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Bifurcation diagrams in a class of Kolmogorov systems

G. Moza, C. Lazureanu, F. Munteanu, C. Sterbeti, A. Florea

Abstract

We study a two-dimensional Kolmogorov system when its two parameters vary in a small neighbourhood of the value $0.$ The local behavior of the system is described in terms of bifurcation diagrams.

Bifurcation diagrams in a class of Kolmogorov systems

Abstract

We study a two-dimensional Kolmogorov system when its two parameters vary in a small neighbourhood of the value The local behavior of the system is described in terms of bifurcation diagrams.
Paper Structure (5 sections, 5 theorems, 46 equations, 9 figures, 2 tables)

This paper contains 5 sections, 5 theorems, 46 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Analytical properties of the system
  3. Bifurcation diagrams
  4. Conclusions
  5. Acknowledgments

Key Result

Lemma 2.5

Let $F,G:D\subset \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be two smooth functions of the form a) $F\left( \mu _{1},\mu _{2}\right) =a\mu _{2}\left( 1+O\left( \left\vert \mu \right\vert \right) \right) +b\mu _{1}\left( 1+O\left( \left\vert \mu \right\vert \right) \right) ,$$ab\neq 0,$ and b) $G\left( ii) if $\Delta =b^{2}-ac>0,$ then $G\left( \mu _{1},\mu _{2}\right) =0$ is a union of two curves pa

Figures (9)

  • Figure 1: Bifurcation diagrams of system (\ref{['s2']}) for: $\theta-\gamma\delta<0,$$\delta >0$ (I), respectively, $\theta-\gamma\delta<0,$$\delta <0,$$\theta<0,$$a>0$ and $-1<\gamma<0$ (II).
  • Figure 2: The behavior of the system (\ref{['s2']}) when $\left( \mu _{1},\mu _{2}\right)$ crosses $T_1$ in the first bifurcation diagram (I).
  • Figure 3: Phase portraits corresponding to different regions of the bifurcation diagrams.
  • Figure 4: Phase portraits corresponding to different regions of the bifurcation diagrams.
  • Figure 5: Bifurcation diagrams of system (\ref{['s2']}) for: $\theta-\gamma\delta<0,$$\delta<0,$$\theta<0,$ and $a>0$ and $\gamma<-1$ (III), respectively, $\theta-\gamma\delta<0,$$\delta<0,$$\theta<0$ and $a<0$ (IV).
  • ...and 4 more figures

Theorems & Definitions (16)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 6 more