Modeling the lowest-cost splitting of a herd of cows by optimizing a cost function

Animals live in groups to defend against predation and to obtain food. However, for some animals-especially ones that spend long periods of time feeding-there are costs if a group chooses to move on before their nutritional needs are satisfied. If the conflict between feeding and keeping up with a group becomes too large, it may be advantageous for some groups of animals to split into subgroups with similar nutritional needs. We model the costs and benefits of splitting in a herd of cows using a cost function that quantifies individual variation in hunger, desire to lie down, and predation risk. We model the costs associated with hunger and lying desire as the standard deviations of individuals within a group, and we model predation risk as an inverse exponential function of the group size. We minimize the cost function over all plausible groups that can arise from a given herd and study the dynamics of group splitting. We examine how the cow dynamics and cost function depend on the parameters in the model and consider two biologically-motivated examples: (1) group switching and group fission in a herd of relatively homogeneous cows, and (2) a herd with an equal number of adult males (larger animals) and adult females (smaller animals).