# Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Hypothesis**

**1**

**(H1).**

**Hypothesis**

**2**

**(H2).**

**Lemma**

**1**

**.**If $p>0,\phantom{\rule{4pt}{0ex}}q>0$ and $\dot{s}\left(\mu \right)\le q-ps\left(\mu \right)$, we have

**Lemma**

**2**

**.**If $s\left(\mu \right)\ge 0$ and ${lim}_{\mu \to \infty}s\left(\mu \right)\le {M}_{s}$ such that

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Remark**

**1.**

## 4. Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The global attractivity, periodic solution and permanence of system (34). Here, we take different initial values. (

**a**) Permanence, periodicity and global attractivity of ${z}_{1}(t)$, ${z}_{2}(t)$ and ${z}_{3}(t)$. (

**b**) Permanence, periodicity and global attractivity of ${z}_{1}(t)$ and ${z}_{2}(t)$. (

**c**) Permanence, periodicity and global attractivity of ${z}_{1}(t)$ and ${z}_{3}(t)$. (

**d**) Permanence, periodicity and global attractivity of ${z}_{2}(t)$ and ${z}_{3}(t)$. (

**e**) Permanence, periodicity and global attractivity of ${z}_{1}(t)$, ${z}_{2}(t)$ and ${z}_{3}(t)$.

**Figure 2.**Dynamical behaviors of system (35). Here, we take different initial values. (

**a**) Permanence and non-global attractivity of ${z}_{1}(t)$. (

**b**) Permanence and non-global attractivity of ${z}_{2}(t)$. (

**c**) Permanence and non-global attractivity of ${z}_{3}(t)$. (

**d**) Permanence and non-global attractivity of ${z}_{1}(t)$ and ${z}_{2}(t)$. (

**e**) Permanence and non-global attractivity of ${z}_{1}(t)$ and ${z}_{3}(t)$. (

**f**) Permanence and non-global attractivity of ${z}_{2}(t)$ and ${z}_{3}(t)$. (

**g**) Permanence and non-global attractivity of ${z}_{1}(t)$, ${z}_{2}(t)$ and ${z}_{3}(t)$.

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**MDPI and ACS Style**

Jiang, Z.; Halik, A.; Muhammadhaji, A.
Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays. *Axioms* **2023**, *12*, 501.
https://doi.org/10.3390/axioms12050501

**AMA Style**

Jiang Z, Halik A, Muhammadhaji A.
Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays. *Axioms*. 2023; 12(5):501.
https://doi.org/10.3390/axioms12050501

**Chicago/Turabian Style**

Jiang, Zhao, Azhar Halik, and Ahmadjan Muhammadhaji.
2023. "Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays" *Axioms* 12, no. 5: 501.
https://doi.org/10.3390/axioms12050501