Volume 2010, Article ID 891812,18pages doi:10.1155/2010/891812
Research Article
Analysis of a Nonautonomous Delayed
Predator-Prey System with a Stage Structure for
the Predator in a Polluted Environment
G. P. Samanta
1, 21Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, 81473 Bratislava, Slovakia
2Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, India
Correspondence should be addressed to G. P. Samanta,g p [email protected] Received 3 July 2009; Accepted 7 February 2010
Academic Editor: Harvinder S. Sidhu
Copyrightq2010 G. P. Samanta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A two-species nonautonomous Lotka-Volterra type model with diffusional migration among the immature predator population, constant delay among the matured predators, and toxicant effect on the immature predators in a nonprotective patch is proposed. The scale of the protective zone among the immature predator population can be regulated through diffusive coefficientsDit,
i 1,2. It is proved that this system is uniformly persistent permanenceunder appropriate conditions. Sufficient conditions are derived to confirm that if this system admits a positive periodic solution, then it is globally asymptotically stable.
1. Introduction
In recent years, many countries have already realized that the pollution of the environment is a very serious and urgent problem. It is well known that with the rapid development of modern industries and agricultures, a large quantity of toxicants and contaminants enter into ecosystems one after another. One of the most important and meaningful questions in mathematical ecology is the permanenceuniformly persistentand extinction of a population in a polluted environment. Organisms are often exposed to a polluted environment and take up toxicants. The change in environment is caused by pollution, affecting the long-term survival of species, humans, and biodiversity of the habitat. The question of the effects of pollutants and toxicants on ecological communities is of tremendous important from both environmental and conservational points of view.
ecosystems. The ability of an aquatic ecosystem to restore itself to the prepollution state is directly related to the compartmental retention times and exchange rates among compartments and the transport rates out of the system. The effect of a toxic substance can be well understood by measuring its presence throughout an aquatic system and by determining the rates of its transport betweencompartments and storage within compartments. An understanding of the compartmental and total system stability of a toxic substance can be achieved with the use of conceptual models1–4.
Acid rain results from certain kinds of air pollution that mix with precipitation, such as rain or fog, then falls to earth as an acidic solution and its major components are oxides of sulfur and nitrogen that are mainly the by-products of coal-burning power plants, copper melting, factory, and automobile emissions. These oxides are chemically changed in the atmosphere and return to the earth as rain, snow, fog, or dust. In the United states, the most recognized form of acid rain results from sulfur dioxide emissions, which are converted into sulfuric acid in the atmosphere. When this is mixed with precipitation and falls to earth, the effect is precisely like pouring a diluted acid solution onto everything it touches. In lakes also, this acidification process can change ecological structures. In this way toxic substances are invaded into the ecological communities5,6.
Other examples are oil pollution in the seas 7, degradation of forests 8, 9, and dumping of toxic waste in rivers and lakes10,11.
In order to protect environment and regulate the release of toxic substances wisely, we must assess the harm of the toxicant to the species exposed to it. By using mathematical models, Hallam and Clark12, Hallam et al.13,14, Hallam and De Luna15, De Luna and Hallam16, Zhien et al.17, Huaping and Zhien18, Freedman and Shukla19, Wang and Ma20, Dubey21, Xiao and Chen22, Ghosh et al.23, Buonomo and Lacitignola24, Li et al.25, Samanta and Maiti26, J. Wang and K. Wang27, He and Wang28,29, Das et al.30, and many others studied the effects of toxic substances on various ecosystems.
The effect of “stage-structure” to the dynamics of predator-prey system is a natural phenomenon and represents the division of a population into immature and mature individuals. As is common, the dynamics-eating habitats, susceptibility to toxicants, and so forth. are often quite different in these two subpopulations. Hence investigation of the effects of such a subdivision on the interaction of species is very much important from the ecological point of view. Stage-structured models have already received much attention by many authors31,32. Many models in mathematical ecology can be formulated as system of differential equations with time delays. The effect of the past history on the stability of system is also an important problem in population ecology. Recently uniformly persistent and stability of a population dynamical system involving time delays have been considered by many authors 33. Dispersal is a ubiquitous phenomenon in the natural world. Its importance in understanding the ecological and evolutionary dynamics of populations is mirrored by a large number of mathematical models34. In order to prevent the destruction of biological resources and to protect the environment, all kinds of measures have been proposed. Establishing a protective patch in mathematical ecology is applied widely. The practical effects of the protective patch on the polluted population is worth investigation.
whose individual members have a life history that takes them through two stages, immature and mature. Moreover, there is a big difference between the immature and mature species in many aspects. For example, for less concentration of C2H4in an environment, the immature
cymbidium is so susceptible that it begins to wither, whereas, the mature cymbidium often has not been affected22,35. Therefore, it is important to consider the effect of stage structure on population growth in a polluted environment. In this paper, we have considered pollution in a nonautonomous predator-prey system together with diffusional migration among the immature predator population between protective and nonprotective patches where there is a constant delay among the matured predators due to the gestation of the predator based on the assumption that mature adult predators can only contribute to the reproduction of the predator biomass and that the change rate of predators depends on the number of the preys and of the predators presented at some previous time. The scale of the protective zone among the immature predator population can be regulated through diffusive coefficientsDit, i
1,2.Here we incorporate stage-structure for the predator and assume that the toxicants have no effect on the mature species. This seems to be reasonable. The main feature of the present paper is to study the asymptotical behaviour of the toxicant-population model with stage structure, diffusional migration, and time delay so as to obtain some conditions under which the population is uniformly persistent. In addition, by constructing an appropriate Lyapunov function, we have derived sufficient conditions to confirm that if the system admits a positive periodic solution, then it is globally asymptotically stable.
2. The Basic Mathematical Model
Our study is based on the hypothesis of a complete spatial homogeneous environment. We incorporate stage-structure for the predator into a nonautonomous Lotka-Volterra type predator-prey model. We consider the following delayed model to describe the dynamics of prey and predator population in a polluted environment. The state variables of the model are xt, y1t, y2t,and y3t, the densities of the prey, immature predator in nonprotective patch with toxicants, immature predator in protective patch without toxicants, and mature predator species at timet,respectively,C0tis the concentration of toxicant in the immature predator organism in nonprotective patchwith toxicantsat timet,and Cetis
the concentration of toxicant in the environment at timet. Here we assume that the toxicants have no effect on the mature species and immature predator in protective patch.
Let us consider the two-species nonautonomous Lotka-Volterra type model with diffusional migration among the immature predator population, constant delay among the matured predators, and toxicant effect on immature predators in nonprotective patch:
dxt dt xt
r1t−a11txt−a12ty3t
,
dy1t
dt b1ty3t−θ1ty1t−δ1ty1t−dtC0ty1t−β1ty
2
1t D1t
y2t−y1t,
dy2t
dt b2ty3t−θ2ty2t−δ2ty2t−β2ty
2
2t D2t
dy3t
dt y3t
−r2t−a22ty3t
a21ty3t−τxt−τ θ1ty1t θ2ty2t,
dC0t
dt ktCet lt−gtC0t−mtC0t, dCet
dt −htCet ut.
2.1
The model is derived under following assumptions.
A1The prey population: the growth of the species is of a Lotka-Volterra nature. At any timet > 0, the birth rate is proportional to the existing prey population with proportional functionr1tanda11tis the intraspecific competition rate function of the prey.r1tanda11tare continuous, bounded, and strictly positive functions in the interval0,∞.
A2The predator population: the predators cannot hunt prey when the predators are infant immature. The mature predators feed only on prey; a12t is the capturing rate function of the predator, a21t/a12t is the conversion rate function of nutrients into the reproduction of the mature predator, and τ ≥ 0 is a constant delay due to the gestation of the mature predator based on the assumption that mature adult predators can only contribute to the reproduction of the predator biomass and that the change rate of predators depends on the number of the preys and of the predators presented at some previous time. To protect the immature predator population, the region Ψ of immature predator population is divided into two patches Ψ1 and Ψ2. Pollution is permitted in Ψ1 and is inhibited in Ψ2. We call Ψ1 and Ψ2 as the nonprotective and protective patches, respectively. Since the difference of densities between patches Ψ1 and
Ψ2 exists, the diffusive migration can occur between these two patches, and D1t, D2t are diffusion coefficient functions of nonprotective immature and protective immature predator species, respectively. At any timet > 0, the rates of transitions from nonprotective immature and protective immature individuals to mature individuals are proportional to the existing nonprotective immature and protective immature populations, respectively, with proportional functions θ1t and θ2t, respectively, and the death rates of mature, nonprotective immature, and protective immature population are proportional to the existing mature, nonprotective immature, and protective immature population, respectively, with proportional functions r2t, δ1t, and δ2t, respectively. The birth rates of nonprotective immature and protective immature population are proportional to the existing mature population with proportional functions b1t and b2t, respectively.a22t,β1t,andβ2tare the intraspecific competition rate functions of the mature, nonprotective immature, and protective immature population, respectively. a12t, a21t, a22t, Dit, θit, r2t, δit, bit, and βit, i
1,2are continuous, bounded, and strictly positive functions in the interval0,∞.
and the third and fourth terms represent the organismal net loss of toxicant due to metabolic processing and other causes. kt denotes environmental toxicant uptake rate per unit mass organism, ltdenotes uptake rate of toxicant in food per unit mass organism, and −gtCot and −mtCot represent the egestion
and depuration rates of the toxicant in the organism, respectively. The first term
−htCeton the right of the sixth equation in system2.1represents the toxicant
loss from the environment itself by biological transformation, chemical hydrolysis, volatilization, microbial degradation, photosynthetic degradation, and so on and the exogenous rate of input of toxicant into the environment is represented by ut. The capacity of the environment is so large that the change of toxicants in the environment that comes from uptake and egestion by the organisms can be neglected 15,36, 37. The coefficients dt, kt, lt, gt, mt, ht,and ut are assumed to be nonnegative, continuous, and bounded functions in the interval
0,∞.
The initial conditions for system2.1take the form
xθ φ1θ, yiθ ψiθ, C0θ φ2θ, Ceθ φ3θ, i1,2,3,
φiθ≥0, ψiθ≥0 i1,2,3, θ∈−τ,0,
x0>0, yi0>0, 0≤C00≤1, 0≤Ce0≤1 i1,2,3,
2.2
where Φ φ1θ, φ2θ, φ3θ, ψ1θ, ψ2θ, ψ3θ ∈ C−τ,0, R6, the Banach space of
continuous functions mapping the interval−τ,0intoR6,whereR6 {x
1, x2, x3, x4, x5, x6:
xi≥0, i1,2,3,4,5,6}.
It is well known by the fundamental theory of functional differential equations38
that system2.1has a unique solutionxt, y1t, y2t, y3t, C0t, Cetsatisfying initial
conditions2.2. Letflinf
t≥0ft, fusupt≥0ft,for a continuous and bounded functionftdefined
on 0, ∞.
In this paper, we always assume that the following holds:
mindl, kl, ll, gl, ml, hl, ul≥0, min
i,j1,2
ril, alij, bli, θli, δli, βil, Dli>0,
maxdu, ku, lu, gu, mu, hu, uu <∞, max
i,j1,2
riu, auij, bui, θiu, δui, βui, Diu<∞.
2.3
Each ofC0tandCetis a concentration, and thus, these variables cannot be greater
than 1. So we should give some conditions, such that
0≤C0t≤1, 0≤Cet≤1, ∀t≥0. 2.4
Theorem 2.1. Every solution of system2.1with initial conditions2.2exists and is unique in the interval0,∞andxt>0, y1t>0, y2t>0, y3t>0, C0t≥0, Cet≥0, for all t≥0.
conditions 2.2 exists and is unique on 0, α, where 0 < α ≤ ∞ 38, chapter 2. From system2.1with initial conditions2.2, we have
xt x0exp
t
0
r1s−a11sxs−a12sy3sds >0, ∀t≥0. 2.5
Next, we show thaty1t > 0 for allt ∈ 0,∞.Otherwise, there exists at1 ∈ 0,∞
such thaty1t1 0, y˙1t1 ≤ 0, and y1t > 0 for all t ∈ 0, t1.Hence there must have y2t>0 for all t∈0, t1.If this statement is not true, then there exists at2∈0, t1such that
y2t2 0 andy2t>0 on 0, t2.We claim thaty3t>0 for all t∈0, t2.If this statement is not true, then there exists at3∈0, t2such thaty3t3 0 andy3t≥0 for allt∈−τ, t3.
Furthermore,
dy3t
dt ≥ −r2ty3t−a22ty
2
3t, ∀t∈0, t3, 2.6
then,y3t≥y30expt0−r2s−a22sy3sds >0,for all
t∈0, t3 ⇒y3t3≥y30exp
t3
0
−r2s−a22sy3sds >0, 2.7
which is a contradiction. Hence,y3t>0 for allt∈0, t2.Integrating the third equation of system2.1from 0 tot2, we have
y2t2 y20exp
−
t2
0
D2s θ2s δ2s β2sy2sds
t2
0
b2uy3u D2uy1uexp
u
t2
D2s θ2s δ2s β2sy2sds
du >0,
2.8
which is a contradiction withy2t2 0. Soy2t>0,for all t∈0, t1and hencey3t>0,for all t∈0, t1. Integrating the second equation of system2.1from 0 tot1,we have
y1t1 y10exp
−
t1
0
D1s θ1s δ1s dsC0s β1sy1sds
t1
0
b1uy3u D1uy2uexp
u
t1
D1s θ1s δ1s
dsC0s β1sy1sds
du >0,
2.9
Also,
C0t e−t0{ms gs}ds
t
0
{ksCes ls}e
s
0{ms gs}dsds C00
≥0, ∀t≥0,
Cet e−
t 0hsds
t
0
uses0hsdsds Ce0
≥0 ∀t≥0.
2.10
This completes the proof.
Theorem 2.2. From system2.1with initial conditions2.2, ifku lu ≤ gl ml, uu ≤hl,then
0≤C0t≤1, 0≤Cet≤1, for allt≥0.
Proof. According toTheorem 2.1, we haveC0t≥0, Cet≥0, for allt≥0. Now we have to
prove thatC0t≤1, Cet≤1, for allt≥0.
If the conclusion is false, then the maximum interval is0, Tsuch that 0 ≤ C0t ≤ 1, 0≤Cet≤1,for allt∈0, Tand at least one of the following cases will arise:
iC0T 1, CeT<1;
iiC0T<1, CeT 1;
iiiC0T CeT 1.
We will prove that none of these cases is true.
i C0T 1, CeT<1; using the conditionku lu≤gl ml,we have
dC0t dt
tT
kTCeT lT−gTC0T−mTC0T
< kT lT−gT−mT
≤ku lu−gl−ml≤0,
2.11
thus there existt1>0,s.t. 0≤C0t≤1, 0≤Cet≤1,for allt∈T, T t1.This contradicts
the definition of the interval0, T.So there is noT such that 0≤C0t≤1, 0≤Cet≤1,for
allt∈0, TwithC0T 1, CeT<1.
With the same reasoning as in casei,for casesiiandiii,as far astwhich keeps 0≤C0t≤1 and 0≤Cet≤1 is concerned, the interval0, Tcan be extended rightwards.
This contradicts the property ofT.Therefore, 0≤C0t≤1, 0 ≤Cet≤1,for allt≥ 0.This
Putting the expression ofCetinC0t, we can expressC0tin terms of some bounded
continuous functions; therefore the system2.1and2.2may be simplified as follows:
dxt dt xt
r1t−a11txt−a12ty3t,
dy1t
dt b1ty3t−θ1ty1t−δ1ty1t−dtC0ty1t−β1ty
2
1t D1t
y2t−y1t,
dy2t
dt b2ty3t−θ2ty2t−δ2ty2t−β2ty
2
2t D2t
y1t−y2t,
dy3t
dt y3t
−r2t−a22ty3t a21ty3t−τxt−τ θ1ty1t θ2ty2t.
2.12
The initial conditions are
xθ φ1θ, yiθ ψiθ i1,2,3,
φ1θ≥0, ψiθ≥0 i1,2,3, θ∈−τ,0,
x0>0, yi0>0 i1,2,3,
2.13
where Φ φ1θ, ψ1θ, ψ2θ, ψ3θ ∈ C−τ,0, R4, the Banach space of continuous
functions mapping the interval −τ,0 intoR4, where R4 {x1, x2, x3, x4 : xi ≥ 0, i
1,2,3,4}.
3. Uniformly Persistent of System
2.12
Here we wish to discuss the uniformly persistent of the system2.12with initial conditions
2.13, which demonstrates how this system will be uniformly persistent, is that, the long-term survivali.e., will not vanish in timeof all components of the system2.12with initial conditions2.13, under some conditions.
Definition 3.1. The system 2.12 is said to be uniformly persistent, that is, the long-term survivalwill not vanish in timeof all components of the system2.12, if there are positive constantsviandwi i1,2,3,4such that
v1≤lim inf
t→ ∞ xt≤lim supt→ ∞ xt≤w1,
v2≤lim inf
t→ ∞ y1t≤lim supt→ ∞ y1t≤w2,
v3 ≤lim inf
t→ ∞ y2t≤lim supt→ ∞ y2t≤w3,
v4≤lim inf
t→ ∞ y3t≤lim supt→ ∞ y3t≤w4
hold for any solutionxt, y1t, y2t, y3tof2.12with initial conditions2.13. Herevi
andwi i1,2,3,4are independent of2.13.
Theorem 3.2see39. Consider the following equation:
˙
xt axt−τ−bxt−cx2t, 3.2
wherea, b, c, τ >0; xt>0, for−τ≤t≤0.One has the following:
Iif a > b,then limt→ ∞xt a−b/c;
IIif a < b,then limt→ ∞xt 0.
Theorem 3.3. LetXt xt, y1t, y2t, y3tTdenote any solution of system2.12and2.13. Suppose that system2.12satisfiesbu
1 > θl1 δl1 dlC0l, b2u> θl2 δl2,andau21r1u> al11r2l−θu1−θu2>0.
Then there exista T3 >0 such that
xt≤M1, yit≤M4 i1,2,3, ∀t≥T3, where M1>
ru 1
al11,
M4> M∗max
⎧ ⎨ ⎩
bu1−θ1l −δ1l −dlCl 0
βl 1
,b
u
2−θl2−δl2
βl 2
,
au21r1u−al11r2l−θ1u−θ2u al
11al22
⎫ ⎬ ⎭.
3.3
Proof. LetM1 > r1u/al11.We have ˙xt≤xtr1t−a11txt≤xtr1u−al11xt.Therefore,
ifx0≤M1, thenxt≤M1, for allt≥0.Ifx0> M1,and let−α1M1r1u−al11M1, α1>0,
then there exist an 1>0,s.t. if t∈0, 1, xt> M1,and we have ˙xt<−α1<0.Therefore,
there exist aT1 > 0 s.t. xt ≤ M1,for all t ≥ T1,where M1 > r1u/al11.We defineVt
max{y1t, y2t, y3t}. Calculating the upper-right derivative ofVtalong the solution of system2.12and2.13, we have the followings:
1ify1t≥y2t≥y3t,ory1t≥y3t≥y2t,then
D Vt y˙1t
≤bu1y1t−θ1ly1t−δ1ly1t−dlC0ly1t−βl1y21t
bu
1−θl1−δl1−dlCl0
y1t−βl
1y21t, ∀t≥0;
3.4
2ify2t≥y1t≥y3t,ory2t≥y3t≥y1t,then
D Vt y˙2t
≤bu2y2t−θl2y2t−δ2ly2t−βl2y22t
bu
2−θl2−δl2
y2t−βl
2y22t, ∀t≥0;
3ify3t≥y1t≥y2t,or,y3t≥y2t≥y1t,then
D Vt y˙3t
≤ −r2ly3t−al22y32t au21M1y3t−τ
θu1 θu2y3t
−r2l −θu1 −θu2y3t−al22y32t au21M1y3t−τ, ∀t≥T1 τ.
3.6
Let
M2 > M1∗max
bu
1 −θl1−δl1−dlCl0
βl1 ,
b2u−θ2l−δ2l βl2
. 3.7
From3.4and3.5, we can obtain the followings:
iif max{y10, y20, y30} ≤M2,then max{y1t, y2t, ytt} ≤M2,for allt≥0; iiif max{y10, y20, y30} > M2, and let−α2 maxM2{b1u−θ1l −δ1l −dlCl0−
βl
1M2}, M2{bu2−θl2−δl2−βl2M2}, α2>0,we consider the following two cases: aV0 y10> M2,
bV0 y20> M2.
Ifa holds, then there exist an 2>0,s.t. ift∈0, 2, Vt y1t> M2,and we have
D Vt y˙1t<−α2<0.
Ifbholds, then there exist an 3>0,s.t. ift∈0, 3, Vt y2t> M2,and we have
D Vt y˙2t<−α2<0.
If3.6holds, then byTheorem 3.2, there existT2≥T1 τ >0,s.t.
y3t< M3, ∀t≥T2≥T1 τ >0,whereM3>
au21r1u−al11r2l−θu1−θu2 al
11al22
sinceM1 can be chosen close enough to
r1u al
11
.
3.8
From the above discussions, we conclude that there existT3≥T2 ≥T1 τ >0,s.t.
xt≤M1, yit≤M4 i1,2,3, ∀t≥T3, whereM1>
ru 1
al11,
M4> M∗max
⎧ ⎨ ⎩
bu1−θ1l −δ1l −dlCl 0
βl 1
,b
u
2−θl2−δl2
βl 2
,
au21r1u−al11r2l−θ1u−θ2u al
11al22
⎫ ⎬ ⎭.
3.9
Theorem 3.4. Suppose that system2.12and2.13satisfies the following conditions:
b1l > θu1 δu1 duCu0, b2l > θu2 δu2, r2l > θu1 θu2, al21r1l > au11r2u−θ1l−θl2,
max
⎧ ⎨ ⎩
bu1−θl1−δl1−dlCl 0
βl 1
,b
u
2 −θ2l−δl2
βl 2
,a
u
21r1u−al11
r2l−θu1−θu2 al
11al22
⎫ ⎬ ⎭
< M4<min
⎧ ⎨ ⎩ rl 1 au 12 , al
21r1l−au11
ru
2 −θl1−θl2
al 21au12
⎫ ⎬ ⎭.
3.10
Then system2.12and2.13is uniformly persistent.
Proof. LetXt xt, y1t, y2t, y2tTbe a solution of2.12and2.13.∴x˙t≥xtrl 1−
au11xt−au12M4,for allt ≥ T3 usingTheorem 3.3andT3 is defined there. Now, M4 < al
21r1l −au11r2u−θl1−θl2/al21au12,and r2l > θu1 θu2 together imply 0 < r2u−θl1−θl2/al21 < r1l−au12M4/au11.Let us choosem1in such a way that
0< r
u
2 −θl1−θ2l
al21 < m1< rl
1−au12M4
au 11
⇒al21m1> r2u−θ1l−θl2. 3.11
IfxT3 ≥ m1,then xt ≥ m1,for allt ≥ T3.IfxT3 < m1 and letμ1 xT3r1l −au11m1 −
au
12M4>0,then there exist an 1>0,s.t.xt< m1,and ˙x1t> μ1>0,for allt∈T3, T3 1.
Therefore, there exist aT4> T3>0,s.t.xt≥m1,for allt≥T4.
DefineVt min{y1t, y2t, y3t}.Then calculating the lower-right derivative of Vtalong the solution of system2.12and2.13, we obtainthe following.
1ify1t≤y2t≤y3t,or,y1t≤y3t≤y2t,then
D Vt y˙1t
≥b1ly1t−θ1uy1t−δu1y1t−duC0uy1t−βu1y21t
bl1−θu1 −δ1u−duCu0y1t−βu1y12t, ∀t≥0;
3.12
2ify2t≤y1t≤y3t,or,y2t≤y3t≤y1t,then
D Vt y˙2t
≥bl2y2t−θu2y2t−δ2uy2t−βu2y22t
bl
2−θu2 −δ2u
y2t−βu
2y22t, ∀t≥0;
3ify3t≤y1t≤y2t,or,y3t≤y2t≤y1t,then
D Vt y˙3t
≥ −r2uy3t−au22y32t al21m1y3t−τ
θl1 θ2ly3t
−r2u−θl1−θl2y3t−au22y32t al21m1y3t−τ, ∀t≥T4 τ.
3.14
Let us choosem2in such a way that 0< m2<min{bl1−θ1u−δu1 −duCu0/βu1, bl2−
θu
2 −δ2u/βu2.From3.12and3.13, we can obtain the followings:
iif V0 min{y10, y20, y30} ≥ m2,then min{y1t, y2t, ytt} ≥ m2,for all
t≥0;
iiifV0 min{y10, y20, y30}< m2,and letμ2 miny10{bl1−θ1u−δu1−duCu0−
βu
1m2}, y20{bl2−θ2u−δ2u−βu2m2}>0,we consider the following two cases: aV0 y10< m2,
bV0 y20< m2.
Ifa holds, then there exist an 2 > 0,s.t. if t ∈ 0, 2, Vt y1t < m2,and we
have D Vt y˙1t> μ2>0.
Ifb holds, then there exist an 3 > 0,s.t. if t ∈ 0, 3, Vt y2t < m2,and we
have D Vt y˙2t> μ2>0.
If3.14holds, then by3.11andTheorem 3.2, there existT5≥T4 τ >0,s.t.
y3t> m3, ∀t≥T5≥T4 τ >0, where, m3<
al21m1−
r2u−θl1−θ2l
au22 . 3.15
From above discussions, we conclude that there existT6 ≥T5≥T4 τ > T3>0,s.t.xt≥m1
and yit≥m4 i1,2,3,for allt≥T6where
0< r
u
2 −θl1−θl2
al 21
< m1<
r1l−au12M4
au11 ,
0< m4<min
⎧ ⎨ ⎩
bl
1−θ1u−δu1−duC0u
βu 1
,b
l
2−θu2−δu2
βu 2
, al
21m1−
ru
2 −θl1−θ2l
au 22
⎫ ⎬ ⎭.
3.16
From the above discussions, we conclude that there exist T6 ≥ T5 ≥ T4 τ > T3 > 0 s.t.
every solution of system2.12and2.13eventually enters and remains in the regionΩ
{x, y1, y2, y3 | m ≤ x, yi ≤ M, i 1,2,3}, for allt ≥ T6, where m min{m1, m4}and
Mmax{M1, M4},andM1is defined inTheorem 3.3. This completes the proof.
instantaneous per capita death rate function of protective immature predator population, C0t concentration of toxicant in the immature predator organism in nonprotective patch at time t, Cet concentration of toxicant in the environment at time t, and dt dose
response parameter of nonprotective immature predator species to the organismal toxicant concentration at timetlead to make the underlying system uniformly persistent.
4. Global Asymptotic Stability of Periodic Solution
From our everyday experience we know that the biological and environmental parameters are subject to fluctuation in time, and the effects of a periodically varying environment have an important selective forces on systems in a fluctuating environment. To investigate this kind of phenomenon, in the model, the coefficients should be periodic functions of time.
Suppose that system2.12 is a periodic system with initial conditions2.13. Here we assume that the periodic system2.12satisfies all conditions ofTheorem 3.4. We derive sufficient conditions for all solutions of system2.12to converge to a periodic solution.
Theorem 4.1 see 40. Suppose that the continuous operator A maps the closed and bounded convex setQ∈Rnonto itself, then the operatorAhas at least one fixed point in setQ.
Theorem 4.2. If periodic system2.12satisfies conditions ofTheorem 3.4, then there is at least one strictly positive periodic solution of system2.12
Proof. From Theorems 3.3, 4.1 and 2 in Teng and Chen41, it is easy to see that there is at least one strictly positive periodic solution of system2.12.
Theorem 4.3. If system 2.12 has a periodic solution, then it is globally asymptotically stable provided lim inft→ ∞Ait>0, i1,2, δl1 dlCl0 2mβl1 Dl1−Du2 >0, δl2 2mβl2 Dl2−Du1 >0, whereA1t al11 −Ma21t τ, A2t r2l 2ma22l −au12−bu1 −b2u−Ma21t τ, m, Mare defined inTheorem 3.4.
Proof. Letut, v1t, v2t, v3tTis a periodic solution of system2.12and2.13. Suppose thatxt, y1t, y2t, y3tTis any positive solution of2.12and2.13. Let
L1t |lnxt−lnut|
3
i1
yit−vit. 4.1
Calculating the upper-right derivative ofL1talong the positive solution of system 2.12
and2.13, it follows that
D L1t
˙ xt xt−
˙ ut ut
sgnxt−ut
3
i1
˙
yit−v˙it
sgnyit−vit
sgnxt−ut−a11txt−ut−a12ty3t−v3t
sgny1t−v1tb1ty3t−v3t−θ1t δ1t dtC0ty1t−v1t
−β1ty2
1t−v21t
sgny2t−v2tb2ty3t−v3t−θ2t δ2ty2t−v2t
−β2ty22t−v22t D2ty1t−v1t−y2t−v2t
sgny3t−v3t−r2ty3t−v3t−a22ty23t−v23t a21ty3t−τ
×xt−τ−ut−τ a21tut−τ
y3t−τ−v3t−τ
θ1ty1t−v1t θ2ty2t−v2t
≤ −al11|xt−ut| −δ1l dlC0l 2mβ1l D1l −D2uy1t−v1t
−δ2l 2mβl2 Dl2−D1uy2t−v2t−r2l 2mal22−au12−bu1−bu2y3t−v3t
a21ty3t−τ|xt−τ−ut−τ| a21tut−τy3t−τ−v3t−τ.
4.2
Let
L2t
t
t−τa21s τy3s|xs−us|ds
t
t−τa21s τus
y3s−v3sds. 4.3
Then,
D L2t a21t τy3t|xt−ut| −a21ty3t−τ|xt−τ−ut−τ|
a21t τuty3t−v3t−a21tut−τy3t−τ−v3t−τ.
4.4
Let
Lt L1t L2t. 4.5
Then,
D Lt≤ −al11−Ma21t τ|xt−ut| −δ1l dlCl0 2mβl1 Dl1−Du2y1t−v1t
−δ2l 2mβl2 D2l −D1uy2t−v2t
−r2l 2mal22−au12−bu1−bu2 −Ma21t τ
y3t−v3t
−A1t|xt−ut| −δl1 dlCl0 2mβl1 Dl1−Du2y1t−v1t
−δ2l 2mβl2 D2l −D1uy2t−v2t−A2ty3t−v3t,
whereA1t al11−Ma21t τ, A2t r2l 2ma22l −au12−bu1−bu2−Ma21t τ.By assumption,
δl
1 dlCl0 2mβl1 D1l−Du2 >0, δ2l 2mβ2l Dl2−Du1 >0,and there existα1 >0, α2>0, T∗≥T6>0, T6is defined inTheorem 3.4, such thatA1t≥α1, A2t≥α2,for allt≥T∗.Integrating both
sides of4.6on the intervalT∗, t,we get
Lt
t
T∗
A1s|xs−us|ds δ1l dlCl0 2mβl1 Dl1−Du2
t
T∗
y1s−v1sds
δl
2 2mβl2 Dl2−Du1
t
T∗
y2s−v2sds
t
T∗
A2sy3s−v3sds≤LT∗
⇒Lt α1
t
T∗
|xs−us|ds δl
1 dlCl0 2mβl1 Dl1−Du2
t
T∗
y1s−v1sds
δl2 2mβl2 Dl2−Du1
t
T∗
y2s−v2sds α2
t
T∗
y3s−v3sds≤LT∗
⇒Ltis bounded on T∗,∞and also
∞
T∗|xs−us|ds <∞
∞
T∗
yis−visds <∞ i1,2,3.
4.7
ByTheorem 3.3,|xs−us|, |yis−vis| i1,2,3are bounded onT∗,∞.On the other
hand, it is easy to see that ˙xt, ˙ut, ˙yit, and ˙vit i 1,2,3 are bounded fort ≥ T∗.
Therefore, |xt−ut|, |yit−vit| i 1,2,3are uniformly continuous onT∗,∞.By
Barbalat’s Lemma42, we conclude that
|xt−ut| −→0, yit−vit−→0 ast−→ ∞, i1,2,3. 4.8
This completes the proof.
From Theorems3.4and4.3we observe that the diffusion coefficient functions and time delay have no effect on the uniformly persistent of the system2.12but they have some effect on the global asymptotic stability of this system.
5. Conclusions
The toxic of polluted population comes from the environment. Suppose that environment of patch 1 is polluted nonprotective region, and patch 2 is the protective region for the immature predator population which is only affected by pollution. In order to protect the existence of the polluted population, we can use the artificial method, the one’s own purification function of the population, toxin in the body of the individuals in nonprotective patch can be removed, and then we will bring them into protective patch. The scale of the protective zone can be regulated through diffusive coefficientsDit, i1,2.
We have obtained the survival threshold of each population of this system. Our mathematical results reveal the fact that lower valuesδ1t instantaneous per capita death rate function of nonprotective immature predator population, δ2t instantaneous per capita death rate function of protective immature predator population,C0t concentration of toxicant in the immature predator organism in nonprotective patch at time t, Cet concentration of toxicant in the environment at timet, anddt dose response parameter of nonprotective immature predator species to the organismal toxicant concentration at timet lead to make the underlying system uniformly persistent. By constructing an appropriate Lyapunov function, we have derived sufficient conditions to confirm that if this system admits a positive periodic solution, then it is globally asymptotically stable. We have observed that the diffusion coefficient functions and time delay have no effect on the uniformly persistent of the system2.12but they have some effect on the global asymptotic stability of this system.
In future work, it would be interesting to expand on simulations by using some current realistic data to estimate the parameters. The collection of data relevant to key parameters by ecologists and environmentalists at a global level would be of significant use from an implementation and policy perspective.
Acknowledgments
The author is very grateful to the anonymous referees and the Editor Dr. Harvinder S. Sidhufor their careful reading, valuable comments, and helpful suggestions, which have helped him to improve the presentation of this work significantly. He is grateful to Professor A.Dvurecenskij, Dr. Vrto, Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovak Republic, and Dr. D. N. Garain, P. G. Department of Mathematics, S. K. M. University, Dumka, Jharkhand, India for their helps and encouragements. He is thankful to Slovak Academic Information AgencySAIA, Bratislava, Slovak Republic for financial support.
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