- Research
- Open Access

# Stability analysis of a host parasite model

- Ünal Ufuktepe
^{1}Email author and - Sinan Kapçak
^{1}

**2013**:79

https://doi.org/10.1186/1687-1847-2013-79

© Ufuktepe and Kapçak; licensee Springer. 2013

**Received: **15 November 2012

**Accepted: **8 March 2013

**Published: **28 March 2013

## Abstract

Host parasite models are similar to host parasitoid models except that the parasite does not necessarily kill the host. Leslie/Gower model (Leslie and Gower in Biometrika 47(3/4):219-234, 1960) played a historical role in ecology. We consider the stability of Misra and Mitra’s model (Misra and Mitra in Comput. Math. Appl. 52:525-538, 2006). We study this system analytically and improve the results of Misra and Mitra (Comput. Math. Appl. 52:525-538, 2006).

**MSC:**39A11, 92D25.

## Keywords

- host-parasite
- predator-prey
- Beddington model
- discrete dynamical systems

## 1 Introduction

*predator-prey*model where the growth or decline is exponential in the absence of the other species in both cases (${x}_{t}$ is the prey population). Choosing $a,c>1$ and $b,d>0$ gives rise to

*mutualism*, where both species get a boost from living together. The mass-action terms ${x}_{t}{y}_{t}$ indicate random encounters between the species with a significant outcome (eating or infecting one another,

*etc.*). One of the earliest realistic models of two-species interaction was developed by Nicholson and Bailey, who applied it to the parasitoid

*Encarsia formosa*and the host

*Trialeurodes vaporariorum*. A parasitoid is a parasite that can live on its own as an adult, which then lays eggs into a host, eventually causing its death. The general host-parasitoid model is

where $f({H}_{t},{P}_{t})$ is the fraction of host population that is not parasitized at time *t*, *r* is the fecundity of the host, and *e* is that of the parasite, that is, the average number of successful eggs laid per individual. Note that the parasitoids die in the absence of the host, and for this reason, they have been successfully used for eradicating insect pests. In the Nicholson-Bailey model, the function *f* is determined, under the assumptions of the mass-action principle and Poisson distribution of the number of encounters, to be $f({H}_{t},{P}_{t})={e}^{-c{P}_{t}}$ (the positive constant *c* is called the ‘searching efficiency’ of the parasitoid).

This system represents the rule by which two discrete, cooperating populations reproduce from one generation to the next. The phase variables ${x}_{n}$ and ${y}_{n}$ denote population sizes during the *n* th generation and the sequence or orbit $\{({x}_{n},{y}_{n}):n=0,1,2,\dots \}$ depicts how the populations evolve over time [3].

where $a,b>0$. This is the population model we will assume for our host species. Hassell *et al.* [2] 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.

Note that such simplifications, including the convention $a=1$ in the Hassell model, lead to the interpretation of the ‘population’ variable as a ‘suitable multiple of the population’.

## 2 Fixed points of the system (1.4)

Now, we can see that the parameter *R* is important for the existence of the fixed points other than the extinction fixed point $(0,0)$. We have the following cases.

Case 1: $R\le 1$.

For this case, we have ${R}^{\frac{1}{b}}-1\le 0$ and $\frac{1}{c}ln[\frac{R}{{(1+{H}^{\ast})}^{b}}]<0$. Hence, there is no exclusion and coexistence fixed point for $R<1$.

Case 2: $R>1$.

When the graph of *F* intersects the horizontal line $z=R$, some fixed points are obtained. Note that $x={R}^{\frac{1}{b}}-1$ is a solution of the equation $F(x)=R$, which corresponds to the fixed point $({R}^{\frac{1}{b}}-1,0)$ of the system (1.4). We investigate if there exist some other intersection points.

*y*-intercept 1 and is monotonically increasing without bound. On the other hand, the function on the left-hand side is monotonically decreasing, has

*y*-intercept $R(1+b/c)>1$, and converges to $R/(1+b)$ as $x\to \mathrm{\infty}$. Thus, there is a unique intersection point which means there exists only one critical point (see Figure 1).

*P*component, say $\overline{P}$, is also positive, we solve ${P}^{\ast}>0$ in equation (2.2) and find

Thus, there exists a positive fixed point only if the *H* component of it must be less than the number ${R}^{\frac{1}{b}}-1$, which means among the two intersection points, the one on the right must be ${R}^{\frac{1}{b}}-1$ and hence ${F}^{\prime}({R}^{\frac{1}{b}}-1)<0$. Solving this inequality, we obtain the condition for the existence of the positive fixed point: $R>{(1+\frac{1}{c})}^{b}$.

Thus, we obtain the following result.

**Theorem 2.1**

*For the system*(1.4),

*the following statements hold true*.

- a.
*If*$R\le 1$,*then the only fixed point is the extinction fixed point*$(0,0)$. - b.
*If*$1<R\le {(1+\frac{1}{c})}^{b},$

*then there exist two fixed points*:

*the extinction fixed point*$(0,0)$

*and the exclusion fixed point*$({R}^{\frac{1}{b}}-1,0)$.

- c.
*If*$R>{(1+\frac{1}{c})}^{b},$

*then there exist three fixed points*: *extinction fixed point* $(0,0)$, *exclusion fixed point* $({R}^{\frac{1}{b}}-1,0)$, *and a coexistence fixed point*.

## 3 Stability analysis of the system (1.4)

In this section, the stability of the fixed points is examined.

The eigenvalues for the fixed point $(0,0)$ are ${\lambda}_{1}=R$ and ${\lambda}_{2}=0$. Hence, $(0,0)$ is asymptotically stable if $R<1$. We now consider the exclusion fixed point.

**Theorem 3.1**

*For the system*(1.4),

*the exclusion fixed point*$({R}^{\frac{1}{b}}-1,0)$

*is asymptotically stable if*

*Proof*

where the eigenvalues are ${\lambda}_{1}=1+b(-1+{R}^{-\frac{1}{b}})$ and ${\lambda}_{2}=c(-1+{R}^{\frac{1}{b}})$. Applying the stability conditions $|{\lambda}_{1}|<1$ and $|{\lambda}_{2}|<1$, we obtain the desired result. □

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

## Authors’ Affiliations

## References

- Clark D, Kulenovik MRS: A coupled system of rational difference equations.
*Comput. Math. Appl.*2002, 43: 849–867. 10.1016/S0898-1221(01)00326-1MATHMathSciNetView ArticleGoogle Scholar - Hassell MP, Comins HN: Discrete time models for two-species competition.
*Theor. Popul. Biol.*1976, 9: 202–221. 10.1016/0040-5809(76)90045-9MATHMathSciNetView ArticleGoogle Scholar - Kulenovic MRS, Nurkanovic M: Global asymptotic behavior of a two dimensional system of difference equations modeling cooperation.
*J. Differ. Equ. Appl.*2003, 9(1):149–159.MATHMathSciNetView ArticleGoogle Scholar - Elaydi S:
*An Introduction to Difference Equations*. Springer, Berlin; 2000.Google Scholar - Elaydi S:
*Discrete Chaos: With Applications in Science and Engineering*. 2nd edition. Chapman & Hall/CRC, Boca Raton; 2008.Google Scholar - Selgrade JF, Ziehe M: Convergence to equilibrium in a genetic model with differential viability between the sexes.
*J. Math. Biol.*1987, 25: 477–490. 10.1007/BF00276194MATHMathSciNetView ArticleGoogle Scholar - Smith HL:
*Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems*. Am. Math. Soc., Providence; 1995.MATHGoogle Scholar - Misra JC, Mitra A: Instabilities in single-species and host-parasite systems: period-doubling bifurcations and chaos.
*Comput. Math. Appl.*2006, 52: 525–538. 10.1016/j.camwa.2006.08.026MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.