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This paper discusses a parasitoid-host-parasitoid ecological model and its dynamical behaviors. On the basis of the center manifold theorem and bi-furcation theory, the existence conditions of the flip bifurcation and Neimark-Sacker bifurcation are derived. In the end of the paper, some typical numerical experiments are performed, which illustrate that the theoretical method is effective.

With the increasing application of ecological models, the host-parasitoid models have been extensively explored. The host-parasitoid models are of great significance among the relationships between the biotic populations. Although this kind of model is investigated by many scholars (see [

Xu and Boyce in [

{ H ( t + 1 ) = H ( t ) exp [ r ( 1 − H ( t ) K ) − a [ P ( t ) ] 1 − m ] , P ( t + 1 ) = H ( t ) [ 1 − exp ( − a [ P ( t ) ] 1 − m ) ] , (1)

where H ( t ) is the host population size in generation t, P ( t ) is the parasitoid population size in generation t. The constant a indicates the searching efficiency and m is the interference coefficient. The host grows logistically with the carrying capacity K and the intrinsic growth rate r.

For simplicity, we rewrite the system (1) as the following:

{ x → x e x p [ r ( 1 − x K ) − a y 1 − m ] , y → x [ 1 − e x p ( − a y 1 − m ) ] . (2)

In our paper, the parasitoid-host-parasitoid system (2) is investigated in further details. We mainly focus on its bifurcations and possible chaos qualitatively. Based on the center manifold theorem and bifurcation theory (see [

The layout of this paper is organized as follows: The existence and stability criterion of the equilibria of system (2) are presented in Section 2; Section 3 deals with the flip bifurcation and Neimark-Sacker bifurcation, and derives the existence conditions of the bifurcations; Numerical simulations using MATLAB are presented in Section 4 to illustrate the theoretical results; A brief discussion is carried out in Section 5.

In order to obtain the equilibria of system (2), we need to use the mathematical software. With the aid of Maple program, we get the following three equilibria E 0 ( 0,0 ) , E 1 ( K ,0 ) , E 2 ( x 2 , y 2 ) , where x 2 , y 2 satisfy the following equation:

{ x 2 = x 2 exp [ r ( 1 − x 2 K ) − a ( y 2 ) 1 − m ] , y 2 = x 2 [ 1 − exp ( − a ( y 2 ) 1 − m ) ] . (3)

The qualitative behavior of the system (2) will be investigated. The local dynamics of the system (2) near a fixed point depends on its Jacobian matrix. The Jacobian matrix at the state variable is given by

J ( x , y ) = ( e r − r x K − a y 1 − m − r x K e r − r x K − a y 1 − m − a ( 1 − m ) x y − m e r − r x K − a y 1 − m 1 − e − a y 1 − m a ( 1 − m ) x y − m e − a y 1 − m ) ,

According to J, one can obtain two eigenvalues λ 1 = 0 , λ 2 = e r with given E 0 ( 0,0 ) . From which, one can easily check that E 0 ( 0,0 ) is a stable node ( | e r | < 1 ) . Two eigenvalues at the equilibrium E 1 ( K ,0 ) are λ 1 = 0 , λ 2 = 1 − r , then one can get that E 1 ( K ,0 ) is also a stable node ( | 1 − r | < 1 ) (see [

The following is Jacobian matrix of (3) at E 2 :

J 2 ( x 2 , y 2 ) = ( 1 − r L − H 1 − G G H ) ,

The characteristic equation of matrix J 2 is

λ 2 + P ( x 2 , y 2 ) λ + Q ( x 2 , y 2 ) = 0 , (4)

where

P ( x 2 , y 2 ) = − ( 1 − r L + G H ) , Q ( x 2 , y 2 ) = H − r L G H ,

L = x 2 K , G = e − a y 2 1 − m , H = a ( 1 − m ) x 2 y 2 − m .

From (4), then we have

F ( 1 ) = r L − G H − r G H L + H ,

F ( − 1 ) = 2 − r L + G H − r G H L + H .

In order to disuss the stability of the fixed point E 2 , we also need the following Lemma, which can be easily found from the theorem presented in [

Lemma 2.1. Let F ( λ ) = λ 2 + P λ + Q . Assume that F ( 1 ) > 0 , λ 1 and λ 2 are two roots of F ( λ ) = 0 . Then, we have the following statements:

i) | λ 1 | < 1 , | λ 2 | < 1 if and only if F ( − 1 ) > 0 and Q < 1 ;

ii) | λ 1 | < 1 , | λ 2 | > 1 (or | λ 1 | > 1 and | λ 2 | < 1 ) if and only if F ( − 1 ) < 0 ;

iii) | λ 1 | > 1 , | λ 2 | > 1 if and only if F ( − 1 ) > 0 and Q > 1 ;

iv) λ 1 = − 1 , | λ 2 | ≠ 1 if and only if F ( − 1 ) = 0 and P ≠ 0,2 ;

v) λ 1 , λ 2 are complex and | λ 1 | = | λ 2 | = 1 if and only if P 2 − 4 Q < 0 and Q = 1 .

Let λ 1 and λ 2 be the roots of (2), which are called eigenvalues of the fixed point E 2 ( x 2 , y 2 ) . The fixed point ( x 2 , y 2 ) is a sink or locally asymptotically stable if | λ 1 | < 1 , | λ 2 | < 1 . E 2 ( x 2 , y 2 ) is a source or locally unstable if | λ 1 | > 1 , | λ 2 | > 1 . E 2 ( x 2 , y 2 ) is a saddle if | λ 1 | < 1 and | λ 2 | > 1 (or | λ 1 | > 1 and | λ 2 | < 1 ). The fixed point ( x 2 , y 2 ) is non-hyperbolic if either | λ 1 | = 1 or | λ 2 | = 1 .

From Lemma 2.1, we state the following theorem:

Theorem 2.1. For the positive equilibrium E 2 , we have the following estimates:

i) E 2 ( x 2 , y 2 ) is a sink if the condition hods: r < 2 + G H + H L + G H L and r > H − 1 G H L ;

ii) E 2 ( x 2 , y 2 ) is a source if the condition holds: r < 2 + G H + H L + G H L and r < H − 1 G H L ;

iii) E 2 ( x 2 , y 2 ) is a saddle if the condition holds: r > 2 + G H + H L + G H L ;

iv) E 2 ( x 2 , y 2 ) is non-hyperbolic if either condition (iv.1) or (iv.2) holds:

iv.1) r = 2 + G H + H L + G H L and r ≠ 1 + G H L , r ≠ 3 + G H L ;

iv.2) r = H − 1 G H L and G H − 1 L < r < G H + 3 L .

From the above conclusion, if the term (iv.1) of Theorem 2.1 holds, one can easily find that one of the eigenvalues of E 2 ( x 2 , y 2 ) is -1 and the other is 2 + G H − r L , which is neither 1 nor -1. If the term (iv.2) of Theorem 2.1 holds, then the eigenvalues of E 2 are a pair of complex conjugate numbers whose modulus is 1.

Let

The equilibrium

Let

The equilibrium

Now we mainly center on bifurcations (see [

Select arbitrary parameters

The system (5) has a unique positive fixed point

where

Let

where

We can build an invertible matrix T

Consider the following translation:

then the system (7) becomes

where

and

By the center manifold theorem in [

By simple calculations, one can obtain

Thus, the map is restricted to the center manifold, which is given by

where

If the system (10) goes through a flip bifurcation, the following conditions must hold:

Based on the above analyses and using the bifurcation theorems presented in [

Theorem 3.1. When the parameter

We consider the following system by selecting parameters

Choosing

where

Let

where

The characteristic equation of map (13) at

where

Since parameters

then, we have

Moreover, it is required

So the eigenvalues

Let

Using the following translation:

then the system (13) becomes

where

and

Therefore

To assure that the map (13) passes though Neimark-Sacker bifurcation, we need to let the following discriminatory quantity

where

According to the previous discussions and applying Hopf bifurcation theorems in [

Theorem 3.2. If the parameters

We have gotten the theoretical results of system (2) based on the qualitative theory. In this section, we outline a numerical methods to validate the previous analysis and provide some numerical results by using MATLAB. We draw the diagrams for bifurcation and phase portraits to show new interesting complex

dynamical behaviors. The bifurcation parameters are considered in the following two cases:

Case 1: Varying r in range

From

The phase portraits in

Case 2: We fix

It shows that the Theorem 3.2 holds.

From

The dynamical behaviors of a parasitoid-host-parasitoid system are investigated. The theoretical analyses demonstrate that the system (2) can appear as flip bifurcation and Neimark-Sacker bifurcation. We present the numerical diagrams to validate analytical effectiveness. We also observe many forms of complexities from these diagrams, such as the cascade of period-doubling bifurcation and Neimark-Sacker bifurcation. Hence we can find that the discrete-time models have far richer dynamical behaviors as compared to continuous-time models. All these results are obtained by a simple system of only two maps and we believe that similar results can be achieved by more general systems.

We would like to thank the associate editor and the anonymous referees for their valuable comments and helpful suggestions, which have led to a great improvement of the initial version. We also wish to acknowledge conversations with Yandong Chu of the Department of Mathematics at Lanzhou Jiaotong University regarding this problem. This work is supported by the National Natural Science Foundation of China (No.11161027).

Liu, X.J. and Liu, Y. (2018) Bifurcation and Chaos in a Parasitoid-Host-Parasitoid Model. International Journal of Modern Nonlinear Theory and Application, 7, 1-15. https://doi.org/10.4236/ijmnta.2018.71001