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Stability of equilibriums and bifurcation analysis of two-dimensional autonomous competitive Lotka-Volterra dynamical system

Danijela Branković, Marija Mikić

Abstract

A detailed analysis of the stability of equilibriums and bifurcations of the two-dimensional autonomous competitive Lotka-Volterra dynamical system is performed. Necessary and sufficient conditions are determined for equilibriums (without the origin) to be asymptotically stable or unstable on $\left[0, +\infty\right)^2$. Necessary and sufficient conditions are determined so that the observed dynamical system has no equilibriums in $\left(0, +\infty\right)^2 $. All results are presented in five tables and five figures. We also found that four transcritical bifurcations occur in the observed dynamical system if it is analyzed on $\mathrm{R}^2$.

Stability of equilibriums and bifurcation analysis of two-dimensional autonomous competitive Lotka-Volterra dynamical system

Abstract

A detailed analysis of the stability of equilibriums and bifurcations of the two-dimensional autonomous competitive Lotka-Volterra dynamical system is performed. Necessary and sufficient conditions are determined for equilibriums (without the origin) to be asymptotically stable or unstable on . Necessary and sufficient conditions are determined so that the observed dynamical system has no equilibriums in . All results are presented in five tables and five figures. We also found that four transcritical bifurcations occur in the observed dynamical system if it is analyzed on .

Paper Structure

This paper contains 13 sections, 6 theorems, 12 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Stability analysis of the equilibriums
  3. Case $d_{12} \neq 0$
  4. Case $d_{112} \neq 0$ and $d_{122} \neq 0$
  5. Case $d_{112} \neq 0$ and $d_{122} = 0$
  6. Case $d_{112} = 0$ and $d_{122} \neq 0$
  7. Case $d_{122} = 0$ and $d_{112} = 0$
  8. Case $d_{12} = 0$
  9. Case $d_{122} \neq 0$ or $d_{112} \neq 0$
  10. Case $d_{122} = 0$ and $d_{112} = 0$
  11. Necessary and sufficient conditions for stability of equilibriums
  12. Bifurcation analysis
  13. Conclusion

Key Result

Theorem 3.1

The equilibrium $E_1\left(b_1/a_{11},0\right)$ of the system (2d) is asymptotically stable on $\left[0, +\infty \right)^2$ if and only if one of the following conditions is satisfied:

Figures (5)

  • Figure 1: Phase portraits for the cases $d_{12}>0$, $d_{112}>0$, $d_{122}<0$ and $d_{12}>0$, $d_{112}>0$, $d_{122}>0$, respectively.
  • Figure 2: Phase portraits for the cases $d_{12}>0$, $d_{112}<0$, $d_{122}<0$ and $d_{12}<0$, $d_{112}<0$, $d_{122}>0$, respectively.
  • Figure 3: Phase portraits for the cases $d_{12}>0$, $d_{112}>0$, $d_{122}=0$ and $d_{12}<0$, $d_{112}<0$, $d_{122}=0$, respectively.
  • Figure 4: Phase portraits for the cases $d_{12}>0$, $d_{112}=0$, $d_{122}<0$ and $d_{12}<0$, $d_{112}=0$, $d_{122}>0$, respectively.
  • Figure 5: Case $d_{12}=d_{112}=d_{122}=0$. $\dot{x_1}=x_1 \left(1-x_1-2x_2\right)$, $\dot{x_2}=x_2 \left(2-2x_1-4x_2\right)$.

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6