# Influence of Spatial Dispersal among Species in a Prey–Predator Model with Miniature Predator Groups

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## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model

## 3. Linear Stability Analysis

- ${E}_{0}=(0,0)$,
- ${E}_{a}=(1,0)$, and
- ${E}^{*}=({u}^{*},{v}^{*})$.

- $Tr\left({J}_{{E}^{*}}\right)<0$,
- $Det\left({J}_{{E}^{*}}\right)>0$.

**Theorem**

**1.**

- 1.
- ${u}^{*}(\frac{s{v}^{*}+\delta ({u}^{*}-(\gamma -1){{v}^{*}}^{\gamma})}{{({u}^{*}+{{v}^{*}}^{\gamma})}^{2}}-2)+1<d\delta +\frac{s{v}^{*}}{{u}^{*}+{{v}^{*}}^{\gamma}}$,
- 2.
- $\delta {({u}^{*}+{{v}^{*}}^{\gamma})}^{2}({{v}^{*}}^{\gamma}(ds{v}^{*}+2{{u}^{*}}^{2}(\gamma +2d-1)-{u}^{*}(\gamma +2d-1))+(d-1){{u}^{*}}^{2}(2{u}^{*}-1)+d(2{u}^{*}-1){{v}^{*}}^{2\gamma})>0$.

#### Hopf Bifurcation Analysis

**Theorem**

**2.**

- 1.
- ${\beta}_{1}\left(\overline{d}\right)=0,$
- 2.
- ${\left[Re\left({\sigma}^{\prime}\left(d\right)\right)\right]}_{d=\overline{d}}\ne 0.$

**Proof.**

## 4. Diffusion-Driven Instability

**Theorem**

**3.**

- 1.
- ${a}_{11}+{a}_{22}<0$ and ${a}_{11}{a}_{22}-{a}_{12}{a}_{21}>0,$
- 2.
- ${a}_{11}{d}_{2}+{a}_{22}{d}_{1}>0$,
- 3.
- ${({a}_{11}{d}_{2}+{a}_{22}{d}_{1})}^{2}>4({a}_{11}{a}_{22}-{a}_{21}{a}_{12}){d}_{1}{d}_{2}$

## 5. Weakly Nonlinear Analysis

- The first solution: The stationary state is represented by ${\phi}_{1}={\phi}_{2}={\phi}_{3}=0$, which is stable for $\mu <{\mu}_{2}=0$ and unstable for $\mu >{\mu}_{2}=0$.
- The second solution: The stripe pattern given by ${\phi}_{1}=\sqrt{\frac{\mu}{{b}_{1}}}\ne 0,\phantom{\rule{4pt}{0ex}}{\phi}_{2}={\phi}_{3}=0$ is stable for $\mu >{\mu}_{3}=\frac{{h}^{2}{b}_{1}}{{({b}_{2}-{b}_{1})}^{2}}$ and unstable for $\mu <{\mu}_{3}$.
- The third solution: When $\mu >{\mu}_{1}$, two solutions exist: Hexagon patterns are given by ${\phi}_{1}={\phi}_{2}={\phi}_{3}=\frac{\left|h\right|\pm \sqrt{{h}^{2}+4({b}_{1}+2{b}_{2})\mu}}{2({b}_{1}+2{b}_{2})}$ with $\varphi =0$ or $\pi $, and exist when $\mu >{\mu}_{1}=\frac{-{h}^{2}}{4({b}_{1}+2{b}_{2})}$; the solution ${\phi}^{+}=\frac{\left|h\right|+\sqrt{{h}^{2}+4({b}_{1}+2{b}_{2})\mu}}{2({b}_{1}+2{b}_{2})}$ is stable for $\mu <{\mu}_{4}=\frac{2{b}_{1}+{b}_{2}}{{({b}_{2}-{b}_{1})}^{2}}{h}^{2}$, and ${\phi}^{-}=\frac{\left|h\right|-\sqrt{{h}^{2}+4({b}_{1}+2{b}_{2})\mu}}{2({b}_{1}+2{b}_{2})}$ is unstable.
- The mixed states: When ${\phi}_{1}=\frac{\left|h\right|}{{b}_{2}-{b}_{1}},\phantom{\rule{4pt}{0ex}}{\phi}_{2}={\phi}_{3}=\sqrt{\frac{\mu -{b}_{1}{{\phi}_{1}}^{2}}{{b}_{1}+{b}_{2}}},$ with ${b}_{2}>{b}_{1}$. This exists when $\mu >{\mu}_{3}$ and is always unstable.

**Theorem**

**4.**

- 1.
**Homogeneous solution:**It is stable for $\mu <{\mu}_{2}$ and unstable for $\mu >{\mu}_{2}$.- 2.
**Stripe solution:**It is stable for $\mu >{\mu}_{3}$ and unstable for $\mu <{\mu}_{3}$.- 3.
**Hexagonal solution:**It exists when $\mu >{\mu}_{1}$ and ${\phi}^{+}$ is stable for $\mu <{\mu}_{4}$ and ${\phi}^{-}$ is always unstable.- 4.
**Mixed solution:**It exists but is always unstable when ${b}_{2}>{b}_{1}$ and $\mu >{\mu}_{3}$.

## 6. Numerical Simulations

#### 6.1. Nonspatial Analysis

#### 6.2. Pattern Selection

**Effects of varying ${\mathit{d}}_{\mathbf{1}}$ on pattern formation**

**Effects of varying $\mathit{d}$ on pattern formation**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) The value of the Turing bifurcation parameter corresponding to the diffusion−driven instability. (

**B**) Turing region in the ${d}_{1}-{d}_{2}$ plane. (

**C**) Plot of $Re\left(\sigma \right)$ versus k for varying ${d}_{1}$. (

**D**) Plot of ${\Lambda}_{2}\left({k}^{2}\right)$ versus k for varying ${d}_{1}$. Fixed parameters: $s=2$, $\gamma =\frac{2}{3}$, $\delta =1$, $d=0.3$, and ${d}_{2}=1$.

**Figure 2.**(

**Left**) Time evolution of the population; (

**Right**) phase portrait for system (2). Parameters: $s=2$, $\gamma =\frac{2}{3}$, and $\delta =1$ with varying value of d: (

**A**): $d=0.3$, (

**B**): $d=0.25$, (

**C**): $d=0.2$, and (

**D**): $d=0.12$.

**Figure 3.**The bifurcation diagrams with respect to d for model (2), where blue solid curves represent the stable steady state, limit point cycle (LPC), Hopf bifurcation point (H), and the periodic solution (green color region). With $s=2$, $\gamma =\frac{2}{3}$, $\delta =1$.

**Figure 4.**Bifurcation diagram of amplitude $\phi $ versus $\mu $ with different values of d for model (3). S: strip patterns, ${H}_{0}$: hexagonal patterns ($\varphi =0$), and ${H}_{\pi}$: hexagonal patterns ($\varphi =\pi $). Fixed parameters: $s=2$, $\gamma =\frac{2}{3}$, $\delta =1$, ${d}_{1}=0.02$, and ${d}_{2}=1$.

**Figure 5.**(

**Left**) Time−series evolution of the populations of the model (3). (

**Right**) Two −dimensional Turing patterns for fixed parameters $s=2$, $\gamma =\frac{2}{3}$, $\delta =1$, $d=0.3$, and ${d}_{2}=1$ under varying values of ${d}_{1}$.

**Figure 6.**Two-dimensional Turing patterns stability for fixed parameters $s=2$, $\gamma =\frac{2}{3}$, $\delta =1$, $d=0.3$, and ${d}_{2}=1$.

**Figure 7.**Mean population density of prey and predator: (

**A**) for varying ${d}_{1}$; (

**B**) for varying d with fixed ${d}_{1}=0.02$ and ${d}_{2}=1$.

**Figure 8.**(

**Left**) Time −series evolution of the populations of the model (3). (

**Right**) Two −dimensional Turing patterns. For fixed parameters, $s=2$, $\gamma =\frac{2}{3}$, $\delta =1$, ${d}_{1}=0.02$, and ${d}_{2}=1$ under varying values of d.

Parameter | Biological Meaning |
---|---|

r | is an intrinsic growth rate of prey |

K | is carrying capacity |

c | is attack rate |

$\gamma $ | is Hassell–Varley (H-V) constant |

m | is half-saturation constant |

f | is conversion rate |

D | is death rate of predators |

S.No. | d | ${\mathit{d}}_{1}^{\mathit{T}}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{\mu}}_{1}$ | ${\mathit{\mu}}_{2}$ | ${\mathit{\mu}}_{3}$ | ${\mathit{\mu}}_{4}$ |
---|---|---|---|---|---|---|---|---|

1. | 0.26 | 0.035 | 2930.65 | 5745.24 | −0.000219 | 0 | 0.00468 | 0.0185 |

2. | 0.28 | 0.031 | 3472.19 | 7091.72 | −0.000478 | 0 | 0.00895 | 0.0362 |

3. | 0.3 | 0.027 | 3151.04 | 6871.92 | −0.002559 | 0 | 0.03935 | 0.1645 |

4. | 0.32 | 0.022 | 5040.15 | 10,773.10 | −0.020674 | 0 | 0.33716 | 1.3950 |

S.No. | ${\mathit{d}}_{1}$ | $\mathit{\mu}=\frac{{\mathit{d}}_{1}^{\mathit{T}}-{\mathit{d}}_{1}}{{\mathit{d}}_{1}}$ | Region | Pattern | Figure |
---|---|---|---|---|---|

1. | 0.0265 | 0.01852 | $\mu \in ({\mu}_{2},{\mu}_{3})$ | spots | Figure 5A |

2. | 0.026 | 0.03704 | $\mu \in ({\mu}_{2},{\mu}_{3})$ | spots | Figure 5B |

3. | 0.024 | 0.11111 | $\mu \in ({\mu}_{3},{\mu}_{4})$ | mixed | Figure 5C |

4. | 0.02 | 0.25926 | $\mu >{\mu}_{4}$ | stripes | Figure 5D |

5. | 0.01 | 0.62963 | $\mu >{\mu}_{4}$ | stripes | Figure 5E |

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## Share and Cite

**MDPI and ACS Style**

Shivam; Aljrees, T.; Singh, T.; Varshney, N.; Kumar, M.; Singh, K.U.; Vimal, V.
Influence of Spatial Dispersal among Species in a Prey–Predator Model with Miniature Predator Groups. *Symmetry* **2023**, *15*, 986.
https://doi.org/10.3390/sym15050986

**AMA Style**

Shivam, Aljrees T, Singh T, Varshney N, Kumar M, Singh KU, Vimal V.
Influence of Spatial Dispersal among Species in a Prey–Predator Model with Miniature Predator Groups. *Symmetry*. 2023; 15(5):986.
https://doi.org/10.3390/sym15050986

**Chicago/Turabian Style**

Shivam, Turki Aljrees, Teekam Singh, Neeraj Varshney, Mukesh Kumar, Kamred Udham Singh, and Vrince Vimal.
2023. "Influence of Spatial Dispersal among Species in a Prey–Predator Model with Miniature Predator Groups" *Symmetry* 15, no. 5: 986.
https://doi.org/10.3390/sym15050986