A More Realistic Formulation of Herd Behavior for Interacting Populations

In the ecological literature herd behavior has been considered via either a decreasing response function with the increasing number of the individuals forming the flock, thereby modeling better a group defense mechanism in a predator–prey situation, or via power functions that indicate, in the same context, that the most likely individuals to be attacked are those occupying the peripheral positions in the herd. Other interpretations are possible in different contexts. The latter approach, which for simplicity can be named the square root formulation, provides an interesting mathematical phenomenon, namely the possible disappearance in finite time of the herd population. From the modeling perspective, instead, it suffers from the drawback that for low numbers of individuals, the herd disappears, or better, it coincides with its boundary, and the interactions therefore must involve the whole population. Corrections trying to consider this aspect have been recently proposed, also in the single population environment. In this paper, we propose a novel correction for taking care of the small individual numbers. As a result, the modeling perspective becomes more realistic, but, as it happens also for the previously conceived modifications, the finite time extinction phenomenon is lost.