## Introduction

In a heterogeneous environment, natural selection on a trait can lead to a variety of outcomes, depending on the magnitude and pattern of that heterogeneity and the spatial and temporal distribution and movement of individuals. Consider the following four possible outcomes: (1) multiple genotypes that express a different phenotype matching a particular selective optimum (genetic differentiation); (2) a single genotype that expresses a single intermediate phenotype (jack-of-all-trades); (3) a single genotype that expresses multiple phenotypes in response to environmental cues with each phenotype matching a different selective optimum (phenotypic plasticity); (4) a single genotype with offspring expressing randomly variable phenotypes some of which have high fitness (bet-hedging). (I refer to the process of producing randomly variable offspring as developmental instability and the selective outcome as bet-hedging.) These possibilities are not necessarily mutually exclusive, nor do they exhaust the named list of strategies that combine various aspects of each. For current purposes, however, they suffice to describe the range of possible evolutionary outcomes.

Of that range of possible outcomes, we would like to know which are more likely under particular circumstances. The first two (genetic differentiation and jack-of-all-trades) have been extensively explored, with models that go back to the origins of modern evolutionary theory (Hedrick et al. 1976). Models of the third and fourth (plasticity and bet-hedging) have been examined over the past three decades (Berrigan and Scheiner 2004; Starrfelt and Kokko 2012). This paper is the third in a set of three studying the intersection of phenotypic plasticity, developmental instability, environmental heterogeneity, and life-history strategy. The first paper (Scheiner 2013) examined phenotypic plasticity alone and explored the many ways that environmental variation could be created by the mode and pattern of spatial and temporal heterogeneity in combination with the mode and pattern of movement. The second paper (Scheiner 2014) did the same for developmental instability alone. Those papers set the stage for the current paper that explores the interaction of plasticity and instability. I strongly urge you to first read those papers and Scheiner and Holt (2012) to understand the context of this paper.

For the most part, phenotypic plasticity and developmental instability have been dealt with independently, both in models and empirical studies. In this paper, I explore a simulation model that allows for both outcomes, thus contrasting a deterministic genetic strategy—phenotypic plasticity—with a stochastic strategy—developmental instability. Two previous results stimulated this exploration, one theoretical and one empirical. Regarding theory, one of the unexpected results in our previous simulations (Scheiner and Holt 2012) was selection for hyperplasticity—a reaction norm much steeper than that expected by the environmental gradient. Hyperplasticity was favored when dispersal rates were high, individuals moved prior to selection, and temporal variation was high with a large negative autocorrelation. Under those conditions, there was great uncertainty and variability in the environment at the time of selection such that the optimal strategy was to produce highly variable offspring through plasticity. In effect, plasticity was acting as a form of het-hedging. This result raised the question about what conditions would select for the more classic form of bet-hedging, the production of randomly variable offspring, and how such a strategy might interact with or jointly evolve with plasticity.

The empirical observation was our demonstration that plasticity was positively genetically correlated with developmental instability in *Arabidopsis* (Tonsor et al. 2013). Although not the first such demonstration, it is the most detailed and involves the most traits. DeWitt et al. (1998) proposed that such a linkage could be a limitation on the evolution of plasticity. My model assumes that the same loci determine the amount of plasticity and the amount of instability, linkage by pleiotropy. Even in the absence of pleiotropy, the evolution of instability and plasticity might affect each other. Such interactions hitherto have been unexplored.

In this paper, I ask three questions. (1) Under what conditions is plasticity or instability more likely to evolve when both are possible outcomes? (2) How do the evolution of bet-hedging by developmental instability and phenotypic plasticity interact and trade-off with each other? (3) To what extent does developmental instability act as a limitation on the evolution of phenotypic plasticity when the two are pleiotropically linked?

### Patterns of environmental heterogeneity

The focus of my explorations has been on the myriad ways that environmental variation can combine with organismal biology to produce heterogeneity as perceived by an individual or experienced by a lineage. This myriad can be complex, resulting in a large parameter space to explore. The environment can vary in time and in space, alone or together. It can be uncorrelated or correlated in time and/or in space. Temporal variation can occur within the lifetime of an individual, or across generations. The biology of the organism creates two critical time periods, development (when the phenotype is determined) and selection, and individuals can move before or after selection occurs. That movement can be local (to adjacent demes only), or can be global (to any deme).

Let us deconstruct that complexity. With respect to the evolution of phenotypic plasticity, all variation must be considered relative to when the phenotype is determined, movement occurs, and selection happens (Fig. 1). In my model, phenotypic determination is treated as a developmental stage such that the phenotype is fixed at some point in the life history. Assuming that an organism's phenotype is fixed following development, we can define two scales of temporal variation: (1) Among-generation change in the environment can happen at the time of development or at the time of selection, or both. (2) Within-generation change can happen between the time of development and the time of selection. Similarly, if an organism moves once during its life and selection occurs at a single instance, we can define two life-history patterns: selection then movement, and movement then selection. These two patterns, along with the presence or absence of spatial heterogeneity, define the types of environmental heterogeneity that are relevant to the evolution of phenotypic plasticity. The types relevant to the evolution of developmental instability are a subset of those because the environment plays no role in development; thus, the only environmental change that matters is variation at the time of selection.

I consider three combinations of environmental heterogeneity: (1) temporal variation alone; (2) spatial variation alone; and (3) both temporal and spatial variation. When temporal variation occurs alone, it happens in a single deme once per generation before development and can be independent from one generation to the next or it can be autocorrelated. When spatial variation occurs, it is among demes along a linear gradient producing a spatial autocorrelation. That spatial heterogeneity can be overlaid by temporal variation within demes such that the spatial pattern is a central tendency, that is, present but varying in magnitude and pattern. I consider four broad patterns that combine spatial and temporal variation (Fig. 1).

In the first pattern, the environment of development is fixed among generations, while the environment of selection varies among generations. In the second pattern, the environment varies among generations at the time of development, while the environment of selection is fixed. In the third pattern, the environment changes both at the time of development and at time of selection and those changes are not correlated. In the fourth pattern, environmental change at the time of development carries over to the environment at selection. For each of those four patterns, the temporal variation could be synchronized among demes or occur independently in each deme.

Finally, I consider two different movement patterns: stepping-stone migration and island migration. For stepping-stone migration, movement occurs among nearby demes with the probability of movement decreasing with distance. For island migration, movement occurs among all demes with equal probability, although more complex movement rules are possible. See Scheiner (2013) for a detailed discussion of these patterns and possible ecological scenarios that they represent.

### Phenotypic plasticity and developmental instability alone

My previous explorations of the evolution of plasticity and instability when each was evolving alone found the following. For phenotypic plasticity, among-generation temporal heterogeneity favored plasticity, while within-generation heterogeneity, especially at the time of development, resulted in cue unreliability disfavoring plasticity. In general, spatial variation more strongly favored plasticity than temporal variation, and island migration more strongly favored plasticity than stepping-stone migration.

The pattern of evolutionary response can be quite complex when individuals move in space after development but before selection. Negative correlations among environments between the time of development and selection resulted in seemingly maladaptive reaction norms. When movement occurred before selection, the dispersal rate was high, and temporal variation was large, selection favored hyperplasticity—reaction norms much greater than the fitness optimum. This hyperplasticity was favored because it increases the phenotypic range of a lineage, acting as a form of bet-hedging.

For developmental instability, both temporal and spatial heterogeneity selected for instability. For spatial heterogeneity, the response to selection depended on the life-history strategy and the form and pattern of dispersal with the greatest response for island migration when selection occurred before dispersal. Combining spatial and temporal heterogeneity resulted in substantially more instability than either alone. Positive correlations among generations reduced that response, while synchronizing that temporal change among demes more strongly favored instability but eliminated the synergy between temporal and spatial heterogeneity.

Thus, patterns of environmental heterogeneity can be complex and can interact in unforeseen ways. Temporal and spatial variation do not combine additively but depend on the life-history pattern. For both plasticity and instability, I found higher-order interactions between life-history patterns, dispersal rates, dispersal patterns, and environmental heterogeneity.

### Model structure

My model is an individual-based simulation implemented in Fortran 77 (the computer code is available from Dryad, doi:10.5061/dryad.9 bp88). A summary of parameters is given in Table 1.

Fixed parameters |

Number of nonplastic, plastic, and developmental instability loci = 5 each |

Steepness of the gradient (change in optimum in adjacent demes) = 0.4 units |

Strength of selection within demes (σ) = 2 units |

Mutation rate = 10%/allele/generation |

Mutational effect (standard deviation) = 0.1 units |

Number of generations = 10,000 |

Parameters explored |

Length of the environmental gradient: 1 or 50 demes |

Population size: 1000 or 100 individuals/deme |

Life-history pattern: selection before dispersal versus dispersal before selection |

Timing of environmental change: at the time of development versus selection |

Dispersal pattern: stepping-stone or island |

Dispersal rate (5–84%) |

Magnitude of environmental change (0–50% of the length of the spatial gradient) |

Correlation of environmental change among generations (−0.75–0.75) |

Strength of the pleiotropic effect of plasticity on developmental instability (0–4) |

The genotype of an individual consisted of up to three types of loci: (1) genes with deterministic expression that was independent of the environment (nonplastic loci); (2) genes with deterministic expression that was dependent of the environment (plastic loci); and (3) genes that allowed for random deviation from the deterministic phenotype (instability loci). In some versions of the model, plasticity and developmental instability were determined by the same loci. This combination of loci types allows for all four possible evolutionary outcomes: genetic differentiation, jack-of-all-trades, plasticity, and bet-hedging. For example, adaptation by genetic differentiation would occur when the allelic values of the plastic and instability loci go to zero (i.e., are not expressed). Conversely, adaptation by phenotypic plasticity would occur when the allelic values of the nonplastic determinstic loci and the instability loci go to zero. Intermediate outcomes were also possible in which individuals express the optimal phenotype in a particular environment through nonzero values of any combination of the nonplastic, plastic, and instability loci.

For temporal-only heterogeneity, the metapopulation consisted of a single deme. For spatial heterogeneity, the metapopulation consisted of a linear array of 50 demes. An environmental gradient was created by varying the optimal value of a single trait (phenotype) in a linear fashion along the array from −9.8 to +9.8 arbitrary units at the ends of the gradient, that is, the optimal phenotype in adjacent demes differed by 0.4 units. An individual's phenotype (trait value) was determined by 15 diploid loci: five nonplastic loci, five plastic loci, and five instability loci. The deterministic loci contributed additively to the trait. Allelic values at the plastic loci were multiplied by an environment-dependent quantity before being summed. The effect of the environment (*E*_{i} for deme *i*) on the phenotypic contribution of each unit plastic allelic value varied in a linear fashion, with a slope of 0.04 units [*E*_{i} = 0.04(*i* − 25.5)]. The phenotype of each individual was determined at the time of development as

where *T*_{ij} is the phenotype of the *j*th individual that develops in the *i*th deme, *N*_{ijk} is the allelic value of the *k*th nonplastic allele of that individual, *P*_{ijk} is the allelic value of the *k*th plastic allele, and *R*_{ij} is the developmental instability of that individual. For a given genotype, Σ*N*_{ijk} can also be thought of as the intercept of its reaction norm, or the phenotype of the individual in the absence of plasticity, and the slope of [*E*_{i}Σ*P*_{ijk}] calculated across demes can be thought of as the slope of its reaction norm. For the parameters used here, the optimal reaction norm had a slope of 10.

For developmental instability that was not due to the pleiotropic effects of plasticity, *R*_{ij} was determined by a Gaussian normal deviate with a standard deviation equal to ∑_{k = 1,10}*D*_{ijk}, where *D*_{ijk} is the allelic value of the *k*th instability allele. For instability that was due to pleiotropy, *R*_{ij} = *S*|∑ _{k=1,10}*P*_{ijk}|, where *S* is a Gaussian normal deviate with a standard deviation equal to the scaled effect of plasticity multiplied by the absolute value of the genotypic value of plasticity. Individuals with equal amounts of plasticity but opposite slopes have the same effect on developmental instability.

Life-history events occurred in one of the following two sequences: (1) birth, followed by development (i.e., the phase in the life cycle when the phenotype is determined), then dispersal, selection, and reproduction (denoted as “move first”); or alternatively; (2) birth, development, selection, dispersal, and then reproduction (denote as “select first”). Selection was based on survival with the probability of surviving being a Gaussian function of the difference between an individual's phenotype and the locally optimal phenotype. Fitness (the probability of surviving) was determined as follows:

where *W*_{ijt} is the fitness of the *j*th individual in the *i*th deme in generation *t*,* T*_{ijt} is the phenotype of that individual, *θ*_{it} is the optimal phenotype in that deme, and σ is the strength of selection (selection weakens as σ increases). For all simulations, σ = 2; the length of the spatial gradient across all demes was approximately five times the width of the within-deme selection function.

Temporal variation occurred at one or both of two life-history stages: at the time of development or at the time of selection. This variation could occur once per generation at one of the two stages with the environment remaining fixed at the other stage or could occur at both stages. If the variation occurred at both stages, those changes could be independent or they could carry through both stages. Finally, the changes could be independent among demes, or be synchronized among demes. In the latter case, the optimal phenotype in all demes changed by the same magnitude and direction. That variation could be uncorrelated from one generation to the next or be correlated among generations. If variation was independent among demes, each deme had its own pattern of temporal autocorrelation.

Temporal autocorrelation was simulated using the recursion:

where *θ*_{it} is the environment at either development or selection in the *i*th deme in generation *t*,* E*_{i} is the mean or fixed environment in the *i*th deme (a linear function of *i*), τ is the standard deviation of environmental variation, ρ is the temporal autocorrelation coefficient, and *z*_{it} is a sequence of independent zero-mean, unit-variance Gaussian random deviates. For simulations without temporal variation, *τ* = 0, and for uncorrelated temporal variation, *ρ* = 0. The standard deviation of environmental noise (*τ*) is shown as a percentage of the difference in the optima at the two ends of the gradient. The autocorrelation (*ρ*) varied from −75% to 75%.

Dispersal occurred in one of the two patterns: stepping-stone or island. For the stepping-stone migration pattern, the dispersal probability and the distance moved were determined using a zero-mean Gaussian random number, so that the probability of moving and the average distance moved were correlated (see Fig. 1 of Scheiner and Holt 2012). Increasing the dispersal probability was performed by increasing the variance of the Gaussian so that both more individuals were likely to move, and they were likely to move farther. Individuals that would otherwise disperse beyond the end of the gradient moved to the terminal demes. For the island migration pattern, each individual had a fixed probability of moving. If it moved, it had an equal probability of moving to any of the other demes. For both patterns, dispersal per se had no cost; survival during dispersal was 100%. The dispersal probabilities were the same for the two life-history strategies, *move first* and *select first*; however, the absolute number of individuals dispersing was fewer for the *select first* life-history strategy because of reductions in deme sizes due to selection.

Reproduction was accomplished by assembling pairs of individuals within a deme at random with replacement (allowing for self-fertilization), with each pair producing 1 offspring, then repeating until the carrying capacity of that deme was reached. This procedure assumes soft selection, in that local population size was determined independently of the outcome of selection, and results in individuals competing for offspring. The model assumes that the spatial scale of reproduction and mating matches that of density dependence and the grain of the selective environment.

Each simulation was initialized with 100 individuals being born in each deme, or 1000 individuals for simulations with just temporal variation in a single deme. Those deme and total population sizes (5000 individuals (50 × 100) and 1000 individuals, respectively) were chosen to minimize genetic drift and the effects of population size on among-individual fitness variance while making the simulations computationally tractable. For each individual in the initial generation, allelic values for both plastic and nonplastic loci were chosen independently from the values −2, −1, 0, 1 and 2, with each value being equally likely. (Even though initial values are discrete, due to mutation allelic values are continuous variables after the initial generation; see below.) When new offspring were generated, each allele mutated with a probability of 10%. [Lower mutation rates mainly changed the time-scale over which evolution occurs, rather than the eventual outcome (Scheiner and Holt 2012).] When a mutation occurred, the allelic value was changed by adding a Gaussian deviate (mean of zero and a standard deviation of 0.1 units) to the previous allelic value (i.e., this is an infinite-alleles model). For the developmental instability loci, all alleles began with a value of zero. The probability of mutation and the standard deviation of the Gaussian deviate were the same as for the other loci, but only positive allelic values were retained, negative values were set to zero.

All simulations were run for 10,000 generations to ensure that the equilibrium point (the point after which all calculated quantities showed no further directional trend) was reached. Each parameter combination was replicated 20 times, and the results shown are the means of those replicates. Coefficients of variation of reported parameters were generally low (1–5%). If the metapopulation went extinct, additional realizations were run until 20 successful replications were achieved; for some parameter combinations, the extinction probability was 100% (i.e., no successful replications in 60 runs). Reported outcomes were averaged over successful replications only.

The reaction norm describes how the phenotypic expression of a given genotype varies among environments. The plasticity of a linear reaction norm is best described by its slope. In this model, the slope of the reaction norm is the product of the slope of *E*_{i} across demes and the sum of the values of the plasticity alleles (i.e., the right-hand sum in eq. (1)). For these simulations, as the slope of *E*_{i} was constant, and the final outcome was measured as the average across all demes of the sum of the values of the plasticity alleles for each individual. That is, , where is the mean plasticity of the *i*th deme over all *r* runs, *N* = 100 is the number of individuals per deme, and *P*_{ijn} is the sum of the values of the plasticity alleles of the *j*th individual developing in the *i*th deme in the *n*th run. The overall mean plasticity is the average of across demes and is given by , where *D* is the number of demes. (The order of averaging, over runs within demes first or over demes within runs first, does not affect the final average, because the number of demes is the same for all runs.) The average plasticity was standardized as relative plasticity to the optimal reaction norm (the slope of *E*_{i}) so that a pure plasticity outcome would have a value of 1 and a pure differentiation outcome would have a value of 0. Values outside this range were possible; that is, it was possible to achieve a reaction norm with a slope steeper than the optimal value (>1) or in a direction opposite from the optimal value (<0).