# A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory

## Abstract

**:**

## 1. Introduction

## 2. The Rosenzweig–MacArthur Predator–Prey System

- ${E}_{0}(0,0)$ with eigenvalues ${\lambda}_{1}=b$, ${\lambda}_{2}=-\delta b$;
- ${E}_{1}(1,0)$ with eigenvalues ${\lambda}_{1}=-b-1$, ${\lambda}_{2}=-(b\delta +\delta -c)$;
- ${E}_{2}\left(\right)open="("\; close=")">\frac{b\delta}{c-\delta},\frac{-bc(b\delta +\delta -c)}{{\left(\right)}^{c}}$ with eigenvalues ${\lambda}_{1,2}=\frac{b}{2{\left(\right)}^{c}2}$, where$$A=-\delta (b\delta +\delta -c+bc)$$$$B=\delta {(b\delta +\delta -c+bc)}^{2}+4c{(c-\delta )}^{2}(b\delta +\delta -c)\phantom{\rule{0.166667em}{0ex}}.$$

**Theorem**

**1.**

**Proposition**

**1.**

**Conjecture**

**2.**

## 3. Kosambi–Cartan–Chern (KCC) Geometric Theory and Jacobi Stability

**Theorem**

**3.**

**Theorem**

**4.**

**Definition**

**1.**

## 4. SODE Formulation of the Rosenzweig–MacArthur Predator–Prey System

**Theorem**

**5.**

**Theorem**

**6.**

## 5. Jacobi Stability Analysis of the Rosenzweig–MacArthur Predator–Prey System

**Theorem**

**7.**

**Theorem**

**8.**

**Theorem**

**9.**

**Theorem**

**10.**

**Remark**

**1.**

#### 5.1. Dynamics of the Deviation Vector for the Rosenzweig–MacArthur Predator–Prey System

#### 5.2. Comparison between Lyapunov Stability and Jacobi Stability for Two-Dimensional Systems

**Theorem**

**11.**

## 6. Examples and Discussion

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Case | Conditions | Type of Equilibrium Points |
---|---|---|

1 | $b\delta >c-\delta $ | ${E}_{0}$ saddle, ${E}_{1}$ stable node |

2 | $b\delta =c-\delta $ | ${E}_{0}$ saddle, ${E}_{1}$ saddle node |

3 | $0<b\delta <c-\delta $, $B\ge 0$, $A>0$ | ${E}_{0}$ saddle, ${E}_{1}$ saddle point, |

${E}_{2}$ unstable node | ||

4 | $0<b\delta <c-\delta $, $B\ge 0$, $A<0$ | ${E}_{0}$ saddle, ${E}_{1}$ saddle point, |

${E}_{2}$ stable node | ||

5 | $0<b\delta <c-\delta $, $B<0$, $A>0$ | ${E}_{0}$ saddle, ${E}_{1}$ saddle point, |

${E}_{2}$ unstable focus | ||

6 | $0<b\delta <c-\delta $, $B<0$, $A<0$ | ${E}_{0}$ saddle, ${E}_{1}$ saddle point, |

${E}_{2}$ stable focus | ||

7 | $0<b\delta <c-\delta $, $B<0$, $A=0$ | ${E}_{0}$ saddle, ${E}_{1}$ saddle point, |

${E}_{2}$ weak stable focus (center) |

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Munteanu, F.
A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory. *Symmetry* **2022**, *14*, 1815.
https://doi.org/10.3390/sym14091815

**AMA Style**

Munteanu F.
A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory. *Symmetry*. 2022; 14(9):1815.
https://doi.org/10.3390/sym14091815

**Chicago/Turabian Style**

Munteanu, Florian.
2022. "A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory" *Symmetry* 14, no. 9: 1815.
https://doi.org/10.3390/sym14091815