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. 2018 Nov 14;15(148):20180371.
doi: 10.1098/rsif.2018.0371.

Variability in life-history switch points across and within populations explained by Adaptive Dynamics

Pietro Landi  1 , James R Vonesh  2   3 , Cang Hui  4   5
Affiliations

Variability in life-history switch points across and within populations explained by Adaptive Dynamics

Pietro Landi et al. J R Soc Interface. .

Abstract

Understanding the factors that shape the timing of life-history switch points (SPs; e.g. hatching, metamorphosis and maturation) is a fundamental question in evolutionary ecology. Previous studies examining this question from a fitness optimization perspective have advanced our understanding of why the timing of life-history transitions may vary across populations and environments. However, in nature we also often observe variability among individuals within populations. Optimization theory, which typically predicts a single optimal SP under physiological and environmental constraints for a given environment, cannot explain this variability. Here, we re-examine the evolution of a single life-history SP between juvenile and adult stages from an Adaptive Dynamics (AD) perspective, which explicitly considers the feedback between the dynamics of population and the evolution of life-history strategy. The AD model, although simple in structure, exhibits a diverse range of evolutionary scenarios depending upon demographic and environmental conditions, including the loss of the juvenile stage, a single optimal SP, alternative optimal SPs depending on the initial phenotype, and sympatric coexistence of two SP phenotypes under disruptive selection. Such predictions are consistent with previous optimization approaches in predicting life-history SP variability across environments and between populations, and in addition they also explain within-population variability by sympatric disruptive selection. Thus, our model can be used as a theoretical tool for understanding life-history variability across environments and, especially, within species in the same environment.

Keywords: bistability; disruptive selection; evolution; evolutionary branching; phenotypic variability; time of maturation.

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.
Figure 1.
Schematic of the life-history model. Thick lines: pre-maturation stages (before the time of maturation τ). Normal lines: post-maturation stages (after τ). Dotted line: a portion of overwintering eggs survive and hatch into juveniles in the next year (generation). Juveniles J(t) grow their body size w and survive density-dependent competition with probability σJ(J, τ) until maturation at time τ, when they become A(t) adults. Adults survive until the end of the season with density-dependent probability σA(A, τ) when they lay E(t) eggs with fertility f(w(τ)) proportional to the body size attained at maturation. Eggs survive to the next year (impossible for adults) and hatch into new J(t + 1) juveniles with probability σE(τ) proportional to the duration of the post-maturation phase.
Figure 2.
Figure 2.
Trait-dependent parameters. (a) Size at maturation w(τ) for different values of juvenile initial size w0 (=1.5, dashed; = 1.75, solid lines) and intrinsic growth exponent r (= 1, 5 and 10 from lower to upper curve). (b) Density-dependent juvenile competition survival σJ(J, τ) for different values of competition coefficient cJ (= 1, 2, 5 from upper to lower curve). Similarly for adult competition σA(A, 1 − τ) and cA. (c) Egg survival σE(τ) for different values of egg investment risk exponent ε (= 0.5, 1, and 2 from upper and convex to lower and concave curve). Other parameters: wK = 2 in (a).
Figure 3.
Figure 3.
Pairwise invasibility plots of the innovative trait τ′ in the environment set by the resident trait τ. Grey regions: regions of invasibility, characterized by positive fitness (R0(τ, τ′) > 1). White regions: regions of non-invasibility, characterized by negative fitness (R0(τ, τ′) < 1). Curves: unit-fitness contour (R0(τ, τ′) = 1). The intersections of the unit-fitness curves and the diagonal (marked by dots) are the singular strategies of the trait dynamics, the sign of the fitness in their neighbourhood determines their convergence stability, and the slope of the unit-fitness curve determines their evolutionary stability. (a) One convergence stable ESS (filled dot). (b) One branching point (half-filled dot). (c) One branching point (half-filled dot), one convergence unstable singular strategy (empty dot). (d) One convergence stable boundary strategy. Parameter values: f0σJσAσE = cJ = ε = 1, cA = 0.1, wK = 2, r = 10, and w0 = 0.5 in (a), 1.25 in (b), 1.5 in (c) and 1.75 in (d). Notice that only the product of fertility and survival coefficients matters.
Figure 4.
Figure 4.
Scenarios for time of maturation under different juvenile initial sizes. (a) When the juvenile initial size w0 is small, selection drives the time of maturation towards the internal evolutionary stable strategy (ESS) formula image compromising between growth in the juvenile phase and returns of egg investment in the adult phase (orange interval). When w0 is intermediate selection drives to the branching point turning the population dimorphic (dark green interval, within-population variability) or either to the branching point or to the boundary strategy formula image (dimorphic bistability or conditional disruptive selection region, light green interval of across- and within-population variability). When w0 is high enough, selection drives to direct development (no juvenile phase, red interval). Solid line: convergence stable equilibria (attracting strategies). Dashed line: convergence unstable equilibria (repelling strategies). Thick line: convergence stable but evolutionary unstable equilibria (branching points). TC: transcritical bifurcation. BR: branching bifurcation (B = 0 in (2.6)), turning selection disruptive. SN: saddle-node bifurcation. Parameters as in figure 3. Note that the vertical dotted trajectories represent the monomorphic trajectories of trait evolution under directional selection on the diagonal of figure 3, with arrows pointing in the direction of natural selection. (b) Evolutionary branching of the time of maturation. All initial conditions converge to the branching point (dotted line), where selection becomes disruptive and the two traits diverge and reach a dimorphic equilibrium. Parameters as in figure 3b. (c) Evolutionary branching of the time of maturation. Initial conditions above the convergence unstable equilibrium (dashed line) converge to the branching point (dotted line), where selection becomes disruptive and the two traits diverge and reach a dimorphic equilibrium. Parameters as in figure 3c. The magnified view shows invasibility, i.e. initial exponential rate of divergence shown by the arrow and quantified by B in equation (2.6). B = 2.7 in (b) while B = 9.8 in (c).
Figure 5.
Figure 5.
Changes in the species maturation variability scenarios with respect to juvenile growth, juvenile competition and egg investment returns. Black region: unfeasible parameters (formula image for all values of w0). Grey region: population extinction (formula image). Red region: no juvenile period (direct development). Orange region: monomorphic intermediate maturation strategy. Yellow region: evolutionary bistability between either direct development or intermediate maturation strategy depending on the initial time of maturation (across-population variability, monomorphic populations). Dark green region: (unconditional) disruptive selection (within-population variability). Light green region: conditional disruptive selection or dimorphic bistability (overlap of bistability and disruptive selection), i.e. evolutionary bistability between either direct development or a dimorphic population (across- and within-population variability). (a) Juvenile growth increases the bistability region, while it first increases and then decreases the disruptive selection region. (b) Juvenile competition increases the bistability region but decreases the disruptive selection region. (c) Egg investment returns (inversely related to the egg investment risk exponent ε) increases the bistability region, while first increases and then decreases the disruptive selection region. BR: branching, TC: transcritical, and SN: saddle-node bifurcation curves in the bivariate space (w0, p), with p = r in (a), p = cJ in (b), and p = ε in (c). Dotted line: parameter value as in figure 4a.
Figure 6.
Figure 6.
Hyperbolic tangent, tanh(α(τ′ − τ)).
See this image and copyright information in PMC

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