Dynamics of a modified Nicholson-Bailey host-parasitoid model
- Abdul Qadeer Khan^{1}Email author and
- Muhammad Naeem Qureshi^{1}
https://doi.org/10.1186/s13662-015-0357-2
© Khan and Qureshi; licensee Springer. 2015
Received: 3 October 2014
Accepted: 2 January 2015
Published: 30 January 2015
Abstract
Keywords
MSC
1 Introduction
Many ecological models are governed by differential as well as difference equations. In particular, ecological models with non-overlapping populations are better formulated as discrete dynamical systems compared to the continuous time models. These models have been extensively studied in recent years because of their wide applicability to the study of population dynamics [1, 2]. In fact, in the case of discrete dynamical systems, one has more efficient computational results for numerical simulations and also has rich dynamics as compared to the continuous ones. In recent years, several papers have been published on the mathematical models of biology that discuss the system of difference equations generated from the associated system of differential equations as well as the associated numerical methods [3, 4]. In mathematical biology, the model such as the host-parasitoid has attracted many researchers during the last few decades. Usually, the biologists believe that a unique, positive, locally asymptotically stable equilibrium point in an ecological system is very important [5]. Therefore, it is pertinent to find conditions which may guarantee the global stability of a positive equilibrium point, if it exist, for the given system. See [6] for introduction to mathematical models in biological sciences.
- (i)
Hosts are distributed at random, at density \(x_{n}\) per unit area in generation n.
- (ii)
Parasitoids search at random and independently, each having an ‘area of discovery’ a, and lay an egg in each host found.
- (iii)
Each parasitized host gives rise to one new parasitoid in generation \(n+1\).
- (iv)
Each unparasitized host gives rise to \(b>1\) new hosts in generation \(n + 1\).
Here, \(x_{n}\) and \(y_{n}\) represent the densities of the host and parasitoid population at year n. b 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 \(e^{-ay_{n}}\), where a is a proportionality constant. Similarly, the probability to become infected is then given by \(1-e^{-ay_{n}}\). The parameter c describes the number of parasitoids that hatch from an infected host.
Now, assume that the host has bounded dynamics in absence of parasitoid, i.e., has self-regulation (density dependence). For example, assume host dynamics are inherently logistic (e.g., the Beverton-Holt model).
In this paper our aim is to study the dynamics of system (1). More precisely, we investigate the boundedness character, existence and uniqueness of a positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point, and the rate of convergence of positive solutions of system (1). To investigate the dynamics we shall use standard results from theory of rational difference equations. However, we shall state the results that we employ and refer the interested readers for a systematic study of rational difference equations to [7–21] and the references therein. In Refs. [22–26] qualitative behavior of some biological models is discussed. The rest of the paper is organized as follows. In Section 2 the required known results about linearized stability are given. Section 3 discusses the boundedness character of the model. Section 4 is about the existence and uniqueness of the positive equilibrium point. It also contains the local stability of the equilibrium point. Section 5 discusses the global behavior of the equilibrium point. Whereas Section 6 is about the rate of convergence and Section 7 gives the numerical examples of the proved results. In the last section a brief conclusion is given.
2 Linearized stability
Definition 2.1
- (i)
An equilibrium point \((\bar{x},\bar{y})\) is said to be stable if for every \(\varepsilon> 0\), there exists \(\delta> 0\) such that for every initial condition \((x_{0},y_{0})\), \(\|(x_{0},y_{0}) - (\bar{x}, \bar{y})\| <\delta\) implies \(\|(x_{n}, y_{n}) - (\bar{x}, \bar{y})\| < \varepsilon\) for all \(n > 0\), where \(\|\cdot\|\) is the usual Euclidian norm in \(\mathbb{R}^{2}\).
- (ii)
An equilibrium point \((\bar{x},\bar{y})\) is said to be unstable if it is not stable.
- (iii)
An equilibrium point \((\bar{x},\bar{y})\) is said to be asymptotically stable if there exists \(\eta> 0\) such that \(\|(x_{0},y_{0}) - (\bar{x}, \bar{y})\| <\eta\) and \((x_{n},y_{n})\to(\bar{x},\bar{y})\) as \(n\to\infty\).
- (iv)
An equilibrium point \((\bar{x},\bar{y})\) is called a global attractor if \((x_{n},y_{n})\to(\bar{x},\bar{y})\) as \(n\to\infty\).
- (v)
An equilibrium point \((\bar{x},\bar{y})\) is called an asymptotic global attractor if it is a global attractor and stable.
Definition 2.2
Lemma 2.3
[7]
- (i)
If both roots of Equation (4) lie in the open unit disk \(|\lambda|<1\), then the equilibrium point \((\bar{x},\bar{y})\) is locally asymptotically stable.
- (ii)
If at least one of the roots of Equation (4) has absolute value greater than one, then the equilibrium point \((\bar{x},\bar{y})\) is unstable.
- (iii)A necessary and sufficient condition for both roots of Equation (4) to lie inside the open disk \(|\lambda|<1\) isIn this case the locally asymptotically stable equilibrium \((\bar {x},\bar{y})\) is also called a sink.$$|p|<1-q<2. $$
- (iv)A necessary and sufficient condition for both roots of Equation (4) to have absolute value greater than one isIn this case \((\bar{x},\bar{y})\) is a repeller.$$|q|>1,\qquad|p|<|1-q|. $$
- (v)A necessary and sufficient condition for one root of Equation (4) to have absolute value greater than one and for the other to have absolute value less than one isIn this case the unstable equilibrium \((\bar{x},\bar{y})\) is called a saddle point.$$p^{2}+4q>0,\qquad|p|>|1-q|. $$
- (vi)A necessary and sufficient condition for a root of Equation (4) to have absolute value equal to one isIn this case the equilibrium \((\bar{x},\bar{y})\) is called a non-hyperbolic point.$$|p|=|1-q|. $$
3 Boundedness
The following theorem shows that every positive solution \(\{ (x_{n},y_{n})\}_{n=0}^{\infty}\) of system (1) is bounded.
Theorem 3.1
Every positive solution \(\{(x_{n},y_{n})\}_{n=0}^{\infty}\) of system (1) is bounded.
Proof
Theorem 3.2
Let \(\{(x_{n},y_{n})\}\) be a positive solution of system (1). Then \([0,\frac{b}{d} ]\times[0 ,\frac{ bc}{ d} ]\) is an invariant set for system (1).
Proof
It follows from induction. □
4 Existence and uniqueness of a positive equilibrium point and local stability
The following theorem shows the existence and uniqueness of a positive equilibrium point of system (1).
Theorem 4.1
If \(b>1\) and \(d<\frac{ac}{b\ln(\frac{1+b}{b} )}\), then system (1) has a unique positive equilibrium point \((\bar{x}, \bar {y})\) in \([0,\frac{b}{d} ]\times[0 ,\frac{ bc}{ d} ]\).
Proof
Theorem 4.2
- (i)The unique positive equilibrium point of system (1) is locally asymptotically stable if and only if$$\begin{aligned}& \frac{be^{acr(bdr+1-b)} (acr(1+bdr)^{2}+1 )}{(1+bdr)^{2}} \\& \quad <1-\frac {ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}} <2. \end{aligned}$$
- (ii)The unique positive equilibrium point is a repeller if and only ifand$$\biggl\vert \frac{ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}}\biggr\vert >1 $$$$\begin{aligned}& \frac{be^{acr(bdr+1-b)} (acr(1+bdr)^{2}+1 )}{(1+bdr)^{2}} \\& \quad < \biggl\vert 1-\frac{ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}}\biggr\vert . \end{aligned}$$
- (iii)The unique positive equilibrium point is a saddle point if and only ifand$$\begin{aligned}& \biggl(\frac{be^{acr(bdr+1-b)} (acr(1+bdr)^{2}+1 )}{(1+bdr)^{2}} \biggr)^{2} \\& \quad {}+ 4 \biggl(\frac{ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}} \biggr)>0 \end{aligned}$$$$\begin{aligned}& \frac{be^{acr(bdr+1-b)} (acr(1+bdr)^{2}+1 )}{(1+bdr)^{2}} \\& \quad > \biggl\vert 1-\frac{ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}}\biggr\vert . \end{aligned}$$
- (iv)The unique positive equilibrium point is non-hyperbolic if and only if$$\begin{aligned}& \frac{be^{acr(bdr+1-b)} (acr(1+bdr)^{2}+1 )}{(1+bdr)^{2}} \\& \quad = \biggl\vert 1-\frac{ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}}\biggr\vert . \end{aligned}$$
Proof
5 Global character
Lemma 5.1
[7]
- (i)
\(f(x,y)\) is non-decreasing in x and non-increasing in y.
- (ii)
\(g(x,y)\) is non-decreasing in both arguments.
- (iii)If \((m_{1},M_{1},m_{2},M_{2})\in I^{2}\times J^{2}\) is a solution of the systemsuch that \(m_{1}=M_{1}\) and \(m_{2}=M_{2}\), then there exists exactly one equilibrium point \((\bar{x},\bar{y})\) of system (2) such that \(\lim_{n\to\infty}(x_{n},y_{n})=(\bar {x},\bar{y})\).$$\begin{aligned}& m_{1}= f(m_{1},M_{2}),\qquad M_{1}=f(M_{1},m_{2}), \\& m_{2} = g(m_{1},m_{2}),\qquad M_{2}=g(M_{1},M_{2}) \end{aligned}$$
Theorem 5.2
Assume that \(ac+d>abc\), then the unique positive equilibrium point \((\bar{x},\bar{y})\) in \([0,\frac{b}{d} ]\times[0 ,\frac { bc}{ d} ]\) of system (1) is a global attractor.
Proof
Lemma 5.3
6 The rate of convergence
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of system (1).
Proposition 6.1
(Perron’s theorem [28])
Proposition 6.2
[28]
Using proposition (6.1), one has following result.
Theorem 6.3
7 Examples
In order to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference equations (1). All plots in this section are drawn with Mathematica.
Example 1
Example 2
Example 3
Example 4
Example 5
8 Conclusion
This work is related to the qualitative behavior of the modified Nicholson-Bailey host-parasitoid model. We have investigated the existence and uniqueness of positive steady-state of system (1). Under certain parametric conditions, the boundedness of positive solutions is proved. Moreover, we have shown that the unique positive equilibrium \((\bar {x}, \bar{y})\) in the \([0,\frac{b}{d} ]\times[0 ,\frac { bc}{ d} ]\) point of system (1) is locally asymptotically stable if and only if \(\frac{be^{acr(bdr+1-b)} (acr(1+bdr)^{2}+1 )}{(1+bdr)^{2}}<1-\frac{ab^{2}cre^{acr(bdr+1-b)} (bdr(e^{acr(bdr+1-b)}-1)-1 )}{(1+bdr)^{2}} <2\) hold true. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. Furthermore, the rate of convergence of positive solutions of (1) which converge to its unique positive equilibrium point is demonstrated. Finally, some numerical examples are provided to support our theoretical results. These examples are experimental verification of our theoretical discussions.
Declarations
Acknowledgements
The authors thank the main editor and anonymous referees for their valuable comments and suggestions that led to the improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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