Effects of composite and coordinated interactions between populations of particle-like individuals involving forces and internal states — A statistical-mechanical study

In this analogue modeling of interacting species we consider in particular interactions we refer to as composite interactions between particles called “daphnicles” and “food particles” that evolve on a third particle component in the background. Other force interactions are also taken account of but they play a smaller role in this study. The states of daphnicles and food particles are not only given by their position- and velocity-variables but also by variables of internal properties they have, which we shall call “saturation” for daphnicles and “nutrition” for food particles. The “laws” of variations of all these variables, where the rates of change of saturation and nutrition take place in the central region of the force interaction range where daphnicles and food particles come close together in their movements, are coupled in the composite interactions we consider. Due to this coupling of the “outer” force interaction to the “inner” saturation–nutrition interaction the inner interaction can be raised or diminished and in this way the distribution of say saturation of daphnicles may be regulated. From an extended Liouville equation for a system of a large number of particles where in particular interactions like these are baked in on the level of individual particles we aim to derive equations for daphnicle and food particle distribution functions on a kinetic level; this derivation constitutes the main part of the paper. Proper equations on a coarser grained level of description are then found quite easily: this “mixed” kinetic/moment level is in between the kinetic- and the full moment (macroscopic) levels such that distribution of saturation and nutrition still play a role and the equations still retain essential characteristics of the composite interactions taking place on the microscopic level, besides other force interactions. Though the terminology used suites modeling of biological systems and the results may throw some light on earlier modeling of daphnia behavior, the modeling is also believed to have relevance to systems of interacting species outside this area. The paper gives some examples and results of particular interaction models at the end