# Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models

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## Abstract

**:**

## 1. Introduction

- The UR and HP models individually can exhibit chaotic dynamics for the same parameter regimes. However, they will synchronize when coupled accordingly.
- This synchronization will occur only for small-to-moderate initial conditions.
- For larger initial conditions, the UR and HP models will not synchronize. This is shown numerically and analytically.
- For small initial conditions, the modified UR and HP models will synchronize, but for larger initial conditions, it is numerically seen that they will not synchronize.
- Thus, we reaffirm that the synchronization of three species’ food chains with different top-down control (differently behaving top predators) is caused solely by the top predator.

## 2. Generalized Synchronization Using the OPCL Coupling Method

**Definition 1.**

**Definition 2.**

**Remark 1.**

## 3. Model Systems

#### 3.1. Upadhyay–Rai (UR) Model

#### 3.2. Hastings–Powell (HP) Model

## 4. Generalized Synchronization of the UR Model and HP Model Using the OPCL Coupling Method

## 5. Numerical Results

#### 5.1. Chaos in the UR Model and HP Model for Small Initial Data

#### 5.2. GS for the UR Model and HP Model for Small Initial Data

## 6. Possible Causes of a Lack of Synchronization

**Theorem 1.**

**Remark 2.**

**Proof.**

**Remark 3.**

**Theorem 2.**

**Proof.**

#### No Synchronization for the UR Model or HP Model for Large Initial Data

**Remark 4.**

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Step 1.**- Identify the driver and response systems.
**Step 2.**- Compute the Jacobian of the response system.
**Step 3.**- Construct an H-matrix from the Jacobian and choose values for the H-matrix such that the Routh–Hurwitz criterion is satisfied.
**Step 4.**- Construct the $\alpha $ transformation matrix which ensures the desired goal dynamics.
**Step 5.**- Propose a coupling to achieve the GS state.

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**Figure 1.**Chaotic dynamics with initial data $[{x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right)]=[10,10,10]$ in UR model. (

**a**) Phase diagram showing chaotic attractor in UR model. (

**b**) Time series plot for ${x}_{1}$. (

**c**) Time series plot for ${x}_{2}$. (

**d**) Time series plot for ${x}_{3}$.

**Figure 2.**Chaotic dynamics with initial data $[{y}_{1}\left(0\right),{y}_{2}\left(0\right),{y}_{3}\left(0\right)]=[10,10,10]$ in HP model. (

**a**) Phase diagram showing chaotic attractor in HP model. (

**b**) Time series plot for ${y}_{1}$. (

**c**) Time series plot for ${y}_{2}$. (

**d**) Time series plot for ${y}_{3}$.

**Figure 3.**Chaotic dynamics with small initial data $[{x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right)]=[0.01,0.01,0.1]$ in UR model. (

**a**) Phase diagram showing chaotic attractor in UR model. (

**b**) Time series plot for ${x}_{1}$. (

**c**) Time series plot for ${x}_{2}$. (

**d**) Time series plot for ${x}_{3}$.

**Figure 4.**Chaotic dynamics with small initial data $[{y}_{1}\left(0\right),{y}_{2}\left(0\right),{y}_{3}\left(0\right)]=[0.01,0.01,0.01]$ in HP model. (

**a**) Phase diagram showing chaotic attractor in HP model. (

**b**) Time series plot for ${y}_{1}$. (

**c**) Time series plot for ${y}_{2}$. (

**d**) Time series plot for ${y}_{3}$.

**Figure 5.**GS with small initial data. (

**a**) ${y}_{1}$ − ${g}_{1}$ plot showing a 1:1 correlation. (

**b**) ${y}_{2}$ − ${g}_{2}$ plot showing a 1:1 correlation. (

**c**) ${y}_{3}$ − ${g}_{3}$ plot showing a 1:1 correlation. (

**d**) Simulation showing time evolution for ${x}_{1}$ and ${y}_{1}$ after transients die out. (

**e**) Simulation showing time evolution for ${g}_{1}$ and ${y}_{1}$ after transients die out. (

**f**) Time series plot showing no blow-up in ${x}_{3}$ after transients die out.

**Figure 6.**No GS with large initial data. (

**a**) ${y}_{1}$ − ${g}_{1}$ plot showing no 1:1 correlation. (

**b**) ${y}_{2}$ − ${g}_{2}$ plot showing no 1:1 correlation. (

**c**) ${y}_{3}$ − ${g}_{3}$ plot showing no 1:1 correlation. (

**d**) Simulation showing time evolution for ${x}_{1}$ and ${y}_{1}$. (

**e**) Simulation showing time evolution for ${g}_{1}$ and ${y}_{1}$. (

**f**) Time series plot showing blow-up occurring in ${x}_{3}$ and estimated at time ${T}^{*}$$\approx 0.00048$.

**Figure 7.**GS for modified UR model and HP model with initial data $[{x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right)]=[0.0055,0.0014,0.0015]$ and $[{y}_{1}\left(0\right),{y}_{2}\left(0\right),{y}_{3}\left(0\right)]=[0.0026,0.0084,0.0025]$. (

**a**) Phase plot for ${x}_{1}$ and ${x}_{2}$. (

**b**) Phase plot for ${y}_{1}$ and ${y}_{2}$. (

**c**) Time series plot for ${x}_{3}$ and ${y}_{3}$. (

**d**) ${y}_{1}$ − ${g}_{1}$ plot showing 1:1 correlation. (

**e**) ${y}_{2}$ − ${g}_{2}$ plot showing 1:1 correlation. (

**f**) ${y}_{3}$ − ${g}_{3}$ plot showing 1:1 correlation.

**Figure 8.**GS for modified UR model and HP model with initial data $[{x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right)]=[0.001,0.001,0.001]$ and $[{y}_{1}\left(0\right),{y}_{2}\left(0\right),{y}_{3}\left(0\right)]=[0.001,0.001,0.001]$. (

**a**) Phase plot for ${x}_{1}$ and ${x}_{2}$. (

**b**) Phase plot for ${y}_{1}$ and ${y}_{2}$. (

**c**) Time series plot for ${x}_{3}$ and ${y}_{3}$. (

**d**) ${y}_{1}$ − ${g}_{1}$ plot showing 1:1 correlation. (

**e**) ${y}_{2}$ − ${g}_{2}$ plot showing 1:1 correlation. (

**f**) ${y}_{3}$ − ${g}_{3}$ plot showing 1:1 correlation.

**Figure 9.**No synchronization for modified UR model and HP model with initial data $[{x}_{1}\left(0\right),{x}_{2}\left(0\right),{x}_{3}\left(0\right)]=[1000,1000,1000]$ and $[{y}_{1}\left(0\right),{y}_{2}\left(0\right),{y}_{3}\left(0\right)]=[1000,1000,1000]$. (

**a**) ${y}_{1}$ − ${g}_{1}$ plot showing 1:1 correlation. (

**b**) ${y}_{2}$ − ${g}_{2}$ plot showing no 1:1 correlation. (

**c**) ${y}_{3}$ − ${g}_{3}$ plot showing no 1:1 correlation. (

**d**) Simulation showing time evolution for ${x}_{1}$ − ${y}_{1}$. (

**e**) Time series plot for ${y}_{1}$ − ${g}_{1}$. (

**f**) Time series plot showing blow-up occurring in ${y}_{3}$ and estimated at time ${T}^{*}$$\approx 0.1008$.

Symbols | Description |
---|---|

${x}_{1}$ | prey |

${x}_{2}$ | middle predator |

${x}_{3}$ | top predator |

${a}_{1}$ | intrinsic growth rate of prey |

${b}_{1}$ | measure of competition among prey |

${a}_{2}$ | intrinsic death rate of ${x}_{2}$ in the absence of food ${x}_{1}$ only |

D, ${D}_{1}$ | measure of the level of protection offered to the prey by the environment |

${D}_{2}$ | value of ${x}_{2}$ at which its per capita removal rate becomes ${w}_{2}/2$ |

${D}_{3}$ | Loss of ${x}_{3}$ due to lack of favorite food ${x}_{2}$ |

c | growth rate of ${x}_{3}$ via sexual reproduction |

$w,{w}_{i}^{\prime}s$ | maximum value that per capital rate can attain |

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## Share and Cite

**MDPI and ACS Style**

Takyi, E.M.; Parshad, R.D.; Upadhyay, R.K.; Rai, V. Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models. *Algorithms* **2023**, *16*, 180.
https://doi.org/10.3390/a16040180

**AMA Style**

Takyi EM, Parshad RD, Upadhyay RK, Rai V. Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models. *Algorithms*. 2023; 16(4):180.
https://doi.org/10.3390/a16040180

**Chicago/Turabian Style**

Takyi, Eric M., Rana D. Parshad, Ranjit Kumar Upadhyay, and Vikas Rai. 2023. "Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models" *Algorithms* 16, no. 4: 180.
https://doi.org/10.3390/a16040180