Competition among Predators and Allee Effect on Prey, Their Influence on a Gause-Type Predation Model

Interference or competition among predators (CAP) has often been ruled out in depredation models, although there are varied mathematical forms to describe and incorporate it into this interaction. In this work, we present the most known of these descriptions and one of them will be used in a modified Volterra model. Moreover, of this ecological phenomenon, a simple and strong Allee effect affecting the prey population will be considered in the relationship. An important feature of the new model is to have until two positive equilibrium points, to the difference with the Volterra model (without Allee effect); hence different and interesting dynamic situations appear in the system. Conditions for the existence and local stability of equilibria are determined. The boundedness of solutions, the existence of a limit cycle and a separatrix curve are also proven. Besides, the main properties of the model are examined from an ecological point of view. To make a comparative discussion of our results, an Appendix is added with the main properties of models, in which neither the Allee effect nor the competition among predators is considered. Some simulations are shown to endorse our results.


Introduction
Usually, the analysis of predator-prey models considers different ecological phenomena affecting either one population or both populations. These phenomena can have strong consequences in the relationship between them and modifying the dynamical properties of a system describing it [1,2].
Such is the case of an Allee effect affecting the prey population or else, the competition among predators (CAP) [3]; nonetheless, there are not enough studies to determine the real impact of these two phenomena in that dynamical relationship, when both act simultaneously in the interaction.
Moreover, different mathematical formulations have been given for each of these phenomena; as it has been shown in previous works [3], diverse mathematical forms for the same phenomenon can produce changes in the properties of the system describing the interaction [1]. Then, a comparative study using another modeling for these phenomena must be also realized in the future. To determine the influence of these mathematical expressions in the dynamics of the systems is an interesting objective to the modelers.
So, in this work a modified Volterra model [4] is analyzed, in which the functional response is linear, assuming that (i) the prey growth rate is affected by a strong Allee effect [5] and (ii) there exists self-interference (interference) or competition among predators (CAP) [6].

Interference or Competition among Predators.
The struggle among living things, for food, space, etc., is well known (competition in the survival of the fittest [7]); particularly, the competition among predators for prey or other resources [6] is one of them.
Obviously, intraspecific competitive interactions between individual predators can affect the birth and death rates 2 Mathematical Problems in Engineering of their whole population [4]. Furthermore, antagonistic interactions may also affect predator efficiency in finding and killing prey [4], which can imply the modification of the predator functional response on the models. Different behaviors of predator influencing the interrelation between prey and predator can be assumed, for example, behavior typical of territorial animals where individuals "waste time" in direct contests, thereby decreasing the time each could otherwise devote to foraging (searching for or handling prey); spatially aggregated predators and prey and so on [8].
Diverse mathematical forms to express the competition (or interference) among predators (CAP) have been proposed in the ecological literature and described in the following way: (I) The early expression for CAP was formulated independently in 1975 by J. A. Beddington [9] and D. DeAngelis and coworkers [10], proposing a new functional response. They modify the hyperbolic Holling type II functional response, by adding a term in the denominator, obtaining a new functional response, dependent of two interacting populations and given by ℎ ( , ) = + + , with , , , > 0, where and stand for the prey population size and the predator population size, respectively. The term in the denominator expresses the mutual interference among predators [11].
Although this function combines the hyperbolic response with self-interference among predators, [12] made a deduction involving prey refuges use instead of the CAP.
(II) A second form to express the competence among predators was formulated by Herbert I. Freedman in 1979 [13], modifying the assumption of the usual models that the total prey death rate is the predator functional response times the number of predators [13]. He proposed the function ℎ ( , ) = ℎ ( ) , (2) with being the mutual interference constant such that 0 < ≤ 1, and ℎ( ) is the prey-dependent functional response of predator.
Following C. W. Clark [14], the function ( ) = expresses the congestion among fishing vessels (the men as predators) harvesting a fish school, resulting in decreasing of catch-rates.
(III) The third form to describe the CAP is given by a negative quadratic term added into the predator growth equation [6]. Hence, the function takes the form ( ) = − − 2 , with , and > 0. ( In this case, it is assumed that the predator population be reduced by other causes as the size of the habitat suitable for the predator to live and reproduce there [6]. In our work, we model the CAP with this last form presented in the Bazykin's book [6]; moreover, we consider the linear functional response independent of predator density (i.e., only prey dependent), which means that any single predator affects the prey population growth rate independently of its conspecifics [15].
We note that if in the function is added a quadratic positive term, i.e., ( ) = − + 2 , with and > 0, it has the description of the cooperation or collaboration among predators [16]. This is also a frequent social behavior in nature, used by some predators to enhance their efficiency in the consumption of prey [16].

The Allee Effect.
It is well known that any mechanism describing a positive correlation between a component of individual fitness and the population size of conspecifics can be named as an Allee effect [17][18][19].
This phenomenon, called after the American ecologist Warder Clayde Allee , is also known under different names; it has also been named as density dependence or positive density dependence and other names in Population Dynamics [17,20] or depensation in Fisheries Sciences [14,20].
Populations can exhibit Allee effect dynamics due to a wide range of biological phenomena, for example, reduced antipredator vigilance, social thermoregulation, genetic drift, mating difficulty, reduced antipredator defense, and deficient feeding at low densities; however, several other causes may lead to this phenomenon (see [21] or [5]).
The strong Allee effect implies the existence of a threshold population level > 0 [6,23,24], under which the population becomes extinct. This requires that the population growth rate be negative for population sizes minor than .
Many continuous time equations have been used to model the Allee effect [23], although most of them are topologically equivalent [25]; i.e., solutions have the same qualitative behavior.
The most familiar equation is described by where = ( ) indicates the population size of a species in an environmental [24]. The parameters and are positives; meanwhile can be positive, negative or zero. It has, respectively, a strong ( > 0), a weak ( < 0), or special ( = 0) weak Allee effect.
By ecological reason ≪ since is a population threshold under the growth population rate is negative [17,18]. We note that if < − there is not an Allee effect.
Diverse ecological research suggests that two or more Allee effects can lead to mechanisms acting simultaneously on a single population (see [21]); the combined influence of some of these phenomena is known as multiple (double) Allee effect [21]. In these cases, most complicated equations are proposed [3,26], such as which was presented in [23]. However, when they are considered in predator-prey models can produce changes in the dynamical of the system, particularly on the number of limit cycles of the system [3,26], and the existence of separatrix curves for the trajectories.
This article is organized as follows: The model and general settings are presented in Section 2; in Section 3 we obtain the main results on the boundedness of trajectories, the local nature of equilibrium points and in the last Section the ecological implications of our findings are given.
The obtained results will be compared with the predatorprey model in which competition among predators is not considered, partially studied in the book by Kot [24]; also it will be compared with the model considering double Allee effect [5,21], without self-interference among predators analyzed in [1,2].

The Model
Using the simplest way to describe the Allee effect and expressing the action of many predators in the interaction, considering CAP (self-interference) as in the logistic growth rate. Thus, the model is described by the bidimensional system of Kolmogorov type [27]: where = ( ) and = ( ) represent the prey and predator population size, respectively for ≥ 0 (measured in number of individuals, density by area or volume unit or biomass); the parameters are all positives, i.e., = ( , , , , , , ) ∈ R 7 + , with − < ≪ .
In this work, we consider only > 0, i.e., the prey population is affected by a strong Allee effect.
The parameters have the following ecological meanings: is the intrinsic prey growth rate or biotic potential is the prey environmental carrying capacity > 0 is the strong Allee effect threshold or the minimum of viable prey population is the quantity of prey that can be eaten by a predator in each time unit is the efficiency with which predators convert consumed prey into new predators is the natural death rate of predators is the level of the struggle among predators. Particularly, the term − 2 represents interspecific density-restricted effects on the predators [6,28]. System (6) describes a generalized Gause-type predatorprey model [27], which does not obey the mass action principle [29]. It is of Kolmogorov type [27], defined on The equilibrium points are (0, 0), ( , 0) and the points ( , ) lie on the isoclines (1 − / )( − ) − = 0 and − − = 0. It must be remembered that if = 0, the obtained system is partially studied in the book by M. Kot [24]. Assuming that the prey is affected simoustanly by two Allee effects, the modified system is analyzed in [1], showing a richer dynamics than the model studied in Kot [24].
Setting / = 0, we find that equation (6) generates a slanted predator isocline that passes through the point (0, − / ), out of the first quadrant. With a predator isocline of this type, two positive equilibrium points can appear, one of multiplicity two or none (see Figure 1).
The positive slope of the predator isocline reflects direct density dependence and could have a stabilizing influence. Nonetheless, we prove the unique positive equilibrium point may be unstable, generating a stable limit cycle.
With the objective of to make a comparative study, we summarize in the Appendix, the properties of two related models with the model described by the system (6).
(i) Volterra model with competition among predators is studied in [6] and given by with = ( , , , , , ) ∈ R 6 + . (ii) The Volterra model with Allee effect on prey is studied in [24], described by

Main Results
To simplify the calculations, following the methodology of [1,2,26], we make a reparameterization of the vector field considering the change of variables and a time rescaling given by a diffeomorphism [30,31] :

Proposition 1. Topologically equivalent systems System (6) is topologically equivalent to
Proof. Using the change of variables given by = and = ( / )V, replacing in (6), we have / = ( / ) and / = ( / )( V/ ) obtaining a new system given by or, By means of the time rescaling given by = and by using the chain rule, it follows that Renaming the parameters by = / , = / , = / and = / this becomes system (8).
and det ( , V, ) = / > 0. Then, the diffeomorphism is a smooth change of variables with a rescaling of the time preserving the time orientation. Hence, from (6) we obtain a qualitatively (topologically) equivalent vector field = ∘ , which has the form , where ( , V) and ( , V) are the right sides of system (11). Clearly, the associated second order differential equations system is the Kolmogorov type polynomial (11) [27].
Our study is divided into three cases: when ∈]0, 1[, the special case = 0 and when < 0.
which can has two real positive roots, if > . (16) becomes Mathematical Problems in Engineering 5 which has a unique real positive root, for any value of the parameter .
(c) If < 0, the factor − of equation (16) is negative. Then, any being the sign of the factor ( + 1) − 1, equation (9) has a unique positive solution . This implies a significant difference between the cases of strong and weak Allee effect.
Besides, we note that there is a difference in the dynamics of system (11) with the case in which = 0, i.e., when the CAP does not exist [24].
The number of the positive equilibria of equation (16) can be resumed in the following proposition.
Proof. It is immediate.
Therefore, the number of positive equilibrium points follows from the lemma above, and the different cases obtained are displayed in Table 1.
So, in the cases  (ii) if < 0, then and it exists a unique solution of (9).
To determine the nature local of the equilibrium points we will use the Jacobian matrix given by For system (11) we have the following results:  In system (6), the set Γ = {( , ) ∈ Ω/0 ≤ ≤ , 0 ≤ } is a positively invariant region.

Lemma 6. Boundedness of trajectories
The solutions are bounded.
So, the solutions of system (11) are bounded. Furthermore, there exists the set which is an invariant region, where all the solutions of the system (11) starting in Ω are confined.

Remark 7.
(1) The result established in the above lemma implies the model is well posed [29], i.e., when the prey population size tends to zero, the predator population also tends to zero, since the prey is the unique source of food for predators.
(2) On the other hand, Then, the subregion is a bounded and closed region; i.e., it is a compact region and the Poincaré-Bendixon Theorem applies.

Strong Allee Effect.
In the follow, we assume 0 < ≪ 1. Proof. Evaluating the Jacobian matrix we have that

Nature of Equilibrium Point over the Axis
Then, (0, 0) is an attractor equilibrium point.
Then, according the sign of factor 1− , we have the thesis. Proof. The Jacobian matrix is Therefore, according the sign of factor − , we have the lemma.

Nature of Positive Equilibrium Points.
As there are many cases to study, in the following we consider only a few cases. Then, which sign depends on the sign of Using equation (16) we have We note that is independent of . Proof. Substituting 1 in we have, Then, det det ( , V) < 0. Proof. Analogously the above proof we can demonstrate that > 0; so, the nature of ( 2 , ( 2 − )/ ) depends on the sign of Using the expression for 2 from equation (16) we have tr Then, the sign of tr ( , V) depends on the sign of According the sign of the results follow from Routh-Hurwitz criterion.

Corollary 13. Transversality condition
The system undergoes a Hopf bifurcation with respect to bifurcation parameter around the equilibrium point Proof. If tr ( 2 , V 2 ) = 0, then both eigenvalues will be purely imaginary provided det ( 2 , V 2 ) > 0. Therefore, the Implicit function Theorem assures that a Hopf bifurcation occurs where a periodic orbit is created as the stability of the equilibrium point ( 2 , V 2 ) changes.
(a) As Γ is an invariant region, the orbits cannot cross the straight line = 1 towards the right. By Existence and Uniqueness Theorem, the trajectory determined by the right unstable manifold + ( 1 , V 1 ) cannot meet or intersect the trajectory determined by the superior stable manifold Moreover, the − of the + ( 1 , V 1 ) can lie at point ( , 0) by the Boundedness Lemma or at infinity in the direction of − . On the other hand, the − of the right unstable manifold + ( 1 , V 1 ) can be either (i) the point ( 2 , V 2 ), when this is an attractor, (ii) a stable limit cycle, if ( 2 , V 2 ) is a repeller, or (iii) the point (0, 0). Hence, there is a subset of the parameter space for which obtained. In this case, the same point ( 1 , V 1 ) is the − of the right unstable manifold + ( 1 , V 1 ).
(b) We will analyze the stability of the homoclinic cycle obtained by the breaking of the homoclinic curve determined by the stable manifold + ( 1 , V 1 ) and the unstable manifold Denoting + 1 ( 1 , V 1 ) and − 1 ( 1 , V 1 ) the eigenvalues associated to point ( 1 , V 1 ), where the upper indices correspond to the sign of the respective eigenvalue. We determine the neutrality of the homoclinic cycle considering = 1 [3,6]; then, i.e., Replacing 2 1 = ((( + 1) − 1) 1 − ( − ))/ , we obtain Then, So, the non-infinitesimal limit cycle generated by the breaking of the homoclinic curve and surrounding the equilibrium point ) .
To determine the sign of det ( , ( − )/ ), we consider again the factor from the above Theorem.
The Jacobian matrix in the points ( 2 , ( 2 − )/ ) is given by ) . (75) Then, according to above results (using 2 from equation (16)) det Proof. Immediate since > 0 and the sign of tr ( , V) depends on the sign of According the sign of the results follow from Routh-Hurwitz criterion.
In both cases the systems have at least a unique limit cycle surrounding the unique positive equilibrium point (see Figure 6). Moreover, the points ( , 0) and (1, 0) are saddle points.
The − of the + ( , 0) can be (i) the point (∞, 0) (infinity in the direction of − ), (ii) an unstable limit cycles surrounding the positive equilibrium ( 2 , V 2 ), when this is an attractor.
Since the vector field is continuous with respect to the parameter values, there is a subset of the parameter space for which + ( , 0) intersects + (1, 0) and a heteroclinic curve exists.
This equation defines a surface in the parameter space for which the heteroclinic curve exists.

Remark 22.
(1) We note that a non-infinitesimal limit cycle can be generated by the breaking of the heteroclinic curve [35], which could coincide with the infinitesimal limit cycle generated by the Hopf bifurcation.
(2) The infinitesimal limit cycle generated by the Hopf bifurcation increases until attain the saddle point ( , 0) and after it disappears, when the parameters change a little. Then, the equilibrium point (0, 0) is a almost global attractor [33,34] since all the trajectories, except ( 2 , V 2 ) have that equilibrium as their − .
The Jacobian matrix of system (79) is

Main Properties of System (79).
In this case it has that the equilibriums (a) (1, 0) has the same properties of the case > 0 (above Lemma).

Replacing and simplifying this becomes
and a contradiction is obtained. Then, the nature of the equilibrium ( , ( − )/ ) depends on the sign of corrected to the theorem above. Again, according to the sign of , the results follow from Routh-Hurwitz criterion.
When ( , ( − )/ ) is a repeller it has Then, the transversality condition is fulfilled.

Some Numerical Simulations
In order to reinforce our analytical results, here we present some simulations, for the case > 0. In Figures 3-9 some of different studied cases will be presented, mainly when there exists a unique positive equilibrium.

(A) Existence of a Unique Positive Equilibrium Point
(1) Existence of a unique unstable limit cycle and bistability phenomenon (Figure 3).
(3) Existence of two local attractors and the bi-stability phenomenon ( Figure 5).

Conclusions
In this work, we have analyzed a model considering competition among predators (CAP) and the prey population is affected by an Allee effect. According to the intensity of Allee effect, three cases were studied.
By means of a diffeomorphism [30], we analyzed the topologically equivalent system (11) to the original one, depending only on four parameters; conditions for the existence of positive equilibrium points and their nature were established in some cases.
When > 0 ( > 0), it has a strong Allee effect; one of the main mathematical consequences of the assumption of the existence of CAP is the apparition of a slanted isocline; this straight line can generate up to two positive equilibrium points when it intersects with the prey isocline.
Because it is assumed that the prey population be affected by the Allee effect, it can be proved that the equilibrium point (0, 0) is always an attractor for all parameter values; meanwhile the nature of equilibria ( , 0) and (1, 0) depends on the relation between with 1 and ( , and in the original system).
The point ( , 0) associated with the Allee effect determines a separatrix curve Σ, determined by the stable manifold + ( , 0), which divides the phase plane into two regions. The trajectories having initial conditions above this curve ; meanwhile, those that lie below the separatrix can have a positive equilibrium point or a stable limit cycle as their − . This implies that there exists a great possibility of the prey population going to extinction, although the ratio preypredator is high (many preys and few predators); thus, for the same set of parameter values, both populations can coexist, oscillating around specific population sizes or prey population can be depleted.
An important result was the determination of a subset of the parameter values, for which there exist two positive equilibrium points ( 1 , V 1 ) and ( 2 , V 2 ), the first of them being always a saddle point. The other equilibrium can be an attractor, a repeller or a weak focus, depending on the sign of the trace of the Jacobian matrix evaluated there, since its determinant is positive.
Then, the existence of a homoclinic curve determined by the stable and unstable manifolds of the positive saddle point ( 1 , V 1 ), encircling the second positive equilibrium point ( 2 , V 2 ), was proved. This means that the existence of the bi-stability phenomenon and population sizes can oscillate surround of the point ( 2 , V 2 ) or they can go to extinction, since the equilibrium (0, 0) is a local attractor.
The studied system when > 0 is undoubtedly rather sensitive to disturbances to others proposed in the ecological literature as the Rosenzweig-MacArthur model [4,15], being an important question to ecologists and requiring a careful management in applied contexts of conservation and fisheries [14,20]. The assumption that competition among predators originates a model with a more complex dynamics is fulfilled; as we have seen the interaction can have two or more attractors for the same set of parameter values, even existing the possibility of extinction of both populations. When = 0 ( = 0), it has a special weak Allee effect; in system (11) the equilibriums (0, 0) and ( , 0) are coincident; the collapsed equilibrium has parabolic and hyperbolic sectors, generated by a separatrix curve Σ 0 , dividing the trajectories in the phase plane.
As in the case > 0, there also exists a great possibility of the two populations going to extinction. This is because point (0, 0) is an attractor for a wide set of trajectories, but for other it acts as a saddle point. For a same parameter constraint, depending on the predator population sizes, both populations go to extinction or coexist on a long time, as happens with the strong Allee effect.
When −1 < < 0 (− < < 0) the system has dynamics more simple that the case with strong Allee effect, existing a unique positive equilibrium point. There no exists a separatrix curve and the equilibrium (0, 0) is saddle point, implying that the interacting species coexist either in a fix population sizes or in oscillating population sizes.
Unlike what happens with other models in which there is a marked difference between cases of strong and weak Allee effect [3], in the model studied here there exist minor differences among both cases.
Furthermore, all the obtained results imply a great difference with the well-known Volterra model, which has a unique global attractor equilibrium point at the first quadrant [3]. Also, system (6) has a different dynamic with the Volterra model incorporating only Allee effect on prey, analyzed partially in the Kot's book [24] (in which CAP is not considered); in that model the vertical predator isocline intersects the respective prey isocline in a unique positive equilibrium point.
An open problem after the obtained results in this work is to determine the uniqueness or non-uniqueness of the stable limit cycle shown in the different cases.
In short, the model considering self-interference among predators or CAP reveals a rich dynamic, which must be taken into account for the conservation of species and the modelers.