## Introduction

Phenotypic plasticity can be described as a physiological or developmental contingency on the environment (Bradshaw, 1965; Schlichting & Pigliucci, 1998; West-Eberhard, 2003). In the evolution of adaptive plasticity, the environment has traditionally been seen to play a dual role: first in the determination of the phenotype, and second as a selective agent on the reaction norm and on the phenotype itself. A more precise formulation of plasticity might predicate development not on the selective environment, but rather on the value of one or more environmental cues. Cues may be external environmental signals or internal physiological states – as long as they have some correlation with the selective environment and some causal connection to the phenotype. The variables representing the cue can then be modeled independently of those representing the selective environment, facilitating analysis of the quality of environmental information and the functional consequences of plant responses.

The evolution of optimal reaction norms illustrates the clear distinction between environmental cues and the selective environment. A phenotype with unlimited plasticity might achieve optimality at all times if the plant always matches the appropriate phenotypic state to the changing environmental state – for example, when the plant is able to sense the selective environment directly. However, if no plasticity is possible – either because developmental alternatives are unavailable or because the plant lacks the information necessary to track the environment correctly (Moran, 1992; Getty, 1996) – then the fixed phenotype favored by natural selection may not be one that is optimal for any single environment. Instead, one would expect the favored single phenotype to be a bet-hedging phenotype, one that maximizes long-term fitness given the entire probability distribution of environmental states (Lewontin & Cohen, 1969; Stearns, 1976; Gillespie, 1977; Hopper, 1999). When partial information is available, in the form of environmental cues that reduce uncertainty regarding current or future environmental states, an intermediate solution may arise. If each cue value is associated with a probability distribution of selective environments, then the favored reaction norm would comprise the bet-hedging phenotypes that maximize fitness given each of those distributions. The critical feature of this scenario is the long–term statistical association between environmental cues and selective environments, which has shaped evolutionary responses to cues over time.

We explore such intermediate, bet-hedging plasticity in a simple model of reproductive allocation in annuals. We follow a simple procedure: we take a fitness function relating an allocation schedule and a selective environment to fitness; we define a set of probability distributions of the environmental state; and we present the results of computational searches of the schedule space to find the fitness-maximizing allocation schedules for each distribution. The distributions – and therefore, the search results – can validly be interpreted in two ways. If the distributions are taken to be prior distributions (i.e. historical distributions that represent past selective environments that have shaped the phenotype), the search results describe optimal fixed schedules in environments of different predictabilities – and indeed the search results expand our understanding of fixed allocation in variable environments. If the distributions are taken as posterior environmental distributions (i.e. the distribution for a given year, conditioned on cues available to the plant early in the growing season), the search results describe the component phenotypes of reaction norms which have been optimized for cue quality. Using this approach, we found that cues of lower quality selected for schedules that display limited plasticity among a repertoire of intermediately bet-hedging schedules. Finally, we present the results of simple competition simulations, which confirmed that the most successful allocational reaction norms were those that were identified in the searches as being optimal for the cue-quality used in the simulation. Genotypes with these reaction norms were favored, even when they were in competition with reaction norms either that displayed greater plasticity among more highly specialized phenotypes, or with reaction norms that displayed less plasticity but greater bet-hedging.

### Bayesian plants

Our analysis draws heavily on the concepts of prior and posterior probability distributions, ideas from Bayesian decision theory that describe the role of information in reducing uncertainty on the part of a conscious decision-maker (Robert, 2001). The prior distribution describes the decision-maker's understanding of the possible outcomes of a random process in the absence of some piece of information. The posterior distribution describes the decision-maker's understanding of the outcomes as updated on the basis of that information. While the prior and posterior distributions are commonly explained in terms of conscious prediction and interpretation, consciousness is not essential to their application. In this paper we use *prior* and *posterior* to distinguish the historical distribution of selective environments in the absence of a cue from the conditional distribution modified by a cue available early in the growing season, respectively.

For computational simplicity we only consider the effects of a single cue provided at the outset of the growing season, and do not address the changes in the conditional distribution that could occur due to additional cues, changing internal state of the plant during the season, or variation in the actual date of death of different individuals. We also assume that there are no developmental constraints on reproductive allocation (e.g. in Kozlowski & Wiegert, 1986), so that any conceivable seasonal schedule is allowable in our model. In actual populations, these assumptions would be violated, but we believe that our model is sufficient to address key questions, and the similarity with previous studies in this area (e.g. Paltridge & Denholm, 1974; Cohen, 1976; King & Roughgarden, 1982b) facilitates comparisons of the results. Our primary objective is to demonstrate with a very simple trait how information quality can constrain optimal plasticity through adaptive bet-hedging.

### Quality of environmental information

Models of learning and communication in animals have generally focused on some form of what Koops (2004) calls *reliability*: the probability that some cue value correctly indicates a particular environmental state (e.g. Zahavi, 1975; Charnov, 1976; Green, 1980; McNamara & Houston, 1980; Stephens & Charnov, 1982; Stephens, 1989; Stephens, 1991; Sih, 1992; Hasson, 1994; Nishimura, 1994; Maynard Smith & Harper, 1995; Hughes, 2000; Pierre *et al*., 2003; Koops, 2004). These learning studies are relevant to the study of plant allocation, especially in establishing the importance of information in the evolution of behavioral and life-history traits (e.g. Sih, 1992; Maynard Smith & Harper, 1995) and in laying the Bayesian groundwork for modeling information quality and the estimation of the posterior distribution by sequential sampling of the environment (e.g. Charnov, 1976; Green, 1980; McNamara & Houston, 1980; Stephens & Charnov, 1982; Mangel, 1990). We are aware of one previous study that has applied Bayesian decision theory to plants: Cohen & Mangel (1990) modeled plants that assess the risk of mechanical damage by sequential sampling.

Our work departs from these learning studies in a few important ways. First, we model the reproductive-allocation schedule as a trait that is static within the individual plant's lifetime (i.e. the schedule of daily allocation is set at the outset). As a consequence, the plants in our model cannot make use of the sequential sampling processes that inform learning in animals. We believe a version of sequential sampling to be relevant to the annual plants of our study, in that the prior environmental distribution reflects the selective environment on previous generations, which has shaped the plant's optimal allocation schedule and which represents the ‘best guess’ about the season-length distribution in the absence of additional information. For the posterior distribution, we do not specifically model the nature of environmental cues or mechanisms of plant perception. We simply suppose the existence of a cue, the effect of which – like the effect of sequential sampling in learning animals – is to determine the posterior season-length distribution.

Second, the behavioral traits in the animal-learning studies are generally discrete traits – each optimized for one of the two or several environmental states – and the relevant question is whether to ignore the signal or to respond with the behavior indicated by the environmental state. We believe that few developmental traits in plants are optimized for highly specific sets of environmental conditions. More likely, plant developmental traits often have the effect of bet-hedging over distributions of environments and therefore invite a concept of environmental-information quality that differs from reliability.

We are interested in two related quantities: environmental predictability and, indirectly, environmental-cue quality. Predictability reflects the range of states represented by a distribution and the distribution of probability mass among those states. If *K* is Gaussian-distributed, predictability can be captured in the variance: small variance corresponds to high predictability. In this paper, we use variance as a surrogate for low predictability, because we primarily address Gaussian environmental distributions. Alternatively, predictability could be quantified by Shannon entropy:

where *p*_{k} is the probability of environmental state *k*. Smaller *H* corresponds to greater concentration of probability mass and greater predictability.

Cue quality reflects the increased predictability of the posterior distribution, relative to the prior. We are able to model cue quality very simply in terms of the predictability of posterior distributions; we are thus able to examine the effects of cue reliability, without making specific assumptions about the mechanisms or dynamics underlying the cue itself.

### Reproductive allocation schedules

The timing and form of the transition to reproduction in annuals is a well-modeled developmental trait and has been of particular historical interest to life-history theorists (e.g. Cohen, 1966, 1971, 1976; Paltridge & Denholm, 1974; Denholm, 1975; Vincent & Pulliam, 1980; King & Roughgarden, 1982b). For simplicity, an annual plant can be modeled as two compartments, vegetative and reproductive biomass. On every day of its growing season, an individual allocates newly captured carbon between immediate reproduction and vegetative growth; vegetative structures are used to enhance carbon gain on subsequent days, and thus contribute to future reproductive output. Given this basic tradeoff between current and future reproduction at each point during the season, what is the optimal time-course of allocation?

Cohen (1971) showed that, given a known or constant season length, the schedule that maximizes fitness is a so-called bang-bang schedule: the plant allocates nothing to reproduction until a particular switch date, after which it allocates all photosynthate to reproduction. Bang-bang schedules are specialist phenotypes, differentiated by their switch dates; each schedule maximizes season-long fecundity in a season of a particular length, but would be suboptimal in a longer or shorter season. By contrast, when the season length is extremely uncertain – for example, drawn from a uniform distribution – a schedule of graded allocation is favored, with a period of gradual transition from pure vegetative to pure reproductive allocation (King & Roughgarden, 1982b). That is, it includes an interval of simultaneous allocation both to vegetative growth and to reproduction. Graded allocation is a form of bet-hedging. The fecundity of a graded schedule is necessarily lower than the optimal bang-bang schedule for any one fixed season length. However, the geometric mean fitness over time, which is an appropriate fitness measure for annuals with nonoverlapping generations (Lewontin & Cohen, 1969; Gillespie, 1977), will be higher for the graded schedule than for any one bang-bang schedule. For example, a bang-bang schedule in which the switch to reproduction occurs after the end of a short season has fecundity of zero (and therefore a geometric-mean fitness of zero), and for an annual plant without a seed bank, this results in extinction of the lineage.

Thus we have the optimal allocation strategies given an environment in which the season length is known with certainty, and given one in which all season lengths are equiprobable. Plants in nature are unlikely to face either extreme. The distribution of season lengths (the prior season-length distribution) can vary from a fixed value to a uniform probability distribution, with a range of optimal schedules for each distribution that have not yet been quantitatively explored. In addition, plants may perceive environmental cues that are not selectively relevant in themselves, but by their correlation with other variables they do provide information about selectively relevant conditions. If these cues provide some information about the probability of season length in a given year, then they further enhance the predictability of the environment.

### Fecundity and long-term fitness

Our formula for season-long fecundity in annual plants (Eqn. 1 ) is taken directly from Cohen (1971) and is similar in form to other models of reproductive allocation (e.g. Paltridge & Denholm, 1974; Mirmirani & Oster, 1978; Vincent & Pulliam, 1980; King & Roughgarden, 1982a, 1982b; Iwasa, 1991; Fox, 1992). The plant's structures are divided into two compartments: vegetative and reproductive. For simplicity, we assume that carbohydrate storage and recycling and midseason loss of biomass are negligible (Chiariello & Roughgarden, 1984).

On any day *t* in the growing season, the plant allocates fraction *b _{t}* of the day's photosynthate to reproduction and 1 −

*b*to vegetative growth. The reproductive growth increment

_{t}*l*is proportional to

_{t}*b*and to the size of the vegetative body:

_{t}*l*=

_{t}*b*

_{t}AFv_{t}where *A* is the carbon assimilation rate and *F* is the efficiency of carbon conversion into reproductive structures. Vegetative size *v _{t}* is itself a function of all earlier allocation decisions:

where *v*_{1} is the initial vegetative size and *L* is the vegetative conversion efficiency. If all the *b*_{t} and *v*_{t} are written as vectors **b** and **v**, then over the course of a season of maximum length the season's cumulative fecundity (measured at the start of day κ + 1 is given by:

*R*

_{κ+1}=

**b**·

*F*

**v**

In a season of length *k* ≤ κ, the season-long fecundity is given by:

In an environment of random *k* the population growth rate λ is given by:

where *p** _{k}*is season length

*k*'s probability (Lewontin & Cohen, 1969; Schaffer, 1974; McNamara, 1998). Selection should maximize λ; because Λ, where:

is a monotonically increasing function of λ, selection also maximizes Λ, the population's doubling rate (Kelly, 1956).

In nature, season length can be determined by any selective agent: frost, drought, fire, herbivory, harvest, etc., and it is not necessary that all plants in a population share a common season length. Season length can even reflect edaphic and competitive effects on growth rate, if time is measured in phytochrons rather than in days. The important assumption is that the plant's allocation schedule is determined by a single probability distribution **p** and that **p** be stationary in time, where **p** is the vector of season-length probabilities *p*_{k} over all *k* ≤ κ. Note that **p** may either be the prior distribution, in the case of a plant with no information, or a cue-specific posterior distribution, if the plant has access to a cue.

### Searching the schedule space

We used the population doubling rate of Eqn. 2 as the selection criterion in evolutionary-algorithm (EA) searches to find which allocation schedules **b** were favored under different distributions **p**. EAs search large, potentially rugged solution-spaces for optima by a heuristic process analogous to evolution by natural selection (Holland, 1975; Sumida *et al*., 1990; Mitchell & Taylor, 1999). A large population of candidate solutions to a given optimization problem is generated randomly. In each of many iterations, the population is subjected to selection for fitness under the optimization criterion, and then subjected to some sequence of differential reproduction, mutation and recombination. After many generations the population converges to one or a few solutions that perform well under the criterion.

The solution space of our EA included every possible allocation schedule given κ = 30. Candidate allocation schedules were represented by ordered arrays of 30 loci, each of which took a real-valued number *b _{t}* with 0

*b*1. Initially the 50 individuals of the population were assigned random schedules, with each

_{t}*b*drawn uniformly from [0,1]. In each generation of each search, each individual

_{t}*i*was assigned a relative fitness:

where Λ_{p,i} is the season-long fecundity of schedule *i* under distribution **p** and is given by Eqn. 2. Individual schedules were then ranked by relative fitness and copied into the next generation of the search. The expected number of copies was calculated by the method of Baker (1985). Expected copy number was a function of the schedule's fitness rank:

where *A _{i}* is the expected copy number for schedule

*i*, and min

*A*and max

*A*are, respectively, the expected copy numbers of the worst- and best-ranked schedules (

*rank*

_{min}= 1 and

*rank*

_{max}=

*N*) set here to 0.8 and 1.2. With per-locus mutation rate µ, each locus of every new individual was subjected to an increase or decrease (each with probability 0.5) of its value by 0.05. Finally, all individuals were randomly paired for 1- or 2-crossover recombination, in which the daily allocation values of the two individuals were exchanged at loci to one side of the crossover point (in the case of 1-crossover) or between randomly chosen points (in the case of 2-crossover). In general

*F*and

*L*from Equation 1, and µ, were set to 0.2, 0.3, and 10

^{−3}, respectively. Changing parameter values changed the efficiency of searches but none of the qualitative results. Searches were also implemented using fitness-proportionate selection, in which each candidate schedule's expected number of offspring was proportional to its relative fitness. Fitness-proportionate selection differed from rank selection only in the speed of the search, not in the final results.

Searches continued for 3000 generations, long after consistent directional evolution appeared to have ended by visual inspection of schedule trajectories. In all cases maximum fitness was achieved long before generation 3000.

### Perfect information

Figure 1 illustrates the optimal schedule obtained by the EA for a constant environment in which *k* = 30. The population's mean fitness increases steadily and plateaus by generation 250 (Fig. 2). In generation 1000 (Fig. 1e), all schedules are essentially bang-bang with the allocational switch between days 26 and 27. The minor deviations from pure vegetative or reproductive allocation are due to mutation-selection balance: when mutation is disabled and the search allowed to continue for 500 additional generations, deviations from bang-bang allocation are eliminated (Fig. 1f). We conducted five searches, setting season lengths constant, for every *k* = 1, … , 30. Every search exhibited nearly identical behavior as it approached its respective final bang-bang schedule. Much of the initial random variation disappeared early in every run: schedules with high reproductive allocation early in the season were quickly eliminated. Variation late in the season also disappeared fairly quickly, but there was no strong pattern initially to which late-season strategies survived. Late-season selection early in the search was driven not by selection for late-season allocational trajectories but by genetic hitchhiking, or linkage to strongly selected early selection trajectories (Mitchell, 1996). As the searches progressed, however, all schedules evolved toward the final bang-bang solutions as mutation maintained the diversity available to selection.

### No information

We also conducted searches for the fitness-maximizing schedule when **p** = Uniform(1,30), the uniform distribution with *K* ∈ [1, 30]. Figure 3 shows the progression of all 50 candidate schedules over 1000 generations in a typical search. All final schedules in 10 independent searches shared low but nonzero allocation early in the season, and a steady increase in reproduction as the season progressed to the last days, when allocation approached full reproduction.

One conspicuous feature of the final schedules was the high reproductive allocation on the first day of the season (*c.* 61%). When *k* = 1, a schedule's fecundity is proportional to *b*_{1} itself, and despite the rarity of *k* = 1 (*p*_{1} = 0.033), its effect is dramatic enough to maintain the high *b*_{1}.

### Intermediate information

We used Gaussian season-length distributions to represent different degrees of environmental predictability. Predictability is reflected in the standard deviations σ = 0, 5, 10, 15. All distributions were truncated and subsequently rescaled so that no *K* smaller than 1 or greater than 30 was allowed. All results are from searches conducted over 3000 generations.

In Fig. 4, final schedules are presented for searches under all 120 distributions (4σ-families × 30 modes per family), and under the uniform distribution. All schedules begin with low reproductive allocation and switch, with varying steepness, to high allocation. As the selecting distribution broadens from σ = 0 to σ = 15, the optimal schedule becomes markedly more graded. The onset of reproduction is earlier in the season, and the interval of simultaneous vegetative and reproductive growth is lengthened. Moreover, the optimal schedules for broad distributions begin to resemble the optimal schedule under the uniform distribution of Figs 3 and 4(e). That for the broadest Gaussian distribution even exhibits a slightly elevated *b*_{1}.

Schedules optimized under a common σ but different modal season lengths are increasingly similar with increasing σ. For example, in Fig. 5 the schedules for modal season lengths µ = 15 and µ = 30 are compared among three values of σ. For small σ, the two schedules are strikingly different: the µ = 15 schedule has completed its transition to reproduction by day 12, while the µ = 30 schedule does not even start its transition until after day 15. For σ = 15, by contrast, the plant's reproductive transition occurs roughly from days 5–20, regardless of the modal season length.

### Bimodal distribution

An extreme bimodal distribution, where *p*_{15} = *p*_{30} *=* 0.5, yielded what was essentially two bang-bang schedules appended to each other (Fig. 6). Following a reproductive peak on day 15, the schedule dropped to zero reproduction before peaking again on and immediately before day 30. The first reproductive interval resembled that in the bang-bang schedule for *k* = 15, except that its start was delayed by 2 d. The second interval was indistinguishable from the reproductive interval in the bang-bang schedule for *k* = 30.

The bimodal-selected schedule appears to describe a sort of iteroparity, and it departs from the monotonicity generally associated with analytically derived allocation schedules. King & Roughgarden (1982a) introduced biomass-loss functions to generate optimal schedules with multiple switches. Our result suggests that biomass loss is not necessary: multiple switches in annuals could arise from multimodality of the season-length distribution. Allocation schedules in long-lived perennials, which may suffer heightened risk of mortality in annual cycles (e.g. during winter), could also be modeled using this approach, though the population-level fitness function is complicated by overlapping generations.

### Competitiveness

So far, we have derived the optimal allocation schedules for a variety of environmental distributions, with contrasting predictability and modal season lengths. These solutions represent the schedule favored by natural selection when the distributions are stationary; that is season-length may vary from year to year (when σ > 0), but the prior probability distribution of those season lengths is constant.

If an environmental cue is available, which is correlated with the probable season length in a given year, then over time plants may evolve to respond adaptively to the cue, with reproductive allocation appropriate to the posterior distribution, conditioned on that cue. In our simulations, we assume that the cue is available on day 1, so the plants can exhibit the optimal schedule over the entire season given the posterior season-length distribution. In the absence of a cost of perception, we expect that plants will always enhance long-term fitness by responding to these cues, even if they only provide partial information about the environment, and we used our evolutionary-algorithm environment to confirm this prediction.

Populations were simulated with two allocation types: wild type and mutant type, with 90 and 10 individuals, respectively. In each season, after selecting the actual season length from the uniform distribution, a cue value was drawn from a Gaussian distribution centered on the season length with some standard deviation σ_{c}. Note that by Bayes's rule:

*c*|

*k*} = Pr{

*k*|

*c*} Pr{

*c*}/Pr{

*k*}

Thus assuming that Pr{*c*} ≈ Pr{*k*} = κ^{−1}, choosing the cue value on the basis of the season length is equivalent to modeling the season length's posterior distribution, conditioned on the cue value. At the start of each season, the cue value was available to all allocators. Allocators then exhibited the optimal allocation schedules for the posterior environmental distributions, contingent on the cue value.

The wild types and mutant types chose allocational reaction norms that were found in the EA searches to optimize fitness given **p**_{wt} = Gaussian(µ = *k*, σ = σ_{wt}) and **p**_{mut} = Gaussian(µ= *k*, σ = σ_{c}), respectively. Thus in every simulation, the mutant types allocated appropriately given the posterior season-length predictability, and the wild-types allocated appropriately for some other, incorrect, predictability. Because smaller σ is associated with a greater variety of component schedules (Fig. 5), in every simulation the type with smaller σ allocated according to a steeper reaction norm than the other. We allowed σ_{c} and σ_{wt} to vary in {0,5,10,15,∞}, where σ = ∞ indicates **p** = Uniform (1,30). We did not run simulations for the cases in which σ_{wt} = σ_{c}.

In every generation, each allocator's fitness was calculated, and then offspring were produced for the subsequent generation. For each allocator, offspring shared the parent's allocation behavior (i.e. choosing the optimal reaction norm for **p**_{wt} or **p**_{mut}), and offspring number was proportional to the parent's relative fitness. Each simulation was run until one type reached fixation, and we conducted 500 simulations per combination (σ_{wt}, σ_{c}).

Because in every case the initial mutant frequency was 0.1, the random walk in which the wild-type and mutant strategies had equal fitness would have yielded a mutant fixation rate of 0.1. The observed mutant fixation rate was greater than 0.1 for every combination of σ_{c} and σ_{wt} examined (Table 1). In most cases fixation frequency exceeded 0.6. The notable exception was the set of simulations where σ_{wt} = 2 and σ_{c} = 3. In this case the mutant type achieved fixation more frequently than expected given equal fitness, but the fixation frequency was only 0.26.

Frequency of mutant-type fixation | |||||
---|---|---|---|---|---|

σ_{c} | σ_{wt} = 0 | σ_{wτ} = 1 | σ_{wt} = 2 | σ_{wt} = 3 | σ_{wt} = 4 |

0 | 1.00 | 1.00 | 1.00 | 0.99 | |

1 | 1.00 | 1.00 | 1.00 | 0.99 | |

2 | 1.00 | 0.90 | 0.83 | 0.99 | |

3 | 1.00 | 0.90 | 0.26 | 1.00 | |

4 | 1.00 | 1.00 | 1.00 | 0.60 |

When σ_{wt} = 0, the mutant type almost always reached fixation within 15 generations (Fig. 7). With all other wild-type strategies, the mean time to mutant fixation – as well as the variance in fixation time – increased as the difference between σ_{c} and σ_{wt} decreased. There was no obvious difference between cases in which the mutant types’ reaction norms were steeper than those of the wild-types (right-hand side of Fig. 7) and the broad case in which the wild types’ reaction norms were steeper (left-hand side of Fig. 7).