Abstract
We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant . It is proved that model has two boundary equilibria: , and a unique positive equilibrium under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: . It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.
1. Introduction
The usual framework for the discrete-time host–parasite models is:
where and represent the population size of the host and parasite in successive generations and respectively. The parameter is the host finite rate of increase in the absence of parasites, is the biomass conversion constant and is the function defining the fractional survival of hosts from parasitism. The simplest version of this model is that of Nicholson, and Nicholson and Bailey who explored in depth a model in which the proportion of hosts escaping parasitism is given by the zero term of the Poisson distribution [1–3]:
where are the mean encounters per host. Thus, is the probability of a host will be attacked. Using (2) in (1), one gets
In 2014, Qureshi et al. [4] have investigated the asymptotic behavior of the following Nicholson–Bailey model:
where , and initial conditions are positive real numbers. Further in 2015, Khan and Qureshi [5] have investigated the dynamics of the following modified Nicholson–Bailey model:
where and initial conditions are positive real numbers. Our aim in this paper is to explore the local dynamics along with topological classification and bifurcation analysis of the model (5). First, we make the following rescaling transformations:
then system (5) becomes
For simplicity, we assume that , and then model (7) becomes:
where and .
The rest of the paper is organized as follows: Section 2 deals with the study of existence of equilibria of the model (8). In Section 3, we study the local dynamics and existence of bifurcations about equilibria: of the model. Section 4 deals with the study of Neimark–Sacker bifurcation about of the model (8). Numerical simulations along with discussion are presented in the last Section.
2. Existence of Equilibria of the Discrete-Time Model (8)
In this Section, we study the existence of equilibria of the model (8) in . The results about the existence of equilibria are summarized as follows:
Lemma 1. Discrete-time model (8) has at least two boundary equilibria and the unique positive equilibrium point in . More precisely,(i)For all parametric values and , model (8) has a unique equilibrium point: ;(ii)If then model has boundary equilibrium point: ;(iii)Suppose that and and when , the curve intersect the line at , say. If then there exist a unique such that has a unique positive equilibrium point of (8).
Proof. For finding number of equilibria of the model (8), we have to solve the following system of equations:(i)Let , then equation of system (10) satisfied identically and from equation we obtain . So system (10) has always equilibrium for all parameter values .(ii)Let , then equation of (10) satisfied identically and from equation we obtain . Hence system has boundary equilibrium if .(iii)Now we locate the unique positive equilibrium of (10) in . For this, let , then (10) becomesNow eliminating from (11), one getsDenote,Then the -coordinates of positive equilibria of (8) are the corresponding -coordinates of the point of intersection of and with . By calculating derivative of , one getMoreoverSo, if , then there exists no intersection point of and . This implies that model (8) has no positive equilibria if . And if , then there exists a unique point of intersection of and with (see Figure 1). Therefore, if then (8) has positive equilibrium point and the positive equilibrium point of (8) is unique. We denote it by where is the positive solution of (12).
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3. Local Dynamics and Existence of Bifurcations about Equilibria: , , of the Model (8)
In this Section, we will study the local dynamics of (8) about , and . The Jacobian matrix of (8) about equilibrium becomes
And its characteristic equation is
where
Lemma 2. For equilibrium , the following holds:(i) is a sink if ;(ii) is never source;(iii) is a saddle if ;(iv) is nonhyperbolic if .
From Lemma 2, we can see that one of the eigenvalues about the equilibrium is 1. So fold bifurcation may occurs when parameter vary in the small neighborhood of .
Lemma 3. For , the following holds:(i) is a sink if ;(ii) is never source;(iii) is a saddle if ;(iv) is nonhyperbolic if .
We can easily see that if condition (iv) of Lemma 3 hold then one of the eigenvalues about equilibrium is 1. So fold bifurcation may occur when parameters vary in a small neighborhood of . And we denote the parameters satisfying as
Hereafter, we will investigate the local dynamics of (8) about by using Lemma 2.2 of [6]. The Jacobian matrix of linearized system of (8) about is
where
Moreover eigenvalues of about is given by
where
Hereafter, we will give the topological classification of (8) about according to the sign of .
Lemma 4. For , the following holds:
(i) is Locally Asymptotically Node if(ii) is Unstable Node if(iii) is nonhyperbolic if
Lemma 5. For , following statements holds:(i) is Locally Asymptotically Focus if(ii) is Unstable Focus if(iii) is nonhyperbolic if
If condition (iii) of Lemma 5 holds then we obtain that eigenvalues of are a pair of conjugate complex numbers with modulus one. So Neimark–Sacker bifurcation exists by the variation of parameter in a small neighborhood of . For simplicity, we denote the parameters satisfying as
4. Bifurcation Analysis about of the Model (8)
This Section deals with the study of Neimark–Sacker bifurcation of the model (8) about . Consider parameter in a small neighborhood of , i.e., , where , then (8) becomes:
The characteristic equation of about of (31) is
where
The roots of characteristic equation of about are
Additionally, we required that when , , , which corresponds to . Since and . Thus and hence . So we only require that . By computation, we get
Let then equilibrium of system (8) transform into . By calculating, we obtain
where . Hereafter, when , normal form of system (37) is studied. Expanding (37) up to third order about by Taylor series, we get
where
Now, let
and invertible matrix defined by
Using following translation
(38) gives:
where
In addition,
In order for (43) to undergo Neimark–Sacker bifurcation, it is required that following discriminatory quantity, i.e., (see [6–13]).
where
After calculating, we get
Based on this analysis and Neimark–Sacker bifurcation Theorem discussed in [12, 13], we arrive at the following Theorem:
Theorem 1. If then model (8) undergoes a Neimark–Sacker bifurcation about as the parameters go through . Additionally, attracting (respectively repelling) invariant closed curve bifurcate from if (respectively ).
According to Neimark–Sacker bifurcation discussed in [12, 13], the bifurcation is called supercritical Neimark–Sacker bifurcation if the discriminatory quantity . In the following Section, numerical simulations guarantee that supercritical Neimark–Sacker bifurcation occurs for the model (8). Biologically, attracting closed curve indicates that both parasitoid and host populations will coexist under the periodic or quasi-periodic oscillations with long time.
5. Numerical Simulations and Discussion
This work deals with the study of local dynamics and bifurcation analysis of a discrete-time two-species model in . We proved that the model has two boundary equilibria: and a unique positive equilibrium point under certain parametric conditions. We studied the local dynamics along with topological classification about equilibria: , , and conclusion is presented in Table 1. We proved that about there may exist a fold bifurcation when parameters of (8) are located in the set: . We also proved that if then eigenvalues about are pair of complex conjugate with modulus one and thus in particular supercritical Neimark–Sacker bifurcation occurs under the bifurcation curve:
Biologically, existence of stable closed curves implies that there exist the periodic or quasiperiodic oscillations between host and parasitoid populations. Finally, numerical simulations are provided to verify theoretical discussion. These numerical simulations presented in Figures 2–5 agree with our theoretical discussion. Figure 2 shows that of the model (8) is Locally Asymptotically Focus when , where as presented in Figures 2(a)–2(i) by choosing . But when goes through the bifurcation value , equilibrium of (8) is Unstable Focus. Meanwhile, an attracting closed invariant curve bifurcates from of the model (8) as presented in Figures 3(a)–3(i). Moreover, bifurcation diagrams along with Maximum Lyapunov Exponent in this case, are plotted and drawn in Figure 4. Finally 3D bifurcation diagrams are also plotted and drawn in Figure 5.
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Data Availability
All the data utilized in this article have been included, and the sources from where they were adopted were cited accordingly.
Disclosure
The author declares that he got no funding on any part of this research.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This research is partially supported by the Higher Education Commission of Pakistan.