# Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definitions

#### Fractional Calculus

**Definition**

**1.**

**Definition**

**2.**

## 3. Modeling

#### 3.1. Three-Species Lotka-Volterra Model

#### 3.2. Existence of Solutions

**Lemma**

**1.**

**Proof.**

#### 3.3. Uniqueness of Solutions

## 4. Stability of the Model

#### 4.1. Stability of the Factional-Order System

**Theorem**

**1.**

**Theorem**

**2.**

- (1)
- $\u2225\left(\lambda \left(A\right)\right)\u2225\u2a7e{\displaystyle \frac{1}{1-\alpha}},\lambda \left(A\right)\ne {\displaystyle \frac{1}{1-\alpha}},$
- (2)
- $\mathrm{Re}\left(\lambda \left(A\right)\right)>{\displaystyle \frac{1}{1-\alpha}},$
- (3)
- $\mathrm{Re}\left(\lambda \right(A\left)\right)<0,$
- (4)
- $\left|\mathrm{Im}\left(\lambda \right(A\left)\right)\right|>{\displaystyle \frac{1}{2(1-\alpha )}}.$

**Theorem**

**3.**

- (1)
- $\u2225\lambda \left(J\left({x}^{*}\right)\right)\u2225\u2a7e{\displaystyle \frac{1}{1-\alpha}},\lambda \left(J\left({x}^{*}\right)\right)\ne {\displaystyle \frac{1}{1-\alpha}},$
- (2)
- $\mathrm{Re}\left(\lambda \left(J\left({x}^{*}\right)\right)\right)>{\displaystyle \frac{1}{1-\alpha}},$
- (3)
- $\mathrm{Re}\left(\lambda \left(J\left({x}^{*}\right)\right)\right)<0,$
- (4)
- $\left|\mathrm{Im}\left(\lambda \left(J\left({x}^{*}\right)\right)\right)\right|>{\displaystyle \frac{1}{2(1-\alpha )}}.$

**Proof.**

#### 4.2. Stability of the LV Model with the CF Derivative

#### 4.2.1. The First Equilibrium

#### 4.2.2. The Second Equilibrium

#### 4.2.3. The Third Equilibrium

#### 4.2.4. The Fourth Equilibrium

#### 4.2.5. The Fifth Equilibrium

## 5. Numerical Algorithm

## 6. Numerical Results

#### 6.1. Example 1

#### 6.2. Example 2

#### 6.3. Example 3

#### 6.4. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Simulation model for the interactions of a three-species community represented by Equation (7). The system is isolated for three species and does not depend on interactions with the other species (colorless background).

**Figure 3.**(

**left**) Comparing the behavior of the Caputo and the CF operators for system (7) with the parameters of Example 1, (

**right**) converging to ${\u03f5}_{3}$ with CF operator for $\alpha \le 0.66$ and $({x}_{0},{y}_{0},{z}_{0})=(1.6,1.9,0)$.

**Figure 4.**System (7) with the parameters of Example 2 and $({x}_{0},{y}_{0},{z}_{0})=(2,2,3)$ is asymptotically stable for Caputo (

**left**) and unstable for CF (

**right**).

**Figure 5.**System (7) with the parameters of the Example 3 and $({x}_{0},{y}_{0},{z}_{0})=(0.5,0.1,5)$ is asymptotically stable for both Caputo (

**left**) and CF (

**right**) at ${\u03f5}_{2}$ with different oscillations.

**Figure 6.**System (7) with the parameters of the Example 3 and $({x}_{0},{y}_{0},{z}_{0})=(3,8,0),\alpha =0.4$, t = 200 is unstable for Caputo (

**left**) and asymptotically stable for CF (

**right**).

Equilibrium Points | Conditions |
---|---|

${\epsilon}_{0}$ | Always exists |

${\epsilon}_{1}$ | Always exists |

${\epsilon}_{2}$ | ${a}_{5}\ge 1$ and ${a}_{1}{a}_{6}\ge {a}_{2}({a}_{5}-1)$ |

${\epsilon}_{3}$ | ${a}_{3}\ge 1$ and ${a}_{1}{a}_{4}\ge {a}_{2}({a}_{3}-1)$ |

${\epsilon}_{4}$ | ${a}_{3}\ge 1$, ${a}_{4}({a}_{5}-1)\ge {a}_{6}({a}_{3}-1)$ and ${a}_{4}\ge {\displaystyle \frac{({a}_{2}{a}_{7}-{a}_{6})({a}_{3}-1)}{(1+{a}_{1}{a}_{7}-{a}_{5})}}$ or $(1+{a}_{1}{a}_{7}-{a}_{5})<0$, ${a}_{4}\le {\displaystyle \frac{({a}_{6}-{a}_{2}{a}_{7})({a}_{3}-1)}{(1+{a}_{1}{a}_{7}-{a}_{5})}}$ or $1+{a}_{1}{a}_{7}-{a}_{5}=0$ and ${a}_{6}>{a}_{2}{a}_{7}$ |

Fixed Points | Caputo Derivative | CF Operator |
---|---|---|

${\epsilon}_{0}$ | Always saddle | ${a}_{1}>{\displaystyle \frac{1}{1-\alpha}}$ |

${\epsilon}_{1}$ | ${a}_{1}{a}_{2}<{a}_{2}{a}_{3}-{a}_{2}$ and ${a}_{1}{a}_{2}<{a}_{2}{a}_{3}-{a}_{2}$ | ${a}_{1}{a}_{2}<{a}_{2}{a}_{3}-{a}_{2}$ and ${a}_{1}{a}_{2}<{a}_{2}{a}_{3}-{a}_{2}$ or $\frac{{a}_{1}{a}_{4}-{a}_{2}{a}_{3}}{{a}_{2}}}>{\displaystyle \frac{\alpha}{1-\alpha}$ and $\frac{{a}_{1}a6-{a}_{2}{a}_{5}}{{a}_{2}}}>{\displaystyle \frac{\alpha}{1-\alpha}$ |

${\epsilon}_{2}$ | $\frac{{a}_{5}-1}{{a}_{6}}}<{\displaystyle \frac{{a}_{1}}{{a}_{2}}}<{\displaystyle \frac{{a}_{3}-1}{{a}_{4}}$ | $\frac{{a}_{5}-1}{{a}_{6}}}<{\displaystyle \frac{{a}_{1}}{{a}_{2}}}<{\displaystyle \frac{{a}_{3}-1}{{a}_{4}}$ or $\left[1-{a}_{3}-{\displaystyle \frac{{a}_{4}}{{a}_{6}}}(1-{a}_{5})\right]>{\displaystyle \frac{1}{1-\alpha}}$ and ${\scriptstyle \left[{a}_{2}(1-{a}_{5})\pm \sqrt{{a}_{2}^{2}{(1-{a}_{5})}^{2}+4{a}_{6}(1-{a}_{5})({a}_{1}{a}_{6}+{a}_{2}(1-{a}_{5}))}\right]>\frac{2{a}_{6}}{1-\alpha}}$ |

${\epsilon}_{3}$ | $\frac{{a}_{3}-1}{{a}_{4}}}<{\displaystyle \frac{{a}_{1}}{{a}_{2}}}<{\displaystyle \frac{{a}_{5}-1}{{a}_{6}}$ | $\frac{{a}_{3}-1}{{a}_{4}}}<{\displaystyle \frac{{a}_{1}}{{a}_{2}}}<{\displaystyle \frac{{a}_{5}-1}{{a}_{6}}$ or $1-{a}_{4}-{\displaystyle \frac{{a}_{6}}{{a}_{4}}}(1-{a}_{3})+{\displaystyle \frac{{a}_{7}}{{a}_{4}}}[{a}_{1}{a}_{4}+{a}_{2}(1-{a}_{3})]>{\displaystyle \frac{1}{1-\alpha}}$ and ${\scriptstyle \left[{a}_{2}(1-{a}_{3})\pm \sqrt{{a}_{2}^{2}{(1-{a}_{3})}^{2}+4{a}_{4}(1-{a}_{3})[{a}_{1}{a}_{4}+{a}_{2}(1-{a}_{3})]}\right]>\frac{2{a}_{4}}{1-\alpha}}$ |

${\epsilon}_{4}$ | ${a}_{6}>{\displaystyle \frac{{a}_{2}{a}_{4}({a}_{3}-1)[w+{a}_{2}({a}_{3}-1)]}{w({a}_{2}+{a}_{4})+{a}_{2}{a}_{4}({a}_{3}-1)}}$$\left[w={a}_{4}(1+{a}_{1}{a}_{7}-{a}_{5})+({a}_{6}-{a}_{1}{a}_{7})({a}_{3}-1)\right]$ | ${a}_{6}>{\displaystyle \frac{{a}_{2}{a}_{4}({a}_{3}-1)[w+{a}_{2}({a}_{3}-1)]}{w({a}_{2}+{a}_{4})+{a}_{2}{a}_{4}({a}_{3}-1)}}$ or ${\lambda}_{1},{\lambda}_{2},{\lambda}_{3}>{\displaystyle \frac{1}{(1-\alpha )}}$ (see Equations (48)–(50)) |

**Table 3.**The summary of examples;

**C**,

**CF**, and $U\left(0\right)$ denote Caputo, Caputo-Fabrizio, and initial values, respectively, and the notation ✓ indicates the system is asymptotically stable, while ✗ implies unstability.

Example 1 | C | CF | C | CF | |||

Coefficient | $U\left(0\right)$ | Equilibrium | Eigenvalues | $\alpha =0.98$ | $\alpha \le 0.66$ | ||

${a}_{1}=3$ ${a}_{2}=0.5$ ${a}_{3}=4$ ${a}_{4}=3$ ${a}_{5}=4$ ${a}_{6}=9$ ${a}_{7}=4$ | ${x}_{0}=0.5$ ${y}_{0}=0.9$ ${z}_{0}=0.1$ | ${\epsilon}_{0}(0,0,0)$ | ${\lambda}_{0}(-3,-3,3)$ | ✗ | ✗ | ✗ | ✓ |

${\epsilon}_{1}(6,0,0)$ | ${\lambda}_{1}(-3,15,51)$ | ✗ | ✗ | ✗ | ✓ | ||

${\epsilon}_{2}(0.33,0,2.83)$ | ${\lambda}_{2}(-0.083-2.914i,-0.083+2.914i,-2)$ | ✓ | ✓ | ✓ | ✓ | ||

${\epsilon}_{3}(1,2.5,0)$ | ${\lambda}_{3}(-0.25-2.727i,-0.25+2.727i,16)$ | ✗ | ✗ | ✗ | ✓ | ||

${\epsilon}_{4}(1,-1.5,4)$ Not Acceptable | ${\lambda}_{4}(-1.239-5.904i,-1.239+5.904i,1.978)$ | ✗ | ✗ | ✗ | ✗ | ||

Example 2 | C | CF | |||||

Coefficient | $U\left(0\right)$ | Equilibrium | Eigenvalues | $\alpha =0.6$ | |||

${a}_{1}=3$ ${a}_{2}=0.5$ ${a}_{3}=4$ ${a}_{4}=3$ ${a}_{5}=14$ ${a}_{6}=9$ ${a}_{7}=4$ | ${x}_{0}=2$ ${y}_{0}=2$ ${z}_{0}=3$ | ${\epsilon}_{0}(0,0,0)$ | ${\lambda}_{0}(-13,-3,3)$ | ✗ | ✓ | ||

${\epsilon}_{1}(6,0,0)$ | ${\lambda}_{1}(-3,15,41)$ | ✗ | ✓ | ||||

${\epsilon}_{2}(1.44,0,2.28)$ | ${\lambda}_{2}(-0.361-5.429i,-0.361+5.429i,1.333)$ | ✗ | ✗ | ||||

${\epsilon}_{3}(1,2.5,0)$ | ${\lambda}_{3}(-0.25-2.727i,-0.25+2.727i,6)$ | ✗ | ✓ | ||||

${\epsilon}_{4}(1,1,1.5)$ | ${\lambda}_{4}(0.276-4.123i,0.276+4.123i,-1.053)$ | ✓ | ✗ | ||||

Example 3 | C | CF | |||||

Coefficient | $U\left(0\right)$ | Equilibrium | Eigenvalues | $\alpha =0.4$ | |||

${a}_{1}=8$ ${a}_{2}=0.5$ ${a}_{2}=4$ ${a}_{4}=1$ ${a}_{5}=7$ ${a}_{6}=9$ ${a}_{7}=4$ | ${x}_{0}=0.5$ ${y}_{0}=0.1$ ${z}_{0}=5$ | ${\epsilon}_{0}(0,0,0)$ | ${\lambda}_{0}(-6,-3,8)$ | ✗ | ✓ | ||

${\epsilon}_{1}(160,0,0)$ | ${\lambda}_{1}(-8,157,1434)$ | ✗ | ✓ | ||||

${\epsilon}_{2}(0.666,0,7.966)$ | ${\lambda}_{2}(-0.016-6.913i,-0.016+6.913i,-2.333)$ | ✓ | ✓ | ||||

${\epsilon}_{3}(3,7.85,0)$ | ${\lambda}_{3}(-0.075-4.852i,-0.075+4.852i,52.4)$ | ✗ | ✓ | ||||

${\epsilon}_{4}(3,-5.25,13.1)$ Not Acceptable | ${\lambda}_{4}(-1.274-18.50i,-1.274+18.50i,2.398)$ | ✗ | ✓ |

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**MDPI and ACS Style**

Khalighi, M.; Eftekhari, L.; Hosseinpour, S.; Lahti, L.
Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators. *Symmetry* **2021**, *13*, 368.
https://doi.org/10.3390/sym13030368

**AMA Style**

Khalighi M, Eftekhari L, Hosseinpour S, Lahti L.
Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators. *Symmetry*. 2021; 13(3):368.
https://doi.org/10.3390/sym13030368

**Chicago/Turabian Style**

Khalighi, Moein, Leila Eftekhari, Soleiman Hosseinpour, and Leo Lahti.
2021. "Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators" *Symmetry* 13, no. 3: 368.
https://doi.org/10.3390/sym13030368