ω and thus for the evolution of both inter-temporal helping and harming. Here, helping corresponds to posthumous altruism and harming to posthumous spite since an individual can never obtain direct benefits from its own trait effects on the environment, and these effects are only felt by the next generation (as generations are non-overlapping under a Wright-Fisher process).
Panels A-B: Regions of dispersal parameters leading to the evolution of altruism, Ω > 0 (in white), or of spite, Ω < 0 (in gray) for an example in a 2D lattice model (with D1 = D2 = 13 and N = 50) under a Wright-Fisher life-cycle with fecundity effects (with Ω computed from Eq 34). Panel A: Combination of dispersal probability of the evolving species m = 1 − m0 (x-axis) and of the commons d = 1 − d0 (y-axis) for different levels of environmental decay ϵ in different shades of gray (ϵ = 0.1, 0.5, 1) with expected dispersal distance fixed ( and
). This shows that spite is favoured by high levels of dispersal and environmental decay. Panel B: Combination of expected dispersal distance of the evolving species
(x-axis) and of the commons
(y-axis) for different levels of environmental decay ϵ in different shades of gray (see panel A for the legend) with dispersal probability fixed (m = 0.98 and d = 1). This shows that spite is favoured by dispersal asymmetry between the evolving species and the commons. Panels C-D: Evolution of spite in individual-based simulations under a Wright-Fisher life-cycle with fecundity effects (with D1 = D2 = 13, N = 50, m = 0.3,
, d = 1,
, B = 2, αB = 1, C = 1, αC = 4, P(z) = Nz; for mutation, the trait mutates during reproduction with probability 10−4, in which case a normally distributed deviation with mean 0 and standard deviation 10−2 is added to the parental trait value). Panel C shows the average trait z in the population; panel D shows the average commons level or environmental variable n (with simulations in full lines—see S1 Code—and analytical prediction in dashed lines—from Eq 32 for z and 27 for n). See S1 Data for how to generate these figures using Mathematica.
More generally, numerical explorations of Ω (Fig 5A and 5B) indicate that spite tends to be favoured by: (i) high levels of dispersal in the evolving species; (ii) high levels of movement of the commons; (iii) high environmental decay ϵ; and (iv) significant differences in the dispersal distance of the species and of the commons (e.g. when individuals disperse short distances but the commons move far away from their original patch). These conditions lead to a negative association between gene lineages and the commons these lineages modify. Conversely, altruism tends to be favoured when dispersal and movement are weak, environmental decay is low, and the distributions of the species’ dispersal and of the commons’ movement are similar (Fig 5A and 5B, white region). In fact, under weak dispersal and movement (so that m0 = 1 − m and d0 = 1 − d with m and d close to zero), and regardless of the dispersal and movement distributions, we have
(35)
which is always positive when m and d are sufficiently small (see Appendix G.2 in S1 Text for a derivation of Eq 35). Under these assumptions, an individual’s lineage and the commons originating from its patch will be strongly and locally associated.
To see the effects of isolation by distance on social evolution, we compare in Fig 6A the singular strategy z* for a 2D lattice where the evolving species and the commons follow a binomial model for distance travelled (detailed in Appendix G.5 in S1 Text), with the singular strategy z* under the island model (where dispersal and commons movement are uniform). This comparison illustrates how isolation by distance allows for a wider range of evolved social behaviours as both altruism (z* > 0) and spite (z* < 0) can be favoured by selection depending on the distance dispersed. In contrast, only altruism is favoured by selection under the island model in Fig 6A (compare black full and dashed lines). Fig 6A also shows how selection may favour the evolution of more exaggerated social behaviours under isolation by distance (i.e. larger |z*|). These differences are down to the fact that isolation by distance allows for a wider range in the covariance among genetic and commons movement (recall Eq 33). We also compared the singular strategy z* found by substituting Eq (34) into Eq (32) with results from individual-based simulations. In these simulations, each offspring mutates with probability 10−4, in which case a normally distributed deviation with mean 0 and standard deviation 10−2 is added to the parental trait value. The only difference between these simulations and our analytical scenario is thus that multiple alleles can segregate due to mutation (rather than just two alleles under a trait substitution sequence, see [7] for finite populations). Nevertheless, we find an excellent fit between the convergence stable z* and the mean population trait value in simulations (Fig 6B).
Panel A: Singular trait value z* from Eq 32 for m = 0.75 (in black) and m = 0.2 (in gray). Other parameters: D1 = D2 = 13, so that D = 169, N = 50, , d = 0.99, B = 2, αB = 1, C = 1, αC = 4, P(z) = Nz. Dashed lines show the singular trait value with the same parameters but under the island model of dispersal for both the species and the commons (from Eq 32 with Ω given by eq. (A-126 in S1 Text) derived in Appendix G.3). Panel B: Observed vs. predicted equilibrium trait value in individual-based simulations running for 20 000 generations under different expected dispersal distance of the commons
leading to altruism (z > 0) and spite (z < 0). Other parameters: same as in Fig 5C and 5D. The prediction is shown as a dashed line (from Eq 32) with grey region around for twice the standard deviation obtained from the stationary phenotypic distribution (from eq. A-136 in S1 Text). Observed values of the trait average in the population are shown as black dots for the average from generation 5 000 to 20 000, with error bars for standard deviation over the same 15 000 generations. Simulations were initialised at the predicted convergence stable trait value. S1 Code for simulation code and S2 Data for simulation results.
The case where payoff influences survival rather than fecundity ( and
in Eq 17) is illustrated in Fig 7 (the expression of Ω for this case in terms of characteristic dispersal functions can be found in Appendix G.4 in S1 Text). This analysis reveals that harming tends to be favoured when baseline survival
is low, especially when environmental decay is also low (Fig 7C). This is because, otherwise, an individual may harm itself in the future. But apart from this, selection is not fundamentally different when payoff influences survival rather than fecundity in this model (i.e. under a birth-death vs. death-birth process).
Panels A-B: Scaled relatedness κk,t in a 1D lattice model under survival effects (from Box 3 with in panel A and
in panel B; other parameters: same as in Fig 4A). These panels show that genetic value decays away from the focal deme especially quickly when baseline survival is high (compare panels A and B). Otherwise, these profiles of scaled relatedness are similar to to those in Fig 4A, which suggests that selection acts similarly when the trait affects survival or fecundity. Panel C: Parameter region where selection favours the evolution of helping (Ω > 0) or harming (Ω < 0) under survival effects with adult survival probability
on the x-axis and environmental decay ϵ on the y-axis (Ω computed from Eq 31 using eq. A-129 in S1 Text; other parameters: same as in Fig 4B, i.e. under long-range movement of the commons). See S1 Data for how to generate these figures using Mathematica.
4 Discussion
Our analyses characterise in two main ways the selection gradient on a trait that impacts the deterministic dynamics of environmental state variables that can be abiotic or biotic, which in turn feed back on individual survival and reproduction under isolation by distance.
First, we showed how selection on a trait due to its environmental effects can be understood in terms of how a focal actor influences the fitness of all future individuals via a modification to the environmental state variables that these individuals experience (Eqs 11 and II.D for the case of multiple environmental variables). The relevant trait-driven environmental modifications are formalized as extended phenotypic effects that quantify how a trait change in an actor individual in the present affects the environmental state variables in all patches at all future times (the ek,t effects, Eq 14). While extended phenotypic effects are typically thought to benefit the actor or related contemporaries directly [57, 58], these effects in our model are all indirect, carrying over in space and time, thus influencing the fitness of future carriers of the actor’s trait when dispersal is limited. The associations between environmental and genetic variation that are necessary for selection to target trait-driven environmental modifications are given by the product between the extended environmental effects ek,t and the relatedness coefficients Rk,t (see Eq 11), both of which can be efficiently computed using Fourier transforms (Eqs 14 and II.E and II.F for a multivariate environment). These gene-environment associations indicate that selection favours traits or behaviours with environmental effects such that, when expressed by a focal individual, the environmental effects increase (resp. decrease) the fitness of future individuals that are more (resp. less) related to the focal than other individuals at that same future generation.
The second version of the selection gradient that we derived is based on the extra assumption that interactions between individuals are mediated through a payoff function (see Eq 17), as in most of traditional evolutionary game theory (e.g. [8, 59, 60]). Selection on a trait due to its environmental effects can still be viewed from an actor-centered perspective, but this time at the payoff rather than at the fitness level. Specifically, selection can be quantified in terms of how a focal individual influences the payoff of all future individuals via modifications to the environment these individuals experience, weighted by the relatedness between these individuals and the focal, now scaled to take competition into account (the term proportional to se(z) in Eq 18, with scaled relatedness given in Eqs 19 and 20). The concept of scaled relatedness is useful because it summarizes in a single quantity (here one for each spatial and temporal distance) all the consequences of interactions among related individuals for indirect selection [7, 52, 53, 61]. That is, scaled relatedness balances, on the one hand, the positive effects of boosting the reproductive success of relatives in a particular spatial position, with, on the other hand, the negative effects of increasing competition for these relatives by affecting the reproduction and survival of others across the habitat. The increase of kin competition can be strong enough to offset the indirect benefits of social behaviour when social interactions occur among contemporaries (and generations do not overlap, e.g. [5, 55]). Because the strength of kin competition depends on the specifics of the life cycle (such as whether generations overlap or not, or whether payoff influences fecundity or survival), the evolution of direct social interactions is sensitive to such assumptions (see [52] for a review). This is notably the case under isolation by distance, where the evolution of altruism crucially depends on whether reproduction is modelled as a “death-birth” or a “birth-death” process (e.g. [62–64]). The “death-birth” process is akin to iteroparous reproduction with fecundity effects, and the resulting life cycle can sustain altruism; the “birth-death” process is akin to iteroparous reproduction with survival effects, and the life cycle inhibits altruism via increased kin competition (e.g. [15, 17, 18, 56]).
In contrast, we have found that in our model of environmentally mediated social interactions through a lasting commons, whether selection favours the evolution of helping or harming depends weakly on whether payoff influences survival or fecundity. There are two explanations for this. The first is that, because of environmental legacy, the effects on recipients are felt in the distant future, which decreases the competition among the focal’s own offspring [41, 65]. The second explanation is that, in our model, individuals and their environmental effects can move in space independently, further dissociating the positive and negative effects of interactions among relatives. This decoupling between benefits and costs means that natural selection can readily favor either posthumous altruism or posthumous spite in our model with non-overlapping generations. Which of these behaviours evolves depends on whether the combination of dispersal pattern and commons movement cause environmental effects to fall predominantly on individuals that are more or less related than average in the future.
Our findings on environmentally mediated posthumous spite merit further discussion as existing models suggest that the conditions for the evolution of spite are more restrictive than those for altruism. By spite, we refer here to a trait or behaviour whose expression decreases the individual fitness of both its actor and recipients. This is a strong form of spite, reducing the actor’s fitness (chapter 7 in [5]), which contrasts with the more commonly explored scenarios of weak spite (where the behaviour directly increases the actor’s fitness, e.g. [66, 67]). The evolution of strong spite typically relies on the existence of mechanisms by which individuals can evaluate their relatedness with social partners and thus behave according to some kin or type-recognition mechanism (e.g. [68–72]). By contrast, in our model spite is indiscriminate: an individual deteriorates the environment in the future without paying attention to recipients’ identities, even if this comes at a cost in the present. In the absence of generational overlap (e.g. Wright-Fisher model), such spite is entirely posthumous. With this in mind, it is noteworthy that spite can evolve even when local populations are not small (e.g. of local size 50, Fig 5). More broadly, our results illustrate how environmentally mediated social interactions under isolation by distance can evolve to be as relevant for fitness as direct social interactions.
The two main assumptions of our model are that fitness and environmental effects are homogeneous in space and time, and that environmental dynamics are deterministic. These assumptions are common to previous models interested in environmental or ecological changes in space that are evolutionarily driven, and particularly to those where individuals produce an environmental commons that moves according to a diffusion process (e.g. [73–77]). These models further assume a separation of time scales between demography and the commons such that in between the reproduction, death, or dispersal of any individual in the entire population, the dynamics of the commons reaches a stable distribution across the landscape. By contrast, here we have assumed that the mutation process, rather than the demographic process, is slow compared to environmental dynamics. Reproduction, death, or dispersal can occur on a similar time scale than environmental dynamics in our model, as is usually the case in ecological models (e.g. [78, 79]). As a result, even though environmental dynamics are described by a deterministic system (Eq 9, also as in most ecological models, [78, 79]), realised environmental dynamics fluctuate randomly on a similar time scale than unavoidable genetic fluctuations owing to finite patch size. The next challenge would be to consider a fully stochastic system for the environmental variables (i.e. to extend Eq 9 to the dynamics of a probability distribution). This would be especially useful to investigate the effects of demographic stochasticity in response to trait evolution [25], and to model, for instance, environmentally mediated evolutionary suicide or rescue. Our framework may nevertheless provide a suitable approximation to cases of demographic and environmental stochasticity (with Eq 9 giving the expectation in state variable at the next time step, conditional on the states of at the previous step). This approach has been shown to work well under the island model of dispersal provided that patches were not too small and dispersal not too limited [41]. It would be interesting to investigate how this holds up under isolation by distance. Finally, we considered the evolution of a single trait but our analyses apply directly to the case where multiple traits with environmental effects evolve jointly. To investigate such a case, one would first use our equations to obtain the selection gradient on each trait separately (applying Eq 5 to each trait under focus), and then jointly consider these gradients to perform a standard multidimensional analysis of directional selection (e.g. [80, 81]).
Supporting information
S1 Text. PDF containing Appendices A-G, where we derive the results presented in the main text.
https://doi.org/10.1371/journal.pcbi.1012071.s001
(PDF)
S1 Data. Mathematica Notebook that allows to make Figs 2–5 and 7 (requires Mathematica License).
https://doi.org/10.1371/journal.pcbi.1012071.s002
(NB)
S2 Data. Zipped folder containing: CSV files with simulated data used to make Fig 6B; and Mathematica Notebook that reads CSV files and produces Fig 6B.
https://doi.org/10.1371/journal.pcbi.1012071.s003
(ZIP)
S1 Code. M-file with the code we used for individual based simulations, which lay the bases of Figs 5C, 5D and 6B (requires Mathematica License).
https://doi.org/10.1371/journal.pcbi.1012071.s004
(M)
References
- 1. Estrela S, Libby E, Van Cleve J, Débarre F, Deforet M, Harcombe WR, et al. Environmentally Mediated Social Dilemmas. Trends Ecol Evol. 2019;34(1):6–18. pmid:30415827
- 2. Lehmann L. The adaptive dynamics of niche constructing traits in spatially subdivided populations: evolving posthumous extended phenotypes. Evolution. 2008;62:549–566. pmid:17983464
- 3. Kuijper B, Johnstone RA. The evolution of early-life effects on social behaviour-why should social adversity carry over to the future? Philos Trans R Soc Lond B Biol Sci. 2019;374(1770):20180111. pmid:30966877
- 4.
Frank SA. Foundations of Social Evolution. Princeton, NJ: Princeton University Press; 1998.
- 5.
Rousset F. Genetic Structure and Selection in Subdivided Populations. Princeton, NJ: Princeton University Press; 2004.
- 6. Lehmann L, Rousset F. The genetical theory of social behaviour. Philosophical Transactions of the Royal Society B. 2014;369:1–18. pmid:24686929
- 7. Van Cleve J. Social evolution and genetic interactions in the short and long term. Theoretical Population Biology. 2015;103:2–26. pmid:26003630
- 8.
McNamara JM, Leimar O. Game theory in biology: concepts and frontiers. Oxford University Press; 2020.
- 9. Avila P, Mullon C. Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact. Philosophical Transactions of the Royal Society B. 2023;378:615255. pmid:36934752
- 10. Eshel I. Evolutionary and continuous stability. Journal ot Theoretical Biology. 1983;103:99–111.
- 11. Rousset F, Billiard S. A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal. Journal of Evolutionary Biology. 2000;13:814–825.
- 12. Rousset F. A minimal derivation of convergence stability measures. Journal of Theoretical Biology. 2003;221:665–668. pmid:12713948
- 13. Rogers AR. Group selection by selective emigration: the effects of migration and kin structure. American Naturalist. 1990;135:398–413.
- 14. Taylor PD. Altruism in viscous populations—an inclusive fitness model. Evolutionary Ecology. 1992;6:352–356.
- 15. Irwin AJ, Taylor PD. Evolution of altruism in stepping-stone populations with overlapping generations. Theoretical Population Biology. 2001;60:315–325. pmid:11878832
- 16. Taylor PD, Day T, Wild G. Evolution of cooperation in a finite homogeneous graph. Nature. 2007;447:469–472. pmid:17522682
- 17. Lehmann L, Keller L, Sumpter D. The evolution of helping and harming on graphs: the return of the inclusive fitness effect. Journal of Evolutionary Biology. 2007;20:2284–229. pmid:17956391
- 18. Grafen A, Archetti M. Natural selection of altruism in inelastic viscous homogeneous populations. Journal of Theoretical Biology. 2008;252(4):694–710. pmid:18371985
- 19. Lion S, Gandon S. Habitat saturation and the spatial evolutionary ecology of altruism. Journal of Evolutionary Biology. 2009;22(7):1487–1502. pmid:19519786
- 20. Lion S, Gandon S. Life history, habitat saturation and the evolution of fecundity and survival altruism. Evolution. 2010;64(6):1594–606. pmid:20050913
- 21. Taylor PD. Sex ratio in a stepping-stone population with sex-specific dispersal. Theoretical Population Biology. 1994;45(2):203–218. pmid:8016752
- 22. Comins HN, Hamilton WD, May RM. Evolutionarily stable dispersal strategies. Journal ot Theoretical Biology. 1980;82:205–230. pmid:7374178
- 23. Gandon S, Rousset F. Evolution of stepping-stone dispersal rates. Proceedings of the Royal Society of London Series B-Biological Sciences. 1999;221:2507–2513. pmid:10693822
- 24. Rousset F, Gandon S. Evolution of the distribution of dispersal distance under distance-dependent cost of dispersal. Journal of Evolutionary Biology. 2002;15:515–523.
- 25. Rousset F, Ronce O. Inclusive fitness for traits affecting metapopulation demography. Theoretical Population Biology. 2004;65:127–141. pmid:14766187
- 26. Wright S. Evolution in Mendelian populations. Genetics. 1931;16:97–159. pmid:17246615
- 27. Cadet C, Ferrière R, Metz JAJ, van Baalen M. The evolution of dispersal under demographic stochasticity. American Naturalist. 2003;162(4):427–41. pmid:14582006
- 28. Parvinen K, Dieckmann U, Gyllenberg M, Metz JAJ. Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity. Journal of Evolutionary Biology. 2003;16(1):143–53. pmid:14635889
- 29. Lehmann L, Perrin N, Rousset F. Population demography and the evolution of helping behaviors. Evolution. 2006;60:1137–1151. pmid:16892965
- 30. Wild G, Gardner A, West SA. Adaptation and the evolution of parasite virulence in a connected world. Nature. 2009;459:983–986. pmid:19474791
- 31. Nakamaru M, Matsuda H, Iwasa Y. The evolution of cooperation in a lattice-structured population. Journal of Theoretical Biology. 1997;184:65–81. pmid:9039401
- 32. van Baalen M, Rand AR. The unit of selection in viscous populations and the evolution of altruism. Journal of Theoretical Biology. 1998;193:631–648.
- 33. Le Galliard JF, Ferrière R, Dieckmann U. The adaptive dynamics of altruism in spatially heterogeneous populations. Evolution. 2003;57:1–17. pmid:12643563
- 34. Le Galliard JF, Ferrière R, Dieckmann U. Adaptive Evolution of Social Traits: Origin, Trajectories. American Naturalist. 2005;165:206–224. pmid:15729651
- 35. Nakamaru M, Iwasa Y. The evolution of altruism by costly punishment in lattice-structured populations: score-dependent viability versus score-dependent fertility. Evolutionary Ecology Research. 2005;7:853–870.
- 36. Ohtsuki H, Hauert C, Lieberman E, Nowak MA. A simple rule for the evolution of cooperation on graphs and social networks. Nature. 2006;441:502–505. pmid:16724065
- 37. Lion S, van Baalen M. From infanticide to parental care: Why spatial structure can help adults be good parents. American Naturalist. 2007;170(2):E26–E46. pmid:17874366
- 38. Peña J, Wu B, Arranz J, Traulsen A. Evolutionary Games of Multiplayer Cooperation on Graphs. PLOS Computational Biology. 2016;12(8):1–15. pmid:27513946
- 39. Lion S. Relatedness in spatially structured populations with empty sites: An approach based on spatial moment equations. Journal of Theoretical Biology. 2009;260:121–131. pmid:19505481
- 40. Ellner SP. Pair Approximation for Lattice Models with Multiple Interaction Scales. Journal of Theoretical Biology. 2001;210:435–447. pmid:11403564
- 41. Mullon C, Lehmann L. Eco-evolutionary dynamics in metacommunities: ecological inheritance, helping within species, and harming between species. American Naturalist. 2018;192(6):664–685. pmid:30444662
- 42. Mullon C, Wakano JY, Ohtsuki H. Coevolutionary dynamics of genetic traits and their long-term extended effects under non-random interactions. Journal of Theoretical Biology. 2021;525:110750. pmid:33957155
- 43. Lehmann L, Rousset F. The evolution of social discounting in hierarchically clustered populations. Molecular Ecology. 2012;21:447–471. pmid:21880087
- 44. Taylor PD, Lilicrap T, C D. Inclusive fitness analysis on mathematical groups. Evolution. 2011; p. 849–59. pmid:21044061
- 45. Jansen VAA, Lloyd AL. Local stability analysis of spatially homogeneous solutions of multi-patch systems. J Math Biol. 2000;41(3):232–252. pmid:11072757
- 46. Malécot G. Heterozygosity and relationship in regularly subdivided populations. Theoretical Population Biology. 1975;8:212–241. pmid:1198353
- 47. Sawyer S. Results for the stepping stone model for migration in population genetics. Annals of Probability. 1976;4:699–728.
- 48. Lehmann L. Space-time relatedness and Hamilton’s rule for long-lasting behaviors in viscous populations. American Naturalist. 2010;175(1):136–143. pmid:19911983
- 49. Hamilton WD. Selfish and spiteful behavior in an evolutionary model. Nature. 1970;228:1218–1220. pmid:4395095
- 50. Rousset F. Separation of time scales, fixation probabilities and convergence to evolutionarily stable states under isolation by distance. Theoretical Population Biology. 2006;69:165–179. pmid:16405936
- 51. Ohtsuki H, Nowak MA. Evolutionary games on cycles. Proceedings of the Royal Society of London Series B-Biological Sciences. 2006;273:2249–2256. pmid:16901846
- 52. Lehmann L, Rousset F. How life-history and demography promote or inhibit the evolution of helping behaviors. Philosophical Transactions of the Royal Society B. 2010;365:2599–2617. pmid:20679105
- 53. Peña J, Nöldeke G, Lehmann L. Evolutionary dynamics of collective action in spatially structured populations. Journal of Theoretical Biology. 2015;382:122–136. pmid:26151588
- 54. Alger I, Weibull JW, Lehmann L. Evolution of preferences in structured populations: Genes, guns, and culture. Journal of Economic Theory. 2020;185:1–45.
- 55. Taylor PD. Inclusive fitness in a homogeneous environment. Proceedings of the Royal Society of London Series B-Biological Sciences. 1992;240:299–302.
- 56. Taylor PD, Irwin AJ. Overlapping generations can promote altruistic behavior. Evolution. 2000;54:1135–1141. pmid:11005283
- 57.
Dawkins R. The Extended Phenotype. Oxford: Oxford University Press; 1982.
- 58. Dawkins R. Extended Phenotype—But not too extended. A reply to Laland, Turner and Jablonka. Biology and Philosophy. 2004;19:377–396.
- 59.
Maynard Smith J. Evolution and the Theory of Games. Cambridge: Cambridge University Press; 1982.
- 60.
Weibull JW. Evolutionary Game Theory. Cambridge, MA: MIT Press; 1997.
- 61. Queller DC. Genetic relatedness in viscous populations. Evolutionary Ecology. 1994;8:70–73.
- 62. Ohtsuki H, Nowak MA. Evolutionary stability on graphs. Journal of Theoretical Biology. 2008;251:698–707. pmid:18295801
- 63. Kaveh K, Komarova NL, Kohandel M. The duality of spatial death–birth and birth–death processes and limitations of the isothermal theorem. Royal Society Open Science. 2014;2.
- 64. Débarre F. Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations. Journal of Theoretical Biology. 2017;420:26–35. pmid:28254478
- 65. Lehmann L. The evolution of trans-generational altruism: kin selection meets niche construction. Journal of Evolutionary Biology. 2007;20:181–189. pmid:17210011
- 66.
Hamilton WD. Selection of selfish and altruistic behaviour in some extreme models. In: Eisenberg JF, Dillon WS, editors. Man and Beast: Comparative Social Behavior. Washington, DC: Smithsonian Institutions Press; 1971. p. 59–91.
- 67. Schaffer ME. Evolutionarily stable strategies for a finite population and a variable contest size. Journal of Theoretical Biology. 1988;132(4):469–478. pmid:3226137
- 68. Gardner A, West SA. Spite and the scale of competition. Journal of Evolutionary Biology. 2004;17:1195–1203. pmid:15525404
- 69. Lehmann L, Feldman MW, Rousset F. On the evolution of harming and recognition in finite panmictic and infinite structured populations. Evolution. 2009;63:2896–2913. pmid:19624725
- 70. West SA, Gardner A. Altruism, spite, and greenbeards. Science. 2010;327:1341–1344. pmid:20223978
- 71. Smead R, Forber P. The coevolution of recognition and social behavior. Scientifc Reports. 2016;6:25813. pmid:27225673
- 72. Bruner JP, Smead R. Tag-based spite with correlated interactions. Journal of Theoretical Biology. 2022;540:111052. pmid:35247376
- 73. Borenstein DB, Meir Y, Shaevitz JW, Wingreen NS. Non-Local Interaction via Diffusible Resource Prevents Coexistence of Cooperators and Cheaters in a Lattice Model. PLOS ONE. 2013;8:e63304. pmid:23691017
- 74. Allen B, Gore J, Nowak MA. Spatial dilemmas of diffusible public goods. eLife. 2013;2013. pmid:24347543
- 75. Scheuring I. Diffusive Public Goods and Coexistence of Cooperators and Cheaters on a 1D Lattice. PLOS ONE. 2014;9:e100769. pmid:25025985
- 76. Menon R, Korolev KS. Public good diffusion limits microbial mutualism. Physical Review Letters. 2015;114:168102. pmid:25955075
- 77. Gerlee P, Altrock PM. Persistence of cooperation in diffusive public goods games. Physical Review E. 2019;99:062412. pmid:31330651
- 78.
Case TJ. An Illustrated Guide to Theoretical Ecology. Oxford: Oxford University Press; 2000.
- 79.
Begon M, Townsend CR. Ecology: from Individuals to Ecosystems. 5th ed. West Sussex: Wiley; 2021.
- 80. Brown SP, Taylor PD. Joint evolution of multiple social traits: a kin selection analysis. Proceedings of the Royal Society B: Biological Sciences. 2010;277(1680):415–422. pmid:19828549
- 81. Leimar O. Multidimensional convergence stability. Evolutionary Ecology Research. 2009;11:191–208.
- 82.
Terras A. Fourier analysis on finite groups and applications. 43. Cambridge University Press; 1999.