Expected offspring survival and male allocation to paternal effort; Ensuring self-consistency for the mating cost; Finding the maxima of the fitness function with multiple variables; Emerging association between the expected offspring survival and the cost of paternal care across species with different levels of female promiscuity from Sperm competition games when males invest in paternal care

Sperm competition games investigate how males partition limited resources between pre- and post-copulatory competition. Although extensive research has explored how various aspects of mating systems affect this allocation, male allocation between mating, fertilization and paternal effort has not previously been considered. Yet, paternal care can be energetically expensive and males are generally predicted to adjust their parental effort in response to expected paternity. Here, we incorporate parental effort into sperm competition games, particularly exploring how the relationship between paternal effort and offspring survival affects sperm competition and the relationship between paternity and paternal care. Our results support existing expectations that (i) fertilization effort should increase with female promiscuity and (ii) paternal effort should increase with expected paternity. However, our analyses also reveal that the cost of male care can drive the strength of these patterns. When paternal behaviour is energetically costly, increased allocation to paternal effort constrains allocation to fertilization effort. As paternal care becomes less costly, the association between paternity and paternal effort weakens and may even be absent. By explicitly considering variation in sperm competition and the cost of male care, our model provides an integrative framework for predicting the interaction between paternal care and patterns of paternity.