Hosts and parasitoids are mostly univoltine and have no overlap between successive generations. Therefore, their interactions can be modeled by discrete differences. An early work by Beddington et al. [1] showed that discrete host-parasitoid models can produce a richer set of dynamic patterns than those observed in continuous-time models. More recently, many researchers [2-12] have reported that discrete host-parasitoid models can have very complex dynamics.

However, few studies have explored hyperparasitic systems using mathematical approaches and difference equations. Actually, hyperparasite can play a crucial role in the control of a host-parasitoid interaction if they are successfully established in the community. Furthermore, few works on complex dynamics in parasitic system have considered diapause. In the natural world, many insects which inhabit unpredictable environments display diapause for one year or more, which can be described as prolonged or extra-long diapause [13]. As observed in numerous laboratory and field experiments, diapause is induced by changing responses to temperature, photoperiod, humidity, hormonal treatment and other factors [14- 16]. Thus, incorporating hyperparasite and diapause in parasitic system is more realistic and more practical significance.

In the paper, a hyperparasitic system with prolonged diapause for host is investigated. In the system, we assume that host prolonged diapause occur at larval stage, and parasitoid attack is limited to egg stage before the initiation of host diapause. The parasitoids are physiological “regulators” [17]. In this case, the parasitoid can potentially attack all hosts, but do not undergo prolonged diapause itself. Such behavior has been reported for many ichneumons [18]. Hyperparasite only attacks the parasitoids that parasitize the hosts [19]. Based on the above considerations, the model can be represented by the following difference equation:

where denote the densities of the host, parasitoid, hyperparasite respectively at generation. In the absence of parasitism, the host adapts to Moran-Ricker model [20,21], that is to say, where is the intrinsic growth rate and is the carrying capacity. The function given by Poisson distribution proposed by Nicholson and Bailey [22] stands for the probability that a host escapes parasitism，where is the parasitoid searching efficiency. By analogy with the above, the probability that a parasitoid escapes hyperparasites is. Accordingly, is the hyperparasite searching efficiency. The parameters and represent diapause rate and survival rate of the host, respectively. According to their biological meaning, and are non-negative and less than 1.

In this section, the existence and asymptotic stability analysis of the non-negative equilibrium points of system (1) are investigated. The system has four non-negative equilibrium points which are given by the following statements:

a) The equilibrium point always exists.

b) The positive equilibrium point is given by which exists if and only if

The positive equilibrium point is given by

which exists if and only if

The positive equilibrium point is given by

which exists if and only if

Analysis of the stability of the system (1) close to the above equilibrium points requires that the system is fully specified in terms of densities at time and. For this, we introduce two variables, and, corresponding to the densities of hosts and parasitoids at time respectively. Then the system (1) corresponds to the following form:

Accordingly, the four equilibrium points of the system (1) corresponds to the following forms respectively:

The stabilities of equilibrium points, , and are as the same as these points and respectively. Now we study the linear stability of fixed points in the system (7). The Jacobian matrix at an arbitrary is given by

where

The characteristic equation of is

where

Moreover, an application of the local stability analysis of the system (7), gives the following results:

(1) Substituting the fixed point into the Equation (8), we get

The roots of the Equation (9) are

The modulus of is less than one. Then is local stability if and only if

which yields

(2) Substituting the fixed point into the equation (8), we get

where

.

Several roots of the Equation (11) are . Obviously, the modulus of is less than one. The modulus of is less than one if and only if

Under the conditions of (2) and (12), the stability of is identified by the equation

It follows from the well-known Schur-cohn criterion [23] that the modulus of all roots of the Equation (13) is less than one if and only if

From the inequalities (12) and (14), we obtain the following conditions for the stability of:

Proposition 1. The equilibrium point is locally stable if and only if the following conditions hold:

Proof. From the conditions (15), we can know the inequalities (2) and (12) obviously hold. Let's study the inequalities (14).

1).

2).

When, according to the conditions (15)we obtain.

When, according to the conditions (15), we obtain

.

3).

Obviously,

Therefore, if the conditions (15) are satisfied, the equilibrium point is locally stable.

(3) Substituting the fixed point into the Equation (8), we get

where

Two roots of the Equation (16) are . Obviously, the modulus of is less than one. The modulus of is less than one if and only if

Under the conditions of (4) and (17), the stability of is identified by the equation

By analogy with the above, the modulus of all roots of the Equation (18) is less than one if and only if

are satisfied. Based on the above analysis, we obtain the following sufficient conditions for the stability of.

Proposition 2. The equilibrium point is locally stable if the following conditions hold:

Proof. We only need verify the inequalities (19). According to the condition, we obtain.

Substitute the signs of and, we get,. Substituting the values of, for and rearranging the term we get

According to the condition, we obtain. By analogy with, we get

.

According to the signs of, we obtain. Now, we prove the fourth inequality of (19).

.

According to the conditions and, we obtain. That is to say. At the same time,. The proof is complete.

(4) Substituting the fixed point into the equation (8), we get

where

Under the condition of (6), the stability of is identified by the following equation

According to Schur-cohn criterion [23], the modulus of all roots of the Equation (21) is less than one if and only if

are satisfied. Based on the above analysis, we obtain the following sufficient conditions for the stability of.

Proposition 3. Under the condition of (6), the equilibrium point is locally stable if the following conditions hold:

where

Proof. According to the condition, we obtain. Substitute the signs of, and, we get,. Substitute the values of, and, we get

By the conditions and, we obtain, and. Then, it is easy to verify. And.

According to the condition, we obtain. That is to say. According to the conclusion, we obtain. Now, we prove the fifth inequality of (22).

By the condition, we obtain

The proof is complete.

In this section, we use the bifurcation diagrams, the Maximum Lyapunov exponents, phase portraits and so on to explore the possibilities of dynamical behaviors for system (1).

In this paper, we have proposed and investigated the host-parasitoid-hyperparasite system with prolonged diapause for host. The existence and stability of the nonnegative fixed points are explored. Subsequently, numerical simulations are carried out to exhibit other complex dynamics including stable coexistence, quasi-periodicity, period-doubling bifurcations, and chaotic bands with periodic windows, quasiperiodic attractor and non-unique attractor, intermittent chaos and supertransients and so on. Furthermore, these simulated results are explained according to ecological perspective. From Figure 1, we can know that the system coexists with. That is to say, appropriate diapause rate is better for the stability of the system. Low diapause rate makes the host population suffer from high parasitism risk. High diapause rate goes against the parasitoids growth. These two cases destabilize the system. From Figure 2, we can know that the system is stable with, but the high intrinsic growth rate may destabilize the stable dynamics into more complex dynamic. The host population would increase over carrying capacity with high intrinsic growth rate and then make the whole system lose its stability.