Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.


Introduction
Predator-prey model is one of the basic models between di erent species in real world. As we all know, there are always many species going through two stages, immature and mature, which re ect the di erent characteristics of species at each stage. Therefore, to exhibit the real world phenomenon, stage structure population models are more reasonable than other models. In recent years, numerous papers have considered the predator-prey system with stage structure (see [1][2][3][4][5]).
On the other hand, in general, the consumption of prey by predator throughout its past history governs the present birth rate of the predator, in other words, time delay due to gestation is a common example. Obviously, delay di erential equations exhibit much more complicated dynamics than ordinary di erential equations (see [6][7][8][9][10][11][12][13][14]). For example, Wang and Chen [15] considered the following predator prey system with stage structure for the predator population: x(t) = x(t) r − ax(t − τ ) − µy (t) , y (t) = ηµx(t − τ )y (t − τ ) − βy (t) − dy (t), y (t) = βy (t) − ey (t). (1.1) The authors studied the asymptotic behavior of system (1.1). When a time delay due to gestation of the predators and a time delay from a crowding e ect of the prey are incorporated, we establish conditions for the permanence of the populations and su cient conditions for the existence of globally stable positive equilibrium of system (1.1).
Beddington [16] and DeAngelis et al. [17] established a famous B-D functional response that is a predator dependent functional response. In many cases, predators need to search for food and share or compete for food. Therefore, the stage-structured predator-prey models incorporating Beddington-DeAngelis functional response better re ect the ecology. Chen et al. [18] discussed the stability of the boundary solution of a nonautonomous predator-prey model with the Beddington-DeAngelis functional response, which re ects the dynamics of interacting predators and prey in a uctuating environment. Xia et al. [19] considered stability and traveling waves in a Beddington-DeAngelis type stage-structured predator-prey reaction-di usion systems with nonlocal delays and harvesting. Chen et al. [20] discussed the extinction of a two species nonautonomous competitive system with Beddington-DeAngelis functional response and the e ect of toxic substances. Khajanchi [21] investigated the dynamic behavior of a Beddington-DeAngelis type stage structured predator-prey model:ẋ y (t) = βy (t) − ey (t).
(1.2) By analyzing the above system, conditions for positivity, boundedness, uniform persistence, existence of positive equilibria with their local stability have been established. Also, the author showed the existence of Hopf bifurcation when the conversion parameter k passes the critical value. Finally, the conditions for the occurrence of global stability for the unique interior equilibrium point were derived. In the real world, refuge is a strategy to reduce the risk of predation. It is clear that the existence of refuge can have a signi cant impact on the coexistence of predator species and prey species. In recent years, many papers [22][23][24][25][26][27] have proposed and analyzed predator-prey models incorporating prey refuges. Recently, Wei and Fu [28] discussed the Hopf bifurcation and stability for predator-prey systems with Beddington-DeAngelis type functional response and stage structure for prey incorporating refugė By using the characteristic equations, the local stability of each feasible equilibrium of model (1.3) was discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium was established. Motivated by the works [21] and [28], a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge is investigated in this paper. The proposed model is as follows:ẋ where x(t), y (t) and y (t) denote the densities of prey species, immature predator species and mature predator species at time t, respectively; r is the intrinsic growth rate of prey species; a is the intraspeci c competition rate of prey species; d and e are the death rates of the immature and mature predator species respectively; µ( − m) is the capturing rate of the mature predator; η is the conversion rate of nutrients into the production of predator species; τ denotes the time delay due to the gestation of the mature predator; m ∈ [ , ) is refuge rate to prey; the predator species consumes the prey species with Beddington-DeAngelis functional response incorporating prey refuge µ( −m)x(t)y (t) +b( −m)x(t)+cy (t) , and ηµ( −m)x(t−τ)y (t−τ) +b( −m)x(t−τ)+cy (t−τ) denotes the growth rate of predator which are pregnant at time t − τ.
The initial conditions for system (1.4) take the form The rest of this paper is organized as follows. The boundedness and local stability of the equilibrium and the existence of Hopf bifurcation at positive equilibrium of system (1.4) are derived in the next section. In Section 3, we study the permanence of system (1.4). In Section 4, the global stability of system (1.4) are investigated. In Section 5, the in uence of refuge rate on the densities to predator species and prey species is discussed. We end this paper with some examples and a brie y discussion.

Boundedness, Local stability and Hopf bifurcation
In this section, we study the boundedness and local stability of the equilibrium as well as the existence of Hopf bifurcation at positive equilibrium of system (1.4). It is obvious that solutions of model (1.4) with initial conditions (1.5) are positive for all t ≥ . The result is a direct consequence of Nagumo's theorem [29]. Proof. Let V(t) = ηx(t − τ) + y (t) + y (t), and calculating the derivative of V(t) with respect to t along the positive solution of system (1.4), we havė For a small positive constant s ≤ min{d, e}, Hence there exists a positive constant M = η(s+r) a such thaṫ Thus V(t) is ultimately bounded, that is, each solution z(t) = (x(t), y (t), y (t)) of system (1.4) is ultimately bounded. The proof is complete.

. Equilibria
Obviously, system (1.4) always has a trivial equilibrium E ( , , ) and a predator-extinction equilibrium E (r/a, , ). Further, if the following holds: then model (1.4) has a unique coexistence equilibrium E * (x * , y * , y * ), where Let E = ( x, y , y ) be any arbitrary equilibrium, then the variational matrix of system (1.4) at E is given by and the characteristic equation becomes (2.6) . E = ( , , ) First, we analyze the stability of equilibrium E .
Proof. The characteristic equation (2.6) takes the form at the trivial equilibrium E ( , , ) (2.7) It is readily seen that Eq.(2.7) has a positive root, thus the equilibrium E is always unstable. The proof is complete.

. E = (r/a, , )
After that, we consider the stability of equilibrium E .

Theorem 2.3. If the following holds:
then the predator-extinction equilibrium E (r/a, , ) of system (1.4) is locally asymptotically stable; if (H ) holds, then E is unstable.
Proof. The characteristic equation (2.6) at predator-extinction equilibrium E becomes Clearly, the equation λ + r = has one negative real root, which implies that all other roots of Eq.(2.8) are determined by When τ = , Eq.(2.9) turns to (2.10) According to (H ), we have h + h > . By the Routh-Hurwitz criterion, the boundary equilibrium E is locally asymptotically stable. If (H ) holds, then Eq.(2.10) has at least a positive real root, thus the predatorextinction equilibrium E is unstable.
For τ > , we investigate the existence of purely imaginary roots of (2.9). If iω (ω > ) is a solution of (2.9) if and only if ω satis es Separating the real and imaginary parts, we obtain and h + h > , then h − h > . Hence (2.9) has no positive real roots. By Theorem 3.4.1 in [30], if (H ) holds, then all the roots of (2.9) have negative real parts for all τ ≥ , this implies that the boundary equilibrium E is locally asymptotically stable for all τ ≥ . The proof is complete.
Separating the real and imaginary parts, we have Obviously, if r < r and (H ) hold, then p −q > , this implies that (2.16) has no positive real roots. Therefore, by Theorem 3.4.1 in [30], if r < r and (H ) are satis ed, then all the roots of (2.16) have negative real parts for all τ ≥ . Hence the positive equilibrium E * = (x * , y * , y * ) is locally asymptotically stable for all τ ≥ .
If r < r < min{r , r } holds, which implies p − q < , then there exists a unique positive root ω satisfying (2.16). From (2.15), we have (2.18) By Theorem 3.4.1 in Kuang [30], we see that if p − q < hold, then E * remains stable for τ < τ := τ . We now claim that This shows that there exists at least one eigenvalue with a positive real part for τ > τ . Moreover, the conditions for the existence of a Hopf bifurcation [31] are then satis ed yielding a periodic solution. To this end, di erentiating Eq.(2.13) with respect to τ, it follows that Hence, a direct calculation shows that Thus, the transversal condition holds and a Hopf bifurcation occurs at ω = ω , τ = τ . Now, let us summarize our results as follows: Obviously, we generalize the conclusion of boundary equilibrium in [21], and show that prey refuge a ect the stability of the boundary equilibrium. Further, the global stability of the boundary equilibrium will be studied in Section 4 .
(ii) Notice that the conditions of positive equilibrium E * is locally asymptotically stable in [21] is very complicated. Let τ = and m = in model (1.4), compare Theorem 2.4 of this paper with Lemma 4 of [21], we nd that the conditions of our positive equilibrium locally stable are more extensive and concise than that of [21].

Permanence
Let (x(t), y (t), y (t)) be any positive solution of system (1.4) with initial conditions (1.5). From the rst equation of system (1.4), it follows thaṫ Then for above ε > su ciently small there exists a T > such that if t > T , x(t) ≤ L + ε. We derive from the second and the third equations of system (1.4) that for t > T + τ, Consider the following auxiliary equations: The proof is complete.
Let (x(t), y (t), y (t)) be any positive solution of system (1.4) with initial conditions (1.5). For above ε > su ciently small, it follows from Lemma 3.1 that there exists a T > such that if t > T , y (t) ≤ L + ε, y (t) ≤ L + ε. Hence, we derive from the rst equation of system (1.4) that for t > T , According to condition (3.9) and similar to the proof of Lemma 3.1, we have For above ε > su ciently small, there exists a T ≥ T such that if t > T , x(t) ≥ l − ε. Therefore, it follows from the second and the third equations of system (1.4) that for t > T + τ, y (t) = βy (t) − ey (t). (3.12) Consider the following auxiliary equations: The proof is complete.
As a direct corollary of Lemmas 3.1 and 3.2 we have the following theorem.

Global stability
In this section, we study the global stability of the predator-extinction equilibrium E and the global attractivity of the coexistence equilibrium E * of system (1.4). The strategy of proofs is to use Lyapunov functionals and the LaSalle invariance principle.

The influence of prey refuge
In this section, we investigate the in uence of prey refuge. Under the condition (H ), let us compute the derivative along the positive equilibrium E * with respect to m, that is Due to the existence of E * , which implies that x * is a strictly increasing function of m, that is, increasing the constant amount of prey refuge m leads to the increase of prey densities. When < m < − R , dy * i dm > (i = , ), it then yields that both y * and y * are strictly increasing functions on m ∈ ( , − R ); when − R < m < − R , dy * i dm < (i = , ), which implies that both y * and y * are strictly decreasing functions on m ∈ ( − R , − R ); when m = − R , the predator species reaches its maximum, and when m = − R , the predator species goes to extinction.   Note that ηβ( + bx * ) = .
. By Theorem 2.4, when τ < τ , E * is locally asymptotically stable (see Fig. ); when τ > τ , the positive equilibrium E * of model (1.4) is unstable, which yields a periodic solution (see Fig. ). E * is globally attractive (see Fig. ). ), by calculation, we can obtain m < − , r < min{r , r }, then E * is locally asymptotically stable when τ = . Our simulations show that the constant prey refuge m plays an important role on the coexistence of prey-predator population (see Figs. and ). When m is small and increasing, the predator density increases, due to the fact that predators have su cient preys available for predation, even though the refuge increases (see Fig. ). But, as m crosses its threshold value, the predator density decreases with increasing m, the predators was unable to catch preys to sustain themselves and ultimately goes to extinction due to starvation (see Fig. ). > . = τ , µ = , E * = ( . , . , . ) of system (1.4) is locally asymptotically stable, system (1.4) undergoes a Hopf bifurcation at E * when τ .

Conclusion
In this paper we investigate the in uence of prey refuge on the dynamics of a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay. Su cient conditions are derived to ensure the predator-extinction and the locally asymptotically stability of positive equilibrium. Compared with [21] we get more precise conditions. Also, we nd that time delay can cause a stable equilibrium to become unstable one, even Hopf bifurcation to occur, when time delay passes through some critical values. Furthermore, the persistence is investigated. After that, we study the global stability of the predatorextinction and positive equilibrium by constructing some suitable Lyapunov functionals.   to . = − R , the predator species will decrease.
Also, we discuss the in uence of prey refuge on the densities of predator species and prey species. When prey refuge m in the interval ( , − R ), the density of predators will increase with prey refuge m, due to predator species having su cient food for their predation with su ciently small prey refuge m. Predator population attains its maximum when the prey refuge m = − R . In case of larger values of m (m > − R ), this implies that predators species are less likely to catch prey, and the predator species deceases with the increasing of prey refuge m. Eventually, the predator population will be extinct when prey refuge m = − R .