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A hyperparasitic system with prolonged diapause for host is investigated. It is assumed that host prolonged diapause occur at larval stage, and parasitoid attack is limited to egg stage before the initiation of host diapause. Such behavior has been reported for many ichneumons. Hyperparasite only attacks the parasitoids that parasitize the hosts. Hyperparasitic system is often used in biological control. The existence and stability of nonnegative fixed points are explored. Numerical simulations are carried out to explore the global dynamics of the system, which demonstrate appropriate prolonged diapause rate and appropriate intrinsic growth rate can stabilize the system. The reasons are explained according to the ecological perspective. Furthermore, many other complexities which include quasi-periodicity, period-doubling bifurcations leading to chaos, chaotic attractor, intermittent and supertransients are observed.

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. [

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 [

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” [

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 [

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 [

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 [

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 the section, a one-dimensional bifurcation analysis is carried out to investigate the overall dynamic behavior of the system. One-dimensional bifurcation diagrams give information about the dependence of the dynamics on a certain parameter. The analysis is expected to reveal the type of attractor to which the dynamics will ultimately settle down after passing an initial transient phase and within which the trajectory will then remain forever [

1) Varying in the range, and keeping other parameters fixed as below:

, , , ,. (23)

2) Varying in the range , and keeping other parameters fixed as below:

, , , ,. (24)

The Maximum Lyapunov exponents have been proved to be the most useful dynamic diagnostic tool for chaotic systems. It is the average exponential rate of divergence or convergence of nearby orbits in phase space [

amplitude standing for quasi-periodicity, which are the same as in the range. As increases from to, the Maximum Lyapunov exponents are negative, corresponding to a stable coexistence of the system. When is slightly increased beyond, Most of the Maximum Lyapunov exponents are positive and few are negative. So there exist period windows in the chaotic band.

As can be seen from

the same as in the range. A typical chaotic attractor is presented in

From

Intermittency as illustrated in

interest is tens or hundreds of generations while supertransients can persist thousands of generations or even longer. In

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