Table 1.
Key general symbols.
Fig 1.
A model for environmentally mediated social interactions in space and time.
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.
Fig 2.
Dispersal distribution, relatedness, and scaled relatedness in a 1D lattice model under short and long-range dispersal.
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.
Fig 3.
Dispersal distribution, relatedness, and scaled relatedness in a 2D lattice model under short and long-range dispersal.
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.
Fig 4.
Extended phenotypic effects in a 1D lattice model under short and long-range movement of the commons.
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.
Fig 5.
Selection favours altruism or spite depending on dispersal of the evolving species and the commons in a 2D lattice model.
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.
Fig 6.
Isolation by distance allows for the evolution of a wider range of social behaviours than the island model under environmental feedback.
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.
Fig 7.
Scaled relatedness and selection under survival effects in a 1D lattice model.
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.