# Stability and bifurcation analysis in a host–parasitoid model with Hassell growth function

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## Abstract

In this paper, a discrete-time host–parasitoid model with Hassell growth function for the host is considered. The sufficient conditions for the existence of the equilibrium points are obtained and a local stability analysis of the model is performed. By using the bifurcation theory it is shown that the system undergoes a Neimark–Sacker bifurcation. In addition, bifurcation diagrams and phase portraits of the model are given.

## Keywords

Equilibrium point Host–parasitoid model Local stability Population model## MSC

39A28 39A30 92B05## 1 Introduction

Discrete models have been applied most readily to groups such as an insect population where there is a rather natural division of time into discrete generations. A model which has received considerable attention from experimental and theoretical biologists is the host–parasitoid system [1]. In mathematical biology, the host–parasitoid interactions are very popular subjects since they are important to address the natural enemy of an insect pest. Parasitoids are insect species of which larvae develop as parasites on other insect species. Parasitoid larvae usually kill their host (sometimes the host is paralyzed by the ovipositing parasitoid female) whereas adult parasitoids are free-living insects. Parasitoids and their hosts often have synchronized life-cycles, e.g., both have one generation per year (monovoltinous).

*t*, respectively. The functions

*F*and

*G*give the details of the host–parasitoid interactions. Many authors have investigated various models considering different functions derived from biological facts. The simplest version of the host–parasitoid interactions is the Nicholson–Bailey model given as follows [5]:

*Encarsia formosa*, and the host,

*Trialeurodes vaporariorum*[6].

*R*is the intrinsic growth rate,

*a*is a scaling parameter affecting the equilibrium population size, and

*b*incorporates density-dependent effects such as intra-specific competition. Hassell et al. [13] collected

*R*and

*b*values for about two dozen species from field and laboratory observations and noted that the majority of these cases were within the stable region [15]. In this work, we give conditions for the existence of the equilibrium points of the model (5) and discuss the linear stability of these equilibrium points. By using the bifurcation theory we obtain sufficient conditions for the direction and existence of the Neimark–Sacker bifurcation. All theoretical results obtained are supported with numerical simulations. The bifurcation diagrams and phase portraits are given.

## 2 Equilibrium points and stability analysis of the model (3)

In this section, we will give conditions for existence of equilibrium points of the model (5) and discuss the local stability conditions of these equilibrium points.

### Theorem 2.1

*For the system*(5),

*the following statements hold true*.

- (a)
*There exists an extinction equilibrium point*\(( 0,0 )\). - (b)
*If*\(R\geq1\),*then there exist two equilibrium points as*\(( 0,0 ) \)*and*\(( H_{1}^{\ast},0 )\). - (c)
*If*\(0<\frac{ (1+aH^{\ast} )^{b}}{R}e^{mH^{\ast}}<1\)*then there exists an equilibrium point*\(( H^{\ast},P^{\ast} )\).

### Proof

*f*intersects the horizontal line \(w=R\), we take equilibrium points. Notice that

*f*is a continuous function, \(f(0)=1\), \(f^{{\prime}}(x)>0\), \(\lim_{x\rightarrow \infty}f(x)=\infty\) and when \(R\geq1\), there is a unique intersection point. In Fig. 1, we give the graphs equation (8) with some values of the parameters.

*F*intersects the horizontal line \(z=R\), we can obtain some equilibrium points. By solving \(F^{\prime}(x)=0\), we obtain the following equation:

Since \(F(0)=1,F^{{\prime}}(0)>0\), \(F(x)\longrightarrow0\) as \(x\rightarrow \infty\), the critical point is a local maximum (Fig. 2(b)). □

To determine stability conditions of the discrete system, we can use the following lemma comprising what are called the Schur–Cohn criteria.

### Lemma 2.2

([17])

*The characteristic polynomial*, \(p(\lambda)=\lambda^{2}+p_{1}\lambda+p_{0}\)

*has all its roots inside the unit open disk if and only if*

### Theorem 2.3

*For the model*(5),

*the following statements hold true*.

- (a)
*If*\(|R|<1\),*equilibrium point*\((0,0)\)*is locally asymptotically stable*. - (b)
*If*\(\vert -e^{-mH_{1}^{\ast}}v^{-1-b}R(-v+aH_{1}^{\ast }b+H_{1}^{\ast}mv) \vert <1\)*and*\(H_{1}^{\ast}<\frac{1}{k}\)*then equilibrium point*\((H_{1}^{\ast},0)\)*is locally asymptotically stable*. - (c)
*Suppose that*\(-v+abH^{\ast}+H^{\ast}mv>0\), \(2v-abH^{\ast}+H^{\ast }kv-H^{\ast}mv>0\)*and*\(-1+bH^{\ast}k>0\).*Assume that*$$\begin{aligned} &R_{1}=\frac{e^{H^{\ast}m}v^{b}k(v+abH^{\ast}+H^{\ast }mv)}{ab+kv+mv}, \end{aligned}$$(14)$$\begin{aligned} &R_{2}=\frac{e^{H^{\ast}m}H^{\ast}v^{b}k(-v+abH^{\ast}+H^{\ast }mv)}{2v-abH^{\ast}+H^{\ast}kv-H^{\ast}mv}, \end{aligned}$$(15)$$\begin{aligned} &R_{3}=\frac{e^{H^{\ast}m}H^{\ast2}(v)^{-1+b}k(ab+mv)}{-1+bH^{\ast }k}, \end{aligned}$$(16)*where*\(v=1+aH^{\ast}\).*The positive equilibrium point*\(( H^{\ast },P^{\ast} ) \)*of the system*(5)*is locally asymptotically stable if and only if*\(\max\{R_{1},R_{2}\}< R< R_{3}\).

### Proof

*J*at the coexistence equilibrium point \(( H^{\ast},P^{\ast} ) \) of the following form in which \(P^{\ast }\) is eliminated:

### Example 2.4

## 3 Bifurcation analysis

In this section, we will investigate the existence of the Neimark–Sacker bifurcation for the model (5) and present the conditions and direction of the Neimark–Sacker bifurcation [18, 19, 20, 21, 22, 23, 24, 25].

A Neimark–Sacker bifurcation occurs at a bifurcation point if and only if system (5) satisfies the following conditions: eigenvalue assignment, transversality and nonresonance condition [26, 27]. The following lemma gives the eigenvalue assignment condition for the Neimark–Sacker bifurcation.

### Lemma 3.1

([28])

*A pair of complex conjugate roots of*\(p(\lambda )=\lambda^{2}+p_{1}\lambda+p_{0}\)

*lie on the unit circle if and only if*

### Proposition 3.2

(Eigenvalue assignment)

*Suppose that*\(-v+abH^{\ast}+H^{\ast}mv>0\), \(2v-abH^{\ast}+H^{\ast}kv-H^{\ast }mv>0\)*and*\(-1+bH^{\ast}k>0\). *If*\(\max \{R_{1},R_{2}\}< R=R_{3\text{ }}\)*then the eigenvalue assignment condition of the Neimark–Sacker bifurcation in Lemma *3.1*holds*.

### Proof

From the condition \(R>R_{1}\) we have \(p(1)>0\). On the other hand, the conditions \(-v+abH^{\ast}+H^{\ast}mv>0\), \(2v-abH^{\ast}+H^{\ast }kv-H^{\ast}mv>0\) and \(R>R_{2}\) lead to \(p(-1)>0\). Solving the equation \(D_{1}^{-}=1-p_{0}=0\) with the fact \(-1+bH^{\ast}k>0\), we get \(R=R_{3}\). This completes the proof. □

### Theorem 3.3

*Suppose that*\((H^{\ast},P^{\ast})\)*is the positive equilibrium point of the system* (5). *If Proposition* (3.2) *holds*, \(R\neq\frac{e^{H^{\ast}m}H^{\ast} v^{1+b}k}{-v+abH^{\ast}+H^{\ast }mv}\), \(R\neq\frac{e^{H^{\ast}m}H^{\ast} v^{1+b}k}{-1-v+abH^{\ast}+H^{\ast}mv}\), *and*\(a(0)<0\) (*respectively*\(a(0)>0\)), *then the Neimark–Sacker bifurcation of the system* (5) *at*\(R=R_{3}\)*is supercritical* (*respectively*, *subcritical*) *and there exists a unique closed invariant curve bifurcation from*\((H^{\ast},P^{\ast})\)*for*\(R=R_{3}\), *which is asymptotically stable* (*respectively*, *unstable*).

### Example 3.4

Let us take \(m=1,a=1.1,b=1.15\), \(k=2.2\) and initial condition \((H_{0},P_{0})=(0.5,0.6)\). From solutions of the system (7) and (16), the positive equilibrium point and bifurcation point of the system (5) are obtained: \((H^{\ast},P^{\ast})=(0.76664,0.525223)\) and \(R_{3}=13.81043\), respectively.

*B*and

*C*are defined by

*g*is a complex-valued smooth function). The Taylor expression of g with respect \((z,\overline{z})=(0,0)\) is

### 3.1 Bifurcation analysis of model (4)

In this section, we give stability conditions of coexistence equilibrium points of the model (6) and show that the model (6) undergoes a Neimark–Sacker bifurcation. Also, we obtain the bifurcation diagrams and phase portraits.

### Theorem 3.5

*Suppose that*\(-1-H^{\ast}+bH^{\ast}>0\), \(2+2H^{\ast }-bH^{\ast}+H^{\ast}k+H^{\ast2}k>0\)

*and*\(-1+H^{\ast}k>0\).

*Assume that*

*The positive equilibrium point*\(( H^{\ast},P^{\ast} ) \)

*of the system*(6)

*is locally asymptotically stable if and only if*\(\max \{R_{11},R_{12}\}< R< R_{13}\).

### Example 3.6

### Proposition 3.7

(Eigenvalue assignment)

*Suppose that*\(-1-H^{\ast }+bH^{\ast}>0\), \(2+2H^{\ast}-bH^{\ast}+H^{\ast}k+H^{\ast2}k>0\)*and*\(-1+H^{\ast}k>0\). *If*\(\max\{R_{11},R_{12}\}< R=R_{13}\)*then the eigenvalue assignment condition of the Neimark–Sacker bifurcation holds*.

### Theorem 3.8

*Suppose that*\((H^{\ast},P^{\ast})\)*is positive equilibrium point of the system* (6). *If the Proposition* (3.7) *holds*, \(R\neq\frac{H^{\ast}(1+H^{\ast})^{1+b}k}{-1-H^{\ast}+bH^{\ast}}\), \(R\neq\frac{H^{\ast}(1+H^{\ast})^{1+b}k}{-2-2H+bH}\)*and*\(a(0)<0\) (*respectively*\(a(0)>0\)), *then the Neimark–Sacker bifurcation of the system* (6) *at*\(R=R_{13}\)*is supercritical* (*respectively*, *subcritical*) *and there exists a unique closed invariant curve bifurcation from*\((H^{\ast},P^{\ast})\)*for*\(R=R_{13}\), *which is asymptotically stable* (*respectively*, *unstable*).

### Example 3.9

## 4 Conclusion

This study deals with the stability and bifurcation analysis of a discrete-time host–parasitoid model with Hassell growth function for the host. The existence of the equilibrium points and stability conditions of the system (5) are given. Also, we show that model (5) undergoes a Neimark–Sacker bifurcation by using bifurcation theory.

In the literature, many researchers [29, 30, 31] have reported that discrete host–parasitoid models can have very complex dynamics e.g. exhibit periodic and chaotic dynamics. In the study [29], the authors show that there is a stable coexistence between the host and the parasitoid for a large range of the parameter *r* (intrinsic growth rate for the host population), beyond which the system goes through a quasi-periodicity including a Neimark–Sacker bifurcation. When *r* is slightly increased beyond a threshold value, a chaotic attractor abruptly appears and the periodic attractor disappears. These results are also valid for our system. Figure 4 shows the bifurcation diagram of the system (5) for the parasitoid population and the host population with \(m = 1\), \(a = 1.1\), \(b = 1.15\) and \(k = 2.2\) as the parameter *R* (intrinsic growth rate for the host population) increases. If the parameter *R* reaches 13.2983, then quasi-periodic solutions occur due to the Neimark–Sacker bifurcation.

We note that the model (6) which is studied by [16] is the special case of model (5) for \(m=0\) and \(a=1\). In [16], the authors have not given the stability conditions for coexistence equilibrium point and they have not performed a bifurcation analysis. In this paper, we give stability conditions of the coexistence equilibrium point of model (6) and show that model (6) undergoes a Neimark–Sacker bifurcation, too. Finally, all theoretical results for model (5)–(6) obtained are supported with numerical simulations.

## Notes

### Acknowledgements

The authors would like to thank the referee and the editor for their valuable comments which led to improvement of this work.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors participated in drafting and checking the manuscript, read and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

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