Properties of a System of Diophantine Equations with Implications for Real-World Constrained Branching Processes

We consider an iterative branching process in which an abstract object can subdivide into other objects. The multiplication process may be varied by the occurrence of random "fatal" events in which some of the subsequent objects or states may fail. The process is also constrained to terminate upon reaching a given number of events or alternatively upon reaching a fixed number of iteration steps. A system of diophantine integer-variable equations capable of describing the aforementioned process is proposed. These equations can be applied prospectively to many branching phenomena of physical, biological and demographic nature. The equations, which we call systems of equations S, Q, U can be reformulated into three main classes based on the behavior of the sum of variables with respect to a fixed principal numerical parameter (TC= 'Total Cases'). These systems always admit solutions and these are sought for the three classes. The mathematical properties of the three systems are presented both analytically and graphically, and the software script for calculating numerical solutions is attached. In the case of high TC values, where direct calculation is not possible, special solutions are also sought for the steady state case and the "most probable" case, the latter using statistical mechanics methods. Solutions examples are given for a wide range of TC parameters. We also refer to real-world examples of applications ranging from prey/predator population dynamics to population mortality modeling and 2d lattice space tiling and also tree leaves branching alternatives. The main purpose of the study here proposed is to implement a mathematical frame that can provide tools to be used in the study of real-world applications.