Dynamics of a Predator-Prey System with Mixed Functional Responses

A predator-prey system with two preys and one predator is considered. Especially, two different types of functional responses, Holling type and Beddington-DeAngelis type, are adopted. First, the boundedness of system is showed. Stabilities analysis of system is investigated via some properties about equilibrium points and stabilities of two subsystems without one of the preys of system. Also, persistence conditions of system are found out and some numerical examples are illustrated to substantiate our theoretical results.


Introduction
Ecological systems are mainly characterized by the interaction between species and their surrounding natural environment ( [1]). Among them, two-species continuous time ecological models with one predator and one prey have been studied for several functional responses such as Holling-Tanner type ( [2][3][4]), Beddington-DeAngelis type ( [5][6][7]), and ratio-dependent type ( [8,9]). However, it has been recognized that such two-species ecological models are not sufficient to explain various phenomena observed in nature ( [10][11][12][13]). For this reason, in recent years, ecological models with three and more species have been investigated by many authors in [14][15][16][17][18]. Particularly, in this paper, we will deal with a three-species ecological system with two different preys and one predator.
On the other hand, functional response between two species is known as the relationship between prey and predator. Most three-species systems in [5,10,11,16,18] have the same two functional responses. However, it is reasonable to consider two different functional responses since two preys in the system are different from each other. In fact, if one considers the handling time of the predator to capture the prey, one figures out that the predator has a Holling type-II functional response and if one thinks of the competitions of predators with one another to catch the prey, Beddington-DeAngelis type functional response could be suitable. Thus, in this paper, we consider the following system with two preys and one predator with mixed two functional responses: ( ) = ( ) (− + 1 1 ( ) where 1 ( ), 2 ( ), and ( ) represent the population density of two preys and the predator at time , respectively. The constants ( = 1, 2) are called the intrinsic growth rates, ( = 1, 2) are the coefficients of intraspecific competition, and are the half-saturation constants, ( = 1, 2) are the per capita rate of predation of the predator, is the death rate of the predator, and scales the impact of the predator interference.
The main purpose of this paper is to look into dynamical properties of system (1). In Section 2, the boundedness of system (1), which means the solution of system (1) initiating in the nonnegative octant is bounded, is studied. Stabilities 2 Journal of Applied Mathematics analysis of system (1) is investigated via well-known properties about equilibrium points and the stabilities of two subsystems without one of the preys of system (1). Also, persistence conditions of the main system (1) are found out and some numerical examples are illustrated to substantiate our theoretical results in Section 4. (1) First, let us consider the state space R 3 + = {( , , ) | ≥ 0, ≥ 0, ≥ 0}. It is easy to see that the functions in the right-hand side of system (1) are continuous and have continuous partial derivatives on R 3 + . Moreover elementary calculations yield the fact that they are Lipschizian on R 3 + . Thus the solution of system (1) with nonnegative initial condition exists and is unique. In addition, the solution of system (1) initiating in the nonnegative octant is bounded as shown in the following theorem.
It is easy to see that if 1 (0) > 0, then 1 ( ) > 0 for all > 0. The same is true for 2 and components. Therefore, we conclude clearly that the first octant R 3 + is an invariant domain of system (1). Now, we will discuss conditions that render certain species extinct. According to system (1), even if one of the preys is extinct, predator species could survive since the predator has two preys. However, the higher the death rate of the predator is, the higher the possibility of predator extinction is. Thus the following theorem indicates that if the death rate of the predator is less than a certain value depending on the growth rate of two preys, then the predator will not face extinction.

Stability Analysis of System
Kolmogorov's theorem in [19] assumes the existence of either a stable equilibrium point or a stable limit cycle behavior in the positive quadrant of phase space of a twodimensional (2D) dynamical system, provided certain conditions are satisfied.
In fact, it is easy to see that the subsystem (7) is a Kolmogorov system under the following condition: For this reason, from now on, we assume that subsystem (7) satisfies condition (8). By applying the local stability analysis ([20]) to Kolmogorov system (7) we have the following results.
and it is a locally asymptotically stable point if the following condition holds: Journal of Applied Mathematics 3 Moreover, if the condition < 1 ( 1 − 1 )/( 1 + 1 ) holds, the solution of subsystem (7) approaches to a stable limit cycle even though the system is not a Kolmogorov system. Secondly, we focus on another subsystem of system (1) when the first prey ( 1 ) is absent as follows: Subsystem (11) is a Kolmogorov system if the following condition is satisfied: Simple calculation yields that there exist at most three nonnegative equilibrium points of subsystem (11). Moreover, the stability of such equilibrium points can be studied by applying the local stability analysis to subsystem (11) as the previous case. Thus we summarize results about local stability as follows.
In [15], the authors have investigated the local stability of the equilibrium point 22 .
Theorem 3 (see [15]). The positive equilibrium point 22 = ( , ) of Kolmogorov system (11) is locally asymptotically stable if one of the following sets of conditions is satisfied: However, the solution of subsystem (11) approaches to a stable limit cycle for 1 Now, we turn our concerns on system (1) to investigate the existence and local stability of the equilibrium points of the system. In fact, there are at most seven nonnegative equilibrium points of system (1). The existence conditions of them are mentioned in the following lemma.
(3) The positive equilibrium point 6 = ( * 1 , * 2 , * ) exists in the interior of the first octant if and * 1 satisfies the following equation: Here, Journal of Applied Mathematics Proof. We only consider the existence of the positive equilibrium point 6 . It is easy to see that the equilibrium point exists in the interior of the first octant if and only if there exists a positive solution to the following algebraic nonlinear simultaneous equations: From the first and third equations in (18) By using (19) in the second equation of (18) and by elementary calculation we can obtain the following equation: Since the degree of equation (20) is 5, it has at least one real root * 1 . Moreover if condition (14) is satisfied then all values of * 1 , * 2 , and * are positive.
It is worth noting that since predator dies out in the absence of all preys the equilibrium point (0, 0, ) with > 0 does not exist.
In order to discuss the stability of the equilibrium point 6 = ( * 1 , * 2 , * ), let * = (V * ) be the variational matrix at 6 . Then it follows from (21) that * can be written as follows: ) .
Proof. It is from elementary calculation that 1 > 0,  On the other hand, if one of the conditions (25) and (26) is not satisfied, the positive equilibrium point 6 could not be stable. In order to illustrate an example, we take the parameters as follows:

Persistence of System (1)
The term persistence is given to systems in which strict solutions do not approach the boundary of the nonnegative cones as time goes to infinity. Therefore, for the continuous biological system, survival of all interacting species and the persistence are equivalent. In the following theorem, we find out some persistence conditions of system (1). Theorem 9. Suppose that system (1) has no nontrivial periodic solutions in the boundary planes and satisfies the hypothesis of Theorem 3 and condition 1 ( 1 − 1 )/( 1 + 1 ) < < 1 1 /( 1 + 1 ) holds. Then the necessary conditions for the persistence of system (1) are Proof. Note that the boundedness of system (1) is shown in Theorem 1 and 12 is locally stable under Kolmogrov condition (12). Since 12 and 22 are locally stable by the assumptions, the signs of the eigenvalueŝ2 and 1 determine the stability of the equilibrium points 4 = (̂, 0,̂) and 5 = (0, , ). In fact, if there are no nontrivial periodic solutions in the 2 plane and (29) does not hold (i.e., 1 < 0) then there is an orbit in the positive cone, which approaches to 5 . Hence, condition (29) is one of the necessary conditions for the persistence. Similarly, we obtain the other necessary condition (30) for the persistence of system (1) by applying the same method as mentioned above to the equilibrium point 4 . Now, we will use the abstract theorem of Freedman and Waltman [19] to figure out sufficient conditions for the persistence of system (1). In order to do this, consider the growth functions 1 , 2 , and 3 in (18) of system (1). Then it is shown that the following four conditions are satisfied.
Theorem 11. Suppose that conditions (31) and (32) are satisfied and system (1) has a finite number of limit cycles in the 1 plane or in the 2 plane. Then system (1) persists if hold for each of the limit cycles (0, 2 ( ), V 2 ( )) and ( 1 ( ), 0, V 1 ( )) in the 2 plane and in the 1 plane, respectively, if it exists. Here, T means the period of the limit cycle.

(39)
Therefore the proof is complete.

Conclusions and Remarks
In this paper, we have considered a predator-prey system with two preys and one predator with two different types of functional responses, Holling type and Beddington-DeAngelis type. Until now, many researches for two-prey and one-predator systems have dealt with the same functional responses to describe the relationship between prey and predator even if the two preys are different from each other. Thus in this research we adopted two different types of functional responses to model the relationship between two different preys and predator. We investigated stabilities of system about equilibrium points by virtue of stabilities of two subsystems without one of the preys of system. Also, we found out conditions that guarantee that the system is persistent. In addition, some numerical examples are illustrated to substantiate our theoretical results. Generally speaking, if food is abundant the predators do not interfere with each other to get it; otherwise there is intense competition between predators to get food. In order to describe such kind of phenomenon, we use the mixed functional responses. Thus, due to the mixed functional responses, one can see from Theorem 5 that the value of the impact of the predator interference to catch the prey 2 has an effect on the extinction of another prey 1 . Thus even though ecological systems have two different functional responses, they have a variety of dynamical behaviors.