Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses

In this paper, we examine a diffusive predator-prey model with Beddington-DeAngelis functional response and stage structure on prey under homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of immature prey to their maturity. We investigate the dynamics of their permanence and the extinction of the predator, and provide sufficient conditions for the global attractiveness and the locally asymptotical stability of the semi-trivial and coexistence equilibria.

1. Introduction. In this paper, we consider a diffusive predator-prey model with Beddington-DeAngelis functional response and stage structure on prey: ∂u(x, t) ∂t = D u ∆u(x, t) + be −diτ u(x, t−τ )−au 2 (x, t)− mu(x, t)v(x, t) 1+k 1 u(x, t)+k 2 v(x, t) , ∂v(x, t) ∂t = D v ∆v(x, t) + nmu(x, t)v(x, t) 1 + k 1 u(x, t) + k 2 v(x, t) (1.1) where Ω is a bounded domain in R N with smooth boundary ∂Ω, and u i (x, t) and u(x, t) represent the density of the immature and mature prey, respectively, at time t in location x. Furthermore, v(x, t) denotes the density of the mature predator, and D u and D v are the diffusion coefficients of the mature prey and mature predator, respectively. Moreover, ∂/∂ν denotes the outward normal derivative on the boundary ∂Ω of Ω. All constants are positive for their biological sense, and the time delay τ is the time taken for the juvenile prey to develop from birth to maturity. Finally, b is the birth rate of the mature prey, d i is the mortality rate of immature prey, n is the birth rate of the predator, m(1/time) and k 1 (1/prey) describe the effects of the capture rate and handling time on the feeding rate, respectively, k 2 is a constant describing the magnitude of interference among predators(1/predator), e −diτ is the survival rate of each immature prey to become mature prey, a is the death rate of mature prey, and d is the death rate of the predator.
The following assumptions are imposed in (1.1): (a) Immature prey have a time postponement from their birth to maturity. (b) Young prey are raised by their parents and dependent on their nutrition. They can avoid predators by staying in their eggs, nests, or burrows, i.e., they are immobile and cannot breed.
(c) The mature predator cannot attack immature prey; they can only hunt mature prey, because the immature prey cannot move easily and are hidden in small areas. Immature prey appear significantly less than mature prey in nature.
(d) Young prey reach maturity after surviving the immature stage; if the juvenile death rate is not zero, then not all immature prey survive the juvenile stage.
In accordance with assumptions (b) and (c), there is no diffusion of juvenile prey in the first equation of (1.1). For more background information on (1.1), we refer the reader to [15,16] and the references therein.
We also assume that the predator consumes the mature prey with the functional response of the Beddington-DeAngelis type. Furthermore, the following is assumed as a result of the continuity of solutions to (1.1) : From the first equation in (1.1) with initial condition (1.2), we have the following formula: Hence, the immature prey density u i (x, t) is determined by the mature prey variation u(x, t) that is completely determined by the second and third equations of (1.1); therefore, we mainly deal with the following subsystem: For convenience, we denote In this paper, we focus on the following subsystem : (1.5) Considerable results have been obtained for stage-structured predator-prey models with a time-delay term in the functional responses in a heterogeneous /homogeneous environment. Liu and Zhang [16] considered a stage-structured predator-prey model with a Beddington-DeAngelis functional response based on the time taken from birth for prey to reach maturity in a homogeneous environment : , where x m (θ) > 0 is continuous on −τ ≤ θ ≤ 0, and x i (0), x m (0), y(0) > 0. Here x m and y represent the mature prey and predator densities, respectively, and x i is the immature prey density. The results of the permanence, extinction of species, and global stability of equilibria were demonstrated. Liu and Beretta [15] studied a stage structured predator-prey model with a Beddington-DeAngelis-type functional response to consider the time delay between the birth and maturity of a predator. They examined the permanence of two species and showed both analytically and numerically that stability switches occur at the interior equilibrium as the maturation time delay increases. In [11,12], the authors considered a stage structured diffusive predator-prey model with a Beddington-DeAngelis-type functional response based on the duration of time between the birth and maturity of a predator. The global existence of nonnegative solutions, permanence, global stability, local stability and Hopf bifurcation of the equilibria were demonstrated. The main focus of this paper is to examine the asymptotic properties of the diffusive delayed predator-prey model in (1.1) with a Beddington-DeAngelis type functional response under homogeneous Neumann boundary conditions; we assume that the discrete time delay covers the period of time between the birth and maturity of immature prey. We establish threshold dynamics for the permanence and extinction of the predator. In addition, we provide sufficient conditions for the global attractiveness of the semi-trivial and coexistence equilibria. Furthermore, we demonstrate the locally asymptotical stability of (1.1) at equilibria.
The remainder of this paper is organized as follows. In Section 2, we demonstrate the global existence of nonnegative solutions, possible equilibria, and long-term behavior of time-dependent solutions to (1.5), particularly, its uniformly persistent property. In Section 3, we obtain the locally asymptotical stability at the equilibria.
In Section 4, we study the globally asymptotical stability of system (1.5) at semitrivial and coexistent constant steady states. Finally, we give a brief biological interpretation of our results in Section 5.

2.
Permanence. In this section, we examine the global existence and persistence of solutions to (1.5) by using persistence theory [7] and a comparison argument.
We begin with the following existence theorem : Theorem 2.1. For any nonnegative nontrivial initial functions, the system (1.5) has a unique global solution The above theorem can be proven using the upper-lower solution method [21] since the reaction terms in (1.5) satisfy the Lipschitz conditions in a bounded set; thus, we omit its proof.
Observe that system (1.5) has the zero and nonnegative equilibria E 0 = (0, 0) and E 1 = (K, 0) ; the positive equilibrium E * = (u * , v * ) of (1.5) is the unique solution to the following system: where r and K are defined in (1.4). If 0 > 1, then (2.1) has a unique positive constant steady state E * = (u * , v * ). In particular, it can be found that The following lemma [14] will be used in the remainder of this paper.
and be a nonnegative nontrivial solution to the following scalar problem: Next, we study the asymptotic property of (1.1). We obtain the long-term behavior for any nonnegative solution (u, v) of system (1.5) as t → ∞ for all x ∈ Ω.
Proof. It suffices to prove (i) since (ii) follows a similar argument. First, note that lim sup t→∞ u(x, t) ≤ K in Ω follows from the comparison argument of parabolic problems and Lemma 2.
Using this result and the comparison argument of parabolic problems, for an arbitrary positive , there exists . Therefore, by the arbitrariness of , we obtain the desired result.
We now provide the uniform persistence of solutions to system (1.5). We apply persistence theory [7] to our system. Theorem 2.4. If 0 > 1 and k 2 > m r , then for given initial nonnegative functions Note that system (1.5) generates a semiflow S(t) on X + × X + , and X 0 and ∂X 0 are invariant. As a result of Theorem 2.3, S(t) is a dissipative point in X + × X + , while S(t) : X + × X + → X + × X + is compact for each t > τ . Let M 0 = (0, 0) and M 1 = (K, 0). Then the ω-limit sets of the semiflow S(t) on ∂(X 0 ), is A ∂ = {M 0 , M 1 } because there are only two semi-trivial equilibria M 0 and M 1 in ∂X 0 for model (1.5). The disjoint, compact and isolated invariant sets M 0 and M 1 cover A ∂ . From the assumption that 0 > 1, we can show that there exists a δ such that for any (u To finish the proof, we show that Thus, using a comparison argument, u( This is a contradiction since lim t→∞ z(x, t) = K(1 − m/(rk 2 )) uniformly in Ω by Lemma 2.2. Similarly, a contradiction is reached when W s (M 1 ) ∩ (Y 0 × Y 0 ) = ∅ ; specifically, since (u(x, t), v(x, t)) → (K, 0) as t → ∞ uniformly in Ω, for any given 1, there exists a T > 0 such that 3. Local stability of solutions. In this section, we examine the locally asymptotical stability at equilibria E 0 = (0, 0), E 1 = (K, 0), and E * = (u * , v * ). Throughout the remainder of this paper, we use the following notations.
Then the linearized equation of system (1.5) at any feasible equilibrium point E can be expressed by u t = L(ũ,ṽ)u ; specifically, Thus, the characteristic equation of system (1.5) at equilibrium E takes the following form : We now investigate the locally asymptotical stability of (1.5) at E 1 = (K, 0).
Finally, we obtain the following theorem which provide sufficient conditions for the locally asymptotical stability of system (1.5) at E * = (u * , v * ).
then the positive constant equilibrium E * of system (1.5) is locally asymptotically stable.
Proof. We have already seen that E * exists if 0 > 1. The characteristic equation (3.1) of system (1.5) at E * reduces to (3.5) When λ = α + βi for α, β ∈ R, if the real and imaginary parts are separated, we have For simplicity, let

Global stability of solutions.
In this section, we investigate the globally asymptotical stability of the semitrivial solution (K, 0) and positive constant solution (u * , v * ). First, we investigate the behavior of solutions to (1.5) at the semitrivial solution E 1 = (K, 0).  u(x, t), v(x, t)) of system (1.5) uniformly converges to E 1 on Ω as t → ∞. Furthermore, the semitrivial uniform steady state (B, K, 0) of system (1.1) is globally asymptotically stable, where Proof. First, we consider the case when 0 < 1. We can select a sufficiently small positive constant such that nm(K+ ) 1+k1(K+ ) ≤ d. Using the comparison argument of parabolic problems and Lemma 2.2, we obtain lim sup t→∞ u(x, t) ≤ K. Thus, there exists T 1 ∈ (0, ∞) such that u(x, t) ≤ K + in Ω × [T 1 , ∞). Therefore, the second equation of (1.5) satisfies The comparison argument yields This implies that there exists Since the existence of , ∞) yields lim inf t→∞ u(x, t) ≥ K from the arbitrariness of the small positive constant , we can apply the comparison argument once again to obtain lim t→∞ u(x, t) = K in Ω.
Since the last term of (4.7) satisfies under the assumption that ak 2 ≥ mk 1 , we obtain dW dt ≤ 0. Therefore, for any t > 0, W (t) is a Lyapunov function for system (1.5), i.e., for t > 0, W (t) < 0 along their trajectories, except at (u * , v * ) where W (t) = 0. Hence, by Lyapunov function theory, our obtained results imply the global asymptotic stability at E * . 5. Biological interpretations. In this paper, we studied the asymptotic properties of a diffusive delayed predator-prey model with a Beddington-DeAngelis functional response under homogeneous Neumann boundary conditions. The discrete time delay was assumed to cover the time between birth and maturity of prey.
First, we obtained sufficient conditions for the permanence and extinction of predators for system (1.5). Theorem 2.4 established that predator and prey coexist permanently if nmb (ae diτ + k 1 b) > d and k 2 > m b e diτ .
The conditions imply that the predator recruitment rate at the peak of prey abundance which is greater than its death helps the persistence of system (1.5). Furthermore, we found that large magnitude of interference among predators k 2 played a role in the persistence of system (1.5) and that its ability to satisfy the conditions was contingent upon the choice of τ . By Theorem 4.1, the semitrivial solution (K, 0) = ( be −d i τ a , 0) was found to be globally asymptotically stable provided This implies that if the predator death rate d is high or if the predator birth rate n is low, the only predator may be extinct. Predators may be driven into extinction by a decreased prey carrying capacity K = be −diτ , due to either a large prey maturation τ or high juvenile prey morality rate d i . From Theorem 4.2, the coexistence equilibrium (u * , v * ) was found to be globally asymptotically stable provided nmb (ae diτ + k 1 b) > d, k 2 ≥ mk 1 a .

S. LEE AND I. AHN
The conditions imply that sufficiently large mutual interference k 2 between predators has the ability to stabilize system (1.5).