Bifurcation Analysis and Chaos Control in a Discrete-Time Parasite-Host Model

A discrete-time parasite-host systemwith bifurcation is investigated in detail in this paper.The existence and stability of nonnegative fixed points are explored and the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. And we also prove the chaos in the sense of Marotto. The numerical simulations not only illustrate the consistence with the theoretical analysis, but also exhibit other complex dynamical behaviors, such as bifurcation diagrams, Maximum Lyapunov exponents, and phase portraits. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period orbits, attracting invariant cycles and chaotic attractors of the discrete-time parasite-host system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point by using the feedback control method.

We investigate the discrete-time parasite-host system in this paper.In some cases, parasites can reduce host density and induce host population extinction.Ebert et al. [30] formulated the following epidemiological microparasite model: to understand how six microparasites regulate Daphnia populations and drive the populations to extinction, where () and () denote uninfected (susceptible) and infected (infected) hosts densities at time , respectively;  is the maximum per capita birth rate of uninfected hosts; (0 ⩽  ⩽ 1) is the relative fecundity of an infected host;  is the carrying capacity of for the host population;  is the parasiteindependent host background mortality;  is the constant infection rate and  is the parasite-induced excess death rate; all parameters are positive.In this model, the microparasite transmission is assumed via a mass action process, the fecundity of uninfected host is density-dependent, and the fecundity of infected host may be reduced due to being infected compared with that of uninfected host.Here, 0 ≤  ≤ 1 means that the fecundity of infected hosts is reduced, but they still have certain fecundity,  = 0 means that infected hosts completely lose fecundity;  = 1 means that the fecundity of infected hosts is not affected by parasite infection.For model (1), when  > , there is always equilibrium (0, 0) which is a saddle.This implies that extinction of host is impossible when  > , that is to say, when  >  host always persists.But, in this model, the simulation for the stochastic model indicates that extinction of host likely occurs in certain parameter regions.
Applying the forward Euler discrete scheme to system (3), we obtain the following discrete-time parasite-host system as follows: : (   ) where ℎ is the integral step size.The fixed points of system (4) satisfy the following equations: In our paper, we apply the forward Euler Scheme to discrete the parasite-past model and mainly focus on the existence and stability of nonnegative fixed points and flip bifurcation, Neimark-Sacker bifurcation, and possible chaos in the sense of Marotto's definition [32] in the discrete-time parasite-host system by using the center manifold theorem [33] and the bifurcation theory [17][18][19][20][33][34][35][36].When the integral step size ℎ is taken as a bifurcation parameter, the detailed existence conditions of flip bifurcation and Neimark-Sacker bifurcation are given in a very strict mathematical way.Numerical simulations are shown, including maximum Lyapunov exponents, bifurcation diagrams, and phase portraits, to verify theoretical analyses, and display some new and interesting nonlinear dynamical behaviors of the parasite-host system.In particular, one of our investigations demonstrates that the integral step size makes difference with respect to nonlinear dynamical behaviors of the discrete parasite-host system when the integral step size is taken into account as a bifurcation parameter.
The organization of this paper is as follows.In Section 2, the theorem on the existence and stability of fixed points for system (4) is proved.In Section 3, we give the sufficient conditions of existence for flip bifurcation and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory.In Section 4, we first rigorously prove the existence of chaos in the sense of Marotto's definition.Numerical simulations are presented not only to verify the theoretical analysis but also to exhibit other complex dynamics in Section 5.In Section 6, chaos is controlled to an unstable fixed point using the feedback control method.Finally, we conclude this paper with comments and discuss the future work.

The Existence and Stability of Fixed Points
In this section, we discuss the existence and stability of the fixed points.It is clear that the fixed points of system (4) satisfy the following equations: By calculation of the above system, we get the following results: system (4) has three fixed points (0, 0), (1 − , 0) and the positive fixed point ( * ,  * ), where ( * ,  * ) satisfy Now we study the stability of these fixed points.Note that the modules of eigenvalues of the characteristic equation at the fixed point determine the local stability of a fixed point (, ).
The Jacobian matrix  of system (4) at any point (, ) is given by Let and then The characteristic equation of (, ) is where Then we will discuss the local dynamics of three fixed points (0, 0), (1 − , 0) and the positive fixed point ( * ,  * ).In order to discuss the stability of the fixed points of system (4), we get the following existence proposition of the fixed points by simple analysis [17][18][19][20].

Proposition 1.
The eigenvalues of the fixed point (1 − , 0) are (iv) (1 − , 0) is a saddle except for any values of parameters as (i) to (iii).
Case 3. The characteristic equation of the Jacobian matrix  of system (4) at the positive fixed point ( * ,  * ) is written as Let and then Proposition 2. Let ( * ,  * ) be the positive fixed point of ( 4): (i) it is a sink if one of the following conditions holds: (ii) it is a source if one of the following conditions holds: (iii) it is a saddle if the following condition holds: (iv) it is nonhyperbolic if one of the following conditions holds: (iv. ( The fixed point ( * ,  * ) may undergo Neimark-Sacker bifurcation when parameters vary in a small neighborhood of  3 .

Bifurcation Analysis
Based on the above analysis, we will mainly focus on the flip bifurcation of the positive fixed point ( * ,  * ) if parameters vary in a small neighborhood of  1 or  2 and the Neimark-Sacker bifurcation of ( * ,  * ) if parameters vary in a small neighborhood of  3 , respectively, to investigate the local and global stability of system (4).We choose parameter ℎ as a bifurcation parameter for studying the flip bifurcation and Neimark-Sacker bifurcation of ( * ,  * ) by using the center manifold theorem and bifurcation theory in [17][18][19][20][33][34][35][36].
We construct an invertible matrix  as follows: and for (25), use the translation and then the map ( 25) becomes where and From the center manifold theorem [33], we can determine the center manifold   (0, 0, 0) of ( 29) at the fixed point (0, 0) in a small neighborhood of ℎ * = 0, and then we obtain that there exists a center manifold   (0, 0, 0), which can be approximately represented as follows: where ((|x| + |ℎ * |) 3 ) is a function with order at least 3 in the variables, and Therefore, the map (29) which is restricted to the center manifold   (0, 0, 0) is defined: where For map (34) in order to undergo a flip bifurcation, we require that two discriminatory quantities  1 and  2 are not zero, where and On the basis of the above analysis and the theorem of [34][35][36], we obtain the following result.Theorem 3. If  2 ̸ = 0, then map (4) undergoes a flip bifurcation at the fixed point ( * ,  * ) when the parameter ℎ varies in a small neighborhood of ℎ 1 .Moreover, if  2 > 0 (resp.,  2 < 0), the period-2 orbits that bifurcate from ( * ,  * ) are stable (resp., unstable).
and then  is invertible and use the translation for (40); then model (40) becomes the following form: In order to undergo Neimark-Sacker bifurcation for (48), we require that the following discriminatory quantity is not zero [34][35][36]: where Therefore, from the above analysis and the theorem in [34][35][36], we have the following result.

Existence of Marotto's Chaos
In this section, we rigorously prove map (4) possesses chaotic behavior in the sense of Marotto's definition [32].
We first present Marotto's chaos definitions and theorem which are quoted from [32].
For any map  :   →   , and any positive integer , let   represent the composition of  with itself  times.For a differentiable function , let () denote the Jacobian matrix of  evaluated at the point  ∈   , and |()| its determinant.Let   () denote the closed ball in   of radius  centered at the point  and  0  () its interior.Also let ‖‖ be the usual Euclidean norm of  in   .Definition 5. Let  be differentiable in   ( 0 ).The point  0 ∈   is an expanding fixed point of  in   ( 0 ), if ( 0 ) =  0 and all eigenvalues of () exceed 1 in norm for all  ∈   ( 0 ).Definition 6. Assume that  0 is an expanding fixed point of  in   ( 0 ) for some  > 0; then  0 is said to be a snap-back repeller of  if there exists a point  ∈   ( 0 ) with  ̸ =  0 ,   () =  0 and |  ()| ̸ = 0 for some positive integer .
Next, we give specific values of the parameters for illustrating the existence of conditions in Theorem 9.
For Case (ii).System (4) has only one positive fixed point (0.376, 0.2981) when  = 0.2,  = 1.25,  = 0.005, and  = 0.465.After calculation of the positive fixed point of system (4), the Neimark-Sacker bifurcation emerges at ℎ = 1.5569 and its eigenvalues are ,  = 0.6199 ± 0.7847.For ℎ = 1.5569, we have |, | = 1,  = 0.24415 > 0,  = 0.5998.This shows the correctness of Theorem 4. From Figure 3, we observe that the fixed point (0.376, 0.2981) of system (4) is stable for ℎ < 1.5569, that it loses its stability at ℎ = 1.5569, and that an invariant circle appears when the parameter ℎ exceeds 1.5569.The maximum Lyapunov exponents corresponding to Figure 3 are calculated and plotted in these.For ℎ > 1.5669, some Lyapunov exponents are bigger than 0 and some are smaller than 0, so there exist stable fixed point or stable period windows in the chaotic region.In general the positive Lyapunov exponent is considered to be one of the characteristics implying the existence of chaos [37][38][39][40].The phase portraits which are associated with Figure 3 are displayed in Figure 4, which clearly depicts how a smooth invariant circle bifurcates from the stable fixed point (0.376, 0.2981) and periodic orbits and attracting chaotic sets.

Chaos Control
In this section, we apply the state feedback control method [24,41,42] to stabilize chaotic orbits at an unstable fixed point of (4).
Consider the following controlled form of system (4): with the following feedback control law as the control force: where  1 and  2 are the feedback gain and ( * ,  * ) is the positive point of model ( 4).The Jacobian matrix  of the controlled system (71) and (72) evaluated at the point ( * ,  * ) is given by where  11 , Assume that the eigenvalues are given by  1 and  2 ; then and The lines of marginal stability are determined by solving the equation  1 = ±1 and  1  2 = 1.These conditions guarantee that the eigenvalues  1 and  2 have modulus less than 1.

Conclusion
In this paper, we mainly investigated the existence and stability of the nonnegative fixed points of system (4) and    flip bifurcation and Neimark-Sacker bifurcation under some conditions at the unique positive fixed point ( * ,  * ) by using manifold theorem and bifurcation theory and also proved the chaos in the sense of Marotto.Our main results are given in Propositions 1, 2 and Theorems 3, 4, and 9 and numerical simulations in Section 5.In the details of the result, when the integral step size ℎ is chosen as a bifurcation parameter, the discrete-time parasite-host system exists with much richer nonlinear dynamical behaviors.Seen from Figures 1-6, there exist period-1, 2, 4, and 8 orbits, attracting invariant cycles, and even stranger chaotic attractors.These results demonstrate that the integral step size ℎ makes differences corresponding to the local and global stability of the discretetime parasite-host system.Moreover, Figure 1 shows that the system exists in flip bifurcation, which proves the correctness of Theorem 3. In addition, Figure 1 demonstrates the system exists with other complex dynamics including stable coexistence, period-doubling bifurcations.Figures 3  and 5 show that the system exists with Neimark-Sacker bifurcation, which proves the correctness of Theorem 4. The phase portraits corresponding to Figures 1, 3, and 5 are shown in Figures 2, 4, and 6.These figures show that, along with step ℎ increase, the stability properties of the equilibrium will be lost and the flip bifurcation, Neimark-Sacker bifurcation, and chaos phenomenon will appear.Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.
However, in this paper we only present the numerical results.In our future work, in order to investigate the more biological significance of system (4), we expect to obtain some real data to illustrate the validity of our theoretical results and to study the effect of bifurcations and chaos on a discrete parasite-host model.

Figure 1 :
Figure 1: Lyapunov exponents and bifurcation diagram in the ℎ- plane with the parameters given by case (i).

Figure 2 :
Figure 2: Phase portraits for various values of ℎ in Figure 1.

Figure 3 :
Figure 3: Lyapunov exponents and bifurcation diagram in the ℎ- space for ℎ = 1.5-2.6 with the parameters given by case (ii).

Figure 4 :
Figure 4: Phase portraits for various values of ℎ in Figure 3.

Figure 6 :
Figure 6: Phase portraits for various values of ℎ in Figure 5.