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Bifurcation and chaos control in a discretetime predator–prey model with nonlinear saturated incidence rate and parasite interaction
 Waqas Ishaque^{1, 2},
 Qamar Din^{2}Email authorView ORCID ID profile,
 Muhammad Taj^{1} and
 Muhammad Asad Iqbal^{2}
https://doi.org/10.1186/s136620191973z
© The Author(s) 2019
 Received: 12 July 2018
 Accepted: 15 January 2019
 Published: 25 January 2019
Abstract
The dynamical behavior of the predator–prey system is influenced effectively due to the mutual interaction of parasites. Regulations are imposed on biodiversity due to such type of interaction. With implementation of nonlinear saturated incidence rate and piecewise constant argument method of differential equations, a threedimensional discretetime model of prey–predator–parasite type is studied. The existence of equilibria and the local asymptotic stability of these steady states are investigated. Moreover, explicit criteria for a Neimark–Sacker bifurcation and a perioddoubling bifurcation are implemented at positive equilibrium point of the discretetime model. Chaos control is discussed through implementation of a hybrid control technique based on both parameter perturbation and a state feedback strategy. At the end, some numerical simulations are provided to illustrate our theoretical discussion.
Keywords
 Eco–epidemiological model
 Local stability
 Hopf bifurcation
 Flip bifurcation
 Chaos control
1 Introduction
Trophic interactions, keystone species, food webs, competition and biodiversity are greatly affected due to interaction of parasites and consequently the community structure can be reverted. Furthermore, qualitative analysis of prey and predator interaction is a distinguished field for investigation in mathematical ecology. On the other hand, presence of some kind of disease in prey–predator interaction is a prominent topic for investigation, and this is a comparatively new subject in the field of ecoepidemiology. In order to observe the spread of some sort of disease, both aspects for ecology and epidemiology are included in such a field of study. It is important to point out some earlier studies in the area related to mathematical ecology and epidemiology. For this purpose, Hadeler and Freedman [1] studied a prey–predator system in which both types of species (prey and predator) were subjected to parasitism. Freedman [2] investigated a prey–predator model in which the predator population is always treated as primary host whereas the prey population density may be either a secondary host or a primary host for the parasite interaction. Beltrami and Carroll [3] studied a simple mathematical system of trophic type in which a mortality of virus–induced type is included, and the actual bloom patterns for several species are also presented. Venturino [4] investigated prey–predator systems subjected to some sort of disease in prey density. Moreover, Chattopadhyay and Arino [5] examined a three species ecoepidemiological model, that is, the predator population, the infected prey, and the sound prey. Keeping in view the interaction of three species for an ecoepidemiological model consisting of predator density, infected prey and sound prey, a mathematical model was investigated by Chattopadhyay et al. [6]. Furthermore, a Hopf bifurcation was also studied at its positive steady state.
According to survey reports published by US Geological Survey National Wildlife Health Center from 1978 to 2003 many white pelicans were to die out due to type C botulism [14]. In 1996, over 8500 white pelicans had died at Salton Sea in California due to infection of type C botulism. The major cause of that disaster were Tilapia fish which are a key source for C botulism toxins produced in white pelican birds. In addition, the study has indicated that bacterial infections take part in lowering oxygen levels in the tissues of affected fish. Due to lack of oxygen, the fish find oxygen from the ocean surface and this leads to an appropriate environment to produce botulism in the tissues of the infected fish. When pelicans attack on these defenseless fish, the botulism toxins is ingested by pelican birds and as a result avian botulism is produced in their bodies [15].
Moreover, in system (1.2), it is assumed that pelican birds only prey infected Tilapia, but do not capture the healthy Tilapia because the infected fish are feeble and become easier to prey. On the other hand, there is a significant number of infected Tilapia fish present in the Salton Sea and due to their struggle against death these are more unprotected and attractive to the Pelican birds [7]. Meanwhile, healthy Tilapia easily escape and predation becomes difficult.
Moreover, the remaining discussion for this paper is organized as follows. In Sect. 2, we explore the existence of equilibria and conditions for their local asymptotic stability. In Sect. 3, an explicit criterion for Hopf bifurcation is implemented at positive steady state of system (1.3). Section 4 is dedicated to an implementation of an explicit criterion for flip bifurcation at a positive equilibrium of system (1.3). In Sect. 5, a chaos control strategy based on parameter perturbation and a state feedback control methodology is implemented to system (1.3). Finally, some suitable numerical simulations are presented in Sect. 6 for verification of our theoretical discussion.
2 Existence of equilibria and stability
Lemma 2.1
([16])
Lemma 2.2
 (i)Equilibrium point \(E_{1}\) for model (1.3) is a sink and locally asymptotically stable if and only if the following conditions are satisfied:$$ s < 2,\qquad k \alpha < \mu . $$
 (ii)Suppose that \(\mu < k \alpha \), then the equilibrium point \(E_{2}\) for model (1.3) is a sink and locally asymptotically stable if and only if the following conditions are satisfied:where \(\beta _{2}\), \(\beta _{1}\) and \(\beta _{0}\) are written in (2.4).$$ \vert \beta _{2} + \beta _{0} \vert < 1+ \beta _{1},\qquad \vert \beta _{2} 3 \beta _{0} \vert < 3 \beta _{1},\quad \textit{and}\quad \beta _{0}^{2} + \beta _{1}  \beta _{0} \beta _{2} < 1, $$
Lemma 2.3
3 Hopf bifurcation analysis
Here we investigate the parametric conditions for which the positive steady state for discretetime system (1.3) undergoes a Neimark–Sacker (Hopf) bifurcation. For such an investigation an explicit criterion of Hopf bifurcation is used without computing the eigenvalues for the variational matrix of system under consideration (see also [17, 18]). For this purpose, an explicit criterion for Hopf bifurcation is given below.
Lemma 3.1
([19])
 (C1):

We assume that \(\mathbb{D}_{n 1}^{} ( \xi _{0}, u )=0\), \(\mathbb{D}_{n 1}^{+} ( \xi _{0}, u )>0\), \(P_{\xi _{0}} (1)>0\), \((1 )^{n} P_{\xi _{0}} (1)>0\), \(\mathbb{D}_{i}^{ \pm } ( \xi _{0}, u )>0\), for \(i = n 3, n 5, \ldots , 2\) (or 1), when n is odd (or even), this is known as the eigenvalue criterion.
 (C2):

Next, we suppose that \(( \frac{d}{d \xi } ( \mathbb{D}_{n 1}^{} ( \xi , u ) ) )_{\xi = \xi _{0}} \neq 0\), and this is known as the transversality criterion for the Hopf bifurcation.
 (C3):

Finally, we consider that \(\cos ( \frac{2 \pi }{l} ) \neq \varphi \), or the resonance condition \(\cos ( \frac{2 \pi }{l} ) = \varphi \), where \(l =3,4,5,\ldots \) , and \(\varphi =10.5 P_{\xi _{0}} (1) \mathbb{D}_{n 3}^{} ( \xi _{0}, u )/ \mathbb{D}_{n 2}^{+} ( \xi _{0}, u )\), which is called the nonresonance, or resonance criterion, then Hopf bifurcation takes place for the critical value \(\xi _{0}\).
If we take \(n =3\), then the following lemma provides us with the parametric conditions for which model (1.3) undergoes a Hopf bifurcation whenever s is chosen as bifurcation parameter.
Lemma 3.2
Proof
4 Flip bifurcation analysis
Now we investigate the parametric conditions for which a unique positive steady state for discretetime system (1.3) encounters a perioddoubling (flip) bifurcation. For such an investigation, we use an explicit criterion of the perioddoubling (flip) bifurcation without computing the eigenvalues for the Jacobian matrix of a given model. For this purpose, we need the following result.
Lemma 4.1
([20])
 (H1):

\(P_{\xi _{0}} (1)=0\), \(\mathbb{D}_{n1}^{\pm } ( \xi _{0},u)>0\), \(P_{\xi _{0}} (1)>0\), \(\mathbb{D}_{i}^{\pm } ( \xi _{0},u)>0\), \(i=n2,n4,\ldots , 1\) (or 2), when n is even (or odd, respectively), and this is known as the eigenvalue criterion for flip bifurcation.
 (H2):

\(\frac{\sum_{i=1}^{n} (1 )^{ni} \tau '_{i}}{ \sum_{i=1}^{n} (1 )^{ni} (ni+1) \tau _{i1}} \neq 0\), where \(\tau '_{i}\) denotes derivative of \(\tau ( \xi )\) at \(\xi = \xi _{0}\), and this is called the transversality criterion, then a flip bifurcation occurs for the critical value \(\xi _{0}\).
Furthermore, if we take \(n =3\) in Lemma 4.1, then we obtain the following result for model (1.3) providing conditions for a flip bifurcation when s is chosen as the bifurcation parameter.
Lemma 4.2
5 Chaos control
Next, we investigate a chaos control technique for the discretetime model (1.3). Keeping in view the simplicity, a hybrid control methodology is chosen which was firstly proposed by Luo et al. [21]. The hybrid chaos control method consists of a single control parameter which lies inside the open unit interval. Furthermore, the implementation for such a hybrid control strategy is comparatively simple one and it is based on both parameter perturbation and state feedback control strategy. It is worthwhile to mention some other investigations for controlling chaos in discretetime systems and the interested reader is referred to [22–31].
Lemma 5.1
6 Numerical simulation and discussion
Example 6.1
Example 6.2
Example 6.3
7 Concluding remarks
In this paper, the qualitative nature of discretetime system (1.3) is investigated. The discretetime system is obtained with implementation of a piecewise constant argument and bifurcating and chaotic behavior is studied with the help of standard mathematical techniques. The local behavior for equilibria is investigated by implementing the method of linearization. Furthermore, explicit criteria for Neimark–Sacker bifurcation and perioddoubling bifurcation are used for the investigation of the bifurcating behavior about positive equilibrium point. On the other hand, parametric values are also selected in a range observed in an actual predator–prey system, and in this case there is a brighter chance of occurrence of perioddoubling bifurcation in the discretetime model. A hybrid control strategy based on parameter perturbation and state feedback method is used for controlling the fluctuating and chaotic behavior of model (1.3). Moreover, it is noticed that the hybrid chaos control method can stabilize the chaotic orbits more effectively due to the emergence of a perioddoubling (flip) bifurcation as compared to that of a Neimark–Sacker (Hopf) bifurcation. Furthermore, it is investigated that the parameter c, representing the crowding effect in the system, is more appropriate for the emergence of perioddoubling (flip) and Neimark–Sacker (Hopf) bifurcations whenever it is varied in some suitable interval. It is examined that larger values of c may result flip bifurcation, meanwhile smaller values for c may result from a Neimark–Sacker bifurcation.
Declarations
Acknowledgements
The authors are grateful to the referees for their excellent suggestions, which greatly improved the presentation of the paper.
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Funding
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Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
It is declared that none of the authors have any competing interests in this manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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