Global Dynamics of a Predator-prey System with Holling Type Ii Functional Response *

In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The existence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium.


Introduction
Mathematical modeling and computer simulation provide an effective tool in the study of contemporary population ecology [1,2].In population dynamics, the functional response of predator to prey density refers to the change in the density of prey attacked per unit time per predator as the prey density changes [3].In [4], based on experiment, Holling suggested three kinds of functional response for different species to model the phenomena of predation, it seems more reasonable than the standard Lotka-Voltera type predator-prey system.In [5], Bazykin proposed the following predator-prey system: where u(t), v(t) represent the densities of prey and predator population, respectively.System (1) is called Holling type II predator-prey model in the literature.This system is an extension of the familiar Lotka-Volterra system, in which the divisor 1 + αu is missing, i.e., α = 0. α is interpreted as a constant handling time for each prey captured.
In system (1), it is assumed that each individual predator has the same ability to feed on prey.However, in the natural world, many species go through two or more life stages as they proceed from birth to death, and in different stages, they have different reactions to the environment.For example, the immature predators are raised by their parents, and the rate they attack the prey and the reproductive rate can not be ignored.Stage-structured population models have received great attention in recent years (see, e.g., [8][9][10][11][12][13]).In [12], Yu et al. studied the following strengthen type predator-prey model with stage structure where x(t) represents the density of the prey at time t, y 1 (t) and y 2 (t) represent the densities of the immature and the mature predator at time t, respectively; the parameters a, e, r, a 1 , a 2 , r 1 , r 2 and D are positive constants in which a is the intra-specific competition rate of the prey, a 2 /a 1 is the rate of conversing prey into new mature predator, e is the birth rate of new predators, r is the intrinsic growth rate of the prey, r 1 is the death rate of the immature predator and r 2 is the death rate of the mature predator, and D denotes the rate of immature predator becoming mature predator.In system (2), it was assumed that feeding on prey can only make contribution to the increasing of the physique of the predator and does not make contribution to the reproductive ability.In [12], the global asymptotic stability of the coexistence equilibrium was established by constructing suitable Lyapunov functions.
Motivated by the works of Bazykin [5] and Yu et al. [12], in this paper, we are concerned with the effect of functional response and stage structure on the dynamics of a predator-prey system.To this end, we study the following differential equations where a 1 x/( reasonable for biological meaning, and implies that hunting has a reward in the sense of diminishing their mortality rate. It is easy to show that all solutions of system (3) are defined on [0, +∞) and remain positive for all t ≥ 0.
The organization of this paper is as follows.In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (3) is discussed.In Section 3, we present conditions for the permanence of system (3).Further, based on discussions above, by using the theory of monotone flows for three-dimensional competitive system, the existence of orbitally asymptotically stable periodic solution is obtained.In Section 4, by using suitable Lyapunov functions and LaSalle invariance principle, sufficient conditions are derived for the global stability of the predator-extinction equilibrium and the coexistence equilibrium of system (3).A brief remark is given in Section 5 to conclude this work.

Local stability
In this section, we discuss the local stability of each equilibria of system (3) by analyzing the corresponding characteristic equations.It is easy to show that system (3) always has a trivial equilibrium E 0 (0, 0, 0) and a predator-extinction equilibria E 1 (r/a, 0, 0).Furthermore, if the following holds: (H1) a 2 r > (a + rm)[r 2 − eD D+r1 ] > 0, then system (3) has a unique coexistence equilibrium E * (x * , y * 1 , y * 2 ), where We now study the local stability of each of nonnegative equilibria of system (3).By analyzing the characteristic equation of system (3) at the equilibrium E 0 (0, 0, 0), it is easy to show that E 0 is always unstable.
The characteristic equation of system (3) at the equilibrium E 1 (r/a, 0, 0) takes the form where Eq. ( 4) always has a negative real root λ It is easy to show that roots of λ 2 + g 1 λ + g 0 = 0 www.mii.lt/NAhave only negative real parts.Accordingly, the equilibrium E 1 of system ( 3) is locally asymptotically stable.If (H1) holds, Eq. ( 4) has at least one positive real root.Therefore, E 1 is unstable.
The characteristic equation of system (3) at the coexistence equilibrium where It is readily seen that if p 2 > 0, p 1 p 2 − p 0 > 0, by the Routh-Hurwitz theorem, the coexistence equilibrium E * of system ( 3) is locally asymptotically stable; and Based on the discussions above, we have the following result.

Permanence and stable periodic solution
In this section, we establish the permanence of system (3) and the existence of orbitally asymptotically stable periodic solution.Before giving our main results, we need the following lemmas.
Proof.Let (x(t), y 1 (t), y 2 (t)) be any positive solution of system (3).Define where Calculating the derivative of N (t) along positive solutions of (3), it follows that where δ = min{D + r 1 − a 1 D/k, r 2 − ke/a 1 }.Noting that r 2 (D + r 1 ) > eD, it follows from (6) that We derive from (7) that Hence, for ε > 0 sufficiently small, there exists a T > 0 such that, if t > T , This completes the proof.
Let X be a complete metric space.Suppose that X Let T b (t) = T (t)| X0 and let A b be the global attractor for T b (t).The following lemma was introduced by Wang and Chen [10].
[10] Suppose that T (t) satisfies (8) and the following: is isolated and has an acyclic covering M , where Then X 0 is uniform repeller with respect to X 0 , i.e., there is an ε > 0 such that for any x ∈ X 0 , lim inf t→+∞ d(T (t)x, X 0 ) ≥ ε, where d is the distance of T (t)x from X 0 .
We now investigate the permanence of system (3).
Theorem 2. If (H1) holds, then system (3) is permanent. Proof.Define If X 0 = U 1 ∪ U 2 and X 0 = intR 3 + , it is easy to show that X 0 and X 0 are coexistently invariant.Moreover, by Lemma 1, conditions (i) and (ii) of Lemma 2 are clearly satisfied.
Hence, we only need to verify the conditions (iii) and (iv).There are two constant solutions E 0 (0, 0, 0) and E 1 (r/a, 0, 0) in X 0 , corresponding, respectively, to x(t) = y 1 (t) = y 2 (t) = 0 and x(t) = r/a, y 1 (t) = y 2 (t) = 0.If (x(t), y 1 (t), y 2 (t)) is a solution of system (3) initiating from U 1 , then Obviously, if (H1) holds, y 1 (t) → 0 and y 2 (t) → 0 as t → +∞.If (x(t), y 1 (t), y 2 (t)) is a solution of system (3) initiating from U 2 with x(0) > 0, it is easy to see that x(t) → r/a as t → +∞.This shows that if invariant set E 0 and invariant set E 1 are isolated, {E 0 , E 1 } is isolated and is an acyclic covering.It is obvious that E 0 is isolated invariant.The isolated invariance of E 1 will be a consequence of the following proof.
We now prove that W s (E 0 ) ∩ X 0 = φ and W s (E 1 ) ∩ X 0 = φ.We restrict our attention to the second equation, since the proof for the first is simple.Assume the contrary.Then there exists a positive solution (x(t), ỹ1 (t), ỹ2 (t)) of system (3) such that x(t), ỹ1 (t), ỹ2 (t) → r a , 0, 0 , as t → +∞.
Choose ξ > 0 sufficiently small satisfying Let T > 0 be sufficiently large such that r a − ξ < x(t) < r a + ξ for t ≥ T.
Then we have, for t ≥ T , We consider the matrix A ξ defined by Since A ξ admits positive off-diagonal elements, the Perron-Frobenius theorem implies that there is a positive eigenvector υ for the maximum eigenvalue α of A ξ .Moreover, by computing, we see that the maximum eigenvalue α is positive since we have (9).
Consequently, system (3) is permanent.This completes the proof.
In the following, we show that there exists an orbitally asymptotically stable periodic orbit in system (3).Theorem 3. Let (H1) hold.If p 2 p 1 − p 0 < 0, then system (3) has an orbitally asymptotically stable periodic solution.
www.mii.lt/NAIf we write (12) as z = f (z), the Jacobian matrix of f at z is as follows: is the unique equilibrium of system (12).Since p 2 p 1 − p 0 < 0 holds, the analysis above shows that (z * 1 , z * 2 , z * 3 ) is unstable and det J(z * ) < 0. Moveover, since system (3) is permanent, there exists a compact subset B of E such that for each z 0 ∈ E, there exists a T (z 0 ) > 0 such that z(t, z 0 ) ∈ B for all t ≥ T (z 0 ).Hence, by Theorem 1.2 of [15], system (12) has an orbitally asymptotically stable periodic solution.This completes the proof.
In the following, we give one example to illustrate the main results above.

Global stability
In this section, we discuss the global stability of the coexistence equilibrium E * and the predator-extinction equilibrium E 1 of system (3), respectively.The strategy of proofs is to construct suitable Lyapunov functions and use LaSalle invariance principle.
Calculating the derivative of V 2 (t) along positive solutions of system (3), it follows that On substituting r = ax * + a 1 y * 2 /(1 + mx * ) into (18), we derive that Since the arithmetic mean is greater than or equal to the geometric mean, it is clear that and the equality hold only for x r − a(x + x * ) ≤ 0, with equality if and only if x = x * .This implies that if x(t) > r/(2a) for t ≥ T , V 2 (t) ≤ 0, with equality if and only if x = x * , y 1 = y * 1 , y 2 = y * 2 .The largest compact invariant subset in M = {(x, y 1 , y 2 ) | V2 (x, y 1 , y 2 ) = 0} is the singleton {E * }.Therefore, by LaSalle invariance principle, the coexistence equilibrium E * is globally asymptotically stable.This completes the proof.

Conclusion
In this paper, we have studied the global dynamics of a predator-prey model with Holling type II functional response and stage structure.The global stability of the predator-extinction equilibrium E 1 and the coexistence equilibrium E * of system (3) has been established by using the Lyapunov-LaSalle type theorem.By Theorem 5, we see that if (H1) and (H2) hold, the coexistence equilibrium E * is globally asymptotically stable.Biologically, these indicate that when the intrinsic growth rate of the prey, the birth rate of new predators and the rate of immature predator becoming mature predator are large enough, and the death rate of the immature predator and mature predator are small enough, then the prey and the predator population coexist, and the ecological system is therefore permanent.
1 + mx) describes the Holling type II functional response, here a 1 and m represent the effects of capturing rate and handling time, respectively, and r 2 m > a 2 is Nonlinear Anal.Model.Control, 2011, Vol.16, No. 2, 242-253 )