# An Impulsive Periodic Single-Species Logistic System with Diffusion

Authors: Chenxue Yang; Mao Ye; Zijian Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.


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## Body

## 1. Introduction

In the practical world, on the one hand, owing to natural enemy, severe competition, or deterioration of the patch environment, species dispersal in two or more patches becomes one of the most prevalent phenomena of nature. Many empirical works and monographies on population dynamics in a spatial heterogeneous environment have been done (see [1–9] and the references cited therein). On the other hand, many natural and man-made factors (e.g., fire, drought, flooding deforestation, hunting, harvesting, breeding, etc.) always lead to rapid decrease or increase of population number at fixed moment. Such sudden changes can often be characterized mathematically in the form of impulses. With the development of the theory of impulsive differential equations [10], various population dynamical models of impulsive differential equations have been proposed and studied extensively. For example, many important and interesting results on the permanence, persistence, extinction, global stability, the existence of positive periodic solutions, bifurcation and dynamical complexity, and so forth can be found in [11–17] and the references cited therein.Although considerable researches on the dispersal and impulses of species have been reported in the literature, there are few papers that investigate the dynamical behavior of population systems under the circumstances in which both dispersal and impulse exist. However, dispersal species which undergoes impulses is also one of the most prevalent phenomena of nature. In our previous paper [18], an impulsive periodic predator-prey system with diffusion is studied, and some conditions for the permanence, extinction, and existence of a unique globally stable periodic solution are established. In this paper, we will present and study a single-species periodic logistic system with impulses and dispersal in n different patches. Our model takes the form
(1)x˙i(t)=xi(t)(ri(t)-ai(t)xi(t))+∑j=1n‍Dij(t)(xj(t)-xi(t)),t≠tk,xi(tk+)=hikxi(tk),i=1,2,…,n,k=1,2,…,
where ri(t) and ai(t)(i∈I={1,2,…,n}) represent the intrinsic growth rates and the density-dependent coefficients in patch i, respectively. Dij(t)⩾0(i,j∈I) denotes the dispersal rate of the species from patch j to patch i. hikxi(tk)(i∈I) is the regular pulse at time tk of species x in patch i. Throughout this paper, we always assume the following.(C1)
ri(t),ai(t), and Dij(t)(i,j∈I) are continuously periodic functions with common period T, defined on R+=[0,∞) and ai(t)⩾0,D2⩾Dij(t)⩾D1>0(i≠j),Dii(t)≡0 for all i,j∈I and t∈R+.(C2)
hik>0 for all i∈I,k=1,2,… and there exists a positive integer q such that tk+q=tk+T and hi(k+q)=hik for any i∈I.The organization of this paper is as follows. In the next section, some sufficient conditions for the permanence and extinction of system (1) are obtained. In Section 3, conditions for the existence of a unique globally attractively positive periodic solution are also established. Finally, some numerical simulations are proposed to illustrate the feasibility of our results and discuss the model further.


## 2. Permanence and Extinction

In this section, applying inequality estimation technique, we get some sufficient conditions on the permanence and extinction of system (1).Theorem 1.
There exists a positive constantM such that limsupt→∞xi(t)<M for all i∈I if
(2)∫0Ta(t)dt>0,
where a(t)=mini∈I{ai(t)}.Proof.
Lethk=maxi∈I{hik} for any k=1,2,…, then we have hk+q=hk and there exists a positive constant H such that function
(3)|h(t,μ)|=|∑t⩽tk<t+μ‍lnhk|⩽∑k=1q‍|lnhk|⩽H
for all t∈R+ and μ∈[0,T]. Choose r(t)=maxi∈I{ri(t)}, and r(t) is bounded for all t∈R+. Then from conditions (2) and (3), we have two positive constants τ and δ such that
(4)∫0T(r(t)-a(t)τ)dt+∑k=1q‍lnhk<-δ.
Define the functionV(t)=maxi∈I{xi(t)}. For any t∈R+, there is an i=i(t)∈I such that V(t)=xi(t). Calculating the upper-right derivative of V(t), we obtain
(5)D+V(t)⩽xi(t)(ri(t)-ai(t)xi(t))⩽V(t)(r(t)-a(t)V(t)).
When t=tk, we have V(tk+)=max{xi(tk+)}=max{hikxi(tk)}⩽max{hik}max{xi(tk)}=hkV(tk).
Consider the following auxiliary system:(6)D+w(t)=w(t)(r(t)-a(t)w(t)),w(tk+)=hkw(tk),k=1,2,…
with the initial condition V(0)⩽w(0). If there is a constant M0>0 such that
(7)limsupt→∞w(t)<M0
for any positive solution w(t) of system (6), then, according to the comparison theorem of impulsive differential equations [10], we have V(t)⩽w(t) for all t⩾0. Therefore, choose M=M0, and we will finally have limsupt→∞xi(t)⩽limsupt→∞V(t)⩽limsupt→∞w(t)<M for all i∈I.
Next, we will prove that (7) holds. In fact, for any positive solution w(t) of system (6), we only need to consider the following three cases.
Case
1. There is a t0⩾0 such that w(t)⩾τ for all t⩾t0.
Case
2. There is a t0⩾0 such that w(t)⩽τ for all t⩾t0.
Case
3. w(t) is oscillatory about τ for all t⩾0.
We first consider Case 1. Sincew(t)⩾τ for all t⩾t0, then for t=t0+lT, where l is any positive integer, integrating system (6) from t0 to t, from (4) we have
(8)w(t)=w(t0)exp(∫t0t(r(s)-a(s)w(s))ds+∑t0⩽tk<t‍lnhk)⩽w(t0)exp(∑k=1q‍∫t0t0+T(r(s)-a(s)τ)ds+⋯+∫t0+(l-1)Tt0+lT(r(s)-a(s)τ)ds∫t0t0+T(r(s)-a(s)τ)+l∑k=1q‍lnhk∑k=1q‍)⩽w(t0)exp(-lδ).
Hence, w(t)→0 as l→∞, which leads to a contradiction.
Then, we consider Case 3. From the oscillation ofw(t) about τ, we can choose two sequences {ρn} and {ρn*} satisfying 0<ρ1<ρ1*<⋯<ρn<ρn*<⋯ and limn→∞ρn=limn→∞ρn*=∞ such that
(9)w(ρn)⩽τ,w(ρn+)⩾τ,w(ρn*)⩾τ,w(ρn*+)⩽τ,w(t)⩾τ∀t∈(ρn,ρn*),w(t)⩽τ∀t∈(ρn*,ρn+1).
For any t⩾ρ1, if t∈(ρn,ρn*] for some integer n, then we can choose integer l⩾0 and constant 0⩽ν<T such that t=ρn+lT+ν. Since D+w(t)⩽w(t)(r(t)-a(t)τ) for all t∈(ρn,ρn*) and t≠tk, integrating this inequality from ρn to t, by (3) and (4) we obtain
(10)w(t)⩽w(ρn)exp(∫ρnt(r(s)-a(s)τ)ds+∑ρn⩽tk<t‍lnhk)⩽τexp(l(∫0T(r(t)-a(t)τ)dt+∑k=1q‍lnhk)+∫ρnρn+ν(r(t)-a(t)τ)dt+∑ρn⩽tk<ρn+ν‍lnhk)⩽τexp(-lδ+rT+H)⩽τexp(rT+H),
where r=supt∈[0,T]{|r(t)|+a(t)τ}. If there is an integer n such that t∈(ρn*,ρn+1], then we have w(t)⩽τ⩽τexp(rT+H). Therefore, for Case 3 we always have w(t)⩽τexp(rT+H) for all t⩾ρ1.
Lastly, if Case 2 holds, then we directly havew(t)⩽τexp(rT+H) for all t⩾0.
Choose constantM0=τexp(rT+H)+1, then we see that (7) holds. This completes the proof.Remark 2.
It can be seen from Theorem1 that, in one time period T, if the density-dependent coefficient in patch i (i∈I) is strictly greater than zero and the impulsive coefficient hik is bounded in the same time period, the dispersal species x is always ultimately bounded.Theorem 3.
Assume that all conditions of Theorem1 hold. In addition, there is an i0∈I such that
(11)∫0T(ri0(t)-∑j=1n‍Di0j(t))dt+∑k=1q‍lnhi0k>0.
Then system (1) is permanent.Proof.
The ultimate boundedness of system (1) has been proved in Theorem 1. In the following, we mainly prove the permanence of the system; that is, there is a constant m>0 such that
(12)liminft→∞xi(t)>m
for each i∈I and any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1).
From assumptionhi(k+q)=hik, we obtain that there exists a constant H>0 such that function
(13)|hi(t,μ)|=|∑t⩽tk<t+μ‍lnhik|⩽∑k=1q‍|lnhik|⩽H
for any i∈I,t∈R+ and μ∈[0,T].
Fori=i0, by condition (11) and the boundedness of ai0(t), there are two positive constants τ- and δ- such that
(14)∫0T(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)τ-)dt+∑k=1q‍lnhi0k>δ-.
Letx(t)=(x1(t),x2(t),…,xn(t)) be any positive solution of system (1). Since
(15)x˙i0(t)⩾xi0(t)(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)xi0(t)),(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)xi0(t))t≠tk,
by the comparison theorem of impulsive differential equations, we obtain xi0(t)⩾u(t) for all t⩾0, where u(t) is the positive solution of system
(16)u˙(t)=u(t)(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)u(t)),u(tk+)=hi0ku(tk),k=1,2,…
with initial condition u(0)=xi0(0).
In the following, we first prove that there is a constantm->0 such that
(17)liminft→∞u(t)>m-
for any positive solution u(t) of system (16). We only need to consider the following three cases.
Case
1. There is a t-⩾0 such that u(t)⩽τ- for all t⩾t-.
Case
2. There is a t-⩾0 such that u(t)⩾τ- for all t⩾t-.
Case
3. u(t) is oscillatory about τ- for all t⩾0.
For Case 1, lett=t-+lT, where l⩾0 is any integer. From (14), we obtain
(18)u(t)=u(t-)×exp(∑t-⩽tk<t‍lnhi0k∫t-t(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)u(s))∫t-t(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)u(s))ds+∑t-⩽tk<t‍lnhi0k∫t-t(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)u(s)))⩾u(t-)exp(∑k=1q‍lnhi0k∫t-t-+T(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)τ-)ds+⋯+∫t-+(l-1)Tt-+lT(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)τ-∑j=1n‍)ds+l∑k=1q‍lnhi0k∫t-t-+T(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)τ-))⩾u(t-)exp(lδ-).
Hence, u(t)→∞ as l→∞, which leads to a contradiction.
For Case 3, we choose two sequences{ρn} and {ρn*} satisfying 0<ρ1<ρ1*<⋯<ρn<ρn*<⋯ and limn→∞ρn=limn→∞ρn*=∞ such that
(19)u(ρn)⩾τ-,u(ρn+)⩽τ-,u(ρn*)⩽τ-,u(ρn*+)⩾τ-,u(t)⩽τ-∀t∈(ρn,ρn*),u(t)⩾τ-∀t∈(ρn*,ρn+1).
For any t⩾ρ1, if t∈(ρn,ρn*] for some integer n, then we can choose integer l⩾0 and constant 0⩽ν-<T such that t=ρn+lT+ν-. Since, for all t∈(ρn,ρn*) and t≠tk, we have u˙(t)⩾u(t)(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)τ-), integrating this inequality from ρn to t, then from (13) and (14) we obtain
(20)u(t)⩾u(ρn)×exp(∫ρnt(ri0(s)-∑j=1n‍Di0j(s)-ai0(s)τ-)ds+∑ρn⩽tk<t‍lnhi0k)⩾τ-exp(∑ρn⩽tk<ρn+ν-‍l(∫0T(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)τ-)dt+∑k=1q‍lnhi0k∫0T(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)τ-))+∫ρnρn+ν-(ri0(t)-∑j=1n‍Di0j(t)-ai0(t)τ-)dt+∑ρn⩽tk<ρn+ν-‍lnhi0k)⩾τ-exp(lδ--βi0T-H)⩾τ-exp(-βi0T-H),
where βi0=supt∈[0,T]{|ri0(t)|+∑j=1n‍Di0j(t)+ai0(t)τ-}. If there is an integer n such that t∈(ρn*,ρn+1], obviously we have u(t)⩾τ-⩾τ-exp(-βi0T-H). Therefore, for Case 3 we always have u(t)⩾τ-exp(-βi0T-H) for all t⩾ρ1. Let constant m-=τ-exp(-βi0T-H-1). Then m- is independent of any positive solution of system (16) and we finally have that (17) holds.
Lastly, if Case 2 holds, then fromu(t)⩾τ- for all t⩾0, we directly have that (17) holds.
From the fact thatxi0(t)⩾u(t) for all t⩾0, then we have
(21)liminft→∞xi0(t)⩾liminft→∞u(t)>m-.
It follows immediately from (21) that there is a T-0>0 such that ∑i=1n‍xi(t)⩾m- for all t⩾T-0. Then for any i∈I, when t⩾T-0 and t≠tk we have
(22)x˙i(t)=xi(t)(ri(t)-ai(t)xi(t)-∑j=1n‍Dij(t))+∑j=1n‍Dij(t)xj(t)⩾xi(t)(ri(t)-ai(t)xi(t)-∑j=1n‍Dij(t)-D1)+D1m-.
Obviously, there is a constant β>0 and β is independent of any positive solution of system (1), such that for any i∈I, 0⩽xi(t)⩽β, and all t∈R+ the following inequality holds
(23)xi(t)(ri(t)-ai(t)xi(t)-∑j=1n‍Dij(t)-D1)+D1m->β.
Hence, for any i∈I, by (22) we have
(24)x˙i(t)>β,t≠tk,xi(tk+)⩾h~ixi(tk),k=1,2,…
for all 0⩽xi(t)⩽β and t⩾T-0, where h~i=min1⩽k⩽q{hik}>0.
In order to prove that (12) holds, we only need to consider the following three cases.
Case
1. There is a T-1⩾T-0 such that xi(t)⩾β for all t⩾T-1.
Case
2. There is a T-1⩾T-0 such that xi(t)⩽β for all t⩾T-1.
Case
3. xi(t) is oscillatory about β for all t⩾T-0.
Equation (12) is obviously true if Case 1 holds.
For Case 2, there exists an impulsive timetq*⩾T-1 for some integer q*>0. For any t>tq*+1>tq*⩾T-1, there is an integer p⩾q*+1 such that t∈(tp,tp+1], and from system (24) we have
(25)xi(t)>xi(tp+)+β(t-tp)⩾h~ixi(tp)+β(t-tp)⩾h~i[xi(tp-1+)+β(tp-tp-1)]+β(t-tp)>h~iβτ0,
where τ0=min1⩽k⩽q{tk-tk-1}>0. Therefore, we obtain
(26)liminft→∞xi(t)>h~iβτ0
for any i∈I.
Then, we consider Case 3. We choose two sequences{ρn} and {ρn*} satisfying T-0<ρ1<ρ1*<⋯<ρn<ρn*<⋯ and limn→∞ρn=limn→∞ρn*=∞ such that
(27)xi(ρn)⩾β,xi(ρn+)⩽β,xi(ρn*)⩽β,xi(ρn*+)⩾β,xi(t)⩽β∀t∈(ρn,ρn*),xi(t)⩾β∀t∈(ρn*,ρn+1).
For anyt⩾ρ1, if t∈(ρn,ρn*] for some integer n, we first of all show that ρn must be an impulsive time. Otherwise, there exists a positive constant δ such that there is no pulse in the interval (ρn,ρn+δ)⊂(ρn,ρn*]. Then for any t∈(ρn,ρn+δ), from system (24) we have xi(t)>xi(ρn+)+β(t-ρn)>xi(ρn)⩾β, which leads to a contradiction.
If there is only one impulsive time in the interval(ρn,ρn*](which must be ρn), then from system (24) we get
(28)xi(t)>xi(ρn+)+β(t-ρn)⩾h~ixi(ρn)+β(t-ρn)>h~iβ.
If there are at least twice pulses in(ρn,ρn*], then we can denote (ρn,ρn*]=⋃p=p0q*-1‍(tp,tp+1]⋃‍(tq*,ρn*], where tp0=ρn, q*>p0, p0,p and q* are some positive integers. Hence, for any t∈(tp,tp+1], there is xi(t)>xi(tp+)+β(t-tp) from system (24). Moreover, if p=p0, we have
(29)xi(t)>xi(tp+)+β(t-tp)⩾h~ixi(ρn)+β(t-ρn)>h~iβ.
If p>p0, then
(30)xi(t)>xi(tp+)+β(t-tp)⩾h~i[xi(tp-1+)+β(tp-tp-1)]+β(t-tp)>h~iβτ0.
However, when t∈(tq*,ρn*], then
(31)xi(t)>xi(tq*+)+β(t-tq*)⩾h~i[xi(tq*-1+)+β(tq*-tq*-1)]+β(t-tq*)>h~iβτ0.
It follows from (28)–(31) that
(32)xi(t)⩾min{h~iβ,h~iβτ0}
for all t∈(ρn,ρn*] and any i∈I.
If there is an integern such that t∈(ρn*,ρn+1], obviously we have xi(t)⩾β. Therefore, for Case 3 we always have
(33)xi(t)⩾min{h~iβ,h~iβτ0,β}
for all t⩾ρ1 and any i∈I.
From (26) and (33), choose m=mini∈I{h~iβ,h~iβτ0,β}/2. Then we finally have that (12) holds. System (1) is permanent. The proof of Theorem 3 is completed.Remark 4.
It follows from Theorem3 that system (1) is permanent if there is a positive average growth rate (which does not include the dispersal entrance) in one time period T in any patch i0(i0∈I). In paper [4], the authors showed an interesting result that the dispersal species without impulses is permanent in all other patches if it is permanent in a patch. However, we extend this result to a periodic case with impulses.Remark 5.
In paper [18], we studied an impulsive periodic predator-prey system with Holling type III functional response and diffusion, and the paper mainly considers the influences of Holling type functional response and impulses. The conditions of the main result Theorem 3 require the minimum of the coefficients. However, in this paper, we consider a single-species logistic system. Although the predator is not involved in the model, which is simple than the model in [18], a more accurate and reasonable condition is established in the present paper; that is, in dispersal system, species with impulses is permanent in all patches if it is permanent in a patch, which improves the minimum conditions in the previous paper.Theorem 6.
System (1) is extinct if conditions
(34)∫0Ta(t)dt>0,∫0Tγ(t)dt+∑k=1q‍lnhk⩽0
hold, where γ(t)=maxi∈I{ri(t)-∑j=1n‍Dij(t)+∑j=1n‍Dji(t)} for all t∈[0,T], and a(t) and hk are defined in Theorem 1.Proof.
In fact, from (34), for any constant ε>0, there is a positive constant δ~ such that
(35)∫0T(γ(t)-εa(t)n)dt+∑k=1q‍lnhk<-δ~.
Define V(t)=∑i=1n‍xi(t). When t≠tk, calculating the right-upper derivative of V(t), we have
(36)D+V(t)=∑i=1n‍x˙i(t)=∑i=1n‍xi(t)(ri(t)-∑j=1n‍Dij(t)+∑j=1n‍Dji(t)-ai(t)xi(t))⩽γ(t)V(t)-a(t)∑i=1n‍xi2(t)⩽V(t)(γ(t)-a(t)nV(t)).
When t=tk, we obtain V(tk+)=∑i=1n‍xi(tk+)=∑i=1n‍hikxi(tk)⩽maxi∈I{hik}V(tk)=hkV(tk). From this and (35), a similar argument as in the proof of (7), we can obtain V(t)⩽εexp(γ(t)T+H) for t⩾0. Then from the arbitrariness of ε, we obtain V(t)→0 as t→∞. Finally, we have xi(t)→0 as t→∞ for all i∈I. This completes the proof of Theorem 6.Remark 7.
It can be seen from Theorem6 that system (1) is always extinct if, in one time period T, there are a positive density-dependent coefficient and a nonpositive average growth rate (which includes the dispersal entrance) in the time period in each patch i(i∈I).


## 3. Periodic Solutions

In this section, by constructing an appropriate Lyapunov function, sufficient conditions for the existence of the unique globally attractively positiveT-periodic solution of system (1) are established.Letx(t)=(x1(t),x2(t),…,xn(t)) and x*(t)=(x1*(t),x2*(t),…,xn*(t)) be any two positive solutions of system (1). From Theorem 3, we can obtain that there are constants A>0 and B>0 such that
(37)A⩽xi(t),xi*(t)⩽B∀t⩾0,i∈I.Theorem 8.
Suppose all the conditions of Theorem3 hold. Moreover, if
(38)∫0Tβ(t)dt>0,
where β(t)=mini∈I{ai(t)-∑j=1n‍Dji(t)/A}⩾0, t∈[0,T], then system (1) has a unique globally attractively positive T-periodic solution x*(t)=(x1*(t),x2*(t),…,xn*(t)); that is, any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1) satisfies
(39)limt→∞(xi(t)-xi*(t))=0,i∈I.Proof.
Choose Lyapunov functionV(t)=∑i=1n‍|lnxi(t)-lnxi*(t)|. Since for any impulsive time tk we have
(40)V(tk+)=∑i=1n‍|lnxi(tk+)-lnxi*(tk+)|=∑i=1n‍|lnhikxi(tk)-lnhikxi*(tk)|=V(tk),
then V(t) is continuous for all t⩾0. On the other hand, from (37) we can obtain that for any t∈R+ and t≠tk(41)1B|xi(t)-xi*(t)|⩽|lnxi(t)-lnxi*(t)|⩽1A|xi(t)-xi*(t)|.
For any t∈R+ and t≠tk, calculating the derivative of V(t), we obtain
(42)D+V(t)=∑i=1n‍sgn(xi(t)-xi*(t))(x˙i(t)xi(t)-x˙i*(t)xi*(t))⩽∑i=1n‍(-ai(t)|xi(t)-xi*(t)|)+∑i=1n‍∑j=1n‍D-ij(t),
where
(43)D-ij(t)={Dij(t)(xj(t)xi(t)-xj*(t)xi*(t)),xi(t)>xi*(t),Dij(t)(xj*(t)xi*(t)-xj(t)xi(t)),xi(t)<xi*(t).
For all t⩾0, we estimate D-ij(t) under the following two cases.(i)
Ifxi(t)⩾xi*(t), then D-ij(t)⩽Dij(t)/xi(t)(xj(t)-xj*(t))⩽Dij(t)/A|xj(t)-xj*(t)|.(ii)
Ifxi(t)<yi(t), then D-ij(t)⩽Dij(t)/xi*(t)(xj(t)-xj*(t))⩽Dij(t)/A|xj(t)-xj*(t)|.
It follows from the estimation ofD-ij(t) and (41) that
(44)D+V(t)⩽∑i=1n‍(-ai(t)|xi(t)-xi*(t)|)+∑i=1n‍∑j=1n‍Dij(t)A|xj(t)-xj*(t)|⩽-∑i=1n‍(ai(t)-∑j=1n‍Dji(t)A)|xi(t)-xi*(t)|⩽-β(t)AV(t).
From this and condition (38), we have V(t)⩽V(0)exp(-A∫0t‍β(s)ds)→0 as t→∞. Further more, from (41) we have that (39) holds.
Now let us consider the sequence(x1*(mT,z0),x2*(mT,z0),…,xn*(mT,z0))=z(mT,z0), where m=1,2,… and z0=(x1*(0),x2*(0),…,xn*(0)). It is compact in the domain [A,B]n since A⩽xi*(t)⩽B for all t⩾0 and i=1,2,…,n. Let z- be a limit point of this sequence, with z-=limn→∞z(mnT,z0). Then z(T,z-)=z-. Indeed, since z(T,z(mnT,z0))=z(mnT,z(T,z0)) and z(mnT,z(T,z0))-z(mnT,z0)→0 as mn→∞, we get
(45)∥z(T,z-)-z-∥[A,B]n⩽∥z(T,z-)-z(T,z(mnT,z0))∥[A,B]n+∥z(T,z(mnT,z0))-z(mnT,z0)∥[A,B]n+∥z(mnT,z0)-z-∥[A,B]n⟶0,n⟶∞.
The sequence z(mT,z0),m=1,2,… has a unique limit point. On the contrary, let the sequence have two limit points z-=limn→∞z(mnT,z0) and z~=limn→∞z(mnT,z0). Then, taking into account (39) and z~=z(mnT,z~), we have
(46)∥z--z~∥[A,B]n⩽∥z--z(mnT,z0)∥[A,B]n+∥z(mnT,z0)-z~∥[A,B]n⟶0,n⟶∞,
and hence z-=z~. The solution (x1*(t,z-),x2*(t,z-),…,xn*(t,z-)) is the unique periodic solution of system (1). By (39), it is globally attractive. This completes the proof of Theorem 8.


## 4. Numerical Simulation and Discussion

In this paper, we have investigated a class of single-species periodic logistic system with impulses and dispersal inn different patches. By means of inequality estimation technique and Lyapunov function, we gave the criteria for the permanence, extinction, and existence of a unique globally stable positive periodic solution of system (1).In order to testify the validity of our results and present a more in-depth problem for further discussion, we discuss the following two patchesT-periodic dispersal system:
(47)x˙1(t)=x1(t)(r1(t)-a1(t)x1(t))+D12(t)(x2(t)-x1(t)),x˙2(t)=x2(t)(r2(t)-a2(t)x2(t))+D21(t)(x1(t)-x2(t)),t≠tk,x1(tk+)=h1kx1(tk),x2(tk+)=h2kx2(tk),k=1,2,….We taker1(t)=5-|sin(π/2)t|, r2(t)=2.5+0.5cosπt, a1(t)=1.5, a2(t)=0.8, D12(t)=1.2, D21(t)=0.7, h1k=1.2, h2k=0.8, and tk=0.1k, k=1,2,…. Obviously, r(t)=max{r1(t),r2(t)}=5-|sin(π/2)t|, a(t)=min{a1(t),a2(t)}=0.8, hk=max{h1k,h2k}=1.2, and system (47) is periodic with period T=2. For q=20, we have tk+q=tk+T, h1(k+q)=h1k, and h2(k+q)=h2k for all k=1,2,…. It is easy to verify that hk(t,μ) and hik(t,μ)(i=1,2) are bounded for all t∈R+ and μ∈[0,T]. Further more, since
(48)∫0Ta(t)dt=1.6>0,∫0T(r1(t)-D12(t))dt+∑k=1q‍lnh1k=9.9732>0(i0=1),
all the conditions of Theorem 3 are satisfied. Hence, system (47) is permanent. See Figures 1 and 2.Figure 1
The time series of the permanence of speciesx.Figure 2
The phase of the permanence of speciesx.However, if the survival environment of the two patches is austere, the intrinsic growth rates will be negative. Hence, if we taker1(t)=-1-|sin(π/2)t|, r2(t)=-4+0.5cosπt and all other parameters are retained, then we obtain γ(t)=max{r1(t)-D12(t)+D21(t),r2(t)-D21(t)+D12(t)}=-1.5-|sin(π/2)t| and
(49)∫0Tγ(t)dt+∑k=1q‍lnhk=-0.6268<0,
hence conditions of Theorem 6 are satisfied. From Theorem 6 we find that any positive solution of system (47) will be extinct. See Figure 3.The time series and phase of the extinction of speciesx.From the illustrations of the theorems, we note that there is a great difference on the choice of the intrinsic growth ratesri(t)(i=1,2), which guarantee that the system is permanent or extinct. These differences make us want to know what results will be if all the parameters satisfy
(50)∫0T(ri(t)-∑j=12‍Dij(t))dt+∑k=1q‍lnhik⩽0i=1,2,∫0Tγ(t)dt+∑k=1q‍lnhk>0.
For this aim, we choose r1(t)=2.55-5|sin(π/2)t|, r2(t)=2+0.5cosπt and all other parameters are retained; then
(51)∫0T(r1(t)-D12(t))dt+∑k=1q‍lnh1k=-0.0198<0,∫0T(r2(t)-D21(t))dt+∑k=1q‍lnh2k=-1.8629<0,∫0Tγ(t)dt+∑k=1q‍lnhk=11.3732>0,
and all the conditions of Theorems 3 and 6 are not satisfied. But from Figures 4 and 5 we find that the system is permanent.Figure 4
The time series of the permanence of speciesx.Figure 5
The phase of the permanence of speciesx.Furthermore, if we chooser1(t)=-0.5-|sin(π/2)t|, r2(t)=-4+0.5cosπt and keep all other parameters, then we have
(52)∫0T(r1(t)-D12(t))dt+∑k=1q‍lnh1k=-1.0268<0,∫0T(r2(t)-D21(t))dt+∑k=1q‍lnh2k=-12.4629<0,∫0Tγ(t)dt+∑k=1q‍lnhk=1.3732>0,
which do not satisfy conditions of any theorem. But, from Figure 6 we see that any positive solution of system (1) is extinct.The time series and phase of the extinction of speciesx.Remark 9.
Through the above analysis, we realize that there is a little flaw of the finding conditions of the theorems. A challenging problem is to find some sufficient and necessary conditions (if the conditions hold, then the system will be permanent, otherwise, it will be extinct) to guarantee the permanence and extinction of the system.Throughout Figures1–6, we always take the initial condition x(0)=(x1(0),x2(0))=(2,3).



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*Source: 101238-2013-05-09.xml*