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Life history evolution and the origin of multicellularity: differentiation of types, energy constraints, non-linear curvatures of trade-off functions

A fundamental issue discussed in evolutionary biology is the transition from unicellular to multicellular organisms. Here we develop non-robust models provided in [1] and attempt to get robust models investigated how differentiation of types and energy constraints influence on the optimal behavior of colonies with different size. Constructed models show that each large-sized colony with high initial costs of reproduction tends to full specialization, no matter are all cells in this colony identical or are there cells with different types in this colony. The level of type's diversity determines the number of cells specialized in soma. In small - sized colonies with low initial costs of reproduction, when type's diversity is week, an unspecialized state may bring colony some benefits. However, these benefits may be only local and in optimum in the colony some cells would be specialized, others - unspecialized. The amount of specialized cells in small - sized colony depends on the level of type's diversity in this colony. Adding energy constraint, we may receive robust models even in convex case. In optimum, the colony with different types of cells and energy restriction may be indifferent between some optimal patterns of states. Arbitrary chosen cell may be soma or germ in some states or may be unspecialized in other. Moreover, in each optimal state levels of fecundity and viability of each cell lie in limited ranges. This result reflects the fact that some cell in the colony may lose the potential ability to achieve, for example, high level of fecundity, but does not lose the possibility to perform a reproductive function at all. It means that provided model can describe organisms, which represents the intermediate between unspecialized colonies and full-specialized multicellular organisms.

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