Dynamics of a new delayed stage-structured predator-prey model with impulsive diffusion and releasing

In this work, we propose a new delayed stage-structured predator-prey model with impulsive diffusion and releasing. By the stroboscopic map of the discrete dynamical system, we obtain a prey-extinction boundary periodic solution. Furthermore, we prove that the prey-extinction boundary periodic solution is globally attractive. We also prove that the investigated system is permanent by the theory on the delay and impulsive differential equations. Our results indicate that time delay, impulsive diffusion, and impulsive releasing have influence to the dynamical behaviors of the investigated system. The results of this paper also provide a tactical basis for pest management.


Introduction
Many authors [-] and papers [, ] have studied the predator-prey, competitive, and cooperative models. Permanence and extinction are significant concepts of those models which also show many interesting results. However, the stage structure of a species has been considered very little. In the real world, almost all animals have the stage structure of being immature and mature. Recently, [-] studied the stage structure of species with or without time delays. Aiello et al. [] considered a time delayed stage structure of being immature and mature of the population model Dispersal is a ubiquitous phenomenon in the natural world. It is important for us to understand the ecological and evolutionary dynamics of populations mirrored by the large number of mathematical models devoted to it in the scientific literature [-]. If the population dynamics with the effects of spatial heterogeneity is modeled by a diffusion process, most previous papers focused on the population dynamical system modeled by the ordinary differential equations. But in practice, it is often the case that diffusion occurs in regular pulse. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the ex- The organization of this paper is as follows. In the next section, we introduce the model and background concepts. In Section , some important lemmas are presented. In Section , we give the conditions of global attractivity and permanence for system (.). In Section , a brief discussion is given in the last section to conclude this work.

The model
where system (.) is constructed of two patches. x i (t), y i (t) and z i (t) represent the immature prey population, mature prey population, predator population in patch i (i = , ) at time t. It is assumed that birth into the immature prey population is proportional to the existing mature prey population with proportionality constant r i in patch i (i = , ).
τ i represents a constant time to maturity of prey population in patch i (i = , ), that is, immature prey individuals and mature individuals are divided by age τ i in patch i (i = , ).
The natural death rates w i , w i and w i (i = , ) are assumed for the immature prey population, mature prey population, and predator population in patch i (i = , ). β i (i = , ) is the mature prey population capture rate by the predator population in patch i (i = , ). k i (i = , ) is the conversion rate of nutrients into the reproduction of the predator population in patch i (i = , ). The pulse diffusion occurs every τ >  period. The system evolves from its initial state without being further affected by diffusion until the next pulse appears. y i ((n + l)τ ) = y i ((n + l)τ + )y i ((n + l)τ ) where y i ((n + l)τ + ) represents the density of population in the ith patch immediately after the nth diffusion pulse at time t = (n + l)τ , while y i ((n + l)τ ) represents the density of population in the ith patch before the nth diffusion pulse at time t = (n + l)τ (n ∈ Z + ,  < l < ).  < D <  is the dispersal rate of the predator population between two patches. It is assumed here that the net exchange from the jth patch to the ith patch is proportional to the difference y jy i of the predator population densities. The predator population is released with μ i in patch i (i = , ) at moment t = (n + )τ , n ∈ Z + .
Because x i (t) (i = , ) does not affect the other equations of (.), we can simplify system (.) and restrict our attention to the following system:

The lemmas
The solution of (.), denoted by , is a piecewise continuous function X : Obviously the global existence and uniqueness of solutions of (.) are guaranteed by the smoothness properties of f , which denotes the mapping defined by the right side of system (.) (see Lakshmikantham, []). Before we have the the main results, we need to give some lemmas which will be used in the following.
According to the biological meaning, it is assumed that

Lemma . ([]) Let the function m ∈ PC [R + , R] satisfy the inequalities
Now, we show that all solutions of (.) are uniformly ultimately bounded.
When t = (n + l)τ , When t = (n + )τ , By Lemma ., for t ∈ (nτ , (n + )τ ], we have So V (t) is uniformly ultimately bounded. Hence, by the definition of V (t), there exists a The proof is complete.
If y i (t) =  (i = , ), we have the following subsystem of (.): We can easily obtain the analytic solution of (.) between pulses as follows: Considering the third and fourth equations of (.), we have Lemma . The fixed point (z *  , z *  ) of (.) is globally asymptotically stable.
Proof For convenience, we use the notation (z n  , z n  ) = (z  (nτ + ), z  (nτ + )). The linear form of (.) can be written as Obviously, the near dynamics of (z *  , z *  ) is determined by linear system (.). The stability of (z *  , z *  ) is determined by the eigenvalue of M less than . If M satisfies the Jury criterion [], we can know the eigenvalue of M is less than , We can easily know that (z *  , z *  ) is unique fixed point of (.), and From the Jury criterion, (z *  , z *  ) is locally stable, then it is globally asymptotically stable. This completes the proof.

The dynamics
From the above discussion, we know there exists a prey-extinction boundary periodic solution (, z  (t), , z  (t)) of system (.). In this section, we will prove that the preyextinction boundary periodic solution (, z  (t), , z  (t)) of system (.) is globally attractive.

holds, the prey-extinction boundary periodic solution
Proof From (.), we can obtain Then, we can choose ε  sufficiently small such that From the second and fourth equations of system (.), we obtain dz i (t) dt ≥ -w i z i (t) (i = , ). So we consider the following comparison impulsive differential system: In view of Lemma . and (.), we find that the boundary periodic solution of system (.) z * *  e -w  (t-(n+l)τ ) , t ∈ [(n + l)τ , (n + )τ ), is globally asymptotically stable, where z *  and z *  are determined as (.), z * *  and z * *  are defined as (.).
From the second and fourth equations of system (.), we have While (z  (t), z  (t)) and (z  (t), z  (t)) are the solutions of and Therefore, for any ε  >  (ε  is small enough), there exists an integer k  , n > k  such that , for t large enough. This implies z i (t) → z i (t) (i = , ) as t → ∞. This completes the proof.
The next work is to investigate the permanence of system (.). Before starting our theorem, we give the following definition.
Definition . System (.) is said to be permanent if there are constants m, M >  (independent of initial value) and a finite time T  such that, for all solutions (y  (t), z  (t), y  (t), z  (t)) with all initial values y i ( + ) > , z i ( + ) >  (i = , ), m ≤ y i (t) ≤ M, m ≤ z i (t) ≤ M (i = , ) hold for all t ≥ T  . Here T  may depend on the initial values (y  ( + ), z  ( + ), y  ( + ), z  ( + )).
there is a positive constant q such that each positive solution (y  (t), z  (t), y  (t), z  (t)) of (.) satisfies y i (t) ≥ q, for t large enough, where y * i (i = , ) is determined by where z * i (i = , ) and z * * i (i = , ) are defined as (.) and (.), respectively.
Proof The second and fourth equations of (.) can be rewritten as According to (.), Q i (t) (i = , ) is defined as We calculate the derivative of Q i (t) (i = , ) along the solution of (.): Since we can easily see that there exists a sufficiently small ε >  such that We claim that for any t  > , it is impossible that y i (t) < y * i (i = , ) for all t > t  . Suppose that the claim is not valid. Then, there is a t  >  such that y i (t) < y * i (i = , ) for all t > t  . It follows from the first and third equations of (.) that for all t > t  for all t > t  . Set y m i = min t∈[t  ,t  +τ  ] y i (t), we will show that y i (t) ≥ y m i for all t ≥ t  . Suppose the contrary, then there is a T  >  such that y i (t) ≥ y m i for t  ≤ t ≤ t  + τ  + T  , y i (t  + τ  + T  ) = y m i and y i (t  + τ  + T  ) < . Hence, the second and fourth equations of system (.) imply that This is a contradiction. Thus, y i (t) ≥ y m i for all t > t  . As a consequence, then This implies that as t → ∞, Q i (t) → ∞. It is a contradiction to Q i (t) ≤ M( + τ i r i e -w i τ i ). Hence, the claim is complete.
By the claim, we are left to consider two cases. where q i = y * i e -(w i +β i M)τ i (i = , ). We hope to show that y i (t) ≥ q i (i = , ) for all t large enough. The conclusion is evident in the first case. For the second case, let t * >  and ξ >  satisfy y i (t * ) = y i (t * + ξ ) = y * i (i = , ) and y i (t) < y * i (i = , ) for all t * < t < t * + ξ where t * is sufficiently large such that y i (t) > σ i (i = , ) for t * < t < t * + ξ , y i (t) (i = , ) is uniformly continuous. The positive solutions of (.) are ultimately bounded and y i (t) (i = , ) is not affected by impulses. Hence, there is a T ( < t < τ  ) and T is dependent on the choice of t * such that y i (t * ) > y * i  (i = , ) for t * < t < t * + T. If ξ < T, there is nothing to prove. Let us consider the case T < ξ < τ  . Since y i (t) > -(w i + β i M)y i (t) (i = , ) and y i (t * ) = y * i (i = , ), it is clear that y i (t) ≥ q i (i = , ) for t ∈ [t * , t * + τ  ]. Then, proceeding exactly as the proof for the above claim, we see that y i (t) ≥ q i for t ∈ [t * + τ  , t * + ξ ]. Because the kind of interval t ∈ [t * , t * + ξ ] is chosen in an arbitrary way (we only need t * to be large). We conclude that y i (t) ≥ q for all large t. In the second case, in view of the above discussion, the choice of q is independent of the positive solution, and we prove that any positive solution of (.) satisfies y i (t) ≥ q for all sufficiently large t. This completes the proof of the theorem.
Theorem . If min i=, r i e -w i τ iw iβ i z * i e -(w i -k i β i y * i )lτ + z * * i e -(w i +k i β i y * i )(-l)τ > , system (.) is permanent.