LOCAL DYNAMICS OF A THREE-DIMENSIONAL HOST-PARASITE MODEL

In this paper, local dynamics of a three-dimensional host parasite model as a discrete dynamical system has been studied. The existence of fixed points and stability behaviour near these points are investigated. By use of centre manifold theorem we describe the stability of non-hyperbolic fixed points. Some numerical simulations explain our theoretical results in better way.


Introduction
The behavior of population in nature and the interaction between species have been a subject of interest over many years.In recent years, many researchers paid more attention to population models as a discrete dynamical systems [2,3,8].Discrete-time models are actually more reasonable than the continuous time models when populations have non-overlapping generations.Also, by development in mathematical softwares, more accurate numerical results can be obtained from the discrete model, related to the continuous one, since numerical simulation of continuous models are obtained by discretization [5,14,16].A host-parasite model describes the dynamics of two insects: parasitoid and its host.This model is named after the two researchers, Nicholson and Bailey, who developed the model [1].In this model, parasitoids randomly search for hosts population that grows exponentially in the absence of parasitoids.Moreover, both parasitoids and hosts are supposed to be distributed in a non-contiguous method in the environment [15].After Nicholson-Bailey model, many models have been introduced to describe the relationship between host and parasitoid.Citing some examples: Nicholson and Bailey [15] introduced discrete host-parasitoid model of the form Beddington et al. [6] improved the Nicholson-Bailey model by adding the effect of carrying capacity Khan and Qureshi [11] assumed that the host has bounded dynamics in the absence of parasitoid and they studied dynamics of their system.Atabaigi and Akrami [3] investigated stability and bifurcations of host-parasite model Interested readers can go through the references [1,2,4,8,12] for more details.
However, most of the papers in this fields investigated two dimensional models, i.e. a population model with two species.But higher dimensional models are less considered in literature.
Moreover, the host species may be attacked in different developmental stages by a range of parasitoids.For example, Hassell and Waage discussed a winter moth that is parasitized by egg, larval, and pupal parasitoids [10].Zwölfer [18] modelled the interaction of two species of eurytomid parasitoids, Eurytoma serratulae and E. Robusta, attacking a common host species, the knapweed gall fly (Urophora cardui), on creeping thistle (Cirsium arvense).
Motivated by the above discussion, in this paper, we extend the Nicholson-Bailey hostparasitoid model including one host and two parasitoid: where H n is host population and P n , Q n are the two parasitoid populations at n-th generation respectively, r is the intrinsic growth rate of host and a, b are the searching efficiency of the parasitoids.The underlying biological assumptions in this model are as follows: • There is one host and two parasitoids which parasitize the same host.
• Suppose the parasitoid P acts first, followed by Q that acts on the surviving hosts.
Our paper is organized as follows: in the next section we prepare some analytical tools needed in the sequence.Section 3 is devoted to the local dynamic of the model.In this section we introduce our system as a three-dimensional host-parasite model.Then, we focus to determine the fixed points and corresponding stability nature.In the rest, some discussions are presented to check whether a fixed point is stable, asymptotically stable or unstable under some parameter conditions.In the last Section, some numerical simulation are presented to explain our work in better way

Local dynamics of 3 dimensional maps
In this section, we provide a brief review of the local stability of fixed points in three dimensional discrete dynamical systems.Here, consider nonlinear system (2.1) For simplicity, let us write this system as x → F(x), where x = (x, y, z) and we obtain fixed points of (2.1), i.e. x * = F(x * ).We can determined the local stability of fixed point x * by linearization of system (2.1) near x * .In this way, the Jacobian matrix of system (2.1) calculated by The fixed point x * of F is locally asymptotically stable if all the eigenvalues of the Jacobian, JF(x * ) lie inside the unit disk and if at least one of the eigenvalue lies out of unit disk, the fixed point is unstable.These conditions can be expressed in terms of the roots of characteristic polynomials of JF(x * ).Following lemma can be useful in the proof of the local stability of x * .Lemma 2.1.[13, appendix A.1.2]Let the equation be ) , if and only if the following conditions are fulfilled. (2.3) Note that, if some of the eigenvalues of JF(x * ) lie on the unit disk, then the above lemma is failed.In this case, x * is called non-hyperbolic fixed point.For determining the stability of nonhyperbolic fixed points we can use the center manifold theory.By suitable change of variable, one can transfer the x * on the origin and without loss of generality, system (2.1) can be written as where (y, z) ∈ R t × R s , t + s = 3, and all eigenvalues of A lie on the unit disk and all of the eigenvalues of B are off the unit disk.
The following theorem asserts the existence of a centre manifold, on which the dynamics of system (2.4) is given by the map on the centre manifold [17].
Theorem 2.2.[17] There is a C r -center manifold for system (2.4) that can be represented locally as for a sufficiently small δ .Furthermore, the dynamics restricted to M c are given locally by the map y → Ay + f (y, h(y)). (2.5) The next theorem states that the dynamics on the center manifold Mc deter-mines completely the dynamics of system (2.4).

Local dynamics of the model
Consider the following host-parasite model where r, a, b > 0. Here, we investigate fixed points of the model and theirs stabilities.In the following, we provide the boundary fixed points of the three-dimensional map (3.1).

Boundary fixed points
There are three boundary fixed points (0, 0, 0), ( r ln r a(r−1) , ln r a , 0) and ( r ln r b(r−1) , 0, ln r b ).The Jacobian matrix associated with (3.1) is given by At (0, 0, 0) we have We observe that from (3.3), λ 1 = r and λ 2 = λ 3 = 0. Therefore, the origin is asymptotically stable if 0 < r < 1 and unstable if r > 1.According to Lemma 2.1, for determining the local stability of nontrivial fixed points we can state the following proposition.
Proposition 3.1.Fixed point ( r ln r a(r−1) , ln r a , 0) is stable if and only if the following conditions are fulfilled.b a The characteristic polynomial of (3.4) is Now, its suffice to apply conditions of Lemma 2.1 to p(λ ).Since r > 1 we have Note that condition (3.6) implies condition (3.5).
Now by using (3.5) and (3.6) in (3.7) we obtain (3.8) Similarly, by using (3.5) and (3.6) in (3.9) we have c 0 (c This completes the proof. Proof. At ( r ln r b(r−1) , 0, ln r b ) we have The characteristic polynomial of (3.10) is Similar to the previous proposition, its suffice to apply conditions of Lemma 2.1 to p * (λ ).So, Note that condition (3.12) implies condition (3.11). Hence, For the last condition we obtain .
This completes the proof.

Positive fixed points
In this subsection, we search for positive fixed point(s).So, we suppose that xyz = 0, and have Let Then, the positive fixed point is the positive intersection point(s) of the functions f 1 and f 2 .
Moreover, we have f 1 (0) = ln r b and lim y→0 + f 2 (y) = r−1 ar .By simple calculation we have r ln r , the system has no positive fixed point and (c) a < b(r−1) r ln r , the system has a unique positive fixed point.
By similar argument we can show that f 2 (y) > 0 for y ∈ (0, ln r a ].It is easy to see that functions f 1 and f 2 connect to each other at the terminal point of the interval, i.e. f 1 ( ln r a ) = f 2 ( ln r a ) = 0. Therefore, with this information we consider three cases: (1) If a > b(r−1) r ln r the function f 1 lies above f 2 for y ∈ [0, ln r a ), so the system has no positive fixed point (Figure 1 (a)).
(2) If a = b(r−1) r ln r the function f 1 lies above f 2 for y ∈ (0, ln r a ), and two functions reach to each other at the endpoints.So, the system has no positive fixed point (Figure 1 (b)).
(3) If a < b(r−1) r ln r two functions have a unique positive intersection at y * .So, the system has a unique positive fixed point (Figure 1 (c)).
Hence, we summarize the above discussion in the following proposition.Now, we need to determine the stability of the positive fixed point.The Jacobian matrix at p * is Now, we can apply Lemma 1 to prove the following proposition: Proposition 3.4.The fixed point p * in Proposition 3.3 is stable if and only if the following conditions are fulfilled.

1)ay
Remark.Since the positive fixed point coordinates are not explicitly obtained and there are many parameters in Proposition 3.4, it's not easy to have an analyti-cally proof for it.By numerically computing for a set of different parameters, we obtain that the positive fixed point is unstable.In the other hand, by Gauses competitive exclusion principle [9], one of the parasite may be excluded from the system.So, we expected the positive fixed point to be an unstable fixed point.

Non-hyperbolic cases
In this subsection, we state some conditions that the system (3.1) has the non-hyperbolic fixed point.
• Case (1): If b = a(r−1) ln r , then p * 1 = ( r ln r a(r−1) , ln r a , 0) is a non-hyperbolic fixed point.First, we bring the fixed point p * 1 to the origin by linear transformation, i.e. we have The eigenvalues of linear part of system (3.14), are λ 1 = 1 and λ 2,3 = α ± iβ , where . Let then by using linear transformation we can write this system in Jordan form as: where, We will apply center manifold theory to this problem.The center manifold can locally be represented as follows for sufficiently small u.We assume a center manifold of the form By some calculation, the center manifold is given by the graph of h(u), where and k 1 = − a(r−1) 2r ln r (r + 2σ 1 ), k 2 = r−1 ln r a − 1 2 + aσ 1 r .The map on the center manifold is given by Hence, we can summarize above discussion in the following proposition.
• Case (2): If b = ar ln r (r−1) , then p * 2 = ( r ln r b(r−1) , 0, ln r b ) is a non-hyperbolic fixed point.First, we bring the fixed point p * 1 to the origin by linear transformation, i.e. we have The eigenvalues of linear part of system (3.18), are λ 1 = 1 and λ 2,3 = α ± iβ , where . Let where we can write this system in the Jordan form as follow. where, Similar to Case (1), we will apply center manifold theory to this problem.So, the center manifold can locally be represented as follows for sufficiently small u.We consider a center manifold of the form By some calculation, the center manifold is given by the graph of h(u), where where, k = ar ln 2 r 2(r−1) 2 + aδ 1 ln r r−1 + aδ 1 ln r + ar ln 2 r 2(r−1) .The map on the center manifold is given by

Numerical simulation
In this section, we present some phase portrait of model (3.1) to explain the above theoretical analysis.
Figure 2 illustrates the dynamics of model (3.1) for some values of r.When r < 1, three populations get very close to zero.In the other words, the origin is asymptotically stable.When r > 1, the host population increase in amplitude.In this case, two parasite populations y(t) and z(t) extinct but the host population x(t) goes to infinity.

f 2 FIGURE 1 .
FIGURE 1. Graph of functions f 1 (y) and f 2 (y) for: (a) a > b(r−1)r ln r , the system has no positive fixed point, (b) a = b(r−1) r ln r , the system has no positive fixed point and (c) a < b(r−1) r ln r , the system has a unique positive fixed point.