Global Property in a Delayed Periodic Predator-Prey Model with Stage-Structure in Prey and Density-Independence in Predator

and Applied Analysis 3 It is clear that the solution (x 1 (t), x 2 (t)) of system (6) with initial value (x 1 (0), x 2 (0)) > 0 is positive for all t > 0. We further have the following result as a corollary of Lemma 1. Corollary 3. System (6) has a positive ω-periodic solution (x ∗ 1 (t), x ∗ 2 (t)) which is globally asymptotically stable. Remark 4. As a direct consequence of Corollary 3, we see that system (1) has a predator extinction periodic solution (x ∗ 1 (t), x ∗ 2 (t), 0). Remark 5. Obviously, from Remark 2, by increasing coefficients a(t) and c(t), or decreasing coefficients b(t), d(t), and f(t), we can see that x 1 (t) and x 2 (t) will largen. Otherwise, x ∗ 1 (t) and x 2 (t) will decrease. For system (1), we introduce the following basic assumptions:


Introduction
There are many different kinds of two-species predatorprey dynamical models in mathematical ecology.Particularly, two-species predator-prey model with stage-structure have been extensively studied by a large number of papers, see [1][2][3][4][5] and the reference cited therein.The main research topics include the persistence, permanence and extinction of species, the existence and the global asymptotic properties of positive periodic solutions in periodic case, and the global stability of models in general nonautonomous cases.
In [2], Cui and Song studied a periodic predator-prey system with stage-structure.They provided a sufficient and necessary condition to guarantee the permanence of species for the system.In [3], Cui and Takeuchi studied a periodic predator-prey system with stage-structure with function response.They provided a sufficient and necessary condition to guarantee the permanence of species for the system with infinite delay.Some known results are extended to the delay case.
So far, from these done works on the predator-prey model with stage-structure, the authors always assume that the predator is strictly density-dependent, which is much identical with the real biological background.On the other hand, the effect of periodically varying environment plays an important role in the permanence and extinction of species for the system (e.g., seasonal effects of climate, food supply, mating habits, hunting or harvesting seasons, etc.).Thus, the assumptions of periodicity of the parameters and system with time delay are effective ways to characterize and investigate population systems.Owing to many natural and man-made factors such as the low birth rate, high death rate, decreasing habitats, and the hunting of human beings, and the worse ecological system, some predator species become rare and even liable to extinction.For these predator species, we can ignore the effect of density-dependency.Up to now, there are some works on such investigation for the situation of predator density-independence.The authors always assume that the density of predator is proportional to the predation rate, the conversion rate of the immature prey biomass into predator biomass, and the death rate of predator.Predator densityindependece is reasonable to the real ecosystem.
To our knowledge, few scholars consider the delayed periodic predator-prey models with stage-structure in prey and density-independence in predator.In this paper, we consider the following system: ( Our purpose in this paper is to establish sufficient conditions of integrable form for the permanence and extinction of species for system (1).By using the analysis method, the comparison theorem of cooperative system, and the theory of the persistence of dynamical systems, the integral form criteria on the ultimate boundedness, permanence, and extinction are established.The method used in this paper is motivated by the works on the permanence and extinction for periodic predator-prey systems in patchy environment given by Teng and Chen in [5].The organization of this paper is as follows.In the next section, the basic assumptions for system (1), some notations, and lemmas which will be used in the later sections are introduced as the preliminaries.In Section 3, the main results of this paper are stated.In Section 4, the proofs of the main theorems are given.In Section 5, the theoretical results are confirmed by some special examples and the numerical simulations.Finally, a conclusion is given in Section 6.
Let () be a -periodic continuous function defined on  + , we define Consider the following differential equations system: where functions (), (), (), (), and () are positive periodic and continuous defined on  + with common period  > 0. We have the following result.
Let  be a complete metric space with metric .Suppose that  :  →  is a continuous map.For any  ∈ , we denote   () = ( −1 ()) for any integer  > 1 and  1 () = (). is said to be compact in , if for any bounded set  ⊂  set () = {() :  ∈ } is precompact in . is said to be point dissipative if there is a bounded set  0 ⊂  such that for any  ∈ .Consider lim  → ∞  (  () ,  0 ) = 0.
A nonempty set  ⊂  is said to be invariant if () ⊆ . nonempty invariant set  of  is called to be isolated in , if it is the maximal invariant set in a neighborhood of itself.For a nonempty set  of , set   () := { ∈  : lim  → ∞ (  (), ) = 0} is called the stable set of .
Let  and  be two isolated invariant sets; set  is said to be chained to set , written as  → , if there exists a full orbit though some  ∉  ∪  such that () ⊂  and () ⊂ .A finite sequence { 1 , . . .,   } of isolated invariant sets is called a chain, if  1 →  2 → ⋅ ⋅ ⋅ →   , and if   =  1 , the chain is called a cycle.

Main Results
Firstly, concerning the persistence and permanence of species for system (1), we have the following general result.
Remark 8. Let us see the biological meaning of Theorem 7.
In fact, if the predator species is not ultimately bounded, then the population density of predator species will expand unlimitedly.Since the predation rate of predator species for prey species is strictly positive (i.e.,   > 0 in assumption ( 2 )), the prey species will become extinct because of the massive preying by the predator species.Since the survival of predator is absolutely dependent on the prey species, as an opposite result, the predator species will become extinct too.However, if the predation rate () of the predator species is not strictly positive, that is,   = 0, then it cannot lead to extinction when the population density of predator species expands unlimitedly.Therefore, an important open question is whether we can still obtain the boundedness of predator species which is density-independent when   = 0.
Lastly, from Theorems 9 and 7 given by Teng and Chen in [11] on the existence of positive periodic solutions for general Kolmogorov systems with bounded delays, we have the following result.
then system (1) has at least a positive -periodic solution.
Remark 15.In this paper we obtain the existence of the positive periodic solutions for system (1) under the assumption that all parameters are with common periodicity.However, considering all parameters fluctuating in time with the same period is unrealistic, because it will be more realistic if we allow time fluctuations with different period or even nonperiod with some almost periodic environment, which will be more identical with the sound ecosystem.Therefore, there is a very important open question that is whether the same result given in Lemma 1 will be true under the assumption that the parameter in system (1) is almost periodic.
Remark 16.From Remark 5 we know that by increasing coefficients () and () or decreasing coefficients (), (), and (), then  * 1 () and  * 2 () will largen.This shows that by increasing coefficients () and () or decreasing coefficients (), (), and (), we can get that increases.Thus, condition (12) can be changed to condition (10).Therefore, from Theorems 9 and 12, we obtain that predator () will become into the permanence from the quondam extinction.This shows that the stage-structure in the prey (i.e., the birthrate, mortality, density restriction of infancy prey, the transformation from the infancy prey to the maturity prey, and the mortality and density restriction of maturity prey) will bring the effect for the permanence and extinction of the predator.
Remark 17. System (1) is a pure delay system with respect to ().We cannot use the variable without time delay to control the variable with time delay.This shows that it is very difficult to get the global attractivity of system (1).We will discuss this problem in the future.

Remark 18.
An important open question is that what results will be obtained with the condition Is it the permanence of system (1) or the extinction of predator ()?
Proof of Theorem 9. We will use Lemma 6 to prove this theorem.We choose space and sets  0 and  0 are defined by For any  ∈ , let (, ) = ( 1 (, ),  2 (, ), (, )) be the solution of system (1) with initial value  at  = 0. We define continuous map  in Lemma 6 as follows: where   () = ( + , ) with  ∈ [−ℎ, 0].Now, we verify that all the conditions of Lemma 6 will be satisfied for map .It is easy to see that  0 and  0 are positively invariant.From the expression of right side functional (, ) of system (1), we can directly obtain that, for any bounded set  ⊂ , there is a constant () > 0 such that |(, )| ≤ () for all  ≥ 0 and  ∈ .By the Ascoli-Arzela theorem, it implies that map  is compact on ; that is, for any bounded set  ⊂ , set () = {() =   () :  ∈ } is precompact.Moreover, by Theorem 7, we obtain that map  is also point dissipative on .
Obviously, we have   =  0 .Denote by () the -limit set of solution (, ) of system (1) starting at  = 0 with initial value  ∈ .Let From Remark 2, there is a fixed point of map  in   , which is  1 = ( * 1 (0),  * 2 (0), 0).From (10), we can choose a constant  0 > 0 such that By the continuity of solutions with respect to the initial value, for the above given constant  0 > 0, there exists  0 > 0 such that for all  ∈  0 with ‖ −  1 ‖ ≤  0 , it follows that Now, we prove lim sup Suppose the conclusion is not true, then lim sup for some  ∈  0 .Without loss of generality, we can assume that  (  () ,  1 ) <  0 , ∀ ≥ 0.
Therefore, all the conditions of Lemma 6 are satisfied.By Lemma 6 we finally obtain that map  is uniformly persistent with respect to ( 0 ,  0 ).Further, from Theorem 3.1.1given in [10], we can obtain that all positive solutions of system (1) are uniformly persistent.This completes the proof.

Remark 1.
There is an open question: from Figure 3, we see that () of system (1) has more than one periodic solution.So, we cannot get a globally asymptotically stability solution of system (1).Whether we can get a globally asymptotically stability solution of system (1) under some conditions is our future work.
Example 2. In system (67), the coefficients (), ℎ(), (),  1 ,  2 , and (,  1 ()) are given as in Example 1. But, the other coefficients in system (67) are given as the following different values: () = 1 + sin(2), () = 0.2 sin(2) + 0.75, () = 0.25, () = 0.35, and () = 1.We see that coefficients () and () are decreased and coefficients (), (), and () are increased.From Corollary  x 1 (t) x 2 (t) y(t) we easily verify that assumptions ( 1 )-( 3 ) hold.It is easy to verify that condition (10) in Theorem 9 does not hold, but condition (12) in Theorem 12 holds.Therefore, from Theorem 12, we obtain that predator () in system (67) will become into extinction.The numerical simulations of the above results can be seen in Figures 5 and 6  Remark 2. From the numerical simulations given in Examples 1 and 2, we see that the stage-structure in the prey, specially the birthrate, mortality, density restriction of infancy prey, the transformation from the infancy prey to the maturity prey, and the mortality and density restriction of maturity prey, will bring the very obvious effect for the permanence and extinction of the predator.

Conclusions
In the real world, there are many types of interactions between two species.Predator-prey relations are among the most common ecological interactions.
In this paper, we study the global property in a delayed periodic predator-prey model with stage-structure in prey and density-independence in predator.The survival of species in a biological system is one of the most basic and important problems in mathematical biology, and permanence is an important concept when dealing with this problem.Here, by using the analysis method, the comparison theorem of cooperative system, and the theory of the persistence of dynamical systems, we have established the integral form criteria on the ultimate boundedness, the sufficient integral conditions on the permanence and extinction of species.The method used in this paper is motivated by the works on the permanence and extinction for periodic predator-prey systems in patchy environment given by Teng and Chen in [5].The results obtained in this paper are different from the predator-prey system given in [4], where the authors studied the necessary and sufficient integral conditions on permanence and extinction of species for nonautonomous predator-prey systems with infinite delays and predator density dependence.However, in our paper, we have considered the effects of general predator functional response on the survival of species.Therefore, we have modeled a general nonautonomous predator-prey system with finite delays and density independence.Some well-known results on the predator density-dependence are improved and extended to the predator density-independent cases.