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

We study the qualitative behavior of a class of predator-prey models with Beddington-DeAngelistype 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.


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 X and Y are the densities or biomasses of prey and of predators at time T , respectively.f X is the per capita net prey production in the absence of predation, whereas g X, Y 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 h X, Y represents the numerical response of predators measures the growth rate of predators .The function f will be taken either as the Malthusian growth f X rX or as the logistic model f X rX 1 − X/K .The key role in prey-predator models is played by the functional response g Solomon 1949 .Traditionally, it is assumed that the functional response is a function of prey density only prey-dependent feeding, g g X , 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 preydependent 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 g g X, Y .Arditi and Ginzburg 2 have argued that, in many cases, this predator dependence could be simplified as a ratio-dependent model g g X/Y 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 predatorprey model with the Beddington-DeAngelis functional response: and k 1 are the parameters of model and are assumed to be nonnegative with β 1 nontrivial if β 1 0, then the model 1.2 is the same as that in 13 .These parameters are defined as follows: a 1 resp., a 2 describes the growth rate of prey resp., of predator , b 1 measures the strength of competition among individuals of prey's species, m 1 is the maximum value which per capita reduction rate of prey can attain, γ 1 resp., k 1 measures the extent to which environment provides protection to prey resp., to predator , and m 2 has a similar meaning to m 1 .The functional response in 1.2 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 β 1 Y in the first right term equation modeling mutual interference among predators.Hence this kind of type functional response given in 1.2 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 , t a 1 τ, x t b 1 /a 1 X τ , and y t m 2 b 1 /a 1 a 2 Y τ , then Beddington-DeAngelis predator-prey model 1.2 should take the following nondimensional form:

Boundedness of the Model and Existence of a Positively Invariant Attracting Set
We denote by R 2 the nonnegative quadrant, and by Int R 2 the positive quadrant.
Lemma 2.1.Positive quadrant Int R 2 is invariant for system 1.3 .Proof.From system 1.3 , we observe that the boundaries of the nonnegative quadrant R 2 , which are the positives x-axis and y-axis are invariant; this is immediately obvious from the system 1.3 .Therefore, densities x t and y t are positive for all t ≥ 0 if x 0 > 0 and y 0 > 0. Theorem of existence and uniqueness ensures that the positive solutions of the autonomous system 1.3 and the axis cannot intersect.
Next, we will show that, under some assumptions, the solutions of system 1.3 which start in R 2 are ultimately bounded.First, let us give the following comparison lemma.Definition 2.4.A solution φ t, t 0 , x 0 , y 0 of system 1.3 is said to be ultimately bounded with respect to R 2 if there exists a compact region A ∈ R 2 and a finite time T T T t 0 , x 0 , y 0 such that, for any t 0 , x 0 , y 0 ∈ R × R 2 , φ t, t 0 , x 0 , y 0 ∈ A, ∀t ≥ T.

2.7
Theorem 2.5.Let A be the set defined by where

2.9
Then 2 all solutions of 1.3 initiating in R 2 are ultimately bounded with respect to R 2 and eventually enter the attracting set A.
Proof.Let x 0 , y 0 ∈ A, we will show that x t , y t ∈ A for all t ≥ 0. Obviously, from Lemma 2.1, as x 0 , y 0 ∈ A, x t , y t is in Int R 2 .Then, we have to show that for all t ≥ 0, 0 ≤ x t ≤ 1, and 0 ≤ x t y t ≤ L 1 .
1 a First, we prove that for all t ≥ 0, 0 ≤ x t ≤ 1.
We have x > 0 and y > 0 in Int R 2 ; then every solution φ t x t , y t of system 1.3 , which starts in Int R 2 , satisfies the differential inequality dx/dt ≤ 1 − x t x t .This is obvious by considering the first equation of 1.3 .Thus, x t may be compared with solutions of du/dt 1 − u t u t and u 0 x 0 > 0 which is a Bernoulli's equation; then the solution is It follows that every nonnegative solution φ t satisfies b We prove now that, for all t ≥ 0, 0 ≤ x y ≤ L 1 .
We define the function σ t x t y t ; the time derivative of this function is Since all parameters are positive and solutions initiating in R 2 remain in the nonnegative quadrant, then 12 holds for all x and y being nonnegative.Thus, as max we have

2.16
Since in A, x t ≤ 1 for all t ≥ 0, we obtain Moreover, it can be easily verified that max Consequently with Using Lemma 2.2, with

2.27
b For the second result, let ε > 0 be given, and T 1 > 0 exists, such that

2.28
From 2.22 with T T 1 , we get, for all t ≥ T 1 , σ t x t y t

2.31
Then x t y t ≤ L 1 .

2.33
This completes the proof; then we conclude that system 1.3 is dissipative in R 2 .

Linear Stability
First of all, it is easy to verify that this system has three trivial equilibria, belonging to the boundary of R 2 , i.e., at which one or more of populations has zero density or is extinct The other equilibria are defined by the system ay αx βy γ 1 − x, y x k.

3.2
Proposition 3.1.The system 1.3 has a unique interior equilibria P * x * , y * (i.e., x * > 0 and y * > 0) if the following condition is verified: Proof.We introduce the second equation of 3.2 in the first one; then and we obtain The discriminant of this equation

3.6
Differential Equations and Nonlinear Mechanics 9 Therefore, if 3.3 holds, then

3.8
Now, we show, under the condition 3.3 , that one of this equilibriums is not in R 2 ; let and due to 3.3 Differential Equations and Nonlinear Mechanics then the first point The Jacobian matrix is given by 3.17 1 At P 0 0, 0 , The eigenvalues of this matrix are Hence, all parameters are positive; then P 0 0, 0 is an unstable node.

3.20
The eigenvalues are , k is unstable with the positive y-axis as its stable manifold.

3.22
The eigenvalues are Then the equilibrium P 2 1, 0 is a saddle point with the stable manifold being the x-axis.
Around P * x * , y * , the Jacobian matrix takes the form

3.24
The characteristic equation is To simplify, we developed det J P * respecting one variable, from 3.2 ; then

3.30
which implies that det J P * has the same sign of

3.31
We rewrite

3.33
The discriminant is

3.34
We get three cases.
1 If αk < γ, Δ is negative, f x has the same sign of a γ − αk , and we have a γ − αk > a > 0.
Then, det J P * is positive.

Differential Equations and Nonlinear Mechanics
2 If αk > γ, Δ is positive and f x has at list two solutions x 1 and x 2 , then Then, det J P * is positive.
Remark 3.2.From the expression 3.28 , we find that det J P * is positive, if αk ≤ γ, hence the eigenvalues associated to P * have the same sign.
To determine the sign of these eigenvalues, we have − ay * βy * γ .

3.36
From 3.2 , we get

3.38
We obtain the following lemma.
We used the Cardan's method to solve the cubic equation P 3 x 0. Then we consider the equation a 3 x 3 a 2 x 2 a 1 x a 0 0, 3.39 with a 3 α, a 2 − a 2α , a 1 − a b k − α , and a 0 abk.Making the substitution y a 3 x a 2 /3 reduces the equation to the standard form y 3 − py − q 0, 3.40 where p and q depend on a 3 , a 2 , a 1 , and

3.41
Let y u v.

Differential Equations and Nonlinear Mechanics
We get after the identification of the coefficients 3uv p, u 3 v 3 q.

3.44
Then 3.45 and we obtain that u 3 and v 3 are solutions of the quadratic equation

3.46
Then we constitute three cases.

Uniform Permanence
In this section we shall prove the permanence 8, 17-19 , that is, the uniform persistence and dissipativity, of system 1.3 .
The principal notion of persistence theory is uniform persistence or permanence.Before the study of the permanence of system 1.3 , we introduce some necessary definitions.Consider an ODE model for n interacting biological species The system 4.1 is said to be permanent if for each i 1, 2, . . ., n there are constants ε 0 and 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 .

Differential Equations and Nonlinear Mechanics
Suppose that Y is a complete metric space with Y Y 0 ∪ ∂Y 0 for an open set Y 0 .We will choose Y 0 to be the positive cone in R 2 .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 Y under which Y 0 and ∂Y 0 are forward invariant is said to be permanent if it is dissipative and if there is a number ε > 0 such that any trajectory starting in Y 0 will be at least a distance ε from ∂Y 0 for all sufficiently large t.
Let ω ∂Y 0 ∈ ∂Y 0 denote the union of the sets ω u over u ∈ ∂Y 0 .Definition 4.2.The ω-limit set ω ∂Y 0 is said to be isolated if it has a covering Ω N k 1 Ω k of pairwise disjoint sets Ω k which are isolated and invariant with respect to the flow or the semiflow both on ∂Y 0 and on Y Y 0 ∪ ∂Y 0 , M is called an isolated covering .The set ω ∂Y 0 is said to be acyclic if there exists an isolated covering N k 1 Ω k such that no subset of Ω k is a cycle.Theorem 4. 3 Hale and Waltman 1989 .Suppose that a semiflow on Y leaves both Y 0 and ∂Y 0 forward invariant, maps bounded sets in Y to precompact set for t > 0, and it is dissipative.If in addition 1 ω ∂Y 0 is isolated and acyclic; 2 W s Ω k ∩ Y 0 ∅ for all K, where N k 1 Ω k is the isolated covering used in the definition of acyclicity of ∂Y 0 , and W s denote the stable manifold.
Then the semiflow is permanent.
And, we have this theorem. 4.6 Then, system 1.3 is permanent.
Proof.We take Y the strictly positive quadrant of R 2 ; then ω ∂Y 0 consists of the equilibria P 0 0, 0 , P 1 0, k , and P 2 1, 0 .P 0 0, 0 is an unstable node, P 2 1, 0 is saddle point, and its stable manifold is x-axis.If ak ≤ βk γ, P 1 0, k is a saddle point stable along the y-axis and unstable along the x-axis.Then, all trajectories on the axis ox other than P 0 0, 0 approach the point P 2 1, 0 and all trajectories on the axis oy other than P 0 0, 0 approach the point P 1 0, k .It follows from these structural features that the flow in ∂Y 0 is acyclic.So ω ∂Y 0 is isolated and acyclic.The stable manifold of P 2 1, 0 is the x-axis and the stable manifold of P 1 0, k is the y-axis, and we know, from Theorem 2.5, that these stable manifolds cannot intersect the interior of Y 0 .
In this case, Theorem 4.3 implies permanence of the flow defined by 1.3 .

Global Stability
In this section, we shall prove the global stability of system 1.3 by constructing a suitable Lyapunov function.First, we have to show that there exists one interior equilibrium P * x * , y * .The linear analysis shows that if αk ≤ γ and x * < x 0 , then P * x * , y * is locally stable.We prove now that, under some assumptions, this steady state is globally asymptotically stable.
Theorem 5.1.The interior equilibrium P * x * , y * is globally asymptotically stable if Proof.The proof is based on construction of a positive definite Lyapunov function.Let x * k y − y * − y * ln y y * .

5.7
This function is defined and continuous on Int R 2 .We can easily verify that the function V x, y is zero at the equilibrium x * , y * and is positive for all other positive values of x and y, and thus, P * x * , y * is the global minimum of V .
Since the solutions of 1.3 are bounded and ultimately enter the set A, we restrict the study for this set.The time derivative of V 1 and V 2 along the solutions of system 1.

5.23
In A, all solutions satisfy 0 ≤ x ≤ 1, and from 5.1

5.26
It follows that if the hypotheses of Theorem 5.1 are satisfied, then dV/dt < 0 along all trajectories in the first quadrant except x * , y * ; so P * x * , y * is globally asymptotically stable.

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.

Theorem 4 . 4 .
Let us assume the following condition:
We have to prove that, for x 0 , y 0 ∈ R 2 , x t , y t → A when t → ∞.We will show that lim t → ∞ x t ≤ 1 and lim t → ∞ x t y t ≤ L 1 .