On some methodological issues in mathematical modeling of interacting populations

In this paper, we focus on some important aspects of model building. The discussion specifically concerns the case of predator-prey interactions. We introduce here two models whose slight difference lies just in the way predators survive. In the former, they are taken to feed only on the modeled prey, i.e., to be specialists; in the second one, they are generalists, i.e., they can survive on other not explicitly modeled food resources.But our main focus is on the prey, that may disappear due to the Allee effect, if reduced to very low numbers. On the other hand, they also exhibit herd behavior. Our main aim is the discussion of the issues of mathematical modeling of such situation. We show on this example that modeling requires much more than equations patching from different systems.The analysis of the models indicates that the ecosystem may collapse in the case of specialist predators. If the predators have other feeding resources, they instead can thrive. Both types of models exhibit bistability between the prey-free state and coexistence.