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In this paper, we prove the existence of a limit cycle for a given system of differential equations corresponding to an asymmetrical intraguild food web model with functional responses Holling type II for the middle and top predators and logistic grow for the (common) prey. The existence of such limit cycle is guaranteed, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, under certain conditions for the parameters involved in the system.

It is well known that interaction between three species, in which predation and competition occurs, is called intraguild predation (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 [

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

where

where

Consequently, the intraguild predation model that we will study is

For ecological considerations the domain of interest

We now state our main result. We establish the existence of a unique equi- librium point

Theorem 1 (Main result). If the positive parameters involved in system (2) satisfy the conditions

where

equilibrium point of system (2) in

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 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.

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

Theorem 2. Let

being

The next theorem was proved by E. Hopf in 1942 (see [

Theorem 3 (Hopf's Theorem.). Suppose that the

with

then there is a unique two-dimensional center manifold passing through the point

in a neighborhood of the origin which, for

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

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

Lemma 1. Let

Proof.

Comparing to (5) we obtain the result.

Corollary 1. If

Proof. Use that the system (6) is satisfied.

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

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,

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

Notice that the last equation in the system above, is linear with respect to the variable

Taking

From there if

is a solution of (7). Thus proving the lemma.

Now our goal is to determine when the equilibrium point

Theorem 4. If the parameters involved in system (2) satisfy the conditions of Lemma 2 and additionally

then the equilibrium point

and the eigenvalues of the linear approximation of system (2) at

where

Proof. The Jacobian matrix

where

Using Corollary 1 the characteristic polynomial of

if and only if

In this case the value of

Choose

Now taking

Now taking

Now solving Equation (12) for the parameter

Choosing

and the expression for

the equilibrium point is

and from (13) and (14) the eigenvalues of the linear approximation of system (2) at

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

thus, the linear approximation of system (2) at the equilibrium point

pair of pure imaginary eigenvalues if and only if

Applying the Theorem 2 to system (2) at the equilibrium point

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

Lyapunov coefficient

where

Proof. Let

with

The linear part of system (17) at 0 is

It follows immediately from Theorem 4 that the eigenvalues of

The bilinear function

where

The trilinear function

where

The normalized eigenvector

The adjoint eigenvector

Taking into account the formula of the first Lyapunov constant

where

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.

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

librium point of system (2) in the positive octant of

If

and taking into account the above assignments of

follows from Theorem 4 and Remark 5 that if

values of the linear approximation of system (2) at the equilibrium point

are

of

where,

and

Hence,

Moreover, by Theorem 6 the first Lyapunov coefficient of the differential sys- tem (2) at the equilibrium point

from the critical value

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

thus, the unique equilibrium point of system (18) in the positive octant of

Remark 8. Notice that system (18) has, additionally to

For

For

For

As a consequence these equilibrium points are hyperbolic, moreover they are saddle points.

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

system (2) in the positive octant of

bifurcates from the equilibrium

For example, if we consider the parameters values

then the linear approximation of system (2) at

The real part of the complex eigenvalues is

where

and its derivative is

where

If

and

The Lyapunov coefficient is

hence, we have a supercritical Hopf bifurcation, and then the periodic orbit obtained from the bifurcation is stable.

In

dition

Finally, notice that, under the assignations in (20) one has the following:

・

resource, the meso-predator is superior in comparison with the super- predator;

・

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.

Castillo-Santos, F.E., Rosa, M.A.D. and Loreto-Hernández, I. (2017) Existence of a Limit Cycle in an Intraguild Food Web Model with Holling Type II and Logistic Growth for the Common Prey. Applied Mathematics, 8, 358- 376. https://doi.org/10.4236/am.2017.83030