BIFURCATION ANALYSIS OF A STAGE-STRUCTURED PREDATOR-PREY MODEL WITH PREY REFUGE

. A stage-structured predator-prey model with prey refuge is considered. Using the geometric stability switch criteria, we establish stability of the positive equilibrium. Stability and direction of periodic solutions arising from Hopf bifurcations are obtained by using the normal form theory and cen- ter manifold argument. Numerical simulations conﬁrm the above theoretical results.

1. Introduction. Since it takes time for species to mature, delay differential equations are widely adopted to model the corresponding stage structure. For example, a single species model with a stage structure was studied in [1,2], while a predatorprey model with a stage-structure was considered in [19]. For persistence, stability, and bifurcations of stage-structured predator-prey models, we refer the readers to [3,11,14,17,18,23] and their references therein.
As a result of evolution, the refuge phenomenon enriches the interaction between predators and prey. To better understand the influence of refuge on the predatorprey interaction, some mathematical models with prey refuge have been proposed and studied, see [6,8,7,9,10,13,15,16,21,22]. In general, two types of refuge have been considered: (i) the number of prey using refuge is fixed, and (ii) a constant proportion of prey uses refuge [9].
In this paper, we assume that predators feed only on mature prey population, and we consider the following model where x i (t) and x m (t) represent the immature and mature prey population densities at time t, respectively, and y(t) denotes the density of predator population at time t.
In model (1), α > 0 and r 1 > 0 represent the birth and death rates of the immature population, respectively; the death rate of the mature population is of a logistic nature, i.e., it is proportional to square of the population with a proportionality constant β > 0; r 2 > 0 is the harvesting rate; τ is the time to maturity; the term αe −r1τ x m (t − τ ) represents the immature prey individuals who were born at time t − τ and are alive at time t; k > 0 is the rate of conversion of nutrients from the prey to the reproduction of the predator; d > 0 is the death rate of the predator population; ξ > 0 denotes the fixed quantity of mature prey protected by refuges. For continuity of initial conditions of (1), we assume that: According to [20], since the coefficients of our model are delay-dependent, the coefficients of the corresponding characteristic equation are also delay-dependent. Thus the local stability analysis becomes very complicated. In such case, the geometric stability switch criteria and center manifold argument developed in [5] can be applied to analyze the existence of Hopf bifurcation and the direction of periodic solutions branching from Hopf bifurcations.
We organized the rest of this paper as follows. In Section 2, we apply the geometric stability switch criteria to analyze the corresponding characteristic equation to derive the local stability of equilibria and establish the existence of Hopf bifurcation. In Section 3, by using the normal form theory and the center manifold theorem, we study the stability and direction of periodic solutions branching from Hopf bifurcations. In Section 4, we give some numerical simulations to support the analytic results. In the last section, we provide a conclusion to followed by some discussions.
2. Stability of equilibria. Since the variable x i does not appear in the second and the third equations of (1), we only need to consider the following system with the initial conditions Similar to Theorems 3.1 and 4.1 of [9], we can show that the solutions of (3) with (4) are positive and bounded for all t ≥ 0. For system (3), there is always the trivial equilibrium E 0 = (0, 0). There exists a boundary equilibrium E 1 = αe −r 1 τ −r2 β , 0 , provided that α > r 2 and τ ∈ [0,τ ), whereτ = 1 r1 (ln α − ln r 2 ). Moreover, if the positive equilibrium E 2 = (x * m , y * ) exists, then we must have
Remark 1. In the discussion above,τ <τ . Under condition (H1), E 0 and E 1 are unstable. Details about the stability of E 0 and E 1 follow from [9,19].
In the following, we focus on the stability of the unique positive equilibrium by applying the geometric stability switch criteria developed in [5].
Proof. If τ =τ , then it follows from (5) and (7) that It is easy to show that the function σ(τ ) is decreasing in τ for τ ∈ [0,τ ). This completes the proof. Now, we verify that P (λ, τ ) and Q(λ, τ ) satisfy the condition of the geometric stability switch criteria [5].
for each τ has at most a finite number of real zeros; (V) Each positive root w(τ ) of F (w, τ ) = 0 is continuous and differentiable in τ whenever it exists.
Proof. Based on the above argument, we have According to Lemma 2.1, for any τ ∈ [0,τ ), it follows that and (II) is satisfied.
Since P (λ, τ ) is a second-degree polynomial about λ and Q(λ, τ ) is linear in λ, we have lim , we obtain (IV). F (w, τ ) is continuous in w and τ , and differentiable in w, so the implicit function theorem leads to (V).
3. Direction and stability of Hopf bifurcation. In this section, we obtain the conditions under which a family of periodic solutions bifurcate from the positive equilibrium point E 2 at the critical value of τ * . Following the ideas of Hassard et al. [12], we derive the explicit formulae which can determine the properties of the Hopf bifurcation at critical value of τ * by using the normal form and the center manifold theory [4,20]. Throughout this section, we always assume that system (3) undergoes a Hopf bifurcation at the positive equilibrium E 2 at τ = τ * . Let w * i = w(τ * )i be the corresponding purely imaginary root of the characteristic equation at the positive equilibrium.
By using the idea of Hassard [12], we firstly compute the coordinates to describe the center manifold C 0 at µ = 0, which is a local invariant attracting a two-dimensional manifold in C 0 . Let X t be the solution of (23) when µ = 0, and define z(t) = q * , X t .
Then, we have W (t, θ) = W (z,z, θ), where Then on the center manifold C 0 , we have where z andz are local coordinates for center manifold C 0 in the directions of q 1 andq * , respectively. Note that, W is real if X t is real. It is easy to see thaṫ where g(z,z) = g 20 z 2 2 + g 11 zz + g 02z Now, we determine the coefficients W ij in (31). It follows from (30) and (32) and (35) On the other hand, on C 0 , we havė W =W zż + Wzż(W 20 (θ)z + W 11 (θ)z + · · · )(iw * τ * z + g)
Comparing (35) with (40), we have If θ = 0, From (36) and the definition of A(0), we havė Hence, solving the Bernoulli differential equation, we obtain By a similar method, we obtain where U 1 and U 2 are both two-dimensional vectors. It follows from (36) that Notice that, from the definition of A(0), and Hence, from (44)-(48), it follows that Similarly, we have Then, we obtain and So we can calculate g 21 and derive the following values These formulaes give a description of the Hopf bifurcation derived periodic solutions on the center manifold C 0 . (I) If µ 2 > 0 (µ 2 < 0), the Hopf bifurcation is supercritical (subcritical), meaning that bifurcated periodic solutions exist for µ > µ 2 (µ < µ 2 ).
4. Numerical simulations. In this section, we present numerical simulations of system (3) to illustrate our theoretical results.

5.
Conclusions. In this work, a stage-structured predator-prey model with prey refuge has been considered. We have assumed that a fixed number of prey are protected by refuges to escape the predation. The existence and stability of the positive equilibrium E 2 have been derived under some conditions. It was found that Hopf bifurcation may appear when ξ is small. If ξ is large enough, the positive equilibrium is unstable, while the predator population becomes smaller and can even become extinct. Using the approach of Beretta and Kuang [5], we have shown that the positive equilibrium can be destabilized through a Hopf bifurcation. Moreover, we have investigated the stability and direction of periodic solutions branching from Hopf bifurcations by using the normal form theory and the center manifold theorem. Our results show that the prey refuges play an important role in determining the dynamics of the system.