# The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

- (i)
- Where the constant ratio of the host populations is safe within refuge.
- (ii)
- Where the constant number is protected.

## 2. Stability Analysis around Equilibria

**Lemma**

**1.**

- The equilibrium point $\Omega \left({x}^{\star},{y}^{\star}\right)$ is a sink, if and only if $F\left(-1\right)>0$ and $F\left(0\right)<1$.
- The equilibrium point $\Omega \left({x}^{\star},{y}^{\star}\right)$ is a source, if and only if $F\left(-1\right)>0$ and $F\left(0\right)>1$.
- The equilibrium point $\Omega \left({x}^{\star},{y}^{\star}\right)$ is a saddle point, if and only if $F\left(-1\right)<0$.

- The unique steady-state $\left({x}^{\star},{y}^{\star}\right)$ is locally asymptotically stable if and only if:$$\frac{a\left(1+r\right)\left(r{x}_{0}-{x}^{\star}\right){y}^{\star}}{r\left({x}_{0}+r{x}_{0}-2{x}^{\star}\right)}<{e}^{-a{y}^{\star}}-1\mathrm{and}\frac{a\left({x}^{\star}-r{x}_{0}\right){y}^{\star}}{\left({x}^{\star}-{x}_{0}\right)}1-{e}^{-a{y}^{\star}}.$$
- The unique steady-state $\left({x}^{\star},{y}^{\star}\right)$ is unstable if and only if:$$1+\frac{a\left(1+r\right)\left(r{x}_{0}-{x}^{\star}\right){y}^{\star}}{r\left({x}_{0}+r{x}_{0}-2{x}^{\star}\right)}<{e}^{-a{y}^{\star}}\mathrm{and}\frac{a\left({x}^{\star}-r{x}_{0}\right){y}^{\star}}{\left({x}^{\star}-{x}_{0}\right)}\frac{{e}^{a{y}^{\star}}-1}{{e}^{a{y}^{\star}}}.$$
- The unique steady-state $\left({x}^{\star},{y}^{\star}\right)$ is saddle if and only if:$${e}^{-a{y}^{\star}}-1<\frac{a\left(1+r\right)\left(r{x}_{0}-{x}^{\star}\right){y}^{\star}}{r\left({x}_{0}+r{x}_{0}-2{x}^{\star}\right)}.$$
- The eigenvalues of the characteristic polynomial are complex conjugate with magnitude 1, if and only if:$$\left|\frac{\left({x}^{\star}-r{x}_{0}\right)\left({e}^{a{y}^{\star}}\left(r+ay\right)-r\right)}{\left({e}^{a{y}^{\star}}-1\right)r\left({x}^{\star}-{x}_{0}\right)}\right|<2,$$$$r=\frac{{e}^{-a{y}^{\star}}\left({x}^{\star}-{x}_{0}+{e}^{a{y}^{\star}}\left({x}_{0}+{x}^{\star}\left(a{y}^{\star}-1\right)\right)\right)}{a{x}_{0}{y}^{\star}}.$$

## 3. Neimark–Sacker Bifurcation in Model (2)

**Theorem**

**1.**

## 4. Neimark–Sacker Bifurcation in Model (3)

**Theorem**

**2.**

## 5. Chaos Control

## 6. Numerical Simulations

**Example**

**1.**

**Example**

**2.**

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Bifurcation diagrams and MLE of model (2) with a = 0.4, p = 0.85, and r ∈ [2, 6] (

**a**) bifurcation diagram for host (

**b**) bifurcation diagram for parasitoid (

**c**) MLE.

**Figure 3.**Phase portraits of model (2) in x

_{n}y

_{n}-plane for a = 0.4, p = 0.85 and (x

_{0}, y

_{0}) = (8.12, 6.36). (

**a**) r = 4.614946371656745 (

**b**) r = 4.6 (

**c**) r = 4.5 (

**d**) r = 3.5.

**Figure 5.**Bifurcation diagrams and MLE of model (3) when a = 0.2 and H

_{0}= 2.5. (

**a**) bifurcation for host (

**b**) bifurcation in parasitoid (

**c**) MLE.

**Figure 6.**Phase portraits of model (3) in x

_{n}y

_{n}-plane for a = 0.2, H

_{0}= 2.5. (

**a**) r = 4.41524 (

**b**) r = 4.4 (

**c**) r = 4.3 (

**d**) r = 4.45.

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**MDPI and ACS Style**

Shabbir, M.S.; Din, Q.; Alfwzan, W.F.; De la Sen, M.
The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect. *Axioms* **2023**, *12*, 290.
https://doi.org/10.3390/axioms12030290

**AMA Style**

Shabbir MS, Din Q, Alfwzan WF, De la Sen M.
The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect. *Axioms*. 2023; 12(3):290.
https://doi.org/10.3390/axioms12030290

**Chicago/Turabian Style**

Shabbir, Muhammad Sajjad, Qamar Din, Wafa F. Alfwzan, and Manuel De la Sen.
2023. "The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect" *Axioms* 12, no. 3: 290.
https://doi.org/10.3390/axioms12030290