Research Article On a Periodic Time-Dependent Model of Population Dynamics with Stage Structure and Impulsive Effects

We consider a periodic time-dependent predator-prey system with stage structure and impulsive harvesting, in which the prey has a life history that takes them through two stages, immature and mature. A set of sufficient and necessary conditions which guarantee the permanence of the system are obtained. Finally, we give a brief discussion of our results.


Introduction
In the natural world, there are many species whose individual members have a life history that takes them through two stages, immature and mature. In particular, we have mammalian populations and some amphibious animals in mind, which exhibit these two stages. From the view point of mathematics, the description of the stage structure of the population in the life history is also an interesting problem in population dynamics. The permanence and extinction of species are significant concepts for those stage-structured population dynamical systems. Recently, stage structure models have been studied by many authors 1-3 . This is not only because they are much more simple than the models governed by partial differential equations but also because they can exhibit phenomena similar to those of partial differential models 4 , and many important physiological parameters can be incorporated. Much research has been devoted to the models concerning single-species population growth with the stage structure of immature and mature 5, 6 . Two species models with stage structure were investigated by Wang and Chen 1997, Magnusson 1999, Xiao and Chen 2003 , as well as Cui and Song 2004 . Also, Zhang, Chen, and Neumann 2000 proposed the following autonomous stage 2 Discrete Dynamics in Nature and Society structure predator-prey system: where α, β, β 1 , η, η 1 , r, r 1 , r 2 , and k are all positive constants, k is a digesting constant.
On the other hand, since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. So more realistic and interesting models should take into account the seasonality of the changing environment 7, 8 . This motivated Cui and Y. Takeuchi 2006 to consider the following periodically nonautonomous predator-prey model with stage structure for prey: q t are all positive; g t , φ t, x 1 are nonnegative; x 1 and x 2 denote the density of immature and mature population prey , respectively; and y is the density of the predator that only prey on x 1 immature prey . They provided a set of sufficient and necessary conditions to guarantee the permanence of the above system. Systems with impulsive effects describing evolution processes are characterized by the fact that at certain moments of time, they experience a change of state abruptly. Processes of such type are studied in almost every domain of applied science. Impulsive equations 9, 10 have been recently used in population dynamics in relation to impulsive vaccination 11, 12 , population ecology 13, 14 , the chemotherapeutic treatment of disease 15 , birth pulses 16 , as well as the theory of the chemostat 17 .
Let us assume that the predator population is affected by harvesting e.g., fishing or hunting . Further, as we all know that the harvesting does not occur continuously, that is, the harvesting occurs in regular pulses, then let us assume that at some fixed moments, the predator population in system 1.2 is subject to a perturbation which incorporates the proportional decrease. After a perturbation at step τ k > 0 k ∈ N , the size of the population y τ k becomes where y τ k is the size of the predator population at step τ k before the impulsive perturbation, and 0 < u k < 1 represents the rate at which the predator is harvested.
In this article, we extend the model 1.2 to the case when the predator population is omnivorous and affected by impulsive effects, which is governed by the following system: K. Liu and L. Chen Δy t −u k y t , t τ k , k ∈ N, with the initial value conditions in which x 1 and x 2 are the densities of immature and mature prey, respectively; and y is the density of the predator that can prey on x 1 and x 2 . When its favorite food is severely scarce, population y can eat other resources: Δy t y t − y t . Also, there exists a positive integer q such that u k q u k , τ k q τ k ω, Here, x 2 i φ i t, x i , the number of the prey x i consumed per predator in unit time, is called the predator functional response. We assume that there exists a positive constant L such that The last condition in 1.7 implies that as the prey population increases, the consumption rate of prey consumed per predator increases. The birth rate of the immature prey population is proportional to the existing mature prey population with a proportionality function a t . For the immature prey population, the death rate is proportional to the existing immature prey population with a proportionality function b t . The variable parameter d t represents that the immature prey population is density restriction. The transition rate from immature individuals to the mature individuals is assumed to be proportional to the existing immature population with proportionality coefficient c t . The death rate of the mature population is of a logistic nature with proportionality coefficient f t . Also, p i t and h i t i 1, 2 are the coefficients that relate to conversion rates of the immature and, respectively, mature prey biomass into predator biomass. The coefficients in 1.4 are all continuous ω-periodic for t ≥ 0. In fact, a t , b t , c t , d t , f t , p i t , h i t , and q t are all strictly positive, and φ i t, x i is nonnegative i 1, 2 . The organization of this paper is as follows. In Section 2, we provide some preliminary results which will be useful. In Section 3, we investigate the permanence and extinction of system 1.4 by using analysis technique. In the last section, we give a biological example and a brief discussion of our result.

Preliminary results
Before stating and proving our main results, we give the following definitions, notations, and lemmas which will be useful in the following section.
. . , n} and α t be a continuous ω-periodic function defined on 0, ∞ , then we set Discrete Dynamics in Nature and Society After a simple computations, we have in which w is a positive constant. Obviously, the right-hand side of the above inequality is bounded above for all x 1 t , x 2 t , y t ∈ R 3 . Hence, where λ is a positive constant. When t τ k , we get According to Lemma 2.2 in 9, page 23 , we derive that which implies that system 1.4 is dissipative. This completes the proof.
Next, we consider the following two subsystems of system 1.4 : Lemma 2.2 see 18 . The system 2.7 has a positive ω-periodic solution x * 1 t , x * 2 t which is globally asymptotically stable with respect to R 2 .

Lemma 2.3. If the following conditions
hold, then system 2.8 has a unique ω-periodic solution:

2.10
and for every solution y t of system 2.8 , Proof. Let y t 1/z t and obtain the linear nonhomogeneous impulsive equatioṅ

2.12
Let W t, s s≤τ k <t 1/ 1 − u k exp − t s g ξ dξ be the Cauchy matrix for the relevant homogeneous equation, then the solution of 2.12 has the form The solution z t will be ω-periodic if z ω z 0 , or if In view of conditions 2.9 , 2.14 has a unique solution Then, 2.13 has a unique ω-periodic solution Hence, 2.8 has a unique ω-periodic solution Discrete Dynamics in Nature and Society From 2.12 , we derive that Since conditions 2.9 hold, we note that 0 < κ < 1. Then, from Lemma 2.1, the solution of 2.8 is governed by

2.20
in which λ and w are defined in Lemma 2.1. This completes the proof.

3.1
in which x * 1 t , x * 2 t is the positive ω-periodic solution of system 2.7 .
We need the following lemmas to prove Theorem 3.1.
for t ≥ T 0 . According to Lemma 2.2, the following auxiliary equations: has a globally asymptotically stable ω-periodic solution u * 1 t , u * 2 t . Let u 1 t , u 2 t be the solution of 3.6 with u 1 0 , u 2 0 x 1 0 , x 2 0 . By the vector comparison theorem 19 , we obtain According to the global asymptotic stability of u * 1 t , u * 2 t , for any positive constant ε ≤ min t∈ 0,ω {u * i t /3, i 1, 2} , there exists a T 1 > T 0 such that for all t ≥ T 1 , Hence, for all t ≥ T 1 , we derive that Let This completes the proof. Proof. In view of 3.1 , we can choose a positive constant ε 0 such that Consider the following system with a positive parameter μ:

3.16
By Lemma 2.2, system 3.16 has a positive ω-periodic solution x * 1μ t , x * 2μ t , which is globally asymptotically stable. Let x 1μ t , x 2μ t be the solution of 3.16 with initial condition x * 2 t is the positive periodic solution of 2.7 . Hence, for the above ε 0 , there exists T 2 > T 1 such that for t ≥ T 2 , i 1, 2. According to the continuity of the solution in the parameter μ, we have x iμ t → x * i t i 1, 2 uniformly in T 2 , T 2 ω as μ → 0. Hence, for ε 0 > 0, there exists μ 0 μ 0 ε 0 0 < μ 0 < ε 0 such that t ∈ T 2 , T 2 ω , i 1, 2. Thus, from 3.17 and 3.18 , we get t ∈ T 2 , T 2 ω , i 1, 2. Since x * iμ t and x * i t are all ω-periodic, we have Choose a constant μ 1 0 < μ 1 < μ 0 , μ 1 < ε 0 , from 3.20 , we derive Suppose that 3.13 is not true, then for the above ε 0 , there exists a ν ∈ R 3 such that lim sup t→∞ y t < μ 1 , 3.22 K. Liu and L. Chen 9 where x 1 t , x 2 t , y t is the solution of 1.4 with the initial condition x 10 , x 20 , y 0 ν. So there exists a constant T 3 > T 2 such that 3.23 then we derive thatẋ for t ≥ T 3 . Let x 1μ 1 , x 2μ 1 be the solution of 3.16 with μ μ 1 and x 1μ 1 T 3 , x 2μ 1 T 3 x 1 T 3 , x 2 T 3 , then by the vector comparison theorem, we obtain By the global asymptotic stability of x * 1μ 1 t , x * 2μ 1 t , for the given ε 0 > 0, there exists T 4 > T 3 such that and hence, by 3.21 , we get Since 0 < y t < μ 1 < ε 0 together with 1.4 and 1.7 , we havė

3.28
Hence, it follows from Lemma 2.2 in 9, page 23 that

3.30
By 3.14 , we know that y t → ∞ as t → ∞, which leads to a contradiction. This completes the proof.

3.35
Integrating the above inequality from s k q to t k q , for any q ≥ K Obviously, it follows from 3.33 that t k q s k q g t M y q t dt > ln k 1 for q ≥ K k 1 .

3.37
Hence, in view of the periodicity of g t and q t , we get By 3.14 , 3.33 , and 3.38 , there are positive constants T and N 0 such that for k ≥ N 0 , for q ≥ K k 1 , and for all κ > T. However, 3.39 implies that Then, we geṫ

3.43
Let x 1ε 0 , x 2ε 0 be the solution of 3.16 with μ ε 0 and x 1ε 0 T 3 , x 2ε 0 T 3 x 1 T 3 , x 2 T 3 , then by the vector comparison theorem, we obtain 3.44 From lim q→∞ s k q ∞ and Lemmas 3.2 and 3.3, we obtain that for any k, there is a K For μ ε 0 , 3.16 has a globally asymptotically stable positive ω-periodic solution x * 1ε 0 t , x * 2ε 0 t . By the global asymptotic stability of x * 1ε 0 t , x * 2ε 0 t , for the given ε 0 > 0, there exists T 5 > T 4 , and T 5 is dependent of any k and q such that and hence, by 3.21 , we get By 3.44 , there is an N 1 ≥ N 0 such that t k q − s k q ≥ 2T for all k ≥ N 1 and q ≥ K k 2 , where T ≥ T 5 . Hence, from 3.44 and 3.47 , we obtain

3.51
This leads to a contradiction. This completes the proof.
According to Lemmas 3.2-3.5, we can directly prove the sufficient part of the theorem. Next, we are ready to show the necessity of this theorem.
Consider the case of it is then easy to derive that the predator population y t must be extinct because of sterilization and impulsive harvesting. Suppose that

3.53
We will show that lim t→∞ y t 0.

3.54
In fact, from 1.7 and 3.53 , we know that for any given 0 < < 1, there exists 1 > 0 such that 3.55 K. Liu and L. Chen

3.56
for the given 1 > 0, it is easy to show that there exists T 1 > 0 such that 3.58 We now show that there must exist T 2 > T 1 such that y T 2 < . Otherwise, by the last two equations in system 1.4 , we have This implies that ≤ 0, which is a contradiction.
Note that M is bounded. We then show that Otherwise, there exists T 3 > T 2 such that T 3 ∈ T 2 P 1 ω, T 2 P 1 1 ω , in which P 1 ∈ Z {0, 1, 2, . . .}, and y T 3 > exp M ω . By 3.58 , we have exp M ε ω < y T 3 y T 2 which leads to a contradiction. This implies that 3.60 holds. In view of the arbitrariness of , we get that y t → 0 as t → ∞. This completes the proof.
Remark 3.6. From the proof of Theorem 3.1 above, we note that the predator population y in system 1.4 will be extinct provided that or excessive harvesting, that is, 3.63

Discussion
Our result could provide a useful insight into the conservation of beneficial animals, especially rare animals. As an example, we depict the case of Oreolalax omeimontis tadpole, Oreolalax omeimontis, and red-eared slider T. scripta elegans . Oreolalax omeimontis is a rare species of frog found near Mt. Omei in Sichuan China . The red-eared slider is a native of the Mississippi Valley area of the United States 8, 20 . Since the 1970s, large numbers of red-eared sliders have been produced on turtle farms in the USA for the international pet trade. Red-eared Turtles are traded as pet animals and have been introduced to many countries. They are omnivorous and will eat insects, tadpoles, crayfish, shrimp, worms, snails, amphibians and small fish, as well as aquatic plants, and they hardly may be controlled by a natural enemy. In our model, the variables x 1 t and x 2 t represent the density of Oreolalax omeimontis tadpole and Oreolalax omeimontis at time t, respectively. The variable y t describes the density of the red-eared slider at time t. It is well known that Oreolalax omeimontis is beneficial to humans because they eat so many insect pests. Ironically, although red-eared sliders have been widely introduced throughout the world, its detrimental effects have been reported by many researchers 21, 22 . The red-eared slider is one of main enemies of Oreolalax omeimontis. To protect these beneficial and rare toads, we must control the amount of red-eared sliders in the habitat of Oreolalax omeimontis. From a control point of view, our aim is to keep red-eared sliders at an acceptably low level with a minimum use of artificial control measures, not to eradicate all red-eared sliders. Hence, in the above example, the problem of nonextinction of populations becomes a description of reasonable harvesting rates u k k 1, 2 . . . q . According to our main result, system 1.4 is permanent if and only if the growth of the predator y by foraging prey populations x 1 and x 2 plus its intrinsic rate of increase is positive on average during the period ω, and harvesting rates u k k 1, 2, . . . q of red-eared sliders during the period ω are small enough to satisfy that

4.1
These seem to be reasonable from a biological point of view; but it should be noted that condition 3.1 allows the predator to be g 2 i 1 h i φ i t, x * i x * 2 i < 0 for some time intervals among 0 ≤ t ≤ ω. That is, under reasonable harvesting rates, the predator can survive together with prey populations even if the growth rate of the former is negative for some seasons during a period. We hope that our result can be used to help protect beneficial animals found in their habitats.