### Abstract

- Top of page
- Abstract
- Introduction
- Model Structure
- Results
- Discussion
- Conclusion
- Acknowledgments
- Conflict of Interest
- References
- Appendix
- Supporting Information

To understand empirical patterns of phenotypic plasticity, we need to explore the complexities of environmental heterogeneity and how it interacts with cue reliability. I consider both temporal and spatial variation separately and in combination, the timing of temporal variation relative to development, the timing of movement relative to selection, and two different patterns of movement: stepping-stone and island. Among-generation temporal heterogeneity favors plasticity, while within-generation heterogeneity can result in cue unreliability. In general, spatial variation more strongly favors plasticity than temporal variation, and island migration more strongly favors plasticity than stepping-stone migration. Negative correlations among environments between the time of development and selection can result in seemingly maladaptive reaction norms. The effects of higher dispersal rates depend on the life history stage when dispersal occurs and the pattern of environmental heterogeneity. Thus, patterns of environmental heterogeneity can be complex and can interact in unforeseen ways to affect cue reliability. Proper interpretation of patterns of trait plasticity requires consideration of the ecology and biology of the organism. More information on actual cue reliability and the ecological and developmental context of trait plasticity is needed.

### Introduction

- Top of page
- Abstract
- Introduction
- Model Structure
- Results
- Discussion
- Conclusion
- Acknowledgments
- Conflict of Interest
- References
- Appendix
- Supporting Information

We are faced with a paradox. Simple logic tells us that if organisms could always express the phenotype that corresponded to maximum fitness, such individuals should be favored by natural selection. Yet in most instances, instead of such ubiquitous phenotypic plasticity, most species differentiate into individuals with a range of fixed phenotypes (Hereford 2009). This paper is the latest in a recent series (Scheiner and Holt 2012; Scheiner et al. 2012) aimed at resolving that paradox by systematically exploring the propositions of the theory of the evolution of phenotypic plasticity (Appendix). [I urge reading those papers prior to this current missive as the results presented here build on those.] One way to categorize the factors that limit optimal plasticity are those that are external to the organism – patterns of environmental variability perceived by an organism due either to changes in its environment or to its movement – and those that are internal to an organism – costs of plasticity and developmental limitations. This paper addresses external factors. The purpose of this paper is not to provide definitive answers about when plasticity is and is not favored by selection. As you will see, the answer is complex. Rather, the purpose is to explore those complexities and make obvious the range of information that is necessary if empirical patterns are to be properly understood. The model explored here is meant to be general; as such it probes the boundaries of parameter space rather than attempting to provide precise predictions of a specific system.

Selection for phenotypic plasticity requires environmental heterogeneity (theory proposition 1). However, that simple statement belies much complexity, especially with regard to how different patterns of environmental heterogeneity affect cue reliability (proposition 4). The goal of this paper is to unpack that first proposition and examine many different ways that the environment can vary in space and time, to explore how that variability interacts with cue reliability, and how those different patterns of variation and interaction can favor or disfavor adaptation by phenotypic plasticity.

This exploration of the effect of environmental heterogeneity on the evolution of plasticity will sharpen our tests of the theory of the evolution of plasticity. Unfortunately, because of logistical difficulties and complexities, no strong tests of this theory have been performed, by which I mean precise quantitative predictions derived from appropriately designed models. At best, we have comparisons of empirical results with general, qualitative predictions (DeWitt and Scheiner 2004). By better defining types of environmental heterogeneity and exploring the resulting evolutionary outcomes, we can build more precise models and better link theory and data.

#### Patterns of environmental heterogeneity

The environment can vary in time and/or space. That statement is both simple and obvious, yet hides much complexity. With respect to the evolution of plasticity, all variation must be considered relative to when the phenotype is determined, movement occurs, and selection happens. The complexities considered in this paper go well beyond previous models. Adding such complexities is more than an intellectual exercise as I show how these factors can interact in unexpected ways. Empirical studies very frequently fail to adequately describe their environmental context. Thus, an important task of this paper is to provide an overall framework for describing environmental heterogeneity and then to explore a substantial portion of that parameter space. Any given study will occupy a small portion of that space. However, with my framework and model outcomes, that study can be compared with others, and differences and similarities in patterns of plasticity can be understood.

In the model explored here, phenotypic determination is treated as a developmental stage such that the phenotype is fixed at some point in the life history. For some versions of my model, selection occurs immediately after development. Such a scenario can be conceived as selection on a labile trait in which selection occurs once during the organism's life. My model does not examine selection on continuously labile traits in which fitness is summed over multiple trait values. For such traits, as the rate of phenotypic change approaches instantaneous, the constraints on trait evolution become confined to factors that are internal to the organism (e.g., physiological system limitations). Consideration of such factors is outside the scope of this paper.

Assuming that an organism's phenotype is fixed following development, we can define two scales of temporal variation: among-generation and within-generation with change in the environment happening before or after development. 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.

Temporal heterogeneity only, by definition, means that the population exists in a single, uniform deme with no movement among demes. For this scenario, there are two possible patterns of variation: change once per generation before development and change both before and after development but before selection. Those changes may be independent or be correlated. An example of such a pattern of change is the timing of snow melt in the spring determining the environment prior to development and late-summer rainfall determining the environment of selection. Change occurring after development only is not relevant because there must be environmental heterogeneity that affects phenotypic determination (theory proposition 1).

Expanding our consideration from a single deme to multiple demes now allows for the addition of spatial heterogeneity. My model has a fixed, underlying spatial pattern of heterogeneity. That heterogeneity is overlaid by temporal variation within demes such that the spatial pattern is a central tendency, that is, present but varying in magnitude and pattern.

We can define two patterns of spatial variation that represent the ends of a spectrum: gradient and mosaic. In the simplest case, a gradient is a one-dimensional, monotonic change such that distance in space is equivalent to difference in the environment. We can conceive of more complex patterns, such as two-dimensional gradients – or even a three-dimensional gradient in an aquatic environment – and nonmonotonic patterns of change. Or there may be some degree of spatial autocorrelation among demes resulting in an irregular pattern of similarity with distance. At the other extreme, a mosaic pattern of heterogeneity assumes a spatial autocorrelation of zero. My model examines the simplest case of a one-dimensional, linear gradient. In the real world, there is likely to be some amount of spatial autocorrelation in the environment (Urban 2011) with a linear gradient representing the central tendency of that correlation.

These patterns of spatial variation are mirrored by patterns of movement. Evolutionary models typically consider two patterns: stepping-stone migration and island migration. For stepping-stone migration, movement occurs between adjacent demes only or perhaps more distant demes with the probability of movement decreasing with distance. In contrast, for island migration, movement occurs among all demes with equal probability, although more complex movement rules are possible. Thus, from the perspective of an organism, an island migration pattern is equivalent to a mosaic spatial pattern. My model primarily explores the effects of a stepping-stone migration pattern, with some consideration of the island migration pattern.

Discussions of the effects of environmental heterogeneity on the evolution of plasticity often assume an equivalence between temporal variation after development and movement after development. As will be seen in this paper, variation in time does not lead to the same outcomes as variation in space, and their interaction can lead to complex patterns of evolutionary response.

The types and causes of environmental heterogeneity just described can be combined in a variety of ways. In this paper, I ignore subtleties and complexities of patterns of heterogeneity such that the various combinations represent the boundaries of possibilities. I consider three broad patterns that combine spatial and temporal variation (Table 1). In all cases, the environment varies in space along a gradient.

Table 1. Temporal patterns of environmental variation used in various simulations, including the life history stage at which that variation occurred and whether that variation was correlated within a single generation. Across space, the temporal variation could occur independently in each deme, or could be synchronized across all demes.Pattern | Life history stage | Correlation |
---|

Before development | Before selection |
---|

1 | Fixed | Variable | 0 |

2 | Variable | Fixed | 0 |

3 | Variable | Variable | 0–1 |

In the first pattern, the environment of development is fixed among generations, while the environment of selection varies among generations. An example of such a pattern would be plasticity in leaf traits where soil nutrient content determines leaf thickness, and fitness is later determined by temperature and precipitation.

In the second pattern, the environment varies among generations prior to development, while the environment of selection is fixed. An example would be plasticity in the adult size of dragonflies. The larval pond environment determines size at metamorphosis, which might vary from year to year in food availability; size then determines adult survival.

In the third pattern, the environment changes both prior to development and prior to selection. Those changes may or may not be correlated. For example, adult survival of dragonflies might depend on the interaction of size and temperature, with that temperature variation being independent of the conditions that determine larval food availability. In a contrasting example, leaf thickness might be determined by early-summer temperature and precipitation, and fitness determined by late-summer temperature and precipitation, with those climatic variables correlated with each other.

In the most extreme case, environmental change prior to development would carry over to the environment at selection, a correlation of one. For example, in the water flea, *Daphnia*, in the presence of predators, some species develop extended morphologies – head and tail elongations – that decrease predation (e.g., Krueger and Dodson 1981). *Daphnia* have generation times on the order of weeks, so that predator densities may vary among generations but be relatively constant within a generation.

Temporal variation might be synchronized in space. For example, mean temperature decreases with increasing elevation. In a warmer than average summer, an entire mountainside is likely to experience higher temperatures while maintaining that elevational gradient.

#### Previous models of plasticity evolution

The comprehensive exploration of environmental heterogeneity presented in this paper has not been carried out by previous models (Berrigan and Scheiner 2004). All previous models examined temporal and spatial variation separately and as I will show, combining temporal and spatial variation leads to outcomes that are not predictable by either alone. Except for the recent papers in this series (Scheiner and Holt 2012; Scheiner et al. 2012), no model examined the interaction of spatial and temporal variation. Regarding spatial variation, nearly all models considered either only two demes or a spatial mosaic. In models that considered temporal variation, nearly all models had change occurring just once a generation. Most models assumed no change in the environment between when the phenotype is determined and when selection occurs, if they even made explicit development as a separate life history stage. With regard to patterns of movement, nearly all models assumed one of two patterns: a propagule pool that dispersed equally among all demes, or an island migration pattern with dispersal rates less than 100%. Often these movement patterns were only implied.

There are a few notable exceptions to those generalities. The models of de Jong and collaborators (De Jong 1999; Sasaki and De Jong 1999; De Jong and Behera 2002) and Gomulkeiwicz and Kirkpatrick (Gomulkiewicz and Kirkpatrick 1992) included environmental change after development. The models of Chevin and Lande (2011) and Scheiner (1998) assumed a gradient with stepping-stone migration. However, each of those models examined just a single aspect of environmental heterogeneity. As will be seen in this paper, the typical assumptions in models of plasticity evolution – among-generation temporal variation, few demes, an island migration pattern – are those that maximize selection for plasticity. More complex models are needed if we are to understand why adaptive plasticity is less common than we would expect.

### Model Structure

- Top of page
- Abstract
- Introduction
- Model Structure
- Results
- Discussion
- Conclusion
- Acknowledgments
- Conflict of Interest
- References
- Appendix
- Supporting Information

The model was an individual-based simulation (summary of parameters in Table 2) using a gene-based model of adaptation to an environmental gradient. It assumed that in the absence of temporal variation, the optimal phenotype changes in a linear fashion along that gradient, and the phenotypes of individuals can be expressed by a linear reaction norm. Gene expression is either responsive to the environment (plastic loci) or not (nonplastic loci). Adaptation can occur by two routes: genetic differentiation in which the allelic values of the plastic loci go to zero (i.e., are not expressed) or phenotypic plasticity in which the allelic values of the nonplastic loci go to zero. Because the optimal phenotype changes in a linear fashion along the gradient, and the environmental responsiveness of the plastic loci is linear, the plasticity optimum (where the realized trait value in each habitat is at the local optimum) is a possible outcome. Although presented as a dichotomy, intermediate outcomes are possible in which individuals express the optimal phenotype in a particular environment through nonzero values of both the plastic and nonplastic loci.

Table 2. Summary of the model parameters.Fixed parameters |

Number of nonplastic and plastic loci = 5 each |

Length of the environmental gradient = 50 demes |

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

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

Population size = 100 individuals/deme |

Number of generations = 10,000 |

Parameters explored |

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

Timing of environmental change: before versus after development |

Magnitude of environmental change |

Correlation of environmental change within and among generations |

Migration pattern: stepping-stone versus island |

Dispersal rate |

The model was implemented in Fortran 77 (the computer code is available from Dryad). 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 10 diploid loci: five plastic loci and five nonplastic loci. The loci contributed additively to the trait. Allelic values at the plastic loci were multiplied by an environment-dependent quantity before summing all allelic values. The effect of the environment in a particular deme (*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:

- (1)

where *T*_{ij} is the phenotype of the *j*th individual that develops in the *i*th environment (deme), *N*_{ijk} is the allelic value of the *k*th nonplastic allele of that individual, and *P*_{ijk} is the allelic value of the *k*th plastic allele. There was no random component of phenotypic variation. For a given genotype, Σ*N*_{ijk} can also be thought of as the intercept of its reaction norm at the midpoint of the gradient, or the phenotype of the individual in the absence of plasticity, and [slope(*E*_{i})Σ*P*_{ijk}] can be thought of as the slope of its reaction norm.

Life history events occurred in one of 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:

- (2)

where *W*_{ij} is the fitness of the *j*th individual undergoing selection in the *i*th environment, *T*_{ij} is the phenotype of that individual, *θ*_{i} is the optimal phenotype in that environment, σ is the strength of selection (selection weakens as σ increases). Survival was determined by choosing a random number from a uniform distribution [0,1], and the individual died if its fitness was less than that value.

Temporal variation occurred at one or both of two life history stages: after reproduction but before development or after development before selection (Table 1). Depending on the simulation, this variation occurred once per generation at one of the two stages with the environment remaining fixed at the other stage, or occurred at both stages. If the variation occurred at both stages, those changes could be independent or they could be correlated, including a correlation of 100% (i.e., a single change that carried through the entire life cycle). Finally, the changes could be independent among demes (most simulations), or be synchronized among demes. Previous papers (Scheiner and Holt 2012; Scheiner et al. 2012) explored pattern 1 only.

Temporal autocorrelation was simulated using the recursion:

- (3)

where *θ*_{it} is the environment at either development or selection in the *i*th deme in generation *t*,* O*_{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 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 (Scheiner and Holt 2012). Increasing the dispersal probability was carried out 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 migrate beyond the end of the gradient migrated 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%.

Reproduction occurred following viability selection and was accomplished by assembling pairs of individuals within a deme at random with replacement, with each pair producing 1 offspring, then repeating until the carrying capacity of that deme was reached (100 individuals per deme). This procedure assumes soft selection in that local population size was determined independently of the outcome of selection. It also 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. 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. 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).

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 (Scheiner and Holt 2012). 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 (5–20%). If the metapopulation went extinct, additional realizations were run until 20 successful replications were achieved; for some parameter combinations (see results), the extinction probability was 100% (i.e., no successful replications in 60 runs). Reported outcomes were averaged over successful replications only.

The reaction norm is a mathematical function describing how the phenotypic expression of a given genotype varies among environments. The plasticity of a linear reaction norm is best described by the slope of the function. In this model, the slope of the reaction norm is the product of the slope of *E*_{i} 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 identical, 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. Mean plasticity was calculated at each generation.] The average plasticity was standardized to the optimal reaction norm (i.e., relative plasticity) so that a pure plasticity outcome would have a value of 1 and a pure differentiation outcome would have a value of 0 (flat reaction norms). Intermediate values indicate that the average reaction norm had a slope intermediate between the two pure outcomes. 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).