Abstract
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.
1. 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 ([2–4]), Beddington-DeAngelis type ([5–7]), 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 ([10–13]). For this reason, in recent years, ecological models with three and more species have been investigated by many authors in [14–18]. 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: where , and represent the population density of two preys and the predator at time , respectively. The constants are called the intrinsic growth rates, are the coefficients of intraspecific competition, and are the half-saturation constants, 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 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.
2. Boundedness of System (1)
First, let us consider the state space . It is easy to see that the functions in the right-hand side of system (1) are continuous and have continuous partial derivatives on . Moreover elementary calculations yield the fact that they are Lipschizian on . 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.
Theorem 1. The solution of system (1) initiating in is bounded for all .
Proof. Since , we have Define . Then where . So, by comparison theorem, we obtain that for , where and is a constant of integration. Thus for sufficiently large , which means that all species are uniformly bounded for any initial value in .
It is easy to see that if , then for all . The same is true for and components. Therefore, we conclude clearly that the first octant 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.
Theorem 2. A necessary condition for the predator species to survive is
Proof. From the third equation of system (1), we get In the proof of Theorem 1, , is shown. Then and hence , where . Thus if , that is , then . Therefore is a necessary condition for the predator species to survive.
3. Stability Analysis of System (1)
In order to study stabilities of equilibria of system (1), we first take into account a subsystem of system (1) when the second prey is absent as follows:
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 two-dimensional (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.(1)The equilibrium point always exists and is a saddle point.(2)The equilibrium point always exists and is a saddle point under condition (8).(3)The positive equilibrium point exists, where and it is a locally asymptotically stable point if the following condition holds: Moreover, if the condition 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 () 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.(1)The equilibrium point always exists and is a saddle point.(2)The equilibrium point always exists and is also a saddle point under condition (12).(3)The positive equilibrium point exists, where
In [15], the authors have investigated the local stability of the equilibrium point .
Theorem 3 (see [15]). The positive equilibrium point of Kolmogorov system (11) is locally asymptotically stable if one of the following sets of conditions is satisfied:(1),(2) and ,(3) and , with or .
However, the solution of subsystem (11) approaches to a stable limit cycle for . Here , , , and .
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.
Lemma 4. (1) The trivial equilibrium point and one-prey equilibrium points and always exist.
(2) Two-species equilibrium points , , and exist in the interior of positive quadrant of , , and planes, respectively, if the Kolmogorov conditions and hold, where and are given in (9) and (13), respectively.
(3) The positive equilibrium point exists in the interior of the first octant if
where
and satisfies the following equation:
Here,
Proof. We only consider the existence of the positive equilibrium point . 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) we can have 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 . Moreover if condition (14) is satisfied then all values of , and are positive.
It is worth noting that since predator dies out in the absence of all preys the equilibrium point with does not exist.
In order to investigate stabilities of the equilibrium points, we need to consider the variational matrix of system (1). Thus we get the following matrix: where By using the variational matrix (21), local stabilities of system (1) near the equilibrium points are studied in the following theorems.
Theorem 5. (1) The trivial equilibrium point is a hyperbolic saddle point. In fact, near both prey populations are increasing while the predator population is decreasing. And the equilibrium points and are also hyperbolic saddle points.
(2) The equilibrium point is always unstable; actually, a saddle point with locally stable manifold in the plane and with local unstable manifold in the declines if Kolmogorov conditions (8) and (12) hold.
(3) The equilibrium point is stable if and is unstable if .
(4) Assume that hypotheses of Theorem 3 hold; then the equilibrium point is stable if and is unstable if .
Proof. (1) The eigenvalues of the matrix are , and and their eigenvectors are , and . Furthermore, the eigenvalues of the matricies and are and , respectively. Thus the equilibrium points , and are hyperbolic saddle.
(2) The eigenvalues of the matrix are , and . Therefore, since Kolmogorov conditions (8) and (12) are satisfied, the sign of is always positive. Thus the point is unstable.
(3) Now, consider the equilibrium point . The point has the same stability as in the interior of positive coordinate plane . Furthermore, since the equilibrium point is always stable under condition (8), the local stability of the point depends on the sign of the eigenvalue of the -direction.
(4) Similar to the case of the point , the point has the same stability behavior as in the interior of positive coordinate plane . Thus, since the point is locally stable, if one of the conditions of Theorem 3 is satisfied, then the point is locally stable or unstable along the -direction according to the sign of the eigenvalue of the -direction.
Example 6. In this example we simulate system (1) numerically by using Runge-Kutta method of order 4 to substantiate Theorem 5 when the parameters are as follows: Then it follows from Theorem 5 (4) that is stable since . Figure 1 illustrates the phase portrait of system (1) and time series for , and when initial condition is .
In order to discuss the stability of the equilibrium point , let be the variational matrix at . Then it follows from (21) that can be written as follows:
Thus the characteristic equation of the matrix is obtained as , where ,, and .
From the Routh-Hurwitz criterion ([20]), we know that is locally asymptotically stable if and only if , , and are positive. It is not easy to find the conditions and and . However, we give a sufficient condition to guarantee the local stability of the equilibrium point in the following theorem.
Theorem 7. Suppose that the positive equilibrium point exists in the interior of the positive octant. Then is locally asymptotically stable if
Proof. It is from elementary calculation that , , and under conditions (25) and (26).
Example 8. In order to substantiate Theorem 7, we set the parameters as follows:
The point is locally stable since and . The phase portrait of system (1) and time series for , and are shown in Figure 2.
On the other hand, if one of the conditions (25) and (26) is not satisfied, the positive equilibrium point could not be stable. In order to illustrate an example, we take the parameters as follows:
Then we have the point . Since and , the point does not satisfy condition (25) and moreover Figure 3 exhibits numerically that is unstable. As shown in Figure 3 even if the positive point becomes an unstable point a stable limit cycle could occur.
4. 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 holds. Then the necessary conditions for the persistence of system (1) are and the sufficient conditions for the persistence of system (1) are
Proof. Note that the boundedness of system (1) is shown in Theorem 1 and is locally stable under Kolmogrov condition (12). Since and are locally stable by the assumptions, the signs of the eigenvalues and determine the stability of the equilibrium points and . In fact, if there are no nontrivial periodic solutions in the plane and (29) does not hold (i.e., ) then there is an orbit in the positive cone, which approaches to . 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 .
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 , and in (18) of system (1). Then it is shown that the following four conditions are satisfied.(1)Clearly, we have .(2)Each prey population grows up to its carrying capacity in the absence of predators; that is, and and and . Furthermore, the predator population dies out in the absence of preys; that is, consider .(3) and . There exists exactly one point satisfying , .(4)In the absence of each prey species the predator can survive on the other prey. This is always true under the Kolmogrov conditions (8) and (12). There exist uniquely and satisfying . According to Kolmogrov conditions (8) and (12), we can get that and , respectively.(5)It follows from (8), (12), (31), and (32) that the inequalities , , and hold.
Therefore, by Freedman and Waltman theorem ([19]), system (1) persists under the hypotheses.
Example 10. Now let the parameters be as follows: Then it follows from [15, 21] that subsystems (7) and (11) have no periodic solutions in the boundary planes. Furthermore, it is not difficult to see that system (1) satisfies conditions (2) of Theorem 3 and the other hypotheses in Theorem 9. Thus, all species in system (1) can coexist as time goes away. Figure 4 shows a phase portrait and time series of all species of system (1) with initial condition .
Theorem 11. Suppose that conditions (31) and (32) are satisfied and system (1) has a finite number of limit cycles in the plane or in the plane. Then system (1) persists if hold for each of the limit cycles and in the plane and in the plane, respectively, if it exists. Here, T means the period of the limit cycle.
Proof. First, consider a limit cycle in plane. Then the variational matrix about the limit cycle can be written asNow, let be a solution of system (1) with positive initial condition sufficiently close to the limit cycle. It is easily obtained from the variational matrix that is a solution of system with . Thus, we obtain
From Taylor expansion, we get
Thus increases or decreases as the value of is positive or negative, respectively. Since and the limit cycle are the only possible limit in the plane of trajectories with positive initial condition, the trajectories go away from the plane if conditions (31) and (34) are satisfied.
Similar argument can be applied to each limit cycle to obtain the fact that the trajectories go away from the plane if conditions (32) and (35) hold by considering the variational matrix about the limit cycle as follows:Therefore the proof is complete.
5. 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 has an effect on the extinction of another prey . Thus even though ecological systems have two different functional responses, they have a variety of dynamical behaviors.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the presentation of this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01007379).