It is well known that interaction between three species, in which predation and competition occurs, is called intraguild predation (see  ). This kind of in- teraction can take place in a group of species that exploit the same resources in a similar way (see  ). This kind of interaction among the species in an intraguild model is of particular interest. One of the main questions when looking at inter- action of species is whether or not there will be coexistence among them. This is of importance from the ecological point of view. In the intraguild predation model, one can consider two cases, the symmetric which occurs when there is a mutual predation between two species, and the asymmetric that occurs when one species, usually called intraguild predator, always predate the middle species, which is called the intraguild prey. In both cases it is assumed that the corres- ponding species use common foods (see  ).
The criterion to have coexistence in the asymmetric intraguild predation sys- tem seems to be, on one hand, to impose conditions on the intraguild prey, that is, it should be superior at the competition for the resources in comparison with the intraguild predator, and on the other hand, that the intraguild predator should be substantially benefit from the consumption to the intraguild prey in the sense that its most important food source is intermediate species (see  ).
There are some recent papers where food chain models between three species have been studied in which the authors have obtained results about the co- existence of the species by looking at the existence of limit cycles for the corresponding model systems, for instance tritrophic models with linear growth prey (see    ) and logistic growth prey (see  ). These models do not consider predation of the top predator to the resource (the prey). Hence one can see that intraguild predation is a more complex interaction between species that the tritrophic model.
If the growth rate for the resource is linear, we are assuming that the density of the resource is growing exponentially. When it is assumed logistic growth rate for the resource, the corresponding carry capacity implies that the resource density is bounded, which has Ecological sense but it seems to be more difficult to have a coexistence between the species.
In this paper, we are interested in guaranteeing the coexistence of three species forming an intraguild food web model, which is an asymmetrical intraguild predation model with functional response for the middle predator species, and functional responses for the top predator, and logistic grow for the prey. More precisely models with the form:
where represents the density of a prey that gets eaten by a species of density (mesopredator) and a species of density (super-predator), and the species feeds the species . Moreover represents the carry capacity of the prey and represents the growth rate of the prey. The parameters and are positive constants which represent the benefit from the consumption of food and the parameters and represent the mortality rate of the correspon- ding predators. We will consider that the functions and given in (1) are Holling type II, that is
where and are positive constants.
Consequently, the intraguild predation model that we will study is
For ecological considerations the domain of interest is the positive octant of that is
We now state our main result. We establish the existence of a unique equi- librium point for the system (2) in , at this point, , we show that the system exhibits a Hopf’s Bifurcation and the limit cycle given by the bifurcation is stable. All of this is obtained under certain restrictions on the parameters involved in the system.
Theorem 1 (Main result). If the positive parameters involved in system (2) satisfy the conditions
where and then the point is the unique
equilibrium point of system (2) in moreover, we have a Hopf bifurcation in and the limit cycle that bifurcates from the equilibrium of system (2) as
increases from the critical value is stable.
This article is organized as follows.
In Section 2 we provide the reader with the results that allowed us to study the system. In particular we present the version of the well known Hopf’s Bifurcation Theorem.
Section 3 is devoted to study the equilibrium points for our system in the positive octant with the aim of guaranteeing the hypothesis of Hopf’s Bifurcation Theorem. For this, we consider two subsections, the subsection 3.1 in which we show, under certain conditions on the parameters, the existence of an equi- librium point in the positive octant of ; and subsection 3.2 where we show that under certain conditions on the parameters, the eigenvalues for the linear system at the equilibrium point associated to the system given in (2) are which is real and the conjugated pure imaginary, and also the Lyapunov coefficient is computed.
In Section 4 we provide the proof of our main result in this paper. Further- more in Section 5 we provide the reader with a numerical result showing the stable limit cycle of the system.
In order to obtain all the calculations and simulations in this paper, we made use of a routine in the program Mathematica. This allowed us to simplify most of the process needed to obtain our result.
2.1. Lyapunov Coefficient and Hopf Bifurcation
One of the main tools to determine the existence of a stable or unstable limit cycle is the first Lyapunov coefficient. This, in general, is not easy to calculate. To compute the first Lyapunov coefficient of a differential system at an equilibrium point we make use of result by Kuznetsov (see [  , p. 175]) whose statement is given in the following Theorem (cf.  ).
Theorem 2. Let be a differentiable map of class in an open subset of whose third order Taylor approximation of around is
being and bilinear and trilinear forms, respectively. More over, assume that has a pair of purely imaginary eigenvalues . Let be the eigenvector of corresponding to the eigenvalue , normalized so that the hermitian product satisfies being the conjugate vector of . Let be the adjoint eigenvector such that and . If denotes the identity matrix, then the first Lyapunov constant of the system of Ordinary Differential Equations with an equilibrium point at is
The next theorem was proved by E. Hopf in 1942 (see  and for a proof in the bidimensional case see  and the general case see [  , Section 5], and [  , Section 5.4]). This theorem guarantees the existence of a Hopf’s bifurcation at an equilibrium point of a system of ordinary differential equations whenever reaches a critical value .
Theorem 3 (Hopf's Theorem.). Suppose that the -system
with and has a critical point for and that has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real part. Then there is a smooth curve of equilibrium points with and the eigenvalues, and of , which are pure imaginary at vary smoothly with . Furthermore, if
then there is a unique two-dimensional center manifold passing through the point and a smooth transformation of coordinates such that the sys- tem (4) on the center manifold is transformed into the normal form
in a neighborhood of the origin which, for , has a weak focus of multiplicity one at the origin and
is a universal unfolding of this normal form in a neighborhood of the origin on the center manifold. Moreover a periodic solution bifurcates from the point for if or for if . This periodic solution is stable if and unstable if . For the equilibrium point is a locally stable point for and locally unstable point for . For the equilibrium point is locally unstable point for and locally stable point for .
2.2. A Little of Linear Algebra
In this subsection we show a few results from Linear Algebra that allowed us to simplify our calculations in the next sections. This will provide us with a dif- ferent technique to find the eigenvalues of a given matrix.
If is a matrix with , its characteristic polynomial is de- termined by the entries of as a classical computation shows. In fact,
Lemma 1. Let be a matrix with . Then has eigenvalues and with if and only if:
Proof. has the given eigenvalues of and with , if and only if its characteristic polynomial takes the form:
Comparing to (5) we obtain the result.
Corollary 1. If is a matrix with . Then has eigenvalues and with if and only if , where
Proof. Use that the system (6) is satisfied.
3. Equilibrium Points in the Positive Octant
3.1. Existence of an Equilibrium Point in the Positive Octant of
In order to find the equilibrium points and the restrictions in the parameters involved in the system (2) we use a different approach. We think of the equi- librium point as a new three parameters of the system. In this way our system (2) will have as new parameters the values of which we are setting to be positive. This approach differs from the usual method that is applied to find the equilibrium points (See  ).
In the next lemma we proceed to show the existence of an equilibrium point given conditions on the parameters involved in the system of differential equations. Moreover we can guarantee that the equilibrium point will be in .
Lemma 2. Assume that the parameters in the system (2) are given by
where, and then
is an equilibrium point of the system (2) in the region
Proof. The equilibrium points of the system are solutions of the following equations.
By multiplying the above equations by the denumerators (which are always non zero), involved in each corresponding equation we obtain that the equi- librium point must satisfy (8). Correspondingly each solution of (8) must also be an equilibrium point of the system (2)
By taking and (8) reduces to
Notice that the last equation in the system above, is linear with respect to the variable . Solving this equation and substituting this value on the second one we obtain a system of two equations in and where the exponents of and in each equation is 1. From there we can obtain that the solutions of the system (9) are the following.
Taking and where we have:
From there if then all parameters involved in (2) becomes positive and
is a solution of (7). Thus proving the lemma.
3.2. A Pair of Pure Imaginary Eigenvalues and the First Lyapunov Coefficient
Now our goal is to determine when the equilibrium point exhibits a Hopf's bifurcation. In order to show this, we show the existence of parameters where the equilibrium point has a pair of pure imaginary eigenvalues and a negative real eigenvalue. Making use of Hopf’s Theorem, we shall prove the existence of a Hopf bifurcation.
Theorem 4. If the parameters involved in system (2) satisfy the conditions of Lemma 2 and additionally
then the equilibrium point is
and the eigenvalues of the linear approximation of system (2) at are
Proof. The Jacobian matrix of the system (2) at is
Using Corollary 1 the characteristic polynomial of has roots and where
if and only if
In this case the value of is given by:
Now taking we have and with this choices .
Now taking , the expression for simplifies to:
Now solving Equation (12) for the parameter in terms of and we obtain:
and if with we also obtain that, . If we take then hence all expressions of the as- signed parameters of system (2) are simplified:
and the expression for given by (15) simplifies to Thus
the equilibrium point is
and from (13) and (14) the eigenvalues of the linear approximation of system (2) at are given by Equations (10) and (11), which proves the theorem. □
Remark 5. Notice that by Theorem 4 and Subsection 2.2, the characteristic polynomial of the linear approximation of system (2) at the equilibrium point has the form if and only if the fundamental Equation (12) is satisfied, which, in this case, Equation (12) reduces to
thus, the linear approximation of system (2) at the equilibrium point has a
pair of pure imaginary eigenvalues if and only if .
Applying the Theorem 2 to system (2) at the equilibrium point we get the following result.
Theorem 6. If the parameters involved in system (2) satisfy the hypothesis of Lemma 2 and Theorem 4 then the eigenvalues of the linear approximation of system (2) at the equilibrium point
are and where and the first
Lyapunov coefficient of the differential system (2) at the equilibrium point is given by
where y are the positive constants defined by
Proof. Let as in (16) and consider a linear change of variables to translate to the origin of coordinates, after that change, we obtain a differential sys- tem
with and Denote the vector field associated to this differential system. Now, we compute the linear part the bilinear and trilinear forms of the Taylor expansion of the function
The linear part of system (17) at 0 is
It follows immediately from Theorem 4 that the eigenvalues of are
The bilinear function at vectors is given by
The trilinear function at vectors is given by
The normalized eigenvector of corresponding to eigen- value has coordinates
The adjoint eigenvector of the transpose matrix of corresponding to the eigenvalue has coordinates
Taking into account the formula of the first Lyapunov constant of Theorem 2, the values of and we have that the expression of the first Lyapunov coefficient at the equilibrium point is
where y are the positive constants defined by
Remark 7. Notice that with the parameters as in Theroem 4 and Theorem 6 and according with the above result the first Lyapunov coefficient of the system at the given equilibrium point is always negative.
4. Proof of the Main Result
In this section, using the results given in Section 2 and results obtained in Subsection 3.2, we give a proof of our main result given by Theorem 1.
Proof of Theorem 1. If
it follows immediately from Lemma (2), that is an equi-
librium point of system (2) in the positive octant of
and taking into account the above assignments of and it
follows from Theorem 4 and Remark 5 that if the eigen-
values of the linear approximation of system (2) at the equilibrium point
are and the pure imaginary complex numbers where For with the above assignments
of and the real part of the complex eigenvalues and of the linear approximation of system (2) at the equilibrium point (the equilibrium point does not depend of the parameter ), is
Moreover, by Theorem 6 the first Lyapunov coefficient of the differential sys- tem (2) at the equilibrium point is then applying Hopf’s Theorem, (Theorem 3), we have a Hopf’s bifurcation at and that the limit cycle that bifurcates from the equilibrium of system (2) as increases
from the critical value is stable.
Now, taking into account the assignments for parameters given above, the system (2) has the form:
which is in terms of the free positive parameters and We have that the rest of equilibrium points of system (18) are
thus, the unique equilibrium point of system (18) in the positive octant of is and the theorem is proved.
Remark 8. Notice that system (18) has, additionally to , the equilibrium points given by (19). For the eigenvalues of the linear approximation of the system are:
As a consequence these equilibrium points are hyperbolic, moreover they are saddle points.
5. Numerical Result
Theorem 1 guarantees the existence of a Hopf’s bifurcation if we have the fol- lowing assignments for the parameters of system (2):
With these assignments of the parameters the system (2) is in terms of the free positive parameters and the unique equilibrium point of
system (2) in the positive octant of is By Theorem 1, for close enough to and then a limit cycle
bifurcates from the equilibrium of system (2).
For example, if we consider the parameters values
then the linear approximation of system (2) at is
The real part of the complex eigenvalues is
and its derivative is
If then has eigenvalues
The Lyapunov coefficient is
hence, we have a supercritical Hopf bifurcation, and then the periodic orbit obtained from the bifurcation is stable.
In Figure 1, we exhibit the stable limit cycle of differential system (2) with the above parameters values, that is, we show an orbit tending to the local attractor defined by a stable Hopf periodic orbit with and initial con-
Figure 2 shows the same behavior but with different initial condition .
Figure 1. Stable limit cycle and Time series with initial condition .
Figure 2. Stable limit cycle and Time series with initial condition .
Finally, notice that, under the assignations in (20) one has the following:
・ and hence in the competition for the
resource, the meso-predator is superior in comparison with the super- predator;
・ on the corresponding domains, that is
therefore the super-predator is substantially benefit from the consumption to the meso-predator in the sense that its most important food source is in- termediate species.
From the above, one can conclude that our model makes ecological sense.
The first author was partially supported by CONACyT grant number CB- 2014-243722. The authors would like to thank Prof. Gamaliel Blé González and Prof. Víctor Castellanos Vargas for their helpful discussions and corrections in the preparation of this paper.
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