Bifurcation Analysis and Chaos Control in a Discrete-Time Parasite-Host Model
A discrete-time parasite-host system with 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.
In the theory of ecology, population dynamics are generally governed by continuous-time and discrete-time systems. In recent years, more and more attention is being paid to discrete-time population models [1–20]. We can get more accurate numerical simulation results and richer dynamical behaviors from discrete-time models. In addition, many scholars are interested in how to discretize continuous population dynamical models and study their dynamical properties [21–24]. The recent years, a number of articles which investigated the flip bifurcation, fold bifurcation, pitchfork bifurcation and in the sense of Marotto’s chaos of the discrete system were present in [25–29].
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.  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; is the relative fecundity of an infected host; is the carrying capacity of for the host population; is the parasite-independent 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, means that the fecundity of infected hosts is reduced, but they still have certain fecundity, means that infected hosts completely lose fecundity; means that the fecundity of infected hosts is not affected by parasite infection. For model (1), when , there is always equilibrium , 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.
Rescaling system (1) by the following as [21, 31],and removing the bars, then system (1) becomes in the following form:where , , and are positive constants. Applying the forward Euler discrete scheme to system (3), we obtain the following discrete-time parasite-host system as follows:where is the integral step size. The fixed points of system (4) satisfy the following equations:and then
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  in the discrete-time parasite-host system by using the center manifold theorem  and the bifurcation theory [17–20, 33–36]. When the integral step size 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.
2. 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 , 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 byLetand thenThe characteristic equation of iswhere
Then we will discuss the local dynamics of three fixed points , 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 [17–20].
Case 1. For , we haveSince two eigenvalues of are and , then is a saddle.
Case 2. A simple calculation shows the local dynamics of the fixed point .
Proposition 1. The eigenvalues of the fixed point are , :(i) is a source if , and ;(ii) is a sink if , and ;(iii) is not hyperbolic if , and or ;(iv) 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 aswhereLetand then
Proposition 2. Let be the positive fixed point of (4):(i)it is a sink if one of the following conditions holds:(i.1) and ;(i.2) and ;(ii) it is a source if one of the following conditions holds:(ii.1) and ;(ii.2) and ;(ii.3);(iii)it is a saddle if the following condition holds:(iv)it is nonhyperbolic if one of the following conditions holds:(iv.1) and and ;(iv.2) and .From the above discussion, we can easily obtain that if (iv.1) of Proposition 2 holds, then one of the eigenvalues of the positive fixed point is and the other is neither nor .
Let andWhen parameters vary in a small neighborhood of or , there may be flip bifurcation of the fixed point .
Then we can get that the eigenvalues of the positive fixed point are a pair of conjugate complex numbers with modulus 1 if (iv.2) of Proposition 2 holds.
LetThe 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 or and the Neimark-Sacker bifurcation of if parameters vary in a small neighborhood of , respectively, to investigate the local and global stability of system (4). We choose parameter as a bifurcation parameter for studying the flip bifurcation and Neimark-Sacker bifurcation of by using the center manifold theorem and bifurcation theory in [17–20, 33–36].
3.1. Flip Bifurcation
Since , . Choose as a bifurcation parameter, we consider a perturbation of (23) as follows:where , which is a small perturbation parameter.
Let , . Then we transform the fixed point of map (24) into the origin. By calculating we havewhereand .
From the center manifold theorem , we can determine the center manifold of (29) at the fixed point in a small neighborhood of , and then we obtain that there exists a center manifold , which can be approximately represented as follows:where is a function with order at least 3 in the variables, andTherefore, the map (29) which is restricted to the center manifold is defined:where
For map (34) in order to undergo a flip bifurcation, we require that two discriminatory quantities and are not zero, where and
Theorem 3. If , then map (4) undergoes a flip bifurcation at the fixed point when the parameter varies in a small neighborhood of . Moreover, if (resp., ), the period-2 orbits that bifurcate from are stable (resp., unstable).
3.2. Neimark-Sacker Bifurcation
In this section, we can give a similar argument with the flip bifurcation. Taking parameters arbitrarily from , we consider system (4) with , which is described by The map (38) has a unique positive fixed point
Since . We give a perturbation at . Then (38) becomes into the following form:where , which is a small perturbation parameter.
Note that the characteristic equation associated with the linearization of the map (40) at is followingwhere
Since parameters , the eigenvalues of are a pair of complex conjugate numbers and with modulus 1 by Proposition 2, whereand we have
Moreover, it is required that when , , () which is equivalent to . Note that . Thus, We only need to require that , which leads to
Next, we discuss the normal form of (40) at
Thus, some complicated calculation giveswhere
Theorem 4. If condition (45) holds and , then map (4) undergoes Neimark-Sacker bifurcation at the fixed point when the parameter varies in a small neighborhood of . Moreover, if , resp., , then an attracting (resp., repelling) invariant closed curve bifurcates from the fixed point for (resp., .
4. Existence of Marotto’s Chaos
We first present Marotto’s chaos definitions and theorem which are quoted from .
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 its interior. Also let be the usual Euclidean norm of in .
Definition 5. Let be differentiable in . The point is an expanding fixed point of in , if and all eigenvalues of exceed 1 in norm for all .
Definition 6. Assume that is an expanding fixed point of in for some ; then is said to be a snap-back repeller of if there exists a point with , and for some positive integer .
Theorem 7.  If possesses a snap-back repeller, then the map is chaotic. That is, there exist(i)a positive integer such that for each integer , has a point of period ;(ii)a “scrambled set” of , that is, an uncountable set containing no periodic points of such that(a),(b)for every with (c)for every and any periodic point of (iii)an uncountable subset of such that for every Now we theoretically give the condition of existence of chaotic phenomena for map (4) in the sense of Marotto’s definition of chaos.
Suppose be the fixed point of map (4). We firstly give the conditions such that the is a snap-back repeller. The eigenvalues associated with the fixed point are given by where According to Definition 5, we begin to find a neighborhood of in which the norms of conjugate complex eigenvalues exceed 1 for all , which are equivalent to Let
It is easy to see that , where If ,
then the equation has one real root with multiplicity 2 or two real roots denoted as and .
And ; we can get And for and .
Let where Under the conditions and , the equation has one real root with multiplicity 2 or two real roots denoted as and .
And for all and .
Lemma 8. If the conditions (61) and (63) hold, and the or , then .
Moreover, if one of above conditions holds and the -coordinate of fixed point , , satisfies , then is expanding fixed point of (4) in .
Due to Definition 2 of snap-back repeller, we need to find one point such that , , for some positive integer .
In fact, we haveandNow a map has been constructed to map the point to the fixed point after two iterations if there are solutions different from for (64) and (65). By the calculation, the solutions different from for (65) satisfy the following equation:whereSubstituting and into (64) and solving , one gets whereObviously, if the condition in Lemma 8 is satisfied, the solutions of (64) and (65) satisfies (66) and (68). And the condition of , and holds; then is a snap-back repeller in . Thus, the following theorem is established.
Next, we give specific values of the parameters for illustrating the existence of conditions in Theorem 9.