Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model

In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b a ≠ where , a b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.


Introduction
More efficient computational models for numerical simulation can be created by discrete-time models and also these models can present much more plentiful dynamic behaviors when they would be compared with the same type of continuous-time model [1] [2] [3]. In fact, when populations have non-overlapping generations, discrete-time models will be more rational and applicable than the continuous time models [1] [2]. In the 1935, Nicholson and Bailey could develop a discrete-time model for the population dynamics of insect hosts and their parasitoids [2] [4].
In this paper, we consider a particular case of the following general model [3] [4]: is the density-dependent factor [2]. Density-dependent factor affects host populations. For some parameter values the density-dependent factor has a stabilizing impact. If the parameter r is not high, there will exist a locally asymptotically stable equilibrium.
We investigate the stability for b a ≠ . We do the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a for b a ≠ . In the absence of the parasitoid, the dynamics of this non-dimensionalized model are the same as the dynamics which is given by the Ricker map. Finally, we represent the non-dimensionalized model: we use the following method [8].
Here for N k = and 2 r = , we have Sf N < . Therefore, N k = is asymptotically stable because: When r will be more increased, the period-doubling bifurcation goes to chaotic bands. A stable coexistence is observed when 2 r < . The chaotic attractor appears when r is approximately increased from 2.7. Afterward, complex dynamic patterns are observed.
In Figure 1, we see period-doubling bifurcation.
Furthermore, we see an attracting 3-cycle, which undergoes a sequence of pe- Case b a = which is the density-dependent predator-prey model, is studied in [9]: The pair of difference equations were analyzed by Beddington, Free, and Lawton (1975).
The dynamics of this model are complicated. For some values of the parameters, this model has an asymptotically stable equilibrium with complex eigenvalues. When the parameter a increases, this asymptotically stable equilibrium loses its stability and rise to an invariant closed curve. The dynamics of the invariant curve include periodic and quasi-periodic orbits. Eventually, the invariant curve loses its smoothness and breaks up into a chaotic attractor [9].
In model (2), we have one extinction fixed point ( ) ,0 k and one coexistence fixed point.
The Jacobian matrix for system (2), has the following form: To analyze the local asymptotically stability, the Jacobian matrix at Theorem 1. The extinction fixed point ( ) 0,0 of system (2) is a saddle point and unstable.
Proof. The eigenvalues are 1,2 e ,0 r λ = . Since 0 r > , and hence e 1 r > , the eigenvalue e r is greater than 1 that corresponds to a one-dimensional unstable manifold and 2 0 λ = correspond to a one-dimensional stable manifold.
The Jacobian evaluated at the fixed point ( ) ,0 k is given by: Proof. The eigenvalues of ( ) Two issues remain unresolved. They are when 1 1 1 r λ =− = − and 1 ka = .
Here, we apply the center manifold theorem.
We make a change of variables. Let x u k = + and y v = . With this change of variables, ( ) ,0 k coincide with origin and system (2) by the following form, The non-linear terms for above system are the following form, Now we are going to compute the constants α and β . The function h must satisfy the center manifold equation Without considering the exponential terms, we have the following equation, Thus on the center manifold 0 v = , we have the following map The second case is 1 1 r − = − . Jordan form for the case 1 1 r With this change of variables x u k = + and y v = system (2) for 1 1 r − = − by the following form, The non-linear terms for above system are the following form, Now we are going to compute the constants δ and γ . The function 2 h must satisfy the center manifold equation Solving this equation, we have the unique solution 0 δ = and 0 γ = . Thus on the center manifold 0 u = , we have the following map The coexistence equilibrium for system (2) The Jacobian at the coexistence equilibrium point can be simplified at the form: In Figure 2, we choose 0.2 a = , initial host density 11 N = , initial parasitoid density 1 P = and we plot phase portraits for system (2). We see for 0.5 r >

The Case b a ≠
In this section, we investigate rescaling (nondimensionalization) host-parasitoid dynamics (1) with considering b a ≠ .
The bifurcation diagram for system (1) is displayed in Figure 3, shows for 2.5 r = , the Neimark-Sacker bifurcation occurs in system (1).
The phase portraits of system (1) are displayed in Figure 4, shows for We draw the phase portrait for different values of r (see Figure 5).
To nondimensionalized system (1), Replace each of variables with a quantity scaled relative to a characteristic unit of measure to be determined. By rescaling N and P, one of the parameters can be set to be 1 without loss of generality (e.g. The equilibrium solutions and the local asymptotic stability of the model (3) are analyzed.
We study the different types of bifurcation such as Neimark-Sacker, saddle-node and period-doubling bifurcation.

Solutions and Local Asymptotic Stability
In this subsection, we study the equilibrium solutions and the local asymptotic stability of the model (3).
To analysis the local asymptotically stability, Jacobian matrix is calculated and evaluated at equilibrium.  The Jacobian matrix for system (3), has the following form: One equilibria is the zero equilibrium, where 0 x = and 0 y = .
To analyze the local asymptotically stability, the Jacobian matrix at ( ) ( ) The eigenvalues of ( ) Theorem 5. The axial fixed point ( ) 1, 0 of system (3) is asymptotically stable if we have, 0 Proof. It is straightforward using Linearised Stability Theorem [10] and or Jury condition [11]. The Jacobian at the coexistence equilibrium point can be simplify at the form: It will be shown the coexistence equilibrium point is asymptotically stable if the following Theorem is satisfied: Theorem 6. The coexistence equilibrium point in system (3) is asymptotically stable if we have: Proof. This is straightforward using local stability theorems in [10] [11] [12].
The host-parasitoid interaction exhibit simple steady state dynamics but it is not clear what biological mechanisms ensure that the inequalities in this equation holds, since the dynamical outcomes are very sensitive to two parameters b and r (Mukherjee and Das and Kesh). Theorem 7. For coexistence equilibrium point of system (3) we have a stable and an unstable manifolds, when one of the following conditions are satisfied [8]:
Remark 1. The saddle-node bifurcation occurs when the Jacobian matrix has an eigenvalue equal to 1. This is equivalent to saying that we cross the line − from the stability region [4]. The saddle-node bifurcation is the type I intermittency 1 (route to chaos). Hilborn has discussed about types of intermittency in his book [13]. Here we give the necessary conditions for Neimark-Sacker bifurcations in our two models.

Non-Unique Dynamics and Bistability
It is possible that in a discrete-time dynamical system, different types of attractors such as equilibrium point vs. chaotic attractors, periodic vs. quasiperiodic attractors and quasiperiodic vs. chaotic attractors coexist. This phenomenon is called non-unique dynamics and coexistence of attractors which implies that in the bifurcation diagrams, routes to chaos through period doubling bifurcations, is dependent on initial conditions and therefore non-unique [14] [15] [16]. Non-uniqueness induces a problem for predicting and controlling the dynamics in different of engineering and biological problems. The complex dynamics in a host-parasitoid model induce a multiple attractor. We have bistability between two attractors, an interior one and a boundary one. Bistable dynamics is often generated a threshold level below which the parasitoid will die out regardless of the host population levels (Kang; Aarmbruster; Kuang, 2008) [17]. Complex dynamics in a host-parasitoid model can lead to a Then the eigenvalues of J are, ( ) ( ) is a stable periodic orbit of the Ricker model, i.e., the so-called internal stability (Kon 2006) [15],

Conclusion
In this paper, we studied a discrete-time host-parasitoid model which is non-dimensionalized Nicholson-Bailey model. The model shows rich features

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.