An evolutionary model of sensitive periods when the reliability of cues varies across ontogeny

Abstract Sensitive periods are widespread in nature, but their evolution is not well understood. Recent mathematical modeling has illuminated the conditions favoring the evolution of sensitive periods early in ontogeny. However, sensitive periods also exist at later stages of ontogeny, such as adolescence. Here, we present a mathematical model that explores the conditions that favor sensitive periods at later developmental stages. In our model, organisms use environmental cues to incrementally construct a phenotype that matches their environment. Unlike in previous models, the reliability of cues varies across ontogeny. We use stochastic dynamic programming to compute optimal policies for a range of evolutionary ecologies and then simulate developmental trajectories to obtain mature phenotypes. We measure changes in plasticity across ontogeny using study paradigms inspired by empirical research: adoption and cross-fostering. Our results show that sensitive periods only evolve later in ontogeny if the reliability of cues increases across ontogeny. The onset, duration, and offset of sensitive periods—and the magnitude of plasticity—depend on the specific parameter settings. If the reliability of cues decreases across ontogeny, sensitive periods are favored only early in ontogeny. These results are robust across different paradigms suggesting that empirical findings might be comparable despite different experimental designs.


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Additional plots for 20 time steps Maximal cue reliability 0.75 3

Mature phenotypes 5
Fitness of mature phenotypes 6 Phenotypic plasticity and plasticity in belief   to the Euclidean distance between their phenotypes. Grey lines and diamonds depict 'absolute' phenotypic distance, 23 the average distance between the 10,000 focal individuals and their clones at the end of ontogeny (ranging from 0 to 24 20√2, scaled to a 0 to 1 range). Black lines and circles depict 'proportional' distance, the average absolute distance 25 divided by the maximum possible distance following separation.

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Here, we additionally show distributions of mature phenotypes and compare the fitness of the 29 optimal policy with the fitness of two non-plastic strategies, a generalist and a specialist, to get a 30 sense of whether and by how much the optimal policy outperforms simpler strategies. We do 31 not discuss results from these analyses in the main text, as the results are qualitatively similar to 32 those of a model with fixed cue reliabilities (Panchanathan & Frankenhuis, 2016 1 ). The main text 33 focuses on those results that are qualitatively different when cue reliabilities are variable rather 34 than fixed across ontogeny.  Prior probability of ( ) Prior probability of ( , | ) Cue reliability; conditional probability of receiving in at ( , | ) Cue reliability; conditional probability of receiving in at ( | ) Posterior probability of after having sampled ( | ) Posterior probability of after having sampled 149 150 According to the laws of probability it holds that: 151 Further, we assume that , | = , | . 156 We assume that organisms are Bayesian learners, using the fixed distribution of patches as the 157 prior estimate of the environmental state and the time-dependent cue reliabilities to update 158 these estimates. To see how this works, suppose an organism has sampled a specific sequence of 159 cues = { = , = , … = }. 160 According to Bayes' theorem, its posterior estimate after the first cue is: 161 To compute the posteriors ( | ) and ( | ) after the whole sequence of cues, we have to 165 reapply Bayes' theorem for each cue using the previous posterior as the new prior. We denote the mature phenotype at the end of ontogeny by = ( , , Suppose that an organism has sampled a specific sequence of cues, , throughout ontogeny.

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Its posterior estimates ( | ) and ( | ) reflect the probabilities of being in either 195 environmental state at the end of ontogeny. Thus, to compute rewards and penalties, we 196 need to compute the expectation across both environmental states, weighted by how likely 197 each state is as indicated by the posterior estimates at the end of ontogeny. We denote the 198 mapping from phenotypic increments to rewards and penalties by ( ), where can refer 199 to both and , and derive the following expressions for expected rewards and penalties: 203  204  205  206  207  208  209  210  211  212 Lastly, we present the three functional mappings between the realized phenotype and 213 fitness rewards and penalties: 214 215 Returns on fitness -( ) Formula Parameter settings to ensure that maximal rewards and penalties correspond to − 1 linear 217  218  219  220  221  222  223  224  225  226  227  228  229  230  231  232  233  234  235  236  237  238  239  240  241  242  243  244  245  246  247  248  249  250  251  252  253  254  255  correspond to an organism who encounters cues to the correct environment at each time point. 305 We

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Note, that we have changed the term "yoked" to "reciprocal" in the remainder of the SM and the main 383 manuscript. We left the original figure and legend unaltered compared to its original source.