Boundedness and Global Stability for a Predator-Prey System with the Beddington-DeAngelis Functional Response

Abstract

We study the qualitative behavior of a class of predator-prey models with Beddington-DeAngelis-type functional response, primarily from the viewpoint of permanence (uniform persistence). The Beddington-DeAngelis functional response is similar to the Holling type-II functional response but contains a term describing mutual interference by predators. We establish criteria under which we have boundedness of solutions, existence of an attracting set, and global stability of the coexisting interior equilibrium via Lyapunov function.

1. Introduction and Mathematical Model

Standard Lotka-Volterra systems are also known as the predator-prey systems, on which a large body of existing predator-prey theory is built by assuming that the per capita rate of predation depends on the prey numbers only [1]. Recently, the traditional prey-dependent predator-prey models have been challenged by several biologists based on the fact that functional and numerical response over typical ecological time scale sought to depend on the densities of both prey and predator, especially when predators have to search, share, or compete for food. A more suitable general predator-prey model should be based on the ratio-dependent theory [2–4]. This roughly states that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Moreover, as the number of predators often changes slowly (relative to prey number), there is often a competition among the predators, and the per capita rate of predation should therefore depend on the numbers of both prey and predator, most probably and simply on their ratios. These hypotheses are strongly supported by numerous field and laboratory experiments and observations [5–9].

The general model describing the dynamics of prey-predator populations in continuous time can be written as where and are the densities (or biomasses) of prey and of predators at time , respectively. is the per capita net prey production in the absence of predation, whereas is the functional response of predators (the number of preys eaten per predator per unit time). Natural mortality of prey is considered to be negligible compared to mortality due to predation.The function represents the numerical response of predators (measures the growth rate of predators). The function will be taken either as the Malthusian growth or as the logistic model . The key role in prey-predator models is played by the functional response (Solomon 1949). Traditionally, it is assumed that the functional response is a function of prey density only prey-dependent feeding, , without any dependence on predator density [9, 10]. The hypothesis is based on an analogy with the law of mass action in chemistry assuming that prey and predator individuals encounter each other randomly in space and time [11]. Therefore, the prey-dependent model can be applied to systems which are spatially homogeneous and in which the time scale of prey removal by predators is of the same order of magnitude as that of population reproduction [2].

Many questions in predator-prey theory, including the question of interference between predators, revolve around the expression that is used for the functional response . Arditi and Ginzburg [2] have argued that, in many cases, this predator dependence could be simplified as a ratio-dependent model instead of modeling explicitly all conceivable interference mechanisms (and thus adding parameters to the model).

The Beddington-DeAngelis-type functional response performed even better. Although the predator-dependent models that they considered fit those data reasonably well, no single functional response best describes all the data sets. The Beddington-DeAngelis response can be generated by a number of natural mechanisms [5, 12], and because it admits rich but biologically reasonable dynamics [6], it is worthy for us to further study the Beddington-DeAngelis model.

Therefore, it is interesting and important to study the following autonomous predator-prey model with the Beddington-DeAngelis functional response: with the initial values and . The constants , and are the parameters of model and are assumed to be nonnegative with nontrivial (if , then the model (1.1) is the same as that in [13]).

These parameters are defined as follows: (resp., ) describes the growth rate of prey (resp., of predator), measures the strength of competition among individuals of prey’s species, is the maximum value which per capita reduction rate of prey can attain, (resp., ) measures the extent to which environment provides protection to prey (resp., to predator), and has a similar meaning to . The functional response in (1.1) was introduced by Beddington [5] and DeAnglis et al. (1975) in [7]. It is similar to the well-known Holling type-II functional response but has an extra term in the first right term equation modeling mutual interference among predators. Hence this kind of type functional response given in (1.1) is affected by both predator and prey, that is, the so-called predator dependence by Arditi and Ginzburg [2]. Dynamics for the Holling type-II model have been much studied (see, e.g., [13–15]). Then how the mutual interference term affects the dynamic of the whole system is an interesting problem.

Introducing the following scaling (see [16]), , and , then Beddington-DeAngelis predator-prey model (1.1) should take the following nondimensional form: where , and

2. Boundedness of the Model and Existence of a Positively Invariant Attracting Set

We denote by the nonnegative quadrant, and by the positive quadrant.

Lemma 2.1. Positive quadrant is invariant for system .

Proof. From system , we observe that the boundaries of the nonnegative quadrant , which are the positives -axis and -axis are invariant; this is immediately obvious from the system . Therefore, densities and are positive for all if and . Theorem of existence and uniqueness ensures that the positive solutions of the autonomous system and the axis cannot intersect.

Next, we will show that, under some assumptions, the solutions of system which start in are ultimately bounded. First, let us give the following comparison lemma.

Lemma 2.2. Let be an absolutely-continuous function satisfying the differential inequality where ; then

We define the function to be the solution of differential equation and we have the Gronwall’s lemma.

Lemma 2.3. Let satisfy for the linear, scalar equation with and being continuous functions. If satisfies for the inequalities then

Definition 2.4. A solution of system is said to be ultimately bounded with respect to if there exists a compact region and a finite time such that, for any ,

Theorem 2.5. Let be the set defined by where Then(1)is positively invariant; and(2)all solutions of initiating in are ultimately bounded with respect to and eventually enter the attracting set .

Proof. Let , we will show that for all . Obviously, from Lemma 2.1, as is in . Then, we have to show that for all , and (1)(a)First, we prove that for all . We have and in ; then every solution of system , which starts in , satisfies the differential inequality This is obvious by considering the first equation of . Thus, may be compared with solutions of and which is a Bernoulli’s equation; then the solution is , with , where , which implies that It follows that every nonnegative solution satisfies (b)We prove now that, for all We define the function ; the time derivative of this function is  Since all parameters are positive and solutions initiating in remain in the nonnegative quadrant, then  holds for all and being nonnegative. Thus, as  we have  then  which implies that  Since in , for all , we obtain  Moreover, it can be easily verified that  Consequently  with  Using Lemma 2.2, with  then we get  Then, if ,  Hence, for ,  Then (2)We have to prove that, for when . We will show that and (a)The first result, , is obtained directly from (1a) and Lemma 2.2, since solutions of the initial value problem  verify (b)For the second result, let be given, and exists, such that  From with , we get, for all ,  Then  Let be such that  Then  Hence,  This completes the proof; then we conclude that system is dissipative in .

3. Linear Stability

First of all, it is easy to verify that this system has three trivial equilibria, (belonging to the boundary of , i.e., at which one or more of populations has zero density or is extinct)

The other equilibria are defined by the system

Proposition 3.1. The system has a unique interior equilibria (i.e., and ) if the following condition is verified:

Proof. We introduce the second equation of in the first one; then and we obtain The discriminant of this equation Therefore, if holds, then Consequently, is positive, and the system has two other equilibriums and , where Now, we show, under the condition (3.3), that one of this equilibriums is not in ; let then from and due to which implies that(1)if (2)if , it results that is not in , such that then the first point is in

The Jacobian matrix is given by where (1)At ,  The eigenvalues of this matrix are  Hence, all parameters are positive; then is an unstable node.(2)At ,  The eigenvalues are

 then, we have(a)if , is stable node;(b)if, is unstable with the positive-axis as its stable manifold.(3)At , (3)At ,  The eigenvalues are  Then the equilibrium is a saddle point with the stable manifold being the -axis.

Around , the Jacobian matrix takes the form

The characteristic equation is where

From , we get then

We observe that is positive if To simplify, we developed respecting one variable, from ; then which implies that has the same sign of We rewrite Let

The discriminant is

We get three cases.(1)If is negative, has the same sign of , and we have .Then, is positive.(2)If is positive and has at list two solutions and , then(a)if , has the same sign of ;(b)if has sign of (3)If , then Then, is positive.

Remark 3.2. From the expression (3.6), we find that is positive, if , hence the eigenvalues associated to have the same sign.

To determine the sign of these eigenvalues, we have From , we get Let We obtain the following lemma.

Lemma 3.3. If is verified, the interior equilibrium is locally asymptotically stable if and it is unstable if .

We used the Cardan’s method to solve the cubic equation . Then we consider the equation with , and . Making the substitution reduces the equation to the standard form where and depend on , and Let Then We get after the identification of the coefficients Then and we obtain that and are solutions of the quadratic equation Then we constitute three cases.(1)if , then We have then has at list one real root So, we have if , and if .(2)if , then there are three real roots , and , and therefore, then one of them is positive, then(a)if , then if , and if ;(b)if , then if , and if .(3)if , then there are one real root positive and a double root ; we also have if , and if .

4. Uniform Permanence

In this section we shall prove the permanence [8, 17–19], that is, the uniform persistence and dissipativity, of system .

The principal notion of persistence theory is uniform persistence or permanence. Before the study of the permanence of system , we introduce some necessary definitions. Consider an ODE model for n interacting biological species where denotes the density of the ith species. Let denote the solution of (4.1) with componentwise positive initial values. The system (4.1) is said to be weakly persistent if persistent if and uniformly persistent if there is an such that The system is said to be permanent if for each there are constants and such that Clearly, a permanent system is uniformly persistent which in turn is persistent, and persistence implies weak persistence; a dissipative uniformly persistent system is permanent. For further discussion about various definitions of persistence and permanence and their connections, see [18].

Suppose that is a complete metric space with for an open set . We will choose to be the positive cone in . For the following definitions and theorems, one can see [6], and for the proof of the theorem, see [8].

Definition 4.1. A flow or semiflow on under which and are forward invariant is said to be permanent if it is dissipative and if there is a number such that any trajectory starting in will be at least a distance from for all sufficiently large .

Let denote the union of the sets over

Definition 4.2. The -limit set is said to be isolated if it has a covering of pairwise disjoint sets which are isolated and invariant with respect to the flow or the semiflow both on and on , ( is called an isolated covering). The set is said to be acyclic if there exists an isolated covering such that no subset of is a cycle.

Theorem 4.3 (Hale and Waltman 1989). Suppose that a semiflow on leaves both and forward invariant, maps bounded sets in to precompact set for , and it is dissipative. If in addition(1) is isolated and acyclic;(2) for all , where is the isolated covering used in the definition of acyclicity of , and denote the stable manifold.Then the semiflow is permanent.

And, we have this theorem.

Theorem 4.4. Let us assume the following condition: Then, system is permanent.

Proof. We take the strictly positive quadrant of ; then consists of the equilibria , and . is an unstable node, is saddle point, and its stable manifold is -axis. If is a saddle point stable along the -axis and unstable along the -axis.
Then, all trajectories on the axis other than approach the point and all trajectories on the axis other than approach the point . It follows from these structural features that the flow in is acyclic. So is isolated and acyclic. The stable manifold of is the -axis and the stable manifold of is the -axis, and we know, from Theorem 2.5, that these stable manifolds cannot intersect the interior of .
In this case, Theorem 4.3 implies permanence of the flow defined by .

5. Global Stability

In this section, we shall prove the global stability of system by constructing a suitable Lyapunov function. First, we have to show that there exists one interior equilibrium . The linear analysis shows that if and , then is locally stable. We prove now that, under some assumptions, this steady state is globally asymptotically stable.

Theorem 5.1. The interior equilibrium is globally asymptotically stable if

Proof. The proof is based on construction of a positive definite Lyapunov function. Let where This function is defined and continuous on .
We can easily verify that the function is zero at the equilibrium and is positive for all other positive values of and , and thus, is the global minimum of .
Since the solutions of are bounded and ultimately enter the set , we restrict the study for this set. The time derivative of and along the solutions of system is Then and using , we get after simplifing Therefore, computing via and yields The above equation can be written as where From , it is obvious that if the matrix above is positive definite. This matrix is positive definite if only if all upper-left submatrices are positive (Sylvester’s criteria), that is, since , if only if(1);(2)

Proof. It is of (1) So, as is an attracting positively invariant set, where, all solutions satisfy and , then Therefore, if holds, then

Proof. It is of (2) Since (for fixed) then We have We note, using , that for in If holds, then Hence, is strictly decreasing in , with respect to .
Now In , all solutions satisfy , and from (5.1) then, if holds, in . Hence, is strictly decreasing in . This yields for ; that is, using , we get Consequently, due to , It follows that if the hypotheses of Theorem 5.1 are satisfied, then along all trajectories in the first quadrant except ; so is globally asymptotically stable.

6. Conclusion

The Beddington-DeAngelis functional response admits a range of dynamics which include the possibilities of extinction, persistence, and stable or unstable equilibria. The criteria for persistence are the same as for systems with a Holling-type 2 response.

The future research will complete the qualitative analysis by studying the limit cycles of the model. It will also contain the numerical simulations to justify the obtained results.