Analysis of a class of Kolmogorov systems

TL;DR

A two-dimensional Kolmogorov system depending on two independent parameters and having a degenerate condition is studied and local analytical properties of the system are obtained when the parameters vary in a sufficiently small neighborhood of the origin.

Abstract

A two-dimensional Kolmogorov system with two parameters and having a degenerate condition is studied in this work. We obtain local analytical properties of the system when the parameters vary in a sufficiently small neighborhood of the origin. The behavior of the system is described by bifurcation diagrams. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.

Figures (10)

• Figure 1: Bifurcation diagrams corresponding to $N>0,\sigma _{1}<0$ and (G1) $\delta >0,$$2N-\delta \theta _{1}>0,$ (G2) $\delta >0,$$2N-\delta \theta _{1}<0,$ (G3) $\delta <0,$$2N-\delta \theta _{1}>0,$ (G4) $\delta <0,$$2N-\delta \theta _{1}<0,$$2\gamma k_3<\frac{\theta_1^2}{4N},$ and (G5) $\delta <0,$$2N-\delta \theta _{1}<0,$$2\gamma k_3>\frac{\theta_1^2}{4N},$ respectively, (G6) corresponding to $N>0,$$\sigma_1>0,$$\delta>0$ and $\theta_1>0.$
• Figure 2: Bifurcation diagrams corresponding to $N>0,\sigma_1>0$ and (G7) $\delta>0,$$\theta_1<0,$ (G8) $\delta<0,$$\theta_1>0,$ and (G9) $\delta<0,$$\theta_1<0.$
• Figure 3: The phase portraits corresponding to the diagrams G1-G9
• Figure 4: The phase portraits for $(\mu_1,\mu_2)\in\Delta$
• Figure 5: The phase portraits on the left and right of $T_2$ in G1

Theorems & Definitions (23)

• Remark 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7
• Corollary 2.8
• Proposition 2.9
• Theorem 2.10