Trend prey predator model - Analysis of gause model

Any application of PP (Prey - Predator) models based on nonlinear differential equations requires identification of numerical values of all constants. This is often a problem because of severe information shortages. Many PP models are numerically sensitive and/or chaotic. Moreover, complex PP tasks are based on integration of differential equations with (partially) unknown numerical values of relevant constants and vague heuristics, e.g. vaguely described capture rate. These are the main reasons why PP numerical simulations cannot identify all important/relevant features, e.g. attractors. Trend models use just three values namely positive (increasing), zero (constant), negative (decreasing). A multiplication of a trend variable X by a positive constant a is irrelevant, it means that aX = ( + )X= X. This obvious equation is used to eliminate all positive multiplicative constants a from PP mathematical models. A solution of a trend model is represented by a set of scenarios and a set of time transitions among these scenarios. A trend analogy of a quantitative phase portrait is represented by a discrete and finite set of scenarios and transitions. A trend version of the well-known Gause PP model is studied in details. The provably complete set of 41 scenarios and 168 time transitions among them are given. (C) 2019 The Authors. Published by Elsevier B.V