# Asymptotic behavior of a Nicholson-Bailey model

- Muhammad Naeem Qureshi
^{1}, - Abdul Qadeer Khan
^{1}Email author and - Qamar Din
^{1}

**2014**:62

https://doi.org/10.1186/1687-1847-2014-62

© Qureshi et al.; licensee Springer. 2014

**Received: **4 October 2013

**Accepted: **27 January 2014

**Published: **10 February 2014

## Abstract

In this paper, we study the qualitative behavior of a Nicholson-Bailey host-parasitoid model given by

where *R*, *a* and initial conditions ${x}_{0}$, ${y}_{0}$ are positive real numbers. More precisely, we investigate the local asymptotic stability of the unique positive equilibrium point of this system. Some numerical examples are given to verify our theoretical results.

**MSC:**39A10, 40A05.

## Keywords

## 1 Introduction

*n*. The model is given by

where ${x}_{n}$ and ${y}_{n}$ represent the densities of the host and parasitoid populations at year *n*, respectively, *R* is the number of offspring of an unparasitized host surviving to the next year. Assuming random encounter between hosts and parasitoids, the probability that a host escapes parasitism can be approximated by ${e}^{-a\sqrt{{y}_{n}}}$, where *a* is a proportionality constant. The probability to become infected is then given by $1-{e}^{-a\sqrt{{y}_{n}}}$.

For basic theory of difference equations, we refer to [1–4], and for applications of difference equations, we refer to [5, 6]. In literature there are many papers on qualitative behavior of biological models [7–18].

Our aim is to investigate the necessary and sufficient condition for local asymptotic stability of unique positive equilibrium point of (1).

## 2 Linearized stability

*I*,

*J*are some intervals of real numbers. Furthermore, a solution ${\{({x}_{n},{y}_{n})\}}_{n=0}^{\mathrm{\infty}}$ of system (2) is uniquely determined by initial conditions $({x}_{0},{y}_{0})\in I\times J$. An equilibrium point of (2) is a point $(\overline{x},\overline{y})$ that satisfies

**Definition 1**Let $(\overline{x},\overline{y})$ be an equilibrium point of system (2).

- (i)
An equilibrium point $(\overline{x},\overline{y})$ is said to be stable if for every $\epsilon >0$ there exists $\delta >0$ such that for every initial condition $({x}_{0},{y}_{0})$, $\parallel ({x}_{0},{y}_{0})-(\overline{x},\overline{y})\parallel <\delta $ implies $\parallel ({x}_{n},{y}_{n})-(\overline{x},\overline{y})\parallel <\epsilon $ for all $n>0$, where $\parallel \cdot \parallel $ is the usual Euclidian norm in ${\mathbb{R}}^{2}$.

- (ii)
An equilibrium point $(\overline{x},\overline{y})$ is said to be unstable if it is not stable.

- (iii)
An equilibrium point $(\overline{x},\overline{y})$ is said to be asymptotically stable if there exists $\eta >0$ such that $\parallel ({x}_{0},{y}_{0})-(\overline{x},\overline{y})\parallel <\eta $ and $({x}_{n},{y}_{n})\to (\overline{x},\overline{y})$ as $n\to \mathrm{\infty}$.

**Definition 2**Let $(\overline{x},\overline{y})$ be an equilibrium point of a vector map $F(x,y)=(f(x,y),g(x,y))$, where

*f*and

*g*are continuously differentiable functions at $(\overline{x},\overline{y})$. The linearized system of (2) about the equilibrium point $(\overline{x},\overline{y})$ is given by

where ${X}_{n}=\left(\begin{array}{c}{x}_{n}\\ {y}_{n}\end{array}\right)$ and ${F}_{J}$ is the Jacobian matrix of system (2) about the equilibrium point $(\overline{x},\overline{y})$.

is a unique positive equilibrium point of system (1) if and only if $R>1$.

**Lemma 1** [6]

*Assume that* ${X}_{n+1}=F({X}_{n})$, $n=0,1,\dots $ , *is a system of difference equations and* $\overline{X}$ *is the fixed point of* *F*. *If all eigenvalues of the Jacobian matrix* ${J}_{F}$ *about* $\overline{X}$ *lie inside the open unit disk* $|\lambda |<1$, *then* $\overline{X}$ *is locally asymptotically stable*. *If at least one of them has modulus greater than one*, *then* $\overline{X}$ *is unstable*.

*Consider the second*-

*degree polynomial equation*

*where*

*p*

*and*

*q*

*are real numbers*.

- (i)
*A necessary and sufficient condition for both roots of Equation*(3)*to lie inside the open disk*$|\lambda |<1$*is*$|p|<1+q<2.$

*In this case the locally asymptotically stable equilibrium*$(\overline{x},\overline{y})$

*is also called a sink*.

- (ii)
*A necessary and sufficient condition for both roots of Equation*(3)*to have absolute value greater than one is*$|q|>1,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|p|<|1+q|.$

*In this case*$(\overline{x},\overline{y})$

*is a repeller*.

- (iii)
*A necessary and sufficient condition for one root of Equation*(3)*to have absolute value greater than one and for the other to have absolute value less than one is*${p}^{2}-4q>0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|p|>|1+q|.$

*In this case the unstable equilibrium*$(\overline{x},\overline{y})$

*is called a saddle point*.

- (iv)
*A necessary and sufficient condition for a root of Equation*(3)*to have absolute value equal to one is*$|p|=|1+q|.$

*In this case the equilibrium* $(\overline{x},\overline{y})$ *is called a non*-*hyperbolic point*.

## 3 Main results

The following theorem shows the necessary and sufficient condition for local asymptotic stability of the unique positive equilibrium point of (1).

**Theorem 1**

*Assume that*$R>1$,

*then the unique positive equilibrium point*

*of system*(1)

*is locally asymptotically stable if and only if*

*where* ${R}_{0}>1$ *is the root of the function* $F(R):=Rln(R)-2R+2$.

*Proof*The Jacobian matrix of the linearized system of (1) about the fixed point

**Theorem 2**

*Assume that*$R>1$,

*then the following statements are true*.

- (i)
*The unique positive equilibrium point of*(1)*is a repeller if and only if*$R>{R}_{0}$,*where*${R}_{0}>1$*is the root of the function*$F(R)=Rln(R)-2R+2$. - (ii)
*The unique positive equilibrium point of*(1)*cannot be a saddle point for all*$R>1$. - (iii)
*The unique positive equilibrium point of*(1)*cannot be non*-*hyperbolic for all*$R>1$.

*Proof*(i) Assume that $R>1$, then $|q|=\frac{Rln(R)}{2(R-1)}>1$ if and only if

for all $R>1$. Hence, from (ii) of Lemma 2 the unique positive equilibrium point of (1) is a repeller if and only if $R>{R}_{0}$.

for all $R>1$. Therefore, the proof of (ii) follows from (iii) of Lemma 2.

(iii) Obviously, one can prove (iii). □

## 4 Examples

In order to verify our theoretical results, we consider some interesting numerical examples in this section. These examples represent different types of qualitative behavior of system (1). First four examples show that the unique positive equilibrium point of system (1) is locally asymptotically stable. Meanwhile, the last two examples show that the positive equilibrium point of system (1) is unstable.

**Example 1**Consider system (1) with initial conditions ${x}_{0}=3.2$, ${y}_{0}=2.8$. Moreover, choose the parameters $R=3.99$, $a=1.5$. Then system (1) can be written as

with initial conditions ${x}_{0}=3.2$, ${y}_{0}=2.8$.

**Example 2**Consider system (1) with initial conditions ${x}_{0}=1.7$, ${y}_{0}=1.5$. Moreover, choose the parameters $R=4.8$, $a=1.3$. Then system (1) can be written as

with initial conditions ${x}_{0}=1.7$, ${y}_{0}=1.5$.

**Example 3**Consider system (1) with initial conditions ${x}_{0}=1.9$, ${y}_{0}=1.7$. Moreover, choose the parameters $R=4.7$, $a=1.7$. Then system (1) can be written as

with initial conditions ${x}_{0}=1.9$, ${y}_{0}=1.7$.

**Example 4**Consider system (1) with initial conditions ${x}_{0}=4.9$, ${y}_{0}=2.9$. Moreover, choose the parameters $R=4.2$, $a=0.8$. Then system (1) can be written as

with initial conditions ${x}_{0}=4.9$, ${y}_{0}=2.9$.

**Example 5**Consider system (1) with initial conditions ${x}_{0}=2.9$, ${y}_{0}=4.5$. Moreover, choose the parameters $R=4.99$, $a=0.5$. Then system (1) can be written as

with initial conditions ${x}_{0}=2.9$, ${y}_{0}=4.5$.

**Example 6**Consider system (1) with initial conditions ${x}_{0}=5.8$, ${y}_{0}=6.5$. Moreover, choose the parameters $R=5.2$, $a=0.7$. Then system (1) can be written as

with initial conditions ${x}_{0}=5.8$, ${y}_{0}=6.5$.

## 5 Conclusion and future work

This work is related to the qualitative behavior of a discrete-time Nicholson-Bailey model. We proved that system (1) has a unique positive equilibrium point which is locally asymptotically stable. The method of linearization is used to prove the local asymptotic stability of the unique equilibrium point. Linear stability analysis shows that the steady states of system (1) are stable under the condition (4) of Theorem 1, *i.e.*, $1<R<{R}_{0}$, where ${R}_{0}\approx 4.921553634567504$ is the root of the function $F(R)=Rln(R)-2R+2$. Particularly, the condition for local asymptotic stability in population biology is a very interesting mathematical problem. Usually the biologists believe that a unique, positive, locally asymptotically stable equilibrium point in an ecological system is very important in biological point of view. Therefore, it is very important to find conditions which may guarantee the local asymptotic stability of the unique positive equilibrium point of the given system. In the paper, we prove the necessary and sufficient condition for the local asymptotic stability of the unique positive equilibrium point of system (1). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions. The Neimark-Sacker bifurcation of system (1) is our next aim to study.

## Declarations

### Acknowledgements

This work was supported by the Higher Education Commission of Pakistan.

## Authors’ Affiliations

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