Trend prey predator model - Analysis of gause model

Vladimir Serebriakov,Mirko Dohnal

doi:10.1016/j.gecco.2019.e00634

Abstract

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.

Highlights

Most of mathematical models describing physical or natural processes are formulated in terms of nonlinear differential equations, see e.g. Ibragimov and Ibragimov (2009)
Prey predator studies are important tools to solve a broad spectrum of different task e biology, economics, ecology, see e.g. Focardi and Rizzotto (1999)
Many important PP problems described by ONDEs are ill-known and/or very sensitive

Summary

Introduction

Most of mathematical models describing physical or natural processes are formulated in terms of nonlinear differential equations, see e.g. Ibragimov and Ibragimov (2009). If just one component is ignored the results can be totally misleading This is the key reason why PP models have received considerable attention in scientific literature, see e. A set of ordinary and/or partial nonlinear differential equations are frequently used descriptions of unsteady state behaviours of many different PP systems e. Numerical evaluations of ONDEs (ordinary nonlinear differential equations) constants are very often an information-intensive task, see e.g. A paradigmatic three-dimensional Lorenz model has been frequently studied. It is a well-known fact that different sets of numerical values of its constants determine its chaotic/non-chaotic behaviours. Extreme sensitivity of some chaotic models makes it very difficult to characterize completely all relevant features, e.g. attractors, strange attractors, repellers of chaotic behaviours of systems under study.

Trend models

Trend arithmetic operations

Trend transitions

Gause trend model

Findings

Conclusions