Code from: Evolution of phenotypic plasticity owing to migration

Phenotypic plasticity enables organisms to produce better-suited
phenotypes when the environment changes, enhancing fitness under adverse
conditions. Yet responding to environmental cues may provide little use in
a constant environment, where organisms already express optimal
phenotypes. The forces that sustain plasticity and account for its
widespread presence, thus, remain unclear, as plasticity must remain
advantageous to persist. Although typically associated with changing
environments, maintenance of plasticity requires generational turnover
such that parents and offspring regularly encounter different conditions.
Here, we demonstrate that even low number of migrants between locally
adapted populations, in constant environments, can promote the emergence
and persistence of phenotypic plasticity even when plasticity is costly,
never associated with the fittest genotype, and independent of its genetic
architecture. We support this conclusion by exploring the parameter space
of a two-locus, two-deme model using stochastic simulations and analytical
approximations. We derive analytical conditions under which plasticity is
adaptively maintained as a function of selection strength, migration, and
fitness trade-offs. These findings reshape our understanding of the
evolutionary origins of plasticity and offer insight into how maladaptive
traits can invade adapted populations in stable environments.

表型可塑性（phenotypic plasticity）能够让生物体在环境改变时产生更适配的表型，从而在不利条件下提升自身适合度。然而，在环境恒定且生物体已表达最优表型的情境中，响应环境信号几乎无法带来额外益处。因此，维持可塑性并解释其广泛分布的演化动力仍不明确——毕竟可塑性必须持续具备选择优势才能得以保留。尽管表型可塑性通常与多变环境相关联，但维持该特性需要世代更替，使得亲代与子代能够频繁遭遇不同的环境条件。
本研究证实，即便在恒定环境中的本地适应种群之间仅存在少量迁移，也能推动表型可塑性的产生与维持——即使该可塑性存在适合度成本、从未与最优基因型相关联，且与其遗传架构无关。我们通过随机模拟（stochastic simulations）与解析近似（analytical approximations）方法，对双位点两群模型（two-locus, two-deme model）的参数空间展开分析，验证了这一结论。我们推导得出了可塑性被适应性维持的解析条件，该条件是选择强度、迁移水平与适合度权衡的函数。
这些发现重塑了我们对表型可塑性演化起源的认知，并为适应不良性状如何在稳定环境中入侵已适应种群提供了全新的理论视角。