Dynamics of Nonlinear Interactions among Forces with Lagged Effect

Jair Silvério Dos Santos,Katia A G Azevedo

doi:10.4236/am.2017.810104

Abstract

This paper is concerned with the dynamics of the steady state of a two-delayed nonlinear system of functional differential equations. The stability of the steady state together with its dependence on the magnitude of time delays has been examined by means of characteristic equation corresponding to the nonlinear equation. General criteria for stability involving the two-delay equations have been obtained.

Highlights

Differential equations models that incorporate the history of the phenomenum into the model have been extensively considered as model for many problems because of a commum feature to them, which is, the appearance of oscillations, a fact which is very important in the model problems coming from ecology and mathematical economics [1] [2] [3] [4] [5]
Since delay-differential equations share many properties with ordinary differential equations, we have use methods and techniques of geometric dynamical systems theory that have been implemented in functional differential equations to describe the dynamics of flow associated with the system of equations
This paper studies dynamic characteristics of feed-back effects that incorporate the memory of the phenomenum into the model because of a commum feature to them, which is, the appearance of oscillations, a fact which is very important in the model problems coming from mathematical economics or ecology

Summary

Introduction

Differential equations models that incorporate the history of the phenomenum into the model have been extensively considered as model for many problems because of a commum feature to them, which is, the appearance of oscillations, a fact which is very important in the model problems coming from ecology and mathematical economics [1] [2] [3] [4] [5]. We have decided to keep to the model represented by system 1 with σ ≠ r to improve understanding of the combined effects of functional response and time delays on the dynamics of predator-prey systems. Because of the two delays, the analysis on how the roots of (5) locate with respect to the imaginary axis is a classical problem that, besides being important in itself, plays an important role in the study of asymptotic behavior in the theory of delayed differential equations (see [6]). Azevedo analyze the behavior dynamics of System (1) close to equilibrium solution it is necessary to have detailed information about the behavior of the eigenvalue of the linear equation associated to it, and so this problem can be reduced to the fact that all roots of Equation (5) have negative real part. We have described the region K for System (1) in Theorem 1 and found parameters belonging to ∂ ( K ) , which correspond to the point where the equilibrium of System (1) switches from being stable to unstable with periodic oscillations as the parameter σ crosses the value π

Stability

Hopf Bifurcation

Two-Delayed Kaldor-Kalecki Model

Concluding Remarks