## Mathematical models of multicellular life cycle competition

Organisms have evolved into increasingly larger and more complex organisation, reflecting on a series of major evolutionary transitions. Here, we focus on the major evolutionary transition from unicellular ancestors to multicellular organisms. This transition comes with replication conflicts between cell level and organism level. It is hard to detect this transition directly, as it can date back to billions of years ago. Therefore, mathematical models serve as a major tool to understand this transition. We model multicellular organisms considering a stage-structured population, in which the development of organisms is classified into different stages based on cell numbers. Organisms' growth and reproduction are captured by transition probabilities between different stages. For growth, the transition probability is determined by the organisms' current cell numbers and composition. For reproduction, the transition from mature stages to newborn stages represents a reproductive strategy, which is fixed in terms of cell numbers but randomly distributed in terms of cell composition. We assume that organisms adopt the same reproductive strategy in a population. Populations with different reproductive strategies may have different population growth rates. The fastest-growing population is favored by natural selection. Here, we apply evolutionary game theory to model transition probabilities with a frequency-dependent game and a volunteer dilemma game with varying thresholds in turn. By doing so, we solve the questions of the formation of multicellularity and the evolution of reproductive strategy of multicellularity, respectively. Additionally, we also investigate the effect of organism size. Finally, we apply our model to investigate irreversible somatic differentiation (ISD), for which somatic cells only produce somatic cells. The transition probability for growth depends on somatic-contributed functions, which are sampled by the Monte Carlo method. Overall, our mathematical models provide approaches to identify the conditions of cell interaction for the formation, differentiation and reproduction of multicellular organisms.