Dynamical Behaviors of a Stage-Structured Predator-Prey Model with Harvesting Effort and Impulsive Diffusion

We consider a delayed predator-prey model with harvesting effort and impulsive diffusion between two patches. By the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map, we obtain some sufficient conditions on the existence and global attractiveness of predator-eradicated periodic solution for the system. Furthermore, the permanence of the system is derived.The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems.


Introduction
The effect of spatial factors in population dynamics is an interesting topic since dispersal often occurs between patches in ecological environment [1,2].Some models focused on dispersal process in continuous time meaning [3,4].However, many species diffuse only during a single period, and the diffusion often occurs in regular pulses.For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse during other seasons.Since the short-time diffusion is often assumed to be in the form of impulses in the modeling process, some mathematical models on impulsive diffusion have been studied by impulsive differential equations (see, e.g., [5][6][7][8][9][10][11]).In [5], a single species model with impulsive diffusion was initially formulated by where, for the th patch,   is the density of species in the logistic growth; Δ  () =   ( + ) −   ( − ) and   () =   ( − ),   ( + ),   ( − ) represent the right-hand limit and left-hand one at ; the parameters   ,   /  are the intrinsic growth rates and carrying capacities, respectively; the diffusion occurs at impulsive moments ,  = 0, 1, 2, . . .,  > 0;   ∈ (0, 1) is dispersal rate from the th patch to th, ,  = 1, 2,  ̸ = .We assume that the net exchange from the th patch to th patch is proportional to the difference   −   of population densities.
On the other hand, almost all species go through the life stage from the immature to the mature stage.Since it is necessary to spend units of time that the immature becomes the mature, time delay plays important role in stagestructured model.Aiello and Freedman [6] introduced the following stage-structured single species model: where ,  represent the immature and mature populations densities, respectively,  is the birth rate, ,  are the immature and mature death rate,  is the maturation time delay, and the term  − (−) is the number of immature populations who were born at time  −  and survive at time , and this term represents the transformation from the immature to the mature stages.Note that the death number of mature population is of a logistic nature, which is proportional to the square of the population with proportionality constant .
Recently, some stage-structured models with time delays and impulsive diffusion were investigated in [7][8][9][10][11].Jiao et al. [7,9] and Shao et al. [8,10] investigated a class of predatorprey models with prey-impulsive diffusion between two patches in which the predator is subject to stage-structured effects (delayed effects) only in one of the patches.Dhar and Jatav [11] consider a delayed stage-structured predatorprey model with impulsive diffusion, and both of the predators in two patches are subject to stage-structured effects.Many interesting results in the mentioned publications mainly focused on the global attractiveness of the predatoreradicated periodic solution and the permanence of the systems.To manage effectively the species, we should know or estimate how many members the population has at large time in every patch when a species is uniformly permanent.Mathematically, this corresponds to the ultimately low and upper boundedness of solutions of the systems.Although the permanence was derived by the estimate for the ultimately upper boundedness of solutions, there are some errors or negligence on the estimate in [7][8][9][10][11] (see Remarks 4 and 6 below).In addition, according to Aiello and Freedman [6], it may be more reasonable to take the square function of the mature population as the death numbers of the mature than to take the linear one in [7][8][9][10][11].
Motivated by the above discussion, we propose the following stage-structured predator-prey model with generalized functional response, the harvesting effort of the mature predator, and impulsive diffusion between two predators' territories: where for the th patch  = Mathematically, the proposed system (3) generalizes some existing models.For example, (3) is the model in [7] when  1 (⋅) = 0,  2 ( 2 ) =  2 ,   = 0, and  1 ,  1 are removed; (3) becomes the model in [11] when   = 0,   (  ) =     .In this paper, we will mainly investigate the existence and global attractiveness of periodic solution and the permanence for system (3) by employing the impulsive comparison theorem and the discrete dynamical system determined by the stroboscopic map.The obtained results will modify and improve the ones in some existing publications and give the estimate for the ultimately low and upper boundedness of the systems.Some examples and their simulations are given to illustrate the effectiveness of our results.
Note that the variables  1 ,  2 do not appear in the first, third, fifth, and sixth equations in (3).We will simplify the model and need to restrict our attention to the following system: Suppose that  1 () = 0,  2 () = 0; the above subsystem becomes model (1).Integrating and solving the first two equations of (1) between impulsive moments, we have Combining with the impulsive diffusion, we obtain the following stroboscopic map: ) . ( Here  , =   ( + ),   = (  /  )(1 −  −   ) > 0, and We define a map  :  2 + →  2 + such that The set of all iterations of the map is equivalent to the set of all density sequences generated by system (8).
By using the theory of monotone dynamical systems in [12], Hui and Chen [5] gave the following lemma.Lemma 1.There exists a unique positive fixed point  = ( 1 ,  2 ) of map  in (9),   () →  as  → ∞.Furthermore, all nontrivial trajectories of system (1) approach the positive periodic solution ( x1 (), x2 ()) with period , where To obtain our results, we also need the following lemma.

Extinction of the Predator-Eradicated Periodic Solution
It is clear that system (6) has a predator-eradicated periodic solution ( x1 (), 0, x2 (), 0), where the positive periodic functions x1 (), x2 () with period  are given in (10).In the following, we will show that it is globally attractive.
Remark 4.  *  is the maximum of the periodical function x () defined in (10), which is dependent on the sign of the constant 1/  −   /  ,  = 1, 2. However, the authors neglected the details when they derived the global attractivity of predator-eradicated periodic solution in [7,8,10,11].

Permanence
Firstly, we can estimate the ultimately upper boundedness as follows.(6), where  *  is given in (12) and

Lemma 5. If
Proof.It follows from system (6) that By Lemma 1 and the comparison theorem of impulsive differential equation, we have lim sup  → ∞   () ≤ x () ≤  *  .From the second and fourth equations in system (6), It follows from Lemma 2 and (20) that lim sup (24) From the arbitrariness of  0 , we obtain the conclusion.The proof is complete.
Remark 6.In [7][8][9][10][11], the authors gave the estimate of the ultimately upper boundedness of solutions to derive the persistence of the system by constructing the -functions.
Nevertheless, there are errors on the estimate since the constructed -functions do not satisfy ( + ) ≤ () when  = ,  = 1, 2, . ... In Lemma 5, we make a modification for the estimate.
Theorem 7. Let  *  ,  *  be given in Lemma 5 and where Δ  = |max{0, sgn(q  − (  −    (0) *  )/  )}|; (q 1 , q2 ) is a unique positive fixed point of map  defined in (9) with then system ( 6) is uniformly persistent in the following meaning: Proof.We can see that (26) implies (20).Therefore, lim sup  → ∞   () ≤  *  , lim sup  → ∞   () ≤  *  ,  = 1, 2. Furthermore, we can choose  > 0 sufficiently small such that Note that   (  ()) ≤    (0)  () since   ∈ .From system (6) and Theorem 7, there is an  3 such that By Lemma 1 and the comparison theorem, there exists an  4 >  3 such that where and q can be confirmed homoplastically to   .Letting  1 =  * 1 − ,  2 =  * 2 − , from the second and fourth equations in system (6), we have It follows from (28), Lemma 2, and comparison differential system that lim inf So there is an  5 sufficiently large such that  1 () ≥  * 1 > 0,  2 () ≥  * 2 > 0 for all  >  5 .Hence, by Theorem 7 and the above discussion, we get that system (6) is permanent.The proof is complete.Remark 8. Theorem 7 derives the persistence of the system and estimates how many members the population has at large time in every patch.This is important to manage effectively the species, but the estimate may be conservative and further results need to be developed in this direction.

Examples and Their Simulations
In this section, we will give some examples and their simulations to illustrate the effectiveness of the obtained results.
The following example and its simulation show that the conclusions in Theorems 3 and 7 may not hold when their conditions are not satisfied and point out some errors in existing publications.
1, 2,   ,   ,   are prey population density and predator populations density of the immature and the mature at time , respectively;   represents the growth rate from the immature predator to the mature one;   is the death rate of the immature predator;   is the death rate of mature predator, which is of a logistic nature;   is the harvesting rate of the mature population;   is the rate of conversion of nutrients into the reproduction rate of the mature predator;   represents a constant time to maturity;  > 0 is the period of impulsive diffusion; and   ∈ (0, 1) is dispersal rate; the other parameters have the same biological meaning in system (1).In addition,   ( ()) represents the functional response including Holling-type and Ivlev-type ones, which satisfies