Existence of a Limit Cycle in an Intraguild Food Web Model with Holling Type II and Logistic Growth for the Common Prey

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.


Introduction
It is well known that interaction between three species, in which predation and competition occurs, is called intraguild predation (see [1]).This kind of interaction can take place in a group of species that exploit the same resources in a similar way (see [2]).This kind of interaction among the species in an intraguild model is of particular interest.One of the main questions when looking at interaction 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 corresponding species use common foods (see [1]).
The criterion to have coexistence in the asymmetric intraguild predation system 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 [3]).
There are some recent papers where food chain models between three species have been studied in which the authors have obtained results about the coexistence 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 [4] [5] [6]) and logistic growth prey (see [7]).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 ( ) 1 f x for the middle predator species, and functional responses , f x f y for the top predator, and logistic grow for the prey.More precisely models with the form: where x represents the density of a prey that gets eaten by a species of density y (mesopredator) and a species of density z (super-predator), and the species y feeds the species z .Moreover R represents the carry capacity of the prey and ρ represents the growth rate of the prey.The parameters 1 3 , c c and 4 c are positive constants which represent the benefit from the consumption of food and the parameters 2 c and µ represent the mortality rate of the correspon- ding predators.We will consider that the functions 1 2 , f f and 3 f given in (1)   are Holling type II, that is ( ) Consequently, the intraguild predation model that we will study is ( ) For ecological considerations the domain of interest Ω is the positive octant of 3 ,  that is We now state our main result.We establish the existence of a unique equilibrium point 0 p for the system (2) in Ω , at this point, 0 p , 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 1 > 0 k and 0 > 0, x then the point is the unique equilibrium point of system (2) in Ω moreover, we have a Hopf bifurcation in 0 p and the limit cycle that bifurcates from the equilibrium 0 p of system (2) as µ increases from the critical value 0 47, 775, 075 600, 704 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 equilibrium point 0 p in the positive octant of 3  ; and subsection 3.2 where we show that under certain conditions on the parameters, the eigenvalues for the linear system at the equilibrium point 0 p associated to the system given in (2)   are α which is real and iω ± 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.Furthermore 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.

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 ( ) l p of a differential system at an equilibrium point 0 , p we make use of result by Kuznetsov (see [ [8], p. 175]) whose statement is given in the following Theorem (cf.[4]).
Theorem 2. Let : C in an open subset Ω of n  whose third order Taylor approximation of F around being B and C bilinear and trilinear forms, respectively.More over, assume that A has a pair of purely imaginary eigenvalues i ω ± .Let q be the eigenvector of A corresponding to the eigenvalue i ω , normalized so that the hermitian product satisfies 1 q q ⋅ = being q the conjugate vector of q .Let p be the adjoint eigenvector such that T = A p ip ω − and 1 p q ⋅ =.If I denotes the identity matrix, then the first Lyapunov constant ( ) p C q q q p B q A B q q p B q iI A B q q ω ω The next theorem was proved by E. Hopf in 1942 (see [9] and for a proof in the bidimensional case see [10] and the general case see [ [11], 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 ( ) x F x µ =  whenever µ reaches a critical value 0 µ .
Theorem 3 (Hopf's Theorem.).Suppose that the 4 C -system ( ) , , ) ( ) then there is a unique two-dimensional center manifold passing through the point ( ) 0 0 , x µ and a smooth transformation of coordinates such that the sys- tem (4) on the center manifold is transformed into the normal form

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 different technique to find the eigenvalues of a given matrix.
If M is a 3 3 × matrix with 33 0 m = , its characteristic polynomial is de- termined by the entries of M as a classical computation shows.In fact, ) ( ) Proof.M has the given eigenvalues of α ∈  and iω and only if its characteristic polynomial takes the form: Proof.Use that the system ( 6) is satisfied.

Octant of 3 
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 equilibrium point ( ) as a new three parameters of the system.In this way our system (2) will have as new parameters the values of 0 0 0 , , x y z which we are setting to be positive.This approach differs from the usual method that is applied to find the equilibrium points (See [4]).
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 Ω .
Lemma 2. Assume that the parameters in the system (2) are given by ( ) ( ) ( ) is an equilibrium point of the system (2) in the region .Ω 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 equilibrium point must satisfy (8).Correspondingly each solution of (8) must also be an equilibrium point of the system (2) Notice that the last equation in the system above, is linear with respect to the variable 3 a .Solving this equation and substituting this value on the second one we obtain a system of two equations in 1 a and z where the exponents of 1 a and z in each equation is 1.From there we can obtain that the solutions of the system (9) are the following.

A Pair of Pure Imaginary Eigenvalues and the First Lyapunov Coefficient
Now our goal is to determine when the equilibrium point 0 p 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 0 p is and the eigenvalues of the linear approximation of system (2) at 0 p are ) ( ) In this case the value of α is given by: ( ) ) Now solving Equation ( 12) for the parameter µ in terms of  hence all expressions of the assigned parameters of system (2) are simplified: and the expression for µ given by (15) simplifies to 47, 775, 075 .600, 704 ρ µ = Thus the equilibrium point is and from ( 13) and ( 14) the eigenvalues of the linear approximation of system (2) at 0 p 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 ) Applying the Theorem 2 to system (2) at the equilibrium point 0 p 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   16) and consider a linear change of variables to translate 0 p to the origin of coordinates, after that change, we obtain a differential sys- tem ( ) , , X x y z = and ( ) , , , , ,  , , ,  , , ,  F x y z F x y z F x y z F x y z = associated to this differential system.Now, we compute the linear part , A the bilinear , B and trilinear C forms of the Taylor expansion of the function .F The linear part of system (17) at 0 is The trilinear function C at vectors ( z u v w r s t is given by ( ) ( , , , , , , , ,  , ,  C x y z u v w r s  The normalized eigenvector ( ) , , q q q q = of A corresponding to eigen-value iω has coordinates ( ) Theorem 2, the values of , , A B C and , , , q q p we have that the expression of the first Lyapunov coefficient at the equilibrium point 0 p is ( ) ( ) , , , s s s s y 5 s are the positive constants defined by 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.

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 3 .  R the real part of the complex eigenvalues ( ) λ µ and ( ) λ µ of the linear approximation of system (2) at the equilibrium point 0 , p (the equilibrium point 0 p does not depend of the parameter µ ), is ( ) ( ) ( ) which is in terms of the free positive parameters 0 3 1 , , , x b k ρ and .µ We have that the rest of equilibrium points of system (18) are ( ) ( ) 60,861,300 , ,0 , 2 ,0,0 , and 0,0,0 , 5479 30, 019, 441 thus, the unique equilibrium point of system (18) in the positive octant of 3  is 0 p and the theorem is proved. Remark 8. Notice that system (18) has, additionally to 0 p , the equilibrium points given by (19).For 0 µ µ = the eigenvalues of the linear approximation of the system are:

Numerical Result
Theorem 1 guarantees the existence of a Hopf's bifurcation if we have the following assignments for the parameters of system (2): With these assignments of the parameters the system (2) is in terms of the free positive parameters 0 3 1 , , , x b k ρ and , µ the unique equilibrium point of system (2) in the positive octant of 3  is For example, if we consider the parameters values then the linear approximation of system (2) at ( ) The real part of the complex eigenvalues 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     1 and Figure 2), therefore the super-predator is substantially benefit from the consumption to the meso-predator in the sense that its most important food source is intermediate species.
From the above, one can conclude that our model makes ecological sense.
, a b a b a and 3 b are positive constants.

,
Df p µ 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 ( ) p µ with ( )

3 ×D
matrix with 33 0 m = .Then M has eigenvalues α ∈  and iω ± with 0 ω > if and only if ( ) 0 b y a xy b a x z R x b a x b a x b c c x a c x b a y b a x z b b xa a y

, the expression for 2 ω
simplifies to: 0 p as in ( immediately from Theorem 4 that the eigenvalues of A are 5x y z u v w is given by

=
Taking into account the formula of the first Lyapunov constant ( )

If
linear approximation of system (2) at the equilibrium point 0 , p a a a b b c c and , cycle bifurcates from the equilibrium 0 p of system (2).