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Dynamics of a delayed stage-structured model with impulsive harvesting and diffusion
Ecological Complexity 19 (2014) 111–123
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Dynamics of a delayed stage-structured model with impulsive harvesting and diffusion Liu Yang *, Shouming Zhong School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
A R T I C L E I N F O
A B S T R A C T
Article history: Received 3 January 2014 Received in revised form 14 May 2014 Accepted 27 May 2014 Available online
Based on the predator–prey system with delayed stage-structured for preys and impulsive harvesting and impulsive diffusion for predator, an impulsive delayed differential equation to model the process of periodic harvesting and impulsive diffusion at different fixed moments is proposed and investigated. In this model, patches are created by two different prey populations and each prey population is confined to a particular patch while the predator population can impulsively diffuse between two patches. By using comparison theorem of impulsive delayed differential equation and some analysis techniques, sufficient conditions ensuring the existence of preys-extinction periodic solution and the permanence of the system are established. Our analysis reveals that low birth rates of immature preys, high death rates of immature and mature preys, long maturation time of immature preys to mature preys and large preys’ captured rates are the sufficient condition for the preys-extinction. On the contrary, if we largen the birth rates of immature preys, or decrease the death rates of immature and mature preys, or shorten the maturation time of immature preys or decrease the preys’ captured rates, then the system can become permanent under proper predational strategies. These also show that it is feasible to keep the sustainable development of the ecosystem by controlling the critical ecological parameters. Numerical simulations with hypothetical set of parameter values are carried out to consolidate the analytic findings. ß 2014 Elsevier B.V. All rights reserved.
Keywords: Impulsive diffusion Impulsive harvesting Delay Stage-structured Permanence Extinction
1. Introduction Population dispersal in patchy environment is one of the most prevalent subjects in ecology and mathematical ecology. Within each patch, individuals of each species are supposed to be identical and can migrate to other patches. In most previous papers, population dynamics with the effects of heterogeneity modeled by the diffusion process is focused on the dynamical system modeled by ordinary differential equations and delay differential equations (Cui et al., 2004; Xu and Ma, 2008; Cui and Chen, 2001; Zhou et al., 2008; Ding and Han, 2008; Chen et al., 2003; Song and Chen, 1998; Xu et al., 2004). But in reality, dispersal behavior is very intricate and is always perturbed by environmental change and human activities, etc. In fact, it is often the case that diffusion occurs during short-time slots within seasons or within the lifetimes of animals. In order to be in much better agreement with the real ecological process, this short-time scale dispersal is more suitable assumed to be in the form of regular pulses. Taking birds as an example, when winter comes, they will diffuse between patches in search for a better environment,
* Corresponding author. Tel.: +86 15928740515. E-mail addresses: [email protected] (L. Yang), [email protected] (S. Zhong). http://dx.doi.org/10.1016/j.ecocom.2014.05.012 1476-945X/ß 2014 Elsevier B.V. All rights reserved.
but they do not migrate in other season. Thus impulsive diffusion provides a more natural description for this behavior. With the developments and applications of impulsive differential equations, theories of impulsive differential equations have been introduced into population dynamics, and some important studies about impulsive diffusion have been done (Shao, 2010; Jiao et al., 2011, 2011, 2011; Jiao and Cai, 2009a; Hui and Chen, 2005; Wan et al., 2012; Wang et al., 2007; Dong et al., 2007; Zhang et al., 2013; Zhao et al., 2011). In particularly, a single population was considered (Hui and Chen, 2005; Wan et al., 2012; Wang et al., 2007; Dong et al., 2007; Zhang et al., 2013; Zhao et al., 2011). For example, Zhao et al. (2011) studied the following single species model with impulsive diffusion and pulsed harvesting at the different fixed time 9 8 dx1 ðtÞ > 2 > > > = ¼ r x ðtÞ a x ðtÞ 1 1 11 1 > > dt > > > > > dx2 ðtÞ > > > ; > ¼ r 2 x2 ðtÞ > > dt > ) < Dx1 ðtÞ ¼ Ex1 ðtÞ > > > > Dx2 ðtÞ ¼ 0 > > ) > > > Dx1 ðtÞ ¼ Dðx2 ðtÞ x1 ðtÞÞ > > > > > Dx2 ðtÞ ¼ Dðx1 ðtÞ x2 ðtÞÞ :
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
t ¼ ðn þ l 1ÞT t ¼ nT
(1.1)
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and impulsive diffusion at different fixed moments. To formulate the In this model, suppose that the system is composed of two patches connected by diffusion. xi(t) presents the biomass of the population mathematical model, we make the following assumptions: in patch i(i = 1, 2). r1 denotes the intrinsic growth rate in the first patch and r2 is the death rate in the second patch. r1/a11 is the A1 The patches are created by two prey populations, and each prey environment carrying capacity. 0 < E < 1 is the harvesting effort of population is confined to a particular patch while the predator the population in the first patch at t = (n + l 1)T(0 < l < 1). population can diffuse between two patches. Predator has two 0 < D < 1 is the diffusion coefficient. If xi > xj(i 6¼ j, i, j = 1, 2), the different impulsive time. In the first impulsive time, we only population in patch i diffuses into patch j at a rate D which is harvest the predator in the first patch. And in the second proportional to xi xj. T is the impulsive diffusion period. impulsive time, the predator population will migrate from one patch to other patch. Dxi(nT+) = xi(nT+) xi(nT)(i = 1, 2). xi(nT+)(i = 1, 2) represents the density of the subpopulation in patch i after the nth diffusion pulse A2 The prey populations: each prey population have two life stage, at t = nT, while xi(nT)(i = 1, 2) represents the density of the namely immature and mature stages. The birth rates into the subpopulation in patch i immediately before the nth diffusion immature population are proportional to the existing mature pulse at t = nT. By using the stroboscopic map, they obtained the population with a proportionality a1 in patch 1 and a2 in patch existence and globally asymptotical stability of both the trivial 2, respectively; and the death rates of the immature population solution and the positive periodic solution, and the complete are proportional to the existing immature population with a expression for the periodic solution. proportionality v1 in patch 1 and g1 in patch 2, respectively; the On the other hand, from birth to death, many species usually go death rates of the mature population are proportional to the through two life stages, immature and mature. So it is practical to existing mature population with a proportionality v2 in patch 1 introduce stage-structured into prey–predator models (Aiello and and g2 in patch 2, respectively; the intra-specific competition Freedman, 1990; Song and Chen, 2002; Meng et al., 2008; Song and rates of the mature prey populations are proportional to square Xiang, 2006; Chen and You, 2008; Shao and Dai, 2010; Huang et al., of the population with a proportionality d1 in patch 1 and d2 in 2012; Song et al., 2009; Liu et al., 2008; Wang et al., 2009). In patch 2, respectively. additional, as literatures (Li and Kuang, 2001) pointed out that the A3 The predator population: in the absence of the prey populadelay differential equation shows much more complicated dynamtions, the predator population in the first habitat grows ics than ordinary differential equation since time delay could cause a according to the logistic curve with the intrinsic birth rate b1 stable equilibrium to become unstable and cause the population to and density dependence rate a1. Whereas the predator fluctuate. Therefore it is reasonable to introduce delayed stage subpopulation in the second patch will die with the death structure into prey–predator models (Meng et al., 2008; Song and rate b2. The predator populations only feed on the mature preys Xiang, 2006; Chen and You, 2008; Shao and Dai, 2010; Huang et al., following Holling type-I functional response with different 2012; Song et al., 2009; Liu et al., 2008; Wang et al., 2009). Newly, capturing rates c1 in patch 1 and c2 in patch 2, respectively. The population dynamical system involving delayed stage structure and conversion factor for predator population due to consumption impulsive diffusion have been discussed by some authors, see Jiao of prey is ki(i = 1, 2) in the ith (i = 1, 2) patch. (2010), Shao and Li (2013), Jiao et al. (2009b), Dhar and Jatav (2013), Jiao et al. (2010), and references cited therein. Furthermore, in real nature, some lower-order preys can In the natural world, these assumptions are reasonable for establish their own territory and does not interact with other preys, many species whose immature prey population conceal in the cave whereas the predator can diffuse between the territories at a fixed and are raised by their parents; the rate of predator attacking at moment. Therefore in this paper, we consider a three-species (twoimmature prey can be ignored. Considering the above basic prey and a predator) ecological model with impulsive harvesting assumptions, we can derive the following differential equations: 9 8 dx1 ðtÞ > > > > ¼ a1 x2 ðtÞ a1 ev1 t1 x2 ðt t 1 Þ v1 x1 ðtÞ > > > > dt > > > > > > > > dx2 ðtÞ > > v1 t 1 2 > > > ¼ a1 e x2 ðt t 1 Þ v2 x2 ðtÞ d1 x2 ðtÞ c1 x2 ðtÞz1 ðtÞ > > > > > dt > > > > > > > > dy ðtÞ > > 1 g 1 t 2 > > = ¼ a2 y2 ðtÞ a2 e y2 ðt t 2 Þ g 1 y1 ðtÞ > > > dt > t 6¼ nT; t 6¼ ðn þ l 1ÞT > > > > dy ðtÞ > 2 > > ¼ a2 eg 1 t2 y2 ðt t 2 Þ g 2 y2 ðtÞ d2 y22 ðtÞ c2 y2 ðtÞz2 ðtÞ > > > > > > > > dt > > > > > > > dz1 ðtÞ > > > > ¼ z ðtÞ½b a z ðtÞ þ k c x ðtÞz ðtÞ > 1 1 1 1 1 1 2 1 > > > dt > > > > > > > > dz ðtÞ > > 2 > ; > ¼ b2 z2 ðtÞ þ k2 c2 y2 ðtÞz2 ðtÞ > > dt > > 9 > < Dx1 ðtÞ ¼ 0 > > > (1.2) > > Dx2 ðtÞ ¼ 0 > > > > > > > = > > Dy1 ðtÞ ¼ 0 > > t ¼ ðn þ l 1ÞT > > > > Dy2 ðtÞ ¼ 0 > > > > > > > > > Dz1 ðtÞ ¼ pz1 ðtÞ > > > > > ; > > > D z ðtÞ ¼ 0 > 2 > > 9 > > > Dx1 ðtÞ ¼ 0 > > > > > > > > > > Dx2 ðtÞ ¼ 0 > > > > > > = > > Dy1 ðtÞ ¼ 0 > > t ¼ nT > > > > Dy2 ðtÞ ¼ 0 > > > > > > > > Dz ðtÞ ¼ Dðz ðtÞ z ðtÞÞ > > > 1 2 1 > > > ; > > : Dz2 ðtÞ ¼ Dðz1 ðtÞ z2 ðtÞÞ
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
where x1(t), x2(t) represent the densities of immature and mature prey species, respectively, in the first patch at time t; y1(t), y2(t) denote the densities of immature and mature prey species, respectively, in the second patch at time t; zi(t) is the density of predator species in the ith (i = 1, 2) patch at time t. Dxi(t) = xi(t+) xi(t), Dyi(t) = yi(t+) yi(t), Dzi(t) = zi(t+) zi(t)(i = 1, 2). 0 < p < 1 is the harvesting effort of predator in the first patch at t = (n + l 1)T (0 < l < 1 is a positive constant, n 2 N); 0 < D < 1 is the dispersal rate of predator species at t = nT. The pulse diffusion vt occurs every period T. The terms a1e 1 1x2(t t1) and a2eg1t2y2(t t2) represent the number of immature preys in patch 1 and patch 2, respectively, that were born at time t ti(i = 1, 2) which still survive at time t and are transferred from the immature stage to the mature stage at time t. The initial conditions for system (1.2) take the form of xi ðuÞ ¼ fi ðuÞ; yi ðu Þ ¼ ’i ðuÞ; zi ðuÞ ¼ ci ðuÞ fi ð0Þ > 0; ’i ð0Þ > 0; ci ð0Þ > 0; i ¼ 1; 2; u 2 ½t ; 0; where t ¼ maxft 1 ; t 2 g; ðf1 ðuÞ; f2 ðuÞ; ’1 ðuÞ; ’2 ðuÞ; c1 ðuÞ; c2 ðuÞÞ 2 Cð½t ; 0; R6þ0 Þ , C þ , the Banach space of continuous functions mapping the interval [ t, 0] into R6þ0 which is defined as R6þ0 ¼ fðx1 ; x2 ; x3 ; x4 ; x5 ; x6 Þ : xi 0; i ¼ 1; 2; 3; 4; 5; 6g: We can simplify model (1.2) and restrict our attention to the following subsystem:
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
9 dx2 ðtÞ > ¼ a1 ev1 t 1 x2 ðt t 1 Þ v2 x2 ðtÞ d1 x22 ðtÞ c1 x2 ðtÞz1 ðtÞ > > > dt > > > > dy2 ðtÞ > g 1 t 2 2 = ¼ a2 e y2 ðt t 2 Þ g 2 y2 ðtÞ d2 y2 ðtÞ c2 y2 ðtÞz2 ðtÞ > dt > dz1 ðtÞ > > ¼ z1 ðtÞ½b1 a1 z1 ðtÞ þ k1 c1 x2 ðtÞz1 ðtÞ > > > dt > > > > dz2 ðtÞ ; ¼ b2 z2 ðtÞ þ k2 c2 y2 ðtÞz2 ðtÞ dt 9 Dx2 ðtÞ ¼ 0 > > > = Dy2 ðtÞ ¼ 0 > Dz1 ðtÞ ¼ pz1 ðtÞ > > ; Dz2 ðtÞ ¼ 0 9 Dx2 ðtÞ ¼ 0 > > > = Dy2 ðtÞ ¼ 0 > Dz1 ðtÞ ¼ Dðz2 ðtÞ z1 ðtÞÞ > > ; Dz2 ðtÞ ¼ Dðz1 ðtÞ z2 ðtÞÞ
with the initial conditions
ðf2 ðuÞ; ’2 ðu Þ; c1 ðuÞ; c2 ðuÞÞ 2 Cð½t ; 0; R4þ0 Þ; f2 ð0Þ > 0; ’2 ð0Þ > 0; ci ð0Þ > 0; i ¼ 1; 2: The remainder of this paper is arranged as follows. In Section 2, some preliminaries are provided. In Section 3, the global attractivity of preys-extinction periodic solution and the permanence of system (1.3) are investigated. In Section 4, numerical simulations are given to show the complicated dynamics of system (1.3). Finally, a brief discussion is given to conclude this work in Section 5. 2. The preliminaries Let R+ = [0, 1), R6þ ¼ fX 2 R6 jX 0g. Denote f = (f1, f2, f3, f4, f5, f6)T the map defined by the right hand of the first six equations of system (1.2). Let V : Rþ R6þ ! Rþ , then V is said to belong to class V0 if
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(1) V is continuous in ððn 1ÞT; ðn þ l 1ÞT R6þ and ððn þ l 1Þ T; nT R6þ , and for each X 2 R6þ , n 2 N, limðt;yÞ ! ððn1ÞT þ ;XÞ Vðt; yÞ ¼ Vððn 1ÞT þ ; XÞ and limðt;yÞ ! ððnþl1ÞT þ ;XÞ Vðt; yÞ ¼ Vððn þ l 1Þ T þ ; XÞ exist. (2) V is locally Lipschitzian in X.
Definition 2.1. Let V 2 V0, then for ðt; xÞ 2 ððn 1ÞT; ðn þ l 1ÞT R6þ [ ððn þ l 1ÞT; nT R6þ , the upper right derivative of V(t, X) with respect to the impulsive differential system (1.2) is defined as 1 Dþ Vðt; XÞ ¼ limþ sup ½Vðt þ h; X þ hf ðt; xÞÞ Vðt; XÞ: h h!0 The solution of system (1.2) is a piecewise continuous function X : Rþ ! R6þ , X(t) is continuous on ((n 1)T, (n + l 1)T] and ((n + l 1)T, nT], n 2 N and Xððn 1ÞT þ Þ ¼ limt ! ðn1ÞT þ XðtÞ and Xððn þ l 1ÞT þ Þ ¼ limt ! ðnþl1ÞT þ XðtÞ exist. The smoothness properties of f guarantee the global existence and uniqueness of solution of system (1.2), for the details see book (Lakshmikantham et al., 1989; Bainov and Simeonnov, 1989). Definition 2.2. System (1.2) is said to be permanent if there exist positive constants m and M, such that each positive solution (x1(t), x2(t), y1(t), y2(t), z1(t), z2(t)) of system (1.2) with initial conditions satisfies m xi(t) M, m yi(t) M, m zi(t) M(i = 1, 2) as t! 1.
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
(1.3) t ¼ ðn þ l 1ÞT
t ¼ nT
The following lemma is obvious.
Lemma 2.1. Let X(t) is a solution of system (1.2) with X(0+) 0, then X(t) 0 for all t 0. And further X(t) > 0, t > 0 if X(0+) > 0. We will use an important comparison theorem on impulsive differential equation. Lemma 2.2. (Bainov and Simeonov, 1993) Suppose V 2 V0. Assume that 8 þ > < D Vðt; XÞ gðt; Vðt; XÞÞ 1 Vðt; Xðt þ ÞÞ cn ðVðt; XÞÞ > 2 : þ Vðt; Xðt ÞÞ cn ðVðt; XÞÞ
t 6¼ nT; t 6¼ ðn þ l 1ÞT t ¼ ðn þ l 1ÞT t ¼ nT
where g : R+ R+ ! R is continuous in ððn 1ÞT; ðn þ l 1ÞT R6þ and ððn þ l 1ÞT; nT R6þ and for u 2 R+, n 2 N, limðt;vÞ ! ððn1ÞT þ ;uÞ gðt; vÞ ¼ gððn 1ÞT þ ; uÞ and limðt;vÞ ! ððnþl1ÞT þ ;uÞ gðt; vÞ ¼ gððn þ l
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1ÞT þ ; uÞ exist, cn ði ¼ 1; 2Þ: R+ ! R+ is non-decreasing. Let r(t) be maximal solution of the scalar impulsive differential equation 8 duðtÞ > > ¼ gðt; uðtÞÞ > > < dt 1 þ uðt Þ ¼ cn ðuðtÞÞ > 2 > þ > uðt Þ ¼ c > n ðuðtÞÞ : þ uð0 Þ ¼ u0
t 6¼ nT;
Remark 2.1. According to the expression in Zhao et al. (2011), we can derive that z1 and z2 are as follows z1 ¼
t 6¼ ðn þ l 1ÞT
b1 ð1 pÞð1 D ð1 2DÞeb2 T Þ b1 eb1 T þ b1 ð1 DÞeðb1 þb2 ÞT a1 ð1 p þ peb1 ð1lÞT eb1 T Þð1 ð1 DÞeb2 T Þ
t ¼ ðn þ l 1ÞT t ¼ nT z2 ¼ Dz1 +
existing on [0, 1). Then V(0 , X0) u0, implies that V(t, X(t)) r(t), t 0, where X(t) is any solution of system (1.2). Lemma 2.3. (Kuang, 1987) Consider the following delay differential equation
˙ ¼ a1 xðt t Þ a2 xðtÞ; xðtÞ
Here we claim that the expression of z2 (that is the expression of x2 in Zhao et al., 2011) is not right, therefore we revise the expression of z2 as z2 ¼
Dz1 : ð1 DÞ ð1 2DÞeb2 T
Remark 2.2. The above Lemmas show that
and assume that a1, a2, t > 0, a1 < a2 and x(t) > 0 for t t 0. Then p , 1
lim xðtÞ ¼ 0:
t!1
Next, we introduce two important lemmas, which can be seen in Zhao et al. (2011). For subsystem of (1.2), i.e., 9 8 dz1 ðtÞ > > > > = ¼ z ðtÞ½b a z ðtÞ 1 1 1 1 > > dt > > > > > dz ðtÞ > 2 > > ; > ¼ b2 z2 ðtÞ > > dt > ) < Dz1 ðtÞ ¼ pz1 ðtÞ > > > Dz2 ðtÞ ¼ 0 > > > ) > > > Dz1 ðtÞ ¼ Dðz2 ðtÞ z1 ðtÞÞ > > > > > : Dz2 ðtÞ ¼ Dðz1 ðtÞ z2 ðtÞÞ
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
t ¼ ðn þ l 1ÞT t ¼ nT
b2 T
1 ð1 DÞe eb1 T ðð1 DÞ ð1 2DÞeb2 T Þ
, the trivial equilibrium point (0,0) of system (2.1) is globally asymptotically stable. Lemma 2.5. (Zhao et al., 2011) If
0< p<1
is the critical harvesting effort. If p > p*, which denotes the subpopulation in patch 1 is overexploited, then the subpopulation in two patches will become extinction. In reality, if the subpopulation in patch 1 is heavily exploited, it is natural that the subpopulation in patch 1 will go to extinction. Since the subpopulation in patch 2 come from the diffusion of the subpopulation in patch 1, and then the subpopulation in patch 2 will also become extinction.
(2.1)
Lemma 2.4. (Zhao et al., 2011) If p>1
1 ð1 DÞeb2 T DÞ ð1 2DÞeb2 T Þ
eb1 T ðð1
1 ð1 DÞeb2 T eb1 T ðð1 DÞ ð1 2DÞeb2 T Þ
, system (2.1) has a T-periodic positive solution ðz1 ðtÞ; z2 ðtÞÞ, which is asymptotically stable. Where ðz1 ðtÞ; z2 ðtÞÞ can be expressed as:
Remark 2.3. If p < p*, system (2.1) has a asymptotically stable Tperiodic positive solution ðz1 ðtÞ; z2 ðtÞÞ. The result is useful and important because from the biological viewpoint, a globally stable positive periodic state means that the subpopulation in both patches will coexist. Remark 2.4. The positive T-periodic solution will dramatically depend on the harvesting time lT between two diffusion pulses, and the harvesting fraction p of population at each time. Obviously, the number of the population in two patches is decreasing functions with respect to l. From the biological viewpoint, under the condition of insuring population permanence (i.e., p < p*) in order to catch much population, we must harvest it as late as possible. 3. The dynamics In the previous section, we got the existence of the preysextinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ of the system (1.3). It is clear that the attractivity of preys-extinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ of system (1.3) is equivalent to the globally attractivity of the preys-extinction periodic solution
8 8 b1 z1 > > > > > > > < ðb a z Þeb1 ðtðn1ÞTÞ þ a z ; t 2 ððn 1ÞT; ðn þ l 1ÞT > > 1 1 1 1 1 < z ðtÞ ¼ 1 > ð1 pÞb1 z1 > > ; > : > > ðb1 a1 z1 Þeb1 ðtðn1ÞTÞ þ pa1 z1 eb1 ðtðnþl1ÞTÞ þ a1 ð1 pÞz1 > > > : z2 ðtÞ ¼ z2 eb2 ðtðn1ÞTÞ ; t 2 ððn 1ÞT; nT and b1 ð1 pÞð1 D ð1 2DÞeb2 T Þ b1 eb1 T þ b1 ð1 DÞeðb1 þb2 ÞT z1 ¼ a1 ð1 p þ peb1 ð1lÞT eb1 T Þð1 ð1 DÞeb2 T Þ Dz1 z2 ¼ : ð1 DÞ ð1 2DÞeb2 T
t 2 ððn þ l 1ÞT; nT (2.2)
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
ð0; 0; 0; 0; z1 ðtÞ; z2 ðtÞÞ of system (1.2). Therefore we will only devote to model (1.3). That is to say, for any positive solution (x2(t), y2(t), z1(t), z2(t)) of system (1.3), we have x2 ðtÞ ! 0;
y2 ðtÞ ! 0;
z1 ðtÞ ! z1 ðtÞ;
z2 ðtÞ ! z2 ðtÞ;
as t ! 1:
Now we investigate the global attractivity of the preysextinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ of system (1.3). Theorem 3.1. If the following conditions hold: 1 ð1 DÞeb2 T ; DÞ ð1 2DÞeb2 T Þ ðH2 Þ : a1 ev1 t 1 < v2 þ c1 z1 ; ðH1 Þ : p < 1
eb1 T ðð1
ðH3 Þ : a2 eg 1 t2 < g 2 þ c2 z2 eb2 T ; then system (1.3) has a preys-extinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ, which is globally attractive (for proof see Appendix A.) Remark 3.1. Theorem 3.1 shows that when the predator species (z1(t), z2(t)) approach a positive periodic stable state ðz1 ðtÞ; z2 ðtÞÞ under proper harvesting strategies, the prey species x2(t) and y2(t) become extinct under low birth rates of immature preys, or high death rates of immature and mature preys, or long maturation time of immature preys to mature preys, or large preys’ captured rates. Remark 3.2. Theorem 3.1 tells us that if we guarantee that (H1)– (H3) hold, then the prey species in both patches must be extinction and the predator populations in both patches will coexist. Generally, if the prey species is extinct, as a direct result, the predator species should be also extinct as its survival is dependent on the prey species. However, from Theorem 3.1, we can see that the preys in both patches are extinct, whereas the predator populations in two patches are permanent. Why? In fact, the positiveness of the growth rate b1 of predator subpopulation in patch 1 ensures the predator subpopulation in patch 1 can feed on other food to survive besides the prey population x2(t). Since the predator subpopulation in patch 2 come from the diffusion of the predator subpopulation in patch 1, and then the predator subpopulation in patch 2 will be also permanent. Corollary 3.1. If the following conditions hold: 1 ð1 DÞeb2 T ; eb1 T ðð1 DÞ ð1 2DÞeb2 T Þ v1 t 1 < v2 ; ðH2 Þ : a1 e ðH3 Þ : a2 eg 1 t2 < g 2 ;
ðH1 Þ : p > 1
then system (1.3) is extinction. Remark 3.3. Corollary 3.1 reveals that the prey species x2(t) and y2(t) become extinct under low birth rates of immature preys, or high death rates of immature and mature preys, or long maturation time of immature preys to mature preys, then the predator populations in two patches will be also extinct as overexploiting. Theorem 3.1 proves the preys-extinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ is globally attractive, or say, the prey populations in both patches will be extinct totally. Whereas from the point of ecological balance and saving resources, we hope all populations may coexist in an ecosystem. That is to say, the ecosystem should be permanence. From biological point of view, permanence of species maintains biological balance of an ecosystem which implies the survival of all populations of the system and none of them facing extinction or growing infinitely in future time. Mathematically, permanence of a system means that strictly
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positive solutions have lower and upper bound. Concerning the permanence of species for system (1.2), firstly, we will show that all solutions of (1.2) are uniformly ultimately bounded. Theorem 3.2. There exists a constant M > 0 such that xi(t) M, yi(t) M, zi(t) M(i = 1, 2) for each solution of (1.2) with t large enough (for proof see Appendix A.) Remark 3.4. Let us analyze the biological meaning of Theorem 3.2. In reality, if the predator subpopulation in patch 1 is not ultimately bounded, then the density of predator subpopulation in patch 1 will expand unlimitedly. By the diffusion of predator population between two patches, the density of predator subpopulation in patch 2 will also expand unlimited. Since the predation rates of predator subpopulation in two patches for prey species are strictly positive (i.e., c1, c2 > 0), then the prey species in both patches will become extinct because of the massive preying by the predator species. And then the predator population will feed on the other foods in turn. Also, the other foods will be eaten up because of the massive preying. As an opposite result, the predator species in patch 1 can starve to death to lead it to die out. Since the subpopulation in patch 2 come from the diffusion of the subpopulation in patch 1, and then the subpopulation in patch 2 will also become extinction. Finally, we investigate the permanence of the system (1.2). Theorem 3.3. The system (1.2) is permanent if (H1) and the following conditions hold ðH4 Þa1 ev1 t1 > v2 þ d1 M þ c1 q1 ; ðH5 Þa2 eg 1 t 2 > g 2 þ d2 M þ c2 q2 ; where M, qi(i = 1, 2) are defined in Theorem 3.2 and (A.14), respectively (for proof see Appendix A.) Remark 3.5. From Theorem 3.1, we know that, when there are no prey species x2(t) and y2(t), the predator species (z1(t), z2(t)) will reach a positive periodic stable state ðz1 ðtÞ; z2 ðtÞÞ under proper harvesting strategies (p < p*). While there are prey species x2(t) and y2(t), Theorem 3.3 shows that if ðz1 ; z2 Þ can guarantee that prey species x2(t) and y2(t) satisfies conditions (H4)–(H5), then prey species x2(t) and y2(t) will be also permanent. Obviously, z1 ; z2 are decreasing functions with respect to l. From the viewpoint of biology, under the condition of insuring predator population permanence (i.e., p < p*) in order to ensure the permanence of the prey species and catch much the predator population, we should harvest it as late as possible. Remark 3.6. Theorem 3.3 reveals that if we largen the birth rates of immature preys, or decrease the death rates of immature and mature preys, or shorten the maturation time of immature preys, or decrease the preys’ captured rates, then we can control the prey populations to become permanent from the quondam extinction. 4. Numerical analysis In this section, the solution of system (1.3) with initial condition in the first quadrant is obtained numerically for biologically feasible range of parametric value provides two essential dynamical behavior of system (1.3). To study the dynamics, we suppose the system has the following hypothetical parameter values: a1 = 8, v1 = 1, v2 = 1.2, d1 = 1, c1 = 2, a2 = 8, g1 = 1, g2 = 1.2, d2 = 1, c2 = 2, b1 = 1, a1 = 1, k1 = 0.9, b2 = 1, k2 = 0.9, p = 0.2, D = 0.3, T = 1, l = 0.25 and initial values (0.2, 0.2, 0.2, 0.2). By above analysis, we
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
116
(a)
0.18
0.18
0.16
0.16
0.14
0.14
0.12
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0.1
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0.04
0.04
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0.02 20
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t
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40
t
(d)
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(e)
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0.35
0
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0.2 z2(t)
0.45
80
0.28
0.65
z2(t)
z1(t)
0
(c)
0.7
0.1
0.08
0
(b)
0.2
y 2(t)
x2(t)
0.2
0
20
40
t
t
60
80
100
0.2
0.3
0.4
z1 (t)
0.5
0.6
0.7
Fig. 1. Two prey-extinction periodic solution in both patches with t1 = 1.6 and t2 =2. (a) Time-series of x2(t);(b) time-series of y2(t);(c) time-series of z1(t); (d) time-series of z2(t); and (e) the phase portrait of the predator.
know the time delay t1 and t2 are two of the key parameters that directly impact the dynamics of the system. So, in our numerical experiments the only difference between conditions of Figs. 1–4 is the values of t1 and t2. In Fig. 1, we set t1 = 1.6 and t2 = 2. They are observed that these parameter values satisfy the conditions of
Theorem 3.1. Numerical simulations for the global attractivity of preys-extinction periodic solution are shown in Fig. 1. When we choose t1 = 1.6 and t2 = 0.5, then the predator coexists with the prey in the second patch, and the prey is extinct in the first patch (see Fig. 2). If t1 = 0.5 and t2 = 2, then the prey in the second patch
(a)
0.2
(b)
1.6
0.18
1.4
0.16 1.2
0.14 y2(t)
x2(t)
0.12 0.1
1 0.8
0.08 0.06
0.6
0.04 0.4
0.02 0
0
20
40
60
t
(c)
1
z2(t)
z1(t)
1.2
0
20
40
t
(d)
3
1.4
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100
60
80
100
(e)
3
2.5
2.5
2
2 z2(t)
1.6
80
1.5
1.5
0.8 0.6 0.4 0.2
0
20
40
t
60
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1
1
0.5
0.5
0
0
20
40
t
60
80
100
0
0.2
0.4
0.6
0.8
z1(t)
1
1.2
1.4
1.6
Fig. 2. (a) A prey-extinction periodic solution in the first patch with t1 = 1.6 and t2 = 0.5. (a) Time-series of x2(t);(b) time-series of y2(t); (c) time-series of z1(t); (d) time-series of z2(t); and (e) the phase portrait of the predator.
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
(a)
1
117
(b)
0.2 0.18
0.9
0.16 0.8 0.14 0.12 y2(t)
x2(t)
0.7 0.6
0.1 0.08
0.5
0.06 0.4 0.04 0.3 0.2
0.02 0
20
40
60
t
(c)
2.2
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2
0
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z2(t)
z1(t)
1.4 1.2 1
z2(t)
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t
60
80
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100
0
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40
t
60
80
0
100
0.5
1
1.5
z1(t)
2
Fig. 3. (a) A prey-extinction periodic solution in the second patch with t1 = 0.5 and t2 = 2. (a) Time-series of x2(t); (b) time-series of y2(t); (c) time-series of z1(t); (d) time-series of z2(t); (e) and the phase portrait of the predator.
(a)
1
(b)
1.6
0.9
1.4
0.8
1.2
y2(t)
x2(t)
0.7 0.6
1 0.8
0.5 0.6
0.4
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0
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60
t
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80
0
20
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60
t
(d)
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2
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100
80
100
(e)
3
2.5
2.5
2
2 z2(t)
z2(t)
z1(t)
1.5 1.5
1.5
1
0.5
0
0
20
40
t
60
80
100
1
1
0.5
0.5
0
0
20
40
t
60
80
100
0
0
0.5
1
z1(t)
1.5
2
2.5
Fig. 4. (a) Permanence with t1 = 0.5 and t2 = 0.5. (a) Time-series of x2(t); (b) time-series of y2(t); (c) time-series of z1(t); (d) time-series of z2(t); and (e) the phase portrait of the predator.
is extinct, the one in the first patch will coexists with predator (see Fig. 3). Again, If we choose t1 = 0.5, t2 = 0.5, then system (1.3) will satisfy Theorem 3.3. It is obviously that system (1.3) is permanent (see Fig. 4).
5. Conclusions In this work, we have investigated a delayed predator–prey model with stage structure for preys, and focused on impulsive
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diffusion and impulsive harvesting for predator at different fixed moments. In this model, each prey species was divided by two stages and was restricted to a single patch, and predator can diffuse between two patches. We have proved that the preys-extinction periodic solution of system (1.2) is globally asymptotically stable by using the comparison theorem of delayed impulsive differential equations and some analysis techniques. The sufficient conditions of permanence for the considered system were also established and stated. From Theorem 3.1, we can conclude that if the death rates of immature and mature preys, the maturation time of immature preys to mature preys and preys’ captured rates are high, the birth rates of immature preys are low, then system (1.2) admits a unique preys-extinction periodic solution. That is to say, the prey populations in two patches are extinct under these conditions, whereas the predator population will be permanent under proper harvesting strategies. On the contrary, if the predator population in patch 1 is overexploited, then both the preys and predator will be extinct under low birth rates of immature preys, or high death rates of immature and mature preys, or long maturation time of immature preys to mature preys. From biological point of view, we hope to keep biological balance so that all species in an ecosystem can coexist. And then Theorem 3.3 implies that if we largen the birth rates of immature preys, or decrease the death rates of immature and mature preys, or shorten the maturation time of immature preys, or decrease the preys’ captured rates, then the prey populations and predator populations in two patches will coexist under proper harvesting strategies. The results showed that the birth rates of immature preys, the death rates of immature and mature preys, the maturation time of immature preys to mature preys and the preys’ captured rates play an important part
in determining the preys-extinction and permanence of system (1.2). Numerical simulations by choosing the maturation time of immature preys to mature preys as the key parameters were carried out to illustrate the findings of Theorems 3.1 and 3.3. We can observed that if the maturation time of immature preys are long (t1 = 1.6, t2 = 2) or short (t1 = 0.5, t2 = 0.5), then the preys become extinct or preys and predators coexist in both patches. Whereas when the maturation time of immature prey is long in one patch and short in other patch (t1 = 1.6, t2 = 0.5 or t1 = 0.5, t2 = 2), then the prey in one patch become extinct, while it survives in other patch. Similarly, the results can be also extended by considering the critical parameters. On the other hand, our results also show that the best time of harvesting the predator population in patch 1 is at the end of the diffusion period T, that is, before and near the time of the diffusion under the condition of renewable resource. From the viewpoint of biology, under the condition of insuring predator and preys permanence, in order to catch much the predator population, we must harvest it as late as possible. Therefore it is interesting and helpful for people to provide tactical basis for the biological resource management. Acknowledgements The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this paper. This work was supported by the Program for New Century Excellent Talents in University (NCET-10-0097), the NSFC Tianyuan Foundation (Grant No. 11226256) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY13A010010).
Appendix A Proof of Theorem 3.1. Let (x2(t), y2(t), z1(t), z2(t)) be any solution of system (1.3), we can obtain from the third and the fourth equation of system (1.3) that dz1 ðtÞ z1 ðtÞ½b1 a1 z1 ðtÞ; dt dz2 ðtÞ b2 z2 ðtÞ: dt Now we consider the following impulsive differential equations 8 9 du1 ðtÞ > > > > ¼ u ðtÞ½b a u ðtÞ > = 1 1 1 1 > > dt > > t 6¼ nT; t 6¼ ðn þ l 1ÞT > > du2 ðtÞ > > > > ; > > dt ¼ b2 u2 ðtÞ > > > ) < Du1 ðtÞ ¼ pu1 ðtÞ t ¼ ðn þ l 1ÞT > > > > Du2 ðtÞ ¼ 0 > > > ) > > > Du1 ðtÞ ¼ Dðu2 ðtÞ u1 ðtÞÞ > > t ¼ nT > > > > : Du2 ðtÞ ¼ Dðu1 ðtÞ u2 ðtÞÞ
(A.1)
By Lemmas 2.4 and 2.5, the periodic solution ðu1 ðtÞ; u2 ðtÞÞ of (A.1) is globally asymptotically stable. By using the comparison theorem of impulsive differential equation [Lemma 2.2], we have (
z1 ðtÞ u1 ðtÞ ! u1 ðtÞ ¼ z1 ðtÞ z2 ðtÞ u2 ðtÞ ! u2 ðtÞ ¼ z2 ðtÞ
as t! 1, where ðz1 ðtÞ; z2 ðtÞÞ is the solution of system (2.1). Then there exists a positive constant e1 small enough and a positive integer n1 large enough such that z1 ðtÞ > z1 ðtÞ e1 z1 e1 , d1 ; z2 ðtÞ > z2 ðtÞ e1 z2 eb2 T e1 , d2 ; for all nT < t = (n + 1)T,
n > n 1.
(A.2)
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119
Next, form the first equation and the second equation of system (1.3), we have dx2 ðtÞ a1 ev1 t 1 x2 ðt t 1 Þ ðv2 þ c1 d1 Þx2 ðtÞ; dt dy2 ðtÞ a2 eg 1 t 2 y2 ðt t 2 Þ ðg 2 þ c2 d2 Þy2 ðtÞ; dt
(A.3)
for all t > nT + t. On the other hand, by (H2) and (H3), for the above e1 small enough, we have
a1 ev1 t1 < v2 þ c1 d1 ; a2 eg 1 t2 < g 2 þ c2 d2 : Considering the comparison system of (A.3) as follows dv1 ðtÞ ¼ a1 ev1 t1 v1 ðt t 1 Þ ðv2 þ c1 d1 Þv1 ðtÞ; dt dv2 ðtÞ ¼ a2 eg 1 t 2 v2 ðt t 2 Þ ðg 2 þ c2 d2 Þv2 ðtÞ; dt By Lemma 2.3, one can easily obtain that lim vi ðtÞ ¼ 0:ði ¼ 1; 2Þ
t!1
Since x2 ðsÞ ¼ v1 ðsÞ ¼ ’2 ðsÞ > 0 and y2 ðsÞ ¼ v2 ðsÞ ¼ ’4 ðsÞ > 0 for all s 2 [S t, 0], using the comparison theorem, we can obtain that x2(t) ! 0 and y2(t) ! 0 as t! 1. Without loss of generality, we assume that there exists a positive constant e such that 0 < x2 ðtÞ < e;
0 < y2 ðtÞ < e
for
all
t 0:
(A.4)
From the third and the fourth equation of (1.3) and (A.4), we have dz1 ðtÞ z1 ðtÞ½b1 þ k1 c1 e a1 z1 ðtÞ dt dz2 ðtÞ ðb2 k2 c2 eÞz2 ðtÞ: dt Denote b01 ¼ b1 þ k1 c1 e; b02 ¼ b2 k2 c2 e, Then we can consider the following comparison impulsive differential system 8 9 dw1 ðtÞ > > 0 > > ¼ w ðtÞ½b a w ðtÞ > = 1 1 1 1 > > dt > > > > > > dw2 ðtÞ > > ; > ¼ b02 w2 ðtÞ > > dt > > ) < Dw1 ðtÞ ¼ pw1 ðtÞ > > > Dw2 ðtÞ ¼ 0 > > > > ) > > > Dw1 ðtÞ ¼ Dðw2 ðtÞ w1 ðtÞÞ > > > > > > : Dw2 ðtÞ ¼ Dðw1 ðtÞ w2 ðtÞÞ
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
(A.5)
t ¼ ðn þ l 1ÞT
t ¼ nT
By Lemmas 2.4 and 2.5 again, the unique positive periodic solution of system (A.5) is 8 8 b01 w1 > > > > > ; t 2 ððn 1ÞT; ðn þ l 1ÞT > > > 0 a w Þeb01 ðtðn1ÞTÞ þ a w < > ðb > 1 1 1 1 1 > < w1 ðtÞ ¼ > ð1 pÞb01 w1 > > > ; > 0 : 0 > b01 ðtðn1ÞTÞ > ðb1 a1 w1 Þe þ pa1 w1 eðb1 ðtðnþl1ÞTÞ þ a1 ð1 pÞz1 > > > > : 0 w2 ðtÞ ¼ w2 eb2 ðtðn1ÞTÞ ; t 2 ððn 1ÞT; nT
t 2 ððn þ l 1ÞT; nT
where 0
w1 ¼
w2 ¼
0
0
0
b01 ð1 pÞð1 D ð1 2DÞeb2 T Þ b01 eb1 T þ b01 ð1 DÞeðb1 þb2 ÞT a1 ð1 p þ pe Dw1 0
1 D ð1 2DÞeb2 T
b01 ð1lÞT
b02 T
Þð1 ð1 DÞe
Þ
;
:
Hence for arbitrary small e1 > 0 and any sufficiently large integer n2 > n1 such that z1 ðtÞ < w1 ðtÞ þ e1 ; z2 ðtÞ < w2 ðtÞ þ e1 ;
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for t > n2T. Let e ! 0, then w1 ðtÞ ! z1 ðtÞ;
w2 ðtÞ ! z2 ðtÞ
and z1 ðtÞ < z1 ðtÞ þ e1 ;
z2 ðtÞ < z2 ðtÞ þ e1 :
(A.6)
From (A.2) and (A.6), we have z1 ðtÞ ! z1 ðtÞ;
z2 ðtÞ ! z2 ðtÞ:
This ends the proof.
&
Proof of Theorem 3.2. Define V(t, X(t)) such that Vðt; XðtÞÞ ¼ k1 x1 ðtÞ þ k1 x2 ðtÞ þ k2 y1 ðtÞ þ k2 y2 ðtÞ þ z1 ðtÞ þ z2 ðtÞ then V 2 V0. We calculate the upper right derivative of V(t, X) along a solution of system (1.2) and get the following impulsive differential equation 8 þ D VðtÞ þ LVðtÞ ¼ k1 ða1 v2 þ LÞx2 ðtÞ k1 d1 x22 ðtÞ > > > > > > > k1 ðv1 LÞx1 ðtÞ þ k2 ða2 g 2 þ LÞy2 ðtÞ > > > t 6¼ ðn þ l 1ÞT; > < k d y2 ðtÞ k ðg LÞy ðtÞ 2 2 2
> > > > > > > > Vðt þ Þ VðtÞ > > > : Vðt þ Þ VðtÞ
2
t 6¼ nT
1
1
þ ðb1 þ L a1 z1 ðtÞÞz1 ðtÞ ðb2 LÞz2 ðtÞ t ¼ ðn þ l 1ÞT t ¼ nT
Let 0 < L = min {v1, g2, b2}, then D+V(t) + LV(t) is bounded. Select L1 and L2 such that
Dþ VðtÞ L1 VðtÞ þ L2 þ
Vðt Þ VðtÞ
VðtÞ
t 6¼ nT
t ¼ ðn þ l 1ÞT
Vðt þ Þ VðtÞ where L1,
t 6¼ n þ l 1ÞT; t ¼ nT
L2 are two positive constant. According to Lemma 2.2, we have
L2 L2 Vð0þ Þ expðL1 tÞ þ L1 L1
where t 2 ((n S 1)T, nT]. Hence lim VðtÞ
t!1
L2 L1
Therefore V(t, X(t)) is ultimately bounded, and we know that each positive solution of system is uniformly ultimately bounded. This completes the proof. & Proof of Theorem 3.3. Suppose X(t) = (x1(t), x2(t), y1(t), y2(t), z1(t), z2(t)) is any solution of the system (1.2) with X(0+) > 0. From Theorem 3.2, we assume that xi(t) = M, yi(t) = M and zi(t) = M(i = 1, 2) with t I 0. From (A.2), we have z1 ðtÞ > z1 e1 , d1 and z2 ðtÞ > z2 eb2 T e1 , d2 for all t large enough. Thus we only need to find positive constants mi and ni(i = 1, 2) such that xi(t) I mi and yi(t) I ni(i = 1, 2) for t large enough. The second and fourth equation of system (1.2) can be rewritten as dx2 ðtÞ d ¼ ½a1 ev1 t 1 v2 d1 x2 ðtÞ c1 z1 ðtÞx2 ðtÞ a1 ev1 t1 dt dt dy2 ðtÞ d ¼ ½a2 eg 1 t 2 g 2 d2 y2 ðtÞ c2 z2 ðtÞy2 ðtÞ a2 eg 1 t2 dt dt By (A.7), we defined Q1(t) and Q2(t) as Q 1 ðtÞ ¼ x2 ðtÞ þ a1 ev1 t1 Q 2 ðtÞ ¼ y2 ðtÞ þ a2 eg 1 t2
Z
t
Z tt t 1 tt 2
x2 ðsÞds; y2 ðsÞds:
Z Z
t
x2 ðsÞds; tt 1
(A.7)
t
tt 2
y2 ðsÞds
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Then by Theorem 3.2, we have Q 1 ðtÞ M½1 þ a1 t 1 ev1 t1 ;
(A.8)
Q 2 ðtÞ M½1 þ a2 t 2 eg 1 t 2 ; as t! 1. Calculating the derivative of Q1(t) and Q2(t) along the solution of system (1.2), we get dQ 1 ðtÞ ¼ ½a1 ev1 t1 v2 d1 x2 ðtÞ c1 z1 ðtÞx2 ðtÞ; dt dQ 2 ðtÞ ¼ ½a2 eg 1 t2 g 2 d2 y2 ðtÞ c2 z2 ðtÞy2 ðtÞ: dt According Theorem 3.2, one can obtain that
(A.9)
dQ 1 ðtÞ ½a1 ev1 t 1 v2 d1 M c1 z1 ðtÞx2 ðtÞ; dt dQ 2 ðtÞ ½a2 eg 1 t2 g 2 d2 M c2 z2 ðtÞy2 ðtÞ: dt According to hypotheses (H4) and (H5), one can choose a e3 > 0 be small enough such that
a1 ev1 t1 > v2 þ d1 M þ c1 ½q1 þ e3 ; a2 eg 1 t2 > g 2 þ d2 M þ c2 ½q2 þ e3
(A.10)
(A.11)
where q1 and q2 are defined in (A.14). Now, we will prove that for any t0 > 0, it is impossible that x2 ðtÞ < m2 and y2 ðtÞ < n2 for all t > t0. It follows from the fifth and sixth equations of system (1.2) that for all t > t0 dz1 ðtÞ < z1 ðtÞ½b1 þ k1 c1 m2 a1 z1 ðtÞ dt dz2 ðtÞ < ½b2 k2 c2 n2 z2 ðtÞ dt Consider the following auxiliary impulsive equation 8 9 d˜z1 ðtÞ > > > ¼ z˜1 ðtÞ½b1 þ k1 c1 m2 a1 z˜1 ðtÞ > > = > > dt > > t 6¼ nT; t 6¼ ðn þ l 1ÞT > > > > d˜ z ðtÞ > > ; > 2 ¼ ½b k c n ˜ z ðtÞ 2 2 2 2 > 2 > dt > > ) < Dz˜1 ðtÞ ¼ p˜z1 ðtÞ t ¼ ðn þ l 1ÞT > > > > Dz˜2 ðtÞ ¼ 0 > > > ) > > > Dz˜1 ðtÞ ¼ Dð˜z2 ðtÞ z˜1 ðtÞÞ > > t ¼ nT > > > > : Dz˜2 ðtÞ ¼ Dð˜z1 ðtÞ z˜2 ðtÞÞ
(A.12)
By Lemmas 2.4 and 2.5, we obtain the unique global asymptotically periodic solutions of system (A.12) as follows 8 8 b001 z˜1 > > > > > ; t 2 ððn 1ÞT; ðn þ l 1ÞT > > > b00 ðtðn1ÞTÞ 00 < > ðb1 a1 z˜1 Þe 1 þ a1 z˜1 > > < z˜ ðtÞ ¼ 1 > ð1 pÞb001 z˜1 > > > > ; 00 00 : > > ðb001 a1 z˜1 Þeb1 ðtðnþl1ÞTÞ þ pa1 z˜1 eb1 ðtðn1ÞTÞ þ a1 ð1 pÞ˜z1 > > > > : 00 z˜2 ðtÞ ¼ z˜2 eb2 ðtðn1ÞTÞ ; t 2 ððn 1ÞT; nT
t 2 ððn þ l 1ÞT; nT
and 00
z˜1 ¼ z˜2 ¼
00
00
00
b001 ð1 pÞð1 D ð1 2DÞeb2 T Þ b01 eb1 T þ b001 ð1 DÞeðb1 þb2 ÞT b001 ð1lÞT
a1 ð1 p þ pe D˜z1
b002 T
Þð1 ð1 DÞe
Þ
; (A.13)
00 : 1 D ð1 2DÞeb2 T
where b001 ¼ b1 þ k1 c1 m2 ; b002 ¼ b2 k2 c2 n2 . In view of the comparison theorem of impulsive differential equation (Lemma 2.2), we know that there exists t1(> t0 + t1) and t2(> t0 + t2), for sufficiently small e3, we have z1 ðtÞ < z˜1 ðtÞ þ e3
for all t t 1 ;
z2 ðtÞ < z˜2 ðtÞ þ e3
for all t t 2 :
122
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
Thus
z˜1 ðtÞ < ð1 pÞ˜z1 eðb1 þk1 c1 m2 ÞT , q1 ; z˜2 ðtÞ < z˜2
, q2 :
(A.14)
Thus z1 ðtÞ < q1 þ e3 , s 1 ;
for all t t 1 ;
z2 ðtÞ < q2 þ e3 , s 2 ;
for all
t t2 :
(A.15)
From (A.11), we get
a1 ev1 t1 > v2 þ d1 M þ c1 s 1 ; a2 eg 1 t2 > g 2 þ d2 M þ c2 s 2 : Then dQ 1 ðtÞ > ½a1 ev1 t1 v2 d1 M c1 s 1 x2 ðtÞ; dt dQ 2 ðtÞ > ½a2 eg 1 t 2 g 2 d2 M c2 s 2 y2 ðtÞ: dt m m m Let xm 2 ¼ mint 2 ½t1 ;t1 þt 1 x2 ðtÞ and y2 ¼ mint 2 ½t 2 ;t 2 þt 2 y2 ðtÞ. We will show that x2 ðtÞ x2 for all t I t1, y2 ðtÞ y2 for all t I t2. Otherwise, m there are T1, T2 > 0 such that x2 ðtÞ xm for t 2 ½t ; t þ t þ T ; x ðt þ t þ T Þ ¼ x and ((dx (t + t + T ))/(dt)) = 0 and y2 ðtÞ ym 1 1 1 1 2 1 1 1 2 1 1 1 2 2 2 for m t 2 ½t 2 ; t 2 þ t 2 þ T 2 ; y2 ðt 2 þ t 2 þ T 2 Þ ¼ y2 and ((dy2(t2 + t2 + T2))/(dt)) = 0. Then from the second and the fourth equation of system (1.2) and (A.14), we easily verified that
dx2 ðt 1 þ t 1 þ T 1 Þ ¼ a1 ev1 t1 x2 ðt 1 þ T 1 Þ v2 x2 ðt 1 þ t 1 þ T 1 Þ d1 x22 ðt 1 þ t 1 þ T 1 Þ c1 x2 ðt 1 þ t 1 þ T 1 Þz1 ðt 1 þ t 1 þ T 1 Þ dt ½a1 ev1 t 1 v2 d1 M c1 s 1 xm 2 >0 dy2 ðt 2 þ t 2 þ T 2 Þ ¼ a2 eg 1 t2 y2 ðt 2 þ T 2 Þ g 2 y2 ðt 2 þ t 2 þ T 2 Þ d2 y22 ðt 2 þ t 2 þ T 2 Þ c2 y2 ðt 2 þ t 2 þ T 2 Þz2 ðt 2 þ t 2 þ T 2 Þ dt ½a2 eg 1 t 2 g 2 d2 M c2 s 2 ym 2 >0
(A.16)
m which is a contradiction. Hence, we have x2 ðtÞ xm 2 > 0 for all t I t1 and y2 ðtÞ y2 > 0 for all t I t2. Again from (A.15), we have
dQ 1 ðtÞ > ½a1 ev1 t1 v2 d1 M c1 s 1 xm for all t t 1 ; 2; dt dQ 2 ðtÞ > ½a2 eg 1 t 2 g 2 d2 M c2 s 2 yn2 ; for all t t 2 ; dt which implies that Q1(t)! 1 , Q2(t) ! 1 as t! 1. This is a contradiction to (A.8). Therefore, for any positive constant t0, the inequality m x2 ðtÞ < xm 2 and y2 ðtÞ < y2 cannot hold for all t I t0. There are the following two cases. m Case 1. If x2 ðtÞ xm 2 and y2 ðtÞ y2 hold for all t large enough, then our aim reach. Otherwise, we consider case 2. Case 2. If x2(t) is oscillatory about m2 and y2(t) is oscillatory about n2 . Setting m2 ðv2 þd1 m þc1 MÞt1 2 ; m2 e ; 2 n n2 ¼ min 2 ; n2 eðg 2 þd2 n2 þc2 MÞt2 2
m2 ¼ min
In the following, we shall prove that x2(t) I m2 and y2(t) I n2. Let t1 ; t2 > 0. Obviously, there exist two positive constants j1 and j2 such that x2 ðt1 Þ ¼ x2 ðt1 þ j1 Þ ¼ m2 and x2 ðtÞ < m2 for t1 < t < t1 þ j1 ; y2 ðt2 Þ ¼ y2 ðt2 þ j2 Þ ¼ n2 and y2 ðtÞ < n2 for t2 < t < t2 þ j2 ; where t1 ; t2 is sufficiently large so that the inequality z1(t) < q1 holds true for t1 < t < t1 þ j1 and z2(t) < q2 holds true for t2 < t < t2 þ j2 . Since x2(t) and y2(t) are continuous and bounded and are not effected by impulses, we deduce that x2(t) and y2(t) are uniformly continuous. Hence there exist two constants T1, T2(with 0 < T1 < t1, 0 < T2 < t2 and T1 is independent of the choice of t1 , T2 is independent of the choice of t2 , respectively) such that x2 ðt1 Þ > m2 =2 for all t1 < t t1 þ T 1 and y2 ðt2 Þ > n2 =2 for all t2 < t t1 þ T 2 . If j1 = T1 and j2 = T2, our aim is obtained. If T1 < j1 = t1 and T2 < j2 = t2, from the second equation and fourth equation of (1.2), we have that ððdx2 ðtÞÞ=ðdtÞÞ v2 x2 ðtÞ d1 x22 ðtÞ c1 x2 ðtÞz1 ðtÞ; x2 ðt1 Þ ¼ m2 for all t1 < t t1 þ j1 and ððdy2 ðtÞÞ=ðdtÞÞ g 2 y2 ðtÞ d2 y22 ðtÞ c2 y2 ðtÞz2 ðtÞ for all t2 < t t2 þ j2 ; y2 ðt2 Þ ¼ n2 . According to the hypothesis, it is easy to get that ððdx2 ðtÞÞ=ðdtÞÞ ðv2 þ d1 x2 þ c1 MÞx2 ðtÞ for all t1 < t t1 þ j1 t1 þ t 1 and and ððdy2 ðtÞÞ=ðdtÞÞ ðg 2 þ d2 n2 þ c2 MÞy2 ðtÞ for all t2 < t t2 þ j2 t2 þ t 2 . Then we conclude that x2 ðtÞ m2 eðv2 þd1 x2 þc1 MÞt1 and ðg 2 þd2 n2 þc2 MÞt 2 y2 ðtÞ n2 e . It is clear that x2(t) I m2 for all t1 < t t1 þ j1 and y2(t) I n2 for t2 < t t2 þ j2 . If j1 I t1, j2 I t2, then we get that x2(t) I m2 for all t1 < t t1 þ t 1 and y2(t) I n2 for t2 < t t2 þ t 2 . By continuing the same arguments, one can easily obtain that we can obtain x2(t) I m2 for t1 þ t 1 < t t1 þ j1 and y2(t) I n2 for t2 þ t 2 < t t2 þ j2 . Since the interval ½t1 ; t1 þ j1 and ½t2 ; t2 þ j2 are arbitrarily chosen, we get that x2(t) I m2 and y2(t) I n2 for t large enough.
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
123
Next, from the first and the third equation of system (1.2), we get dx1 ðtÞ a1 ðm2 a1 ev1 t1 MÞ v1 x1 ðtÞ dt dy1 ðtÞ ¼ a2 ðn2 a2 eg 1 t 2 MÞ g 1 y1 ðtÞ dt By integration, it is obviously that for sufficiently small e4 > 0,
a1 ðm2 a1 ev1 t1 MÞ e4 , m1 ; v1 a2 ðn2 a2 eg 1 t2 MÞ e4 , n1 : y1 ðtÞ g1 Take m = min {mi, ni, di}, then xi(t) I m, yi(t) I m, zi(t) I m(i = 1, 2) hold true as t! 1. Together with Theorem 3.2, we can easily derive
x1 ðtÞ
that system (1.2) is permanent. The proof is completed.
&
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Ecological Complexity journal homepage: www.elsevier.com/locate/ecocom
Dynamics of a delayed stage-structured model with impulsive harvesting and diffusion Liu Yang *, Shouming Zhong School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
A R T I C L E I N F O
A B S T R A C T
Article history: Received 3 January 2014 Received in revised form 14 May 2014 Accepted 27 May 2014 Available online
Based on the predator–prey system with delayed stage-structured for preys and impulsive harvesting and impulsive diffusion for predator, an impulsive delayed differential equation to model the process of periodic harvesting and impulsive diffusion at different fixed moments is proposed and investigated. In this model, patches are created by two different prey populations and each prey population is confined to a particular patch while the predator population can impulsively diffuse between two patches. By using comparison theorem of impulsive delayed differential equation and some analysis techniques, sufficient conditions ensuring the existence of preys-extinction periodic solution and the permanence of the system are established. Our analysis reveals that low birth rates of immature preys, high death rates of immature and mature preys, long maturation time of immature preys to mature preys and large preys’ captured rates are the sufficient condition for the preys-extinction. On the contrary, if we largen the birth rates of immature preys, or decrease the death rates of immature and mature preys, or shorten the maturation time of immature preys or decrease the preys’ captured rates, then the system can become permanent under proper predational strategies. These also show that it is feasible to keep the sustainable development of the ecosystem by controlling the critical ecological parameters. Numerical simulations with hypothetical set of parameter values are carried out to consolidate the analytic findings. ß 2014 Elsevier B.V. All rights reserved.
Keywords: Impulsive diffusion Impulsive harvesting Delay Stage-structured Permanence Extinction
1. Introduction Population dispersal in patchy environment is one of the most prevalent subjects in ecology and mathematical ecology. Within each patch, individuals of each species are supposed to be identical and can migrate to other patches. In most previous papers, population dynamics with the effects of heterogeneity modeled by the diffusion process is focused on the dynamical system modeled by ordinary differential equations and delay differential equations (Cui et al., 2004; Xu and Ma, 2008; Cui and Chen, 2001; Zhou et al., 2008; Ding and Han, 2008; Chen et al., 2003; Song and Chen, 1998; Xu et al., 2004). But in reality, dispersal behavior is very intricate and is always perturbed by environmental change and human activities, etc. In fact, it is often the case that diffusion occurs during short-time slots within seasons or within the lifetimes of animals. In order to be in much better agreement with the real ecological process, this short-time scale dispersal is more suitable assumed to be in the form of regular pulses. Taking birds as an example, when winter comes, they will diffuse between patches in search for a better environment,
* Corresponding author. Tel.: +86 15928740515. E-mail addresses: [email protected] (L. Yang), [email protected] (S. Zhong). http://dx.doi.org/10.1016/j.ecocom.2014.05.012 1476-945X/ß 2014 Elsevier B.V. All rights reserved.
but they do not migrate in other season. Thus impulsive diffusion provides a more natural description for this behavior. With the developments and applications of impulsive differential equations, theories of impulsive differential equations have been introduced into population dynamics, and some important studies about impulsive diffusion have been done (Shao, 2010; Jiao et al., 2011, 2011, 2011; Jiao and Cai, 2009a; Hui and Chen, 2005; Wan et al., 2012; Wang et al., 2007; Dong et al., 2007; Zhang et al., 2013; Zhao et al., 2011). In particularly, a single population was considered (Hui and Chen, 2005; Wan et al., 2012; Wang et al., 2007; Dong et al., 2007; Zhang et al., 2013; Zhao et al., 2011). For example, Zhao et al. (2011) studied the following single species model with impulsive diffusion and pulsed harvesting at the different fixed time 9 8 dx1 ðtÞ > 2 > > > = ¼ r x ðtÞ a x ðtÞ 1 1 11 1 > > dt > > > > > dx2 ðtÞ > > > ; > ¼ r 2 x2 ðtÞ > > dt > ) < Dx1 ðtÞ ¼ Ex1 ðtÞ > > > > Dx2 ðtÞ ¼ 0 > > ) > > > Dx1 ðtÞ ¼ Dðx2 ðtÞ x1 ðtÞÞ > > > > > Dx2 ðtÞ ¼ Dðx1 ðtÞ x2 ðtÞÞ :
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
t ¼ ðn þ l 1ÞT t ¼ nT
(1.1)
112
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
and impulsive diffusion at different fixed moments. To formulate the In this model, suppose that the system is composed of two patches connected by diffusion. xi(t) presents the biomass of the population mathematical model, we make the following assumptions: in patch i(i = 1, 2). r1 denotes the intrinsic growth rate in the first patch and r2 is the death rate in the second patch. r1/a11 is the A1 The patches are created by two prey populations, and each prey environment carrying capacity. 0 < E < 1 is the harvesting effort of population is confined to a particular patch while the predator the population in the first patch at t = (n + l 1)T(0 < l < 1). population can diffuse between two patches. Predator has two 0 < D < 1 is the diffusion coefficient. If xi > xj(i 6¼ j, i, j = 1, 2), the different impulsive time. In the first impulsive time, we only population in patch i diffuses into patch j at a rate D which is harvest the predator in the first patch. And in the second proportional to xi xj. T is the impulsive diffusion period. impulsive time, the predator population will migrate from one patch to other patch. Dxi(nT+) = xi(nT+) xi(nT)(i = 1, 2). xi(nT+)(i = 1, 2) represents the density of the subpopulation in patch i after the nth diffusion pulse A2 The prey populations: each prey population have two life stage, at t = nT, while xi(nT)(i = 1, 2) represents the density of the namely immature and mature stages. The birth rates into the subpopulation in patch i immediately before the nth diffusion immature population are proportional to the existing mature pulse at t = nT. By using the stroboscopic map, they obtained the population with a proportionality a1 in patch 1 and a2 in patch existence and globally asymptotical stability of both the trivial 2, respectively; and the death rates of the immature population solution and the positive periodic solution, and the complete are proportional to the existing immature population with a expression for the periodic solution. proportionality v1 in patch 1 and g1 in patch 2, respectively; the On the other hand, from birth to death, many species usually go death rates of the mature population are proportional to the through two life stages, immature and mature. So it is practical to existing mature population with a proportionality v2 in patch 1 introduce stage-structured into prey–predator models (Aiello and and g2 in patch 2, respectively; the intra-specific competition Freedman, 1990; Song and Chen, 2002; Meng et al., 2008; Song and rates of the mature prey populations are proportional to square Xiang, 2006; Chen and You, 2008; Shao and Dai, 2010; Huang et al., of the population with a proportionality d1 in patch 1 and d2 in 2012; Song et al., 2009; Liu et al., 2008; Wang et al., 2009). In patch 2, respectively. additional, as literatures (Li and Kuang, 2001) pointed out that the A3 The predator population: in the absence of the prey populadelay differential equation shows much more complicated dynamtions, the predator population in the first habitat grows ics than ordinary differential equation since time delay could cause a according to the logistic curve with the intrinsic birth rate b1 stable equilibrium to become unstable and cause the population to and density dependence rate a1. Whereas the predator fluctuate. Therefore it is reasonable to introduce delayed stage subpopulation in the second patch will die with the death structure into prey–predator models (Meng et al., 2008; Song and rate b2. The predator populations only feed on the mature preys Xiang, 2006; Chen and You, 2008; Shao and Dai, 2010; Huang et al., following Holling type-I functional response with different 2012; Song et al., 2009; Liu et al., 2008; Wang et al., 2009). Newly, capturing rates c1 in patch 1 and c2 in patch 2, respectively. The population dynamical system involving delayed stage structure and conversion factor for predator population due to consumption impulsive diffusion have been discussed by some authors, see Jiao of prey is ki(i = 1, 2) in the ith (i = 1, 2) patch. (2010), Shao and Li (2013), Jiao et al. (2009b), Dhar and Jatav (2013), Jiao et al. (2010), and references cited therein. Furthermore, in real nature, some lower-order preys can In the natural world, these assumptions are reasonable for establish their own territory and does not interact with other preys, many species whose immature prey population conceal in the cave whereas the predator can diffuse between the territories at a fixed and are raised by their parents; the rate of predator attacking at moment. Therefore in this paper, we consider a three-species (twoimmature prey can be ignored. Considering the above basic prey and a predator) ecological model with impulsive harvesting assumptions, we can derive the following differential equations: 9 8 dx1 ðtÞ > > > > ¼ a1 x2 ðtÞ a1 ev1 t1 x2 ðt t 1 Þ v1 x1 ðtÞ > > > > dt > > > > > > > > dx2 ðtÞ > > v1 t 1 2 > > > ¼ a1 e x2 ðt t 1 Þ v2 x2 ðtÞ d1 x2 ðtÞ c1 x2 ðtÞz1 ðtÞ > > > > > dt > > > > > > > > dy ðtÞ > > 1 g 1 t 2 > > = ¼ a2 y2 ðtÞ a2 e y2 ðt t 2 Þ g 1 y1 ðtÞ > > > dt > t 6¼ nT; t 6¼ ðn þ l 1ÞT > > > > dy ðtÞ > 2 > > ¼ a2 eg 1 t2 y2 ðt t 2 Þ g 2 y2 ðtÞ d2 y22 ðtÞ c2 y2 ðtÞz2 ðtÞ > > > > > > > > dt > > > > > > > dz1 ðtÞ > > > > ¼ z ðtÞ½b a z ðtÞ þ k c x ðtÞz ðtÞ > 1 1 1 1 1 1 2 1 > > > dt > > > > > > > > dz ðtÞ > > 2 > ; > ¼ b2 z2 ðtÞ þ k2 c2 y2 ðtÞz2 ðtÞ > > dt > > 9 > < Dx1 ðtÞ ¼ 0 > > > (1.2) > > Dx2 ðtÞ ¼ 0 > > > > > > > = > > Dy1 ðtÞ ¼ 0 > > t ¼ ðn þ l 1ÞT > > > > Dy2 ðtÞ ¼ 0 > > > > > > > > > Dz1 ðtÞ ¼ pz1 ðtÞ > > > > > ; > > > D z ðtÞ ¼ 0 > 2 > > 9 > > > Dx1 ðtÞ ¼ 0 > > > > > > > > > > Dx2 ðtÞ ¼ 0 > > > > > > = > > Dy1 ðtÞ ¼ 0 > > t ¼ nT > > > > Dy2 ðtÞ ¼ 0 > > > > > > > > Dz ðtÞ ¼ Dðz ðtÞ z ðtÞÞ > > > 1 2 1 > > > ; > > : Dz2 ðtÞ ¼ Dðz1 ðtÞ z2 ðtÞÞ
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
where x1(t), x2(t) represent the densities of immature and mature prey species, respectively, in the first patch at time t; y1(t), y2(t) denote the densities of immature and mature prey species, respectively, in the second patch at time t; zi(t) is the density of predator species in the ith (i = 1, 2) patch at time t. Dxi(t) = xi(t+) xi(t), Dyi(t) = yi(t+) yi(t), Dzi(t) = zi(t+) zi(t)(i = 1, 2). 0 < p < 1 is the harvesting effort of predator in the first patch at t = (n + l 1)T (0 < l < 1 is a positive constant, n 2 N); 0 < D < 1 is the dispersal rate of predator species at t = nT. The pulse diffusion vt occurs every period T. The terms a1e 1 1x2(t t1) and a2eg1t2y2(t t2) represent the number of immature preys in patch 1 and patch 2, respectively, that were born at time t ti(i = 1, 2) which still survive at time t and are transferred from the immature stage to the mature stage at time t. The initial conditions for system (1.2) take the form of xi ðuÞ ¼ fi ðuÞ; yi ðu Þ ¼ ’i ðuÞ; zi ðuÞ ¼ ci ðuÞ fi ð0Þ > 0; ’i ð0Þ > 0; ci ð0Þ > 0; i ¼ 1; 2; u 2 ½t ; 0; where t ¼ maxft 1 ; t 2 g; ðf1 ðuÞ; f2 ðuÞ; ’1 ðuÞ; ’2 ðuÞ; c1 ðuÞ; c2 ðuÞÞ 2 Cð½t ; 0; R6þ0 Þ , C þ , the Banach space of continuous functions mapping the interval [ t, 0] into R6þ0 which is defined as R6þ0 ¼ fðx1 ; x2 ; x3 ; x4 ; x5 ; x6 Þ : xi 0; i ¼ 1; 2; 3; 4; 5; 6g: We can simplify model (1.2) and restrict our attention to the following subsystem:
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
9 dx2 ðtÞ > ¼ a1 ev1 t 1 x2 ðt t 1 Þ v2 x2 ðtÞ d1 x22 ðtÞ c1 x2 ðtÞz1 ðtÞ > > > dt > > > > dy2 ðtÞ > g 1 t 2 2 = ¼ a2 e y2 ðt t 2 Þ g 2 y2 ðtÞ d2 y2 ðtÞ c2 y2 ðtÞz2 ðtÞ > dt > dz1 ðtÞ > > ¼ z1 ðtÞ½b1 a1 z1 ðtÞ þ k1 c1 x2 ðtÞz1 ðtÞ > > > dt > > > > dz2 ðtÞ ; ¼ b2 z2 ðtÞ þ k2 c2 y2 ðtÞz2 ðtÞ dt 9 Dx2 ðtÞ ¼ 0 > > > = Dy2 ðtÞ ¼ 0 > Dz1 ðtÞ ¼ pz1 ðtÞ > > ; Dz2 ðtÞ ¼ 0 9 Dx2 ðtÞ ¼ 0 > > > = Dy2 ðtÞ ¼ 0 > Dz1 ðtÞ ¼ Dðz2 ðtÞ z1 ðtÞÞ > > ; Dz2 ðtÞ ¼ Dðz1 ðtÞ z2 ðtÞÞ
with the initial conditions
ðf2 ðuÞ; ’2 ðu Þ; c1 ðuÞ; c2 ðuÞÞ 2 Cð½t ; 0; R4þ0 Þ; f2 ð0Þ > 0; ’2 ð0Þ > 0; ci ð0Þ > 0; i ¼ 1; 2: The remainder of this paper is arranged as follows. In Section 2, some preliminaries are provided. In Section 3, the global attractivity of preys-extinction periodic solution and the permanence of system (1.3) are investigated. In Section 4, numerical simulations are given to show the complicated dynamics of system (1.3). Finally, a brief discussion is given to conclude this work in Section 5. 2. The preliminaries Let R+ = [0, 1), R6þ ¼ fX 2 R6 jX 0g. Denote f = (f1, f2, f3, f4, f5, f6)T the map defined by the right hand of the first six equations of system (1.2). Let V : Rþ R6þ ! Rþ , then V is said to belong to class V0 if
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(1) V is continuous in ððn 1ÞT; ðn þ l 1ÞT R6þ and ððn þ l 1Þ T; nT R6þ , and for each X 2 R6þ , n 2 N, limðt;yÞ ! ððn1ÞT þ ;XÞ Vðt; yÞ ¼ Vððn 1ÞT þ ; XÞ and limðt;yÞ ! ððnþl1ÞT þ ;XÞ Vðt; yÞ ¼ Vððn þ l 1Þ T þ ; XÞ exist. (2) V is locally Lipschitzian in X.
Definition 2.1. Let V 2 V0, then for ðt; xÞ 2 ððn 1ÞT; ðn þ l 1ÞT R6þ [ ððn þ l 1ÞT; nT R6þ , the upper right derivative of V(t, X) with respect to the impulsive differential system (1.2) is defined as 1 Dþ Vðt; XÞ ¼ limþ sup ½Vðt þ h; X þ hf ðt; xÞÞ Vðt; XÞ: h h!0 The solution of system (1.2) is a piecewise continuous function X : Rþ ! R6þ , X(t) is continuous on ((n 1)T, (n + l 1)T] and ((n + l 1)T, nT], n 2 N and Xððn 1ÞT þ Þ ¼ limt ! ðn1ÞT þ XðtÞ and Xððn þ l 1ÞT þ Þ ¼ limt ! ðnþl1ÞT þ XðtÞ exist. The smoothness properties of f guarantee the global existence and uniqueness of solution of system (1.2), for the details see book (Lakshmikantham et al., 1989; Bainov and Simeonnov, 1989). Definition 2.2. System (1.2) is said to be permanent if there exist positive constants m and M, such that each positive solution (x1(t), x2(t), y1(t), y2(t), z1(t), z2(t)) of system (1.2) with initial conditions satisfies m xi(t) M, m yi(t) M, m zi(t) M(i = 1, 2) as t! 1.
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
(1.3) t ¼ ðn þ l 1ÞT
t ¼ nT
The following lemma is obvious.
Lemma 2.1. Let X(t) is a solution of system (1.2) with X(0+) 0, then X(t) 0 for all t 0. And further X(t) > 0, t > 0 if X(0+) > 0. We will use an important comparison theorem on impulsive differential equation. Lemma 2.2. (Bainov and Simeonov, 1993) Suppose V 2 V0. Assume that 8 þ > < D Vðt; XÞ gðt; Vðt; XÞÞ 1 Vðt; Xðt þ ÞÞ cn ðVðt; XÞÞ > 2 : þ Vðt; Xðt ÞÞ cn ðVðt; XÞÞ
t 6¼ nT; t 6¼ ðn þ l 1ÞT t ¼ ðn þ l 1ÞT t ¼ nT
where g : R+ R+ ! R is continuous in ððn 1ÞT; ðn þ l 1ÞT R6þ and ððn þ l 1ÞT; nT R6þ and for u 2 R+, n 2 N, limðt;vÞ ! ððn1ÞT þ ;uÞ gðt; vÞ ¼ gððn 1ÞT þ ; uÞ and limðt;vÞ ! ððnþl1ÞT þ ;uÞ gðt; vÞ ¼ gððn þ l
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L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123 i
1ÞT þ ; uÞ exist, cn ði ¼ 1; 2Þ: R+ ! R+ is non-decreasing. Let r(t) be maximal solution of the scalar impulsive differential equation 8 duðtÞ > > ¼ gðt; uðtÞÞ > > < dt 1 þ uðt Þ ¼ cn ðuðtÞÞ > 2 > þ > uðt Þ ¼ c > n ðuðtÞÞ : þ uð0 Þ ¼ u0
t 6¼ nT;
Remark 2.1. According to the expression in Zhao et al. (2011), we can derive that z1 and z2 are as follows z1 ¼
t 6¼ ðn þ l 1ÞT
b1 ð1 pÞð1 D ð1 2DÞeb2 T Þ b1 eb1 T þ b1 ð1 DÞeðb1 þb2 ÞT a1 ð1 p þ peb1 ð1lÞT eb1 T Þð1 ð1 DÞeb2 T Þ
t ¼ ðn þ l 1ÞT t ¼ nT z2 ¼ Dz1 +
existing on [0, 1). Then V(0 , X0) u0, implies that V(t, X(t)) r(t), t 0, where X(t) is any solution of system (1.2). Lemma 2.3. (Kuang, 1987) Consider the following delay differential equation
˙ ¼ a1 xðt t Þ a2 xðtÞ; xðtÞ
Here we claim that the expression of z2 (that is the expression of x2 in Zhao et al., 2011) is not right, therefore we revise the expression of z2 as z2 ¼
Dz1 : ð1 DÞ ð1 2DÞeb2 T
Remark 2.2. The above Lemmas show that
and assume that a1, a2, t > 0, a1 < a2 and x(t) > 0 for t t 0. Then p , 1
lim xðtÞ ¼ 0:
t!1
Next, we introduce two important lemmas, which can be seen in Zhao et al. (2011). For subsystem of (1.2), i.e., 9 8 dz1 ðtÞ > > > > = ¼ z ðtÞ½b a z ðtÞ 1 1 1 1 > > dt > > > > > dz ðtÞ > 2 > > ; > ¼ b2 z2 ðtÞ > > dt > ) < Dz1 ðtÞ ¼ pz1 ðtÞ > > > Dz2 ðtÞ ¼ 0 > > > ) > > > Dz1 ðtÞ ¼ Dðz2 ðtÞ z1 ðtÞÞ > > > > > : Dz2 ðtÞ ¼ Dðz1 ðtÞ z2 ðtÞÞ
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
t ¼ ðn þ l 1ÞT t ¼ nT
b2 T
1 ð1 DÞe eb1 T ðð1 DÞ ð1 2DÞeb2 T Þ
, the trivial equilibrium point (0,0) of system (2.1) is globally asymptotically stable. Lemma 2.5. (Zhao et al., 2011) If
0< p<1
is the critical harvesting effort. If p > p*, which denotes the subpopulation in patch 1 is overexploited, then the subpopulation in two patches will become extinction. In reality, if the subpopulation in patch 1 is heavily exploited, it is natural that the subpopulation in patch 1 will go to extinction. Since the subpopulation in patch 2 come from the diffusion of the subpopulation in patch 1, and then the subpopulation in patch 2 will also become extinction.
(2.1)
Lemma 2.4. (Zhao et al., 2011) If p>1
1 ð1 DÞeb2 T DÞ ð1 2DÞeb2 T Þ
eb1 T ðð1
1 ð1 DÞeb2 T eb1 T ðð1 DÞ ð1 2DÞeb2 T Þ
, system (2.1) has a T-periodic positive solution ðz1 ðtÞ; z2 ðtÞÞ, which is asymptotically stable. Where ðz1 ðtÞ; z2 ðtÞÞ can be expressed as:
Remark 2.3. If p < p*, system (2.1) has a asymptotically stable Tperiodic positive solution ðz1 ðtÞ; z2 ðtÞÞ. The result is useful and important because from the biological viewpoint, a globally stable positive periodic state means that the subpopulation in both patches will coexist. Remark 2.4. The positive T-periodic solution will dramatically depend on the harvesting time lT between two diffusion pulses, and the harvesting fraction p of population at each time. Obviously, the number of the population in two patches is decreasing functions with respect to l. From the biological viewpoint, under the condition of insuring population permanence (i.e., p < p*) in order to catch much population, we must harvest it as late as possible. 3. The dynamics In the previous section, we got the existence of the preysextinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ of the system (1.3). It is clear that the attractivity of preys-extinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ of system (1.3) is equivalent to the globally attractivity of the preys-extinction periodic solution
8 8 b1 z1 > > > > > > > < ðb a z Þeb1 ðtðn1ÞTÞ þ a z ; t 2 ððn 1ÞT; ðn þ l 1ÞT > > 1 1 1 1 1 < z ðtÞ ¼ 1 > ð1 pÞb1 z1 > > ; > : > > ðb1 a1 z1 Þeb1 ðtðn1ÞTÞ þ pa1 z1 eb1 ðtðnþl1ÞTÞ þ a1 ð1 pÞz1 > > > : z2 ðtÞ ¼ z2 eb2 ðtðn1ÞTÞ ; t 2 ððn 1ÞT; nT and b1 ð1 pÞð1 D ð1 2DÞeb2 T Þ b1 eb1 T þ b1 ð1 DÞeðb1 þb2 ÞT z1 ¼ a1 ð1 p þ peb1 ð1lÞT eb1 T Þð1 ð1 DÞeb2 T Þ Dz1 z2 ¼ : ð1 DÞ ð1 2DÞeb2 T
t 2 ððn þ l 1ÞT; nT (2.2)
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
ð0; 0; 0; 0; z1 ðtÞ; z2 ðtÞÞ of system (1.2). Therefore we will only devote to model (1.3). That is to say, for any positive solution (x2(t), y2(t), z1(t), z2(t)) of system (1.3), we have x2 ðtÞ ! 0;
y2 ðtÞ ! 0;
z1 ðtÞ ! z1 ðtÞ;
z2 ðtÞ ! z2 ðtÞ;
as t ! 1:
Now we investigate the global attractivity of the preysextinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ of system (1.3). Theorem 3.1. If the following conditions hold: 1 ð1 DÞeb2 T ; DÞ ð1 2DÞeb2 T Þ ðH2 Þ : a1 ev1 t 1 < v2 þ c1 z1 ; ðH1 Þ : p < 1
eb1 T ðð1
ðH3 Þ : a2 eg 1 t2 < g 2 þ c2 z2 eb2 T ; then system (1.3) has a preys-extinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ, which is globally attractive (for proof see Appendix A.) Remark 3.1. Theorem 3.1 shows that when the predator species (z1(t), z2(t)) approach a positive periodic stable state ðz1 ðtÞ; z2 ðtÞÞ under proper harvesting strategies, the prey species x2(t) and y2(t) become extinct under low birth rates of immature preys, or high death rates of immature and mature preys, or long maturation time of immature preys to mature preys, or large preys’ captured rates. Remark 3.2. Theorem 3.1 tells us that if we guarantee that (H1)– (H3) hold, then the prey species in both patches must be extinction and the predator populations in both patches will coexist. Generally, if the prey species is extinct, as a direct result, the predator species should be also extinct as its survival is dependent on the prey species. However, from Theorem 3.1, we can see that the preys in both patches are extinct, whereas the predator populations in two patches are permanent. Why? In fact, the positiveness of the growth rate b1 of predator subpopulation in patch 1 ensures the predator subpopulation in patch 1 can feed on other food to survive besides the prey population x2(t). Since the predator subpopulation in patch 2 come from the diffusion of the predator subpopulation in patch 1, and then the predator subpopulation in patch 2 will be also permanent. Corollary 3.1. If the following conditions hold: 1 ð1 DÞeb2 T ; eb1 T ðð1 DÞ ð1 2DÞeb2 T Þ v1 t 1 < v2 ; ðH2 Þ : a1 e ðH3 Þ : a2 eg 1 t2 < g 2 ;
ðH1 Þ : p > 1
then system (1.3) is extinction. Remark 3.3. Corollary 3.1 reveals that the prey species x2(t) and y2(t) become extinct under low birth rates of immature preys, or high death rates of immature and mature preys, or long maturation time of immature preys to mature preys, then the predator populations in two patches will be also extinct as overexploiting. Theorem 3.1 proves the preys-extinction periodic solution ð0; 0; z1 ðtÞ; z2 ðtÞÞ is globally attractive, or say, the prey populations in both patches will be extinct totally. Whereas from the point of ecological balance and saving resources, we hope all populations may coexist in an ecosystem. That is to say, the ecosystem should be permanence. From biological point of view, permanence of species maintains biological balance of an ecosystem which implies the survival of all populations of the system and none of them facing extinction or growing infinitely in future time. Mathematically, permanence of a system means that strictly
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positive solutions have lower and upper bound. Concerning the permanence of species for system (1.2), firstly, we will show that all solutions of (1.2) are uniformly ultimately bounded. Theorem 3.2. There exists a constant M > 0 such that xi(t) M, yi(t) M, zi(t) M(i = 1, 2) for each solution of (1.2) with t large enough (for proof see Appendix A.) Remark 3.4. Let us analyze the biological meaning of Theorem 3.2. In reality, if the predator subpopulation in patch 1 is not ultimately bounded, then the density of predator subpopulation in patch 1 will expand unlimitedly. By the diffusion of predator population between two patches, the density of predator subpopulation in patch 2 will also expand unlimited. Since the predation rates of predator subpopulation in two patches for prey species are strictly positive (i.e., c1, c2 > 0), then the prey species in both patches will become extinct because of the massive preying by the predator species. And then the predator population will feed on the other foods in turn. Also, the other foods will be eaten up because of the massive preying. As an opposite result, the predator species in patch 1 can starve to death to lead it to die out. Since the subpopulation in patch 2 come from the diffusion of the subpopulation in patch 1, and then the subpopulation in patch 2 will also become extinction. Finally, we investigate the permanence of the system (1.2). Theorem 3.3. The system (1.2) is permanent if (H1) and the following conditions hold ðH4 Þa1 ev1 t1 > v2 þ d1 M þ c1 q1 ; ðH5 Þa2 eg 1 t 2 > g 2 þ d2 M þ c2 q2 ; where M, qi(i = 1, 2) are defined in Theorem 3.2 and (A.14), respectively (for proof see Appendix A.) Remark 3.5. From Theorem 3.1, we know that, when there are no prey species x2(t) and y2(t), the predator species (z1(t), z2(t)) will reach a positive periodic stable state ðz1 ðtÞ; z2 ðtÞÞ under proper harvesting strategies (p < p*). While there are prey species x2(t) and y2(t), Theorem 3.3 shows that if ðz1 ; z2 Þ can guarantee that prey species x2(t) and y2(t) satisfies conditions (H4)–(H5), then prey species x2(t) and y2(t) will be also permanent. Obviously, z1 ; z2 are decreasing functions with respect to l. From the viewpoint of biology, under the condition of insuring predator population permanence (i.e., p < p*) in order to ensure the permanence of the prey species and catch much the predator population, we should harvest it as late as possible. Remark 3.6. Theorem 3.3 reveals that if we largen the birth rates of immature preys, or decrease the death rates of immature and mature preys, or shorten the maturation time of immature preys, or decrease the preys’ captured rates, then we can control the prey populations to become permanent from the quondam extinction. 4. Numerical analysis In this section, the solution of system (1.3) with initial condition in the first quadrant is obtained numerically for biologically feasible range of parametric value provides two essential dynamical behavior of system (1.3). To study the dynamics, we suppose the system has the following hypothetical parameter values: a1 = 8, v1 = 1, v2 = 1.2, d1 = 1, c1 = 2, a2 = 8, g1 = 1, g2 = 1.2, d2 = 1, c2 = 2, b1 = 1, a1 = 1, k1 = 0.9, b2 = 1, k2 = 0.9, p = 0.2, D = 0.3, T = 1, l = 0.25 and initial values (0.2, 0.2, 0.2, 0.2). By above analysis, we
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
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(a)
0.18
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z2(t)
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y 2(t)
x2(t)
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t
60
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z1 (t)
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Fig. 1. Two prey-extinction periodic solution in both patches with t1 = 1.6 and t2 =2. (a) Time-series of x2(t);(b) time-series of y2(t);(c) time-series of z1(t); (d) time-series of z2(t); and (e) the phase portrait of the predator.
know the time delay t1 and t2 are two of the key parameters that directly impact the dynamics of the system. So, in our numerical experiments the only difference between conditions of Figs. 1–4 is the values of t1 and t2. In Fig. 1, we set t1 = 1.6 and t2 = 2. They are observed that these parameter values satisfy the conditions of
Theorem 3.1. Numerical simulations for the global attractivity of preys-extinction periodic solution are shown in Fig. 1. When we choose t1 = 1.6 and t2 = 0.5, then the predator coexists with the prey in the second patch, and the prey is extinct in the first patch (see Fig. 2). If t1 = 0.5 and t2 = 2, then the prey in the second patch
(a)
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x2(t)
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z1(t)
1
1.2
1.4
1.6
Fig. 2. (a) A prey-extinction periodic solution in the first patch with t1 = 1.6 and t2 = 0.5. (a) Time-series of x2(t);(b) time-series of y2(t); (c) time-series of z1(t); (d) time-series of z2(t); and (e) the phase portrait of the predator.
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
(a)
1
117
(b)
0.2 0.18
0.9
0.16 0.8 0.14 0.12 y2(t)
x2(t)
0.7 0.6
0.1 0.08
0.5
0.06 0.4 0.04 0.3 0.2
0.02 0
20
40
60
t
(c)
2.2
80
0
100
20
40
60
t
(d)
0.8
2
0
80
100
(e)
0.8
0.7
0.7
0.6
0.6
0.5
0.5
1.8
z2(t)
z1(t)
1.4 1.2 1
z2(t)
1.6
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.8 0.6 0.4 0.2
0
20
40
t
60
80
0
100
0
20
40
t
60
80
0
100
0.5
1
1.5
z1(t)
2
Fig. 3. (a) A prey-extinction periodic solution in the second patch with t1 = 0.5 and t2 = 2. (a) Time-series of x2(t); (b) time-series of y2(t); (c) time-series of z1(t); (d) time-series of z2(t); (e) and the phase portrait of the predator.
(a)
1
(b)
1.6
0.9
1.4
0.8
1.2
y2(t)
x2(t)
0.7 0.6
1 0.8
0.5 0.6
0.4
0.4
0.3 0.2
0
20
40
60
t
(c)
2.5
80
0
20
40
60
t
(d)
3
2
0.2
100
80
100
(e)
3
2.5
2.5
2
2 z2(t)
z2(t)
z1(t)
1.5 1.5
1.5
1
0.5
0
0
20
40
t
60
80
100
1
1
0.5
0.5
0
0
20
40
t
60
80
100
0
0
0.5
1
z1(t)
1.5
2
2.5
Fig. 4. (a) Permanence with t1 = 0.5 and t2 = 0.5. (a) Time-series of x2(t); (b) time-series of y2(t); (c) time-series of z1(t); (d) time-series of z2(t); and (e) the phase portrait of the predator.
is extinct, the one in the first patch will coexists with predator (see Fig. 3). Again, If we choose t1 = 0.5, t2 = 0.5, then system (1.3) will satisfy Theorem 3.3. It is obviously that system (1.3) is permanent (see Fig. 4).
5. Conclusions In this work, we have investigated a delayed predator–prey model with stage structure for preys, and focused on impulsive
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diffusion and impulsive harvesting for predator at different fixed moments. In this model, each prey species was divided by two stages and was restricted to a single patch, and predator can diffuse between two patches. We have proved that the preys-extinction periodic solution of system (1.2) is globally asymptotically stable by using the comparison theorem of delayed impulsive differential equations and some analysis techniques. The sufficient conditions of permanence for the considered system were also established and stated. From Theorem 3.1, we can conclude that if the death rates of immature and mature preys, the maturation time of immature preys to mature preys and preys’ captured rates are high, the birth rates of immature preys are low, then system (1.2) admits a unique preys-extinction periodic solution. That is to say, the prey populations in two patches are extinct under these conditions, whereas the predator population will be permanent under proper harvesting strategies. On the contrary, if the predator population in patch 1 is overexploited, then both the preys and predator will be extinct under low birth rates of immature preys, or high death rates of immature and mature preys, or long maturation time of immature preys to mature preys. From biological point of view, we hope to keep biological balance so that all species in an ecosystem can coexist. And then Theorem 3.3 implies that if we largen the birth rates of immature preys, or decrease the death rates of immature and mature preys, or shorten the maturation time of immature preys, or decrease the preys’ captured rates, then the prey populations and predator populations in two patches will coexist under proper harvesting strategies. The results showed that the birth rates of immature preys, the death rates of immature and mature preys, the maturation time of immature preys to mature preys and the preys’ captured rates play an important part
in determining the preys-extinction and permanence of system (1.2). Numerical simulations by choosing the maturation time of immature preys to mature preys as the key parameters were carried out to illustrate the findings of Theorems 3.1 and 3.3. We can observed that if the maturation time of immature preys are long (t1 = 1.6, t2 = 2) or short (t1 = 0.5, t2 = 0.5), then the preys become extinct or preys and predators coexist in both patches. Whereas when the maturation time of immature prey is long in one patch and short in other patch (t1 = 1.6, t2 = 0.5 or t1 = 0.5, t2 = 2), then the prey in one patch become extinct, while it survives in other patch. Similarly, the results can be also extended by considering the critical parameters. On the other hand, our results also show that the best time of harvesting the predator population in patch 1 is at the end of the diffusion period T, that is, before and near the time of the diffusion under the condition of renewable resource. From the viewpoint of biology, under the condition of insuring predator and preys permanence, in order to catch much the predator population, we must harvest it as late as possible. Therefore it is interesting and helpful for people to provide tactical basis for the biological resource management. Acknowledgements The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this paper. This work was supported by the Program for New Century Excellent Talents in University (NCET-10-0097), the NSFC Tianyuan Foundation (Grant No. 11226256) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY13A010010).
Appendix A Proof of Theorem 3.1. Let (x2(t), y2(t), z1(t), z2(t)) be any solution of system (1.3), we can obtain from the third and the fourth equation of system (1.3) that dz1 ðtÞ z1 ðtÞ½b1 a1 z1 ðtÞ; dt dz2 ðtÞ b2 z2 ðtÞ: dt Now we consider the following impulsive differential equations 8 9 du1 ðtÞ > > > > ¼ u ðtÞ½b a u ðtÞ > = 1 1 1 1 > > dt > > t 6¼ nT; t 6¼ ðn þ l 1ÞT > > du2 ðtÞ > > > > ; > > dt ¼ b2 u2 ðtÞ > > > ) < Du1 ðtÞ ¼ pu1 ðtÞ t ¼ ðn þ l 1ÞT > > > > Du2 ðtÞ ¼ 0 > > > ) > > > Du1 ðtÞ ¼ Dðu2 ðtÞ u1 ðtÞÞ > > t ¼ nT > > > > : Du2 ðtÞ ¼ Dðu1 ðtÞ u2 ðtÞÞ
(A.1)
By Lemmas 2.4 and 2.5, the periodic solution ðu1 ðtÞ; u2 ðtÞÞ of (A.1) is globally asymptotically stable. By using the comparison theorem of impulsive differential equation [Lemma 2.2], we have (
z1 ðtÞ u1 ðtÞ ! u1 ðtÞ ¼ z1 ðtÞ z2 ðtÞ u2 ðtÞ ! u2 ðtÞ ¼ z2 ðtÞ
as t! 1, where ðz1 ðtÞ; z2 ðtÞÞ is the solution of system (2.1). Then there exists a positive constant e1 small enough and a positive integer n1 large enough such that z1 ðtÞ > z1 ðtÞ e1 z1 e1 , d1 ; z2 ðtÞ > z2 ðtÞ e1 z2 eb2 T e1 , d2 ; for all nT < t = (n + 1)T,
n > n 1.
(A.2)
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
119
Next, form the first equation and the second equation of system (1.3), we have dx2 ðtÞ a1 ev1 t 1 x2 ðt t 1 Þ ðv2 þ c1 d1 Þx2 ðtÞ; dt dy2 ðtÞ a2 eg 1 t 2 y2 ðt t 2 Þ ðg 2 þ c2 d2 Þy2 ðtÞ; dt
(A.3)
for all t > nT + t. On the other hand, by (H2) and (H3), for the above e1 small enough, we have
a1 ev1 t1 < v2 þ c1 d1 ; a2 eg 1 t2 < g 2 þ c2 d2 : Considering the comparison system of (A.3) as follows dv1 ðtÞ ¼ a1 ev1 t1 v1 ðt t 1 Þ ðv2 þ c1 d1 Þv1 ðtÞ; dt dv2 ðtÞ ¼ a2 eg 1 t 2 v2 ðt t 2 Þ ðg 2 þ c2 d2 Þv2 ðtÞ; dt By Lemma 2.3, one can easily obtain that lim vi ðtÞ ¼ 0:ði ¼ 1; 2Þ
t!1
Since x2 ðsÞ ¼ v1 ðsÞ ¼ ’2 ðsÞ > 0 and y2 ðsÞ ¼ v2 ðsÞ ¼ ’4 ðsÞ > 0 for all s 2 [S t, 0], using the comparison theorem, we can obtain that x2(t) ! 0 and y2(t) ! 0 as t! 1. Without loss of generality, we assume that there exists a positive constant e such that 0 < x2 ðtÞ < e;
0 < y2 ðtÞ < e
for
all
t 0:
(A.4)
From the third and the fourth equation of (1.3) and (A.4), we have dz1 ðtÞ z1 ðtÞ½b1 þ k1 c1 e a1 z1 ðtÞ dt dz2 ðtÞ ðb2 k2 c2 eÞz2 ðtÞ: dt Denote b01 ¼ b1 þ k1 c1 e; b02 ¼ b2 k2 c2 e, Then we can consider the following comparison impulsive differential system 8 9 dw1 ðtÞ > > 0 > > ¼ w ðtÞ½b a w ðtÞ > = 1 1 1 1 > > dt > > > > > > dw2 ðtÞ > > ; > ¼ b02 w2 ðtÞ > > dt > > ) < Dw1 ðtÞ ¼ pw1 ðtÞ > > > Dw2 ðtÞ ¼ 0 > > > > ) > > > Dw1 ðtÞ ¼ Dðw2 ðtÞ w1 ðtÞÞ > > > > > > : Dw2 ðtÞ ¼ Dðw1 ðtÞ w2 ðtÞÞ
t 6¼ nT;
t 6¼ ðn þ l 1ÞT
(A.5)
t ¼ ðn þ l 1ÞT
t ¼ nT
By Lemmas 2.4 and 2.5 again, the unique positive periodic solution of system (A.5) is 8 8 b01 w1 > > > > > ; t 2 ððn 1ÞT; ðn þ l 1ÞT > > > 0 a w Þeb01 ðtðn1ÞTÞ þ a w < > ðb > 1 1 1 1 1 > < w1 ðtÞ ¼ > ð1 pÞb01 w1 > > > ; > 0 : 0 > b01 ðtðn1ÞTÞ > ðb1 a1 w1 Þe þ pa1 w1 eðb1 ðtðnþl1ÞTÞ þ a1 ð1 pÞz1 > > > > : 0 w2 ðtÞ ¼ w2 eb2 ðtðn1ÞTÞ ; t 2 ððn 1ÞT; nT
t 2 ððn þ l 1ÞT; nT
where 0
w1 ¼
w2 ¼
0
0
0
b01 ð1 pÞð1 D ð1 2DÞeb2 T Þ b01 eb1 T þ b01 ð1 DÞeðb1 þb2 ÞT a1 ð1 p þ pe Dw1 0
1 D ð1 2DÞeb2 T
b01 ð1lÞT
b02 T
Þð1 ð1 DÞe
Þ
;
:
Hence for arbitrary small e1 > 0 and any sufficiently large integer n2 > n1 such that z1 ðtÞ < w1 ðtÞ þ e1 ; z2 ðtÞ < w2 ðtÞ þ e1 ;
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L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
for t > n2T. Let e ! 0, then w1 ðtÞ ! z1 ðtÞ;
w2 ðtÞ ! z2 ðtÞ
and z1 ðtÞ < z1 ðtÞ þ e1 ;
z2 ðtÞ < z2 ðtÞ þ e1 :
(A.6)
From (A.2) and (A.6), we have z1 ðtÞ ! z1 ðtÞ;
z2 ðtÞ ! z2 ðtÞ:
This ends the proof.
&
Proof of Theorem 3.2. Define V(t, X(t)) such that Vðt; XðtÞÞ ¼ k1 x1 ðtÞ þ k1 x2 ðtÞ þ k2 y1 ðtÞ þ k2 y2 ðtÞ þ z1 ðtÞ þ z2 ðtÞ then V 2 V0. We calculate the upper right derivative of V(t, X) along a solution of system (1.2) and get the following impulsive differential equation 8 þ D VðtÞ þ LVðtÞ ¼ k1 ða1 v2 þ LÞx2 ðtÞ k1 d1 x22 ðtÞ > > > > > > > k1 ðv1 LÞx1 ðtÞ þ k2 ða2 g 2 þ LÞy2 ðtÞ > > > t 6¼ ðn þ l 1ÞT; > < k d y2 ðtÞ k ðg LÞy ðtÞ 2 2 2
> > > > > > > > Vðt þ Þ VðtÞ > > > : Vðt þ Þ VðtÞ
2
t 6¼ nT
1
1
þ ðb1 þ L a1 z1 ðtÞÞz1 ðtÞ ðb2 LÞz2 ðtÞ t ¼ ðn þ l 1ÞT t ¼ nT
Let 0 < L = min {v1, g2, b2}, then D+V(t) + LV(t) is bounded. Select L1 and L2 such that
Dþ VðtÞ L1 VðtÞ þ L2 þ
Vðt Þ VðtÞ
VðtÞ
t 6¼ nT
t ¼ ðn þ l 1ÞT
Vðt þ Þ VðtÞ where L1,
t 6¼ n þ l 1ÞT; t ¼ nT
L2 are two positive constant. According to Lemma 2.2, we have
L2 L2 Vð0þ Þ expðL1 tÞ þ L1 L1
where t 2 ((n S 1)T, nT]. Hence lim VðtÞ
t!1
L2 L1
Therefore V(t, X(t)) is ultimately bounded, and we know that each positive solution of system is uniformly ultimately bounded. This completes the proof. & Proof of Theorem 3.3. Suppose X(t) = (x1(t), x2(t), y1(t), y2(t), z1(t), z2(t)) is any solution of the system (1.2) with X(0+) > 0. From Theorem 3.2, we assume that xi(t) = M, yi(t) = M and zi(t) = M(i = 1, 2) with t I 0. From (A.2), we have z1 ðtÞ > z1 e1 , d1 and z2 ðtÞ > z2 eb2 T e1 , d2 for all t large enough. Thus we only need to find positive constants mi and ni(i = 1, 2) such that xi(t) I mi and yi(t) I ni(i = 1, 2) for t large enough. The second and fourth equation of system (1.2) can be rewritten as dx2 ðtÞ d ¼ ½a1 ev1 t 1 v2 d1 x2 ðtÞ c1 z1 ðtÞx2 ðtÞ a1 ev1 t1 dt dt dy2 ðtÞ d ¼ ½a2 eg 1 t 2 g 2 d2 y2 ðtÞ c2 z2 ðtÞy2 ðtÞ a2 eg 1 t2 dt dt By (A.7), we defined Q1(t) and Q2(t) as Q 1 ðtÞ ¼ x2 ðtÞ þ a1 ev1 t1 Q 2 ðtÞ ¼ y2 ðtÞ þ a2 eg 1 t2
Z
t
Z tt t 1 tt 2
x2 ðsÞds; y2 ðsÞds:
Z Z
t
x2 ðsÞds; tt 1
(A.7)
t
tt 2
y2 ðsÞds
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
121
Then by Theorem 3.2, we have Q 1 ðtÞ M½1 þ a1 t 1 ev1 t1 ;
(A.8)
Q 2 ðtÞ M½1 þ a2 t 2 eg 1 t 2 ; as t! 1. Calculating the derivative of Q1(t) and Q2(t) along the solution of system (1.2), we get dQ 1 ðtÞ ¼ ½a1 ev1 t1 v2 d1 x2 ðtÞ c1 z1 ðtÞx2 ðtÞ; dt dQ 2 ðtÞ ¼ ½a2 eg 1 t2 g 2 d2 y2 ðtÞ c2 z2 ðtÞy2 ðtÞ: dt According Theorem 3.2, one can obtain that
(A.9)
dQ 1 ðtÞ ½a1 ev1 t 1 v2 d1 M c1 z1 ðtÞx2 ðtÞ; dt dQ 2 ðtÞ ½a2 eg 1 t2 g 2 d2 M c2 z2 ðtÞy2 ðtÞ: dt According to hypotheses (H4) and (H5), one can choose a e3 > 0 be small enough such that
a1 ev1 t1 > v2 þ d1 M þ c1 ½q1 þ e3 ; a2 eg 1 t2 > g 2 þ d2 M þ c2 ½q2 þ e3
(A.10)
(A.11)
where q1 and q2 are defined in (A.14). Now, we will prove that for any t0 > 0, it is impossible that x2 ðtÞ < m2 and y2 ðtÞ < n2 for all t > t0. It follows from the fifth and sixth equations of system (1.2) that for all t > t0 dz1 ðtÞ < z1 ðtÞ½b1 þ k1 c1 m2 a1 z1 ðtÞ dt dz2 ðtÞ < ½b2 k2 c2 n2 z2 ðtÞ dt Consider the following auxiliary impulsive equation 8 9 d˜z1 ðtÞ > > > ¼ z˜1 ðtÞ½b1 þ k1 c1 m2 a1 z˜1 ðtÞ > > = > > dt > > t 6¼ nT; t 6¼ ðn þ l 1ÞT > > > > d˜ z ðtÞ > > ; > 2 ¼ ½b k c n ˜ z ðtÞ 2 2 2 2 > 2 > dt > > ) < Dz˜1 ðtÞ ¼ p˜z1 ðtÞ t ¼ ðn þ l 1ÞT > > > > Dz˜2 ðtÞ ¼ 0 > > > ) > > > Dz˜1 ðtÞ ¼ Dð˜z2 ðtÞ z˜1 ðtÞÞ > > t ¼ nT > > > > : Dz˜2 ðtÞ ¼ Dð˜z1 ðtÞ z˜2 ðtÞÞ
(A.12)
By Lemmas 2.4 and 2.5, we obtain the unique global asymptotically periodic solutions of system (A.12) as follows 8 8 b001 z˜1 > > > > > ; t 2 ððn 1ÞT; ðn þ l 1ÞT > > > b00 ðtðn1ÞTÞ 00 < > ðb1 a1 z˜1 Þe 1 þ a1 z˜1 > > < z˜ ðtÞ ¼ 1 > ð1 pÞb001 z˜1 > > > > ; 00 00 : > > ðb001 a1 z˜1 Þeb1 ðtðnþl1ÞTÞ þ pa1 z˜1 eb1 ðtðn1ÞTÞ þ a1 ð1 pÞ˜z1 > > > > : 00 z˜2 ðtÞ ¼ z˜2 eb2 ðtðn1ÞTÞ ; t 2 ððn 1ÞT; nT
t 2 ððn þ l 1ÞT; nT
and 00
z˜1 ¼ z˜2 ¼
00
00
00
b001 ð1 pÞð1 D ð1 2DÞeb2 T Þ b01 eb1 T þ b001 ð1 DÞeðb1 þb2 ÞT b001 ð1lÞT
a1 ð1 p þ pe D˜z1
b002 T
Þð1 ð1 DÞe
Þ
; (A.13)
00 : 1 D ð1 2DÞeb2 T
where b001 ¼ b1 þ k1 c1 m2 ; b002 ¼ b2 k2 c2 n2 . In view of the comparison theorem of impulsive differential equation (Lemma 2.2), we know that there exists t1(> t0 + t1) and t2(> t0 + t2), for sufficiently small e3, we have z1 ðtÞ < z˜1 ðtÞ þ e3
for all t t 1 ;
z2 ðtÞ < z˜2 ðtÞ þ e3
for all t t 2 :
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L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
Thus
z˜1 ðtÞ < ð1 pÞ˜z1 eðb1 þk1 c1 m2 ÞT , q1 ; z˜2 ðtÞ < z˜2
, q2 :
(A.14)
Thus z1 ðtÞ < q1 þ e3 , s 1 ;
for all t t 1 ;
z2 ðtÞ < q2 þ e3 , s 2 ;
for all
t t2 :
(A.15)
From (A.11), we get
a1 ev1 t1 > v2 þ d1 M þ c1 s 1 ; a2 eg 1 t2 > g 2 þ d2 M þ c2 s 2 : Then dQ 1 ðtÞ > ½a1 ev1 t1 v2 d1 M c1 s 1 x2 ðtÞ; dt dQ 2 ðtÞ > ½a2 eg 1 t 2 g 2 d2 M c2 s 2 y2 ðtÞ: dt m m m Let xm 2 ¼ mint 2 ½t1 ;t1 þt 1 x2 ðtÞ and y2 ¼ mint 2 ½t 2 ;t 2 þt 2 y2 ðtÞ. We will show that x2 ðtÞ x2 for all t I t1, y2 ðtÞ y2 for all t I t2. Otherwise, m there are T1, T2 > 0 such that x2 ðtÞ xm for t 2 ½t ; t þ t þ T ; x ðt þ t þ T Þ ¼ x and ((dx (t + t + T ))/(dt)) = 0 and y2 ðtÞ ym 1 1 1 1 2 1 1 1 2 1 1 1 2 2 2 for m t 2 ½t 2 ; t 2 þ t 2 þ T 2 ; y2 ðt 2 þ t 2 þ T 2 Þ ¼ y2 and ((dy2(t2 + t2 + T2))/(dt)) = 0. Then from the second and the fourth equation of system (1.2) and (A.14), we easily verified that
dx2 ðt 1 þ t 1 þ T 1 Þ ¼ a1 ev1 t1 x2 ðt 1 þ T 1 Þ v2 x2 ðt 1 þ t 1 þ T 1 Þ d1 x22 ðt 1 þ t 1 þ T 1 Þ c1 x2 ðt 1 þ t 1 þ T 1 Þz1 ðt 1 þ t 1 þ T 1 Þ dt ½a1 ev1 t 1 v2 d1 M c1 s 1 xm 2 >0 dy2 ðt 2 þ t 2 þ T 2 Þ ¼ a2 eg 1 t2 y2 ðt 2 þ T 2 Þ g 2 y2 ðt 2 þ t 2 þ T 2 Þ d2 y22 ðt 2 þ t 2 þ T 2 Þ c2 y2 ðt 2 þ t 2 þ T 2 Þz2 ðt 2 þ t 2 þ T 2 Þ dt ½a2 eg 1 t 2 g 2 d2 M c2 s 2 ym 2 >0
(A.16)
m which is a contradiction. Hence, we have x2 ðtÞ xm 2 > 0 for all t I t1 and y2 ðtÞ y2 > 0 for all t I t2. Again from (A.15), we have
dQ 1 ðtÞ > ½a1 ev1 t1 v2 d1 M c1 s 1 xm for all t t 1 ; 2; dt dQ 2 ðtÞ > ½a2 eg 1 t 2 g 2 d2 M c2 s 2 yn2 ; for all t t 2 ; dt which implies that Q1(t)! 1 , Q2(t) ! 1 as t! 1. This is a contradiction to (A.8). Therefore, for any positive constant t0, the inequality m x2 ðtÞ < xm 2 and y2 ðtÞ < y2 cannot hold for all t I t0. There are the following two cases. m Case 1. If x2 ðtÞ xm 2 and y2 ðtÞ y2 hold for all t large enough, then our aim reach. Otherwise, we consider case 2. Case 2. If x2(t) is oscillatory about m2 and y2(t) is oscillatory about n2 . Setting m2 ðv2 þd1 m þc1 MÞt1 2 ; m2 e ; 2 n n2 ¼ min 2 ; n2 eðg 2 þd2 n2 þc2 MÞt2 2
m2 ¼ min
In the following, we shall prove that x2(t) I m2 and y2(t) I n2. Let t1 ; t2 > 0. Obviously, there exist two positive constants j1 and j2 such that x2 ðt1 Þ ¼ x2 ðt1 þ j1 Þ ¼ m2 and x2 ðtÞ < m2 for t1 < t < t1 þ j1 ; y2 ðt2 Þ ¼ y2 ðt2 þ j2 Þ ¼ n2 and y2 ðtÞ < n2 for t2 < t < t2 þ j2 ; where t1 ; t2 is sufficiently large so that the inequality z1(t) < q1 holds true for t1 < t < t1 þ j1 and z2(t) < q2 holds true for t2 < t < t2 þ j2 . Since x2(t) and y2(t) are continuous and bounded and are not effected by impulses, we deduce that x2(t) and y2(t) are uniformly continuous. Hence there exist two constants T1, T2(with 0 < T1 < t1, 0 < T2 < t2 and T1 is independent of the choice of t1 , T2 is independent of the choice of t2 , respectively) such that x2 ðt1 Þ > m2 =2 for all t1 < t t1 þ T 1 and y2 ðt2 Þ > n2 =2 for all t2 < t t1 þ T 2 . If j1 = T1 and j2 = T2, our aim is obtained. If T1 < j1 = t1 and T2 < j2 = t2, from the second equation and fourth equation of (1.2), we have that ððdx2 ðtÞÞ=ðdtÞÞ v2 x2 ðtÞ d1 x22 ðtÞ c1 x2 ðtÞz1 ðtÞ; x2 ðt1 Þ ¼ m2 for all t1 < t t1 þ j1 and ððdy2 ðtÞÞ=ðdtÞÞ g 2 y2 ðtÞ d2 y22 ðtÞ c2 y2 ðtÞz2 ðtÞ for all t2 < t t2 þ j2 ; y2 ðt2 Þ ¼ n2 . According to the hypothesis, it is easy to get that ððdx2 ðtÞÞ=ðdtÞÞ ðv2 þ d1 x2 þ c1 MÞx2 ðtÞ for all t1 < t t1 þ j1 t1 þ t 1 and and ððdy2 ðtÞÞ=ðdtÞÞ ðg 2 þ d2 n2 þ c2 MÞy2 ðtÞ for all t2 < t t2 þ j2 t2 þ t 2 . Then we conclude that x2 ðtÞ m2 eðv2 þd1 x2 þc1 MÞt1 and ðg 2 þd2 n2 þc2 MÞt 2 y2 ðtÞ n2 e . It is clear that x2(t) I m2 for all t1 < t t1 þ j1 and y2(t) I n2 for t2 < t t2 þ j2 . If j1 I t1, j2 I t2, then we get that x2(t) I m2 for all t1 < t t1 þ t 1 and y2(t) I n2 for t2 < t t2 þ t 2 . By continuing the same arguments, one can easily obtain that we can obtain x2(t) I m2 for t1 þ t 1 < t t1 þ j1 and y2(t) I n2 for t2 þ t 2 < t t2 þ j2 . Since the interval ½t1 ; t1 þ j1 and ½t2 ; t2 þ j2 are arbitrarily chosen, we get that x2(t) I m2 and y2(t) I n2 for t large enough.
L. Yang, S. Zhong / Ecological Complexity 19 (2014) 111–123
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Next, from the first and the third equation of system (1.2), we get dx1 ðtÞ a1 ðm2 a1 ev1 t1 MÞ v1 x1 ðtÞ dt dy1 ðtÞ ¼ a2 ðn2 a2 eg 1 t 2 MÞ g 1 y1 ðtÞ dt By integration, it is obviously that for sufficiently small e4 > 0,
a1 ðm2 a1 ev1 t1 MÞ e4 , m1 ; v1 a2 ðn2 a2 eg 1 t2 MÞ e4 , n1 : y1 ðtÞ g1 Take m = min {mi, ni, di}, then xi(t) I m, yi(t) I m, zi(t) I m(i = 1, 2) hold true as t! 1. Together with Theorem 3.2, we can easily derive
x1 ðtÞ
that system (1.2) is permanent. The proof is completed.
&
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