Harvesting on a State-Dependent Time Delay Model

Ruijun XIE, Xin ZHANG, Wei ZHANG

Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (1) : 82-96.

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Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (1) : 82-96. DOI: 10.21078/JSSI-2020-082-15
 

Harvesting on a State-Dependent Time Delay Model

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Abstract

In this paper, we propose and analyze a cooperation model with harvesting and statedependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. The main purpose of this article is to obtain the dynamics of our model analytically by controlling the harvesting. We present results on positivity and boundedness of all populations. Criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given by controlling the harvesting. Finally, the global exponentially asymptotical stability criteria of model is obtained by the improved Hanalay inequality.

Key words

state-dependent delay / exponential asymptotic stability / harvesting

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Ruijun XIE , Xin ZHANG , Wei ZHANG. Harvesting on a State-Dependent Time Delay Model. Journal of Systems Science and Information, 2020, 8(1): 82-96 https://doi.org/10.21078/JSSI-2020-082-15

1 Introduction

In recent years, state-dependent delay equations has been hot in research. Many authors investigated state-dependent time delayed population model in [1-14]. Since it is more realistic than constant delay in describing the population dynamics, for example Antarctic whale and seal populations in [15] and the duration of larval development of flies in [16]. Particularly, Lv and Yuan[10] studied the stability of the equilibria of the delay differential equation system
{u1(t)=α1eγ1τ(u1)u1(tτ(u1))β1u12+μ1u1u2,u2(t)=α2eγ2τ(u2)u2(tτ(u2))β2u22+μ2u1u2,u1(θ)=φ1(θ)0,u2(θ)=φ2(θ)0,θ[τM,0].
(1)
The variables ui(t) (i=1,2) denote the densities of cooperative mature species at time t and at position x. the delay τ is the time taken from birth to maturity. The parameters αi and γi (i=1,2) denote the birth rate of mature species i and the death rate of immature species i respectively, and βi measures the competitive effect within mature species i.
Along with the socio-economic development, the ecological resources are faced with being exhausted because of human over-exploitation. However, more and more people realize that the ecological resources not only provide the direct economic value, but also have an extremely vital ecological value. In the last few decades, the research of population resources development and management has been quickly developed and played an important role in the use of biological resources with the long term eco-efficiency and economic benefits. It is well known that harvesting has a strong impact on the dynamic evolution of a population; there has been interest in the modeling of harvesting of biological resources.
In 2015, Al-Omari[6] studied the dynamics of state dependent delay and harvesting on a stage-structured predator-prey model
{x(t)=rx(t)(1x(t)k)ay(t)x(t)h1x(t),y(t)=be(γ+h3)τ(u)y(tτ(u))x(tτ(u))dy(t)h2y(t),yi(t)=by(t)x(t)γyi(t)be(γ+h3)τ(u)y(tτ(u))x(tτ(u))h3yi(t),
where u=yi+y and x(t) and y(t) represent prey and mature predator densities, respectively. yi(t) denotes the immature or juvenile predator density. Here, criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given and global stability of trivial and the boundary equilibria is investigated.
Motivated by the above works, we propose the following cooperation model with state-dependent time delay and harvesting
{v1(t)=α1u1γ1v1α1e(γ1+h~1)τ(u1)u1(tτ(u1))h~1v1,u1(t)=α1e(γ1+h~1)τ(u1)u1(tτ(u1))β1u12+μ1u1u2h1u1,v2(t)=α2u2γ2v2α2e(γ2+h~2)τ(u2)u1(tτ(u2))h~2v2,u2(t)=α2e(γ2+h~2)τ(u2)u2(tτ(u2))β2u22+μ2u1u2h2u2.
(2)
We take the following initial conditions in model (2)
u1(θ)=φ1(θ)0,u2(θ)=φ2(θ)0,θ[τM,0],tR+,
(3)
and
vi0=τ(ui(0))0αiuie(γi+h~i)sds,i=1,2.
The variables vi(t) and ui(t) (i=1,2) denote the densities of cooperative immature and mature species at time t and at position x, respectively. The parameters αi and γi (i=1,2) denote the birth rate of mature species i and the death rate of immature species i respectively, and βi measures the competitive effect within mature species i. The term αieγiτ(ui)ui(tτ(ui)) appearing in both equations represents the rate at time t and position x at which individuals leave the immature and enter the mature class, having just reached maturity. Assume μ1>0 and μ2>0 represent interspecific cooperative effects of u2 on u1 and u1 on u2, respectively. The terms hi,hi~ are the harvesting efforts of the immature and mature species i, respectively. Since we find that the second equation and the fourth equation can be solved, which are independent of the one equation and third equation in model (2), we shall study the dynamics of the cooperation model with state-dependent delay and harvesting
{u1(t)=α1e(γ1+h~1)τ(u1)u1(tτ(u1))β1u12+μ1u1u2h1u1,u2(t)=α2e(γ2+h~2)τ(u2)u2(tτ(u2))β2u22+μ2u1u2h2u2,u1(θ)=φ1(θ)0,u2(θ)=φ2(θ)0,θ[τM,0].
(4)
So we give the following assumptions for model (4):
(A1) τ(ui)0 and 0<τmτ(ui)τM with τm=τ(0) and τM=τ().
(A2) As discussed in [6], for our model to make sense, that is, tτ(ui(t)) is an increasing function of t. There exists L>0, such that τ(u)<L. It seems little is known on harvesting for cooperation model with state-dependent delay, so considering harvesting, state-dependent delay at the same time in a cooperation model is an innovative topic, which is the focus of our research. So we propose a cooperative systems with state-dependent delay and harvesting to discuss the harvesting problem. Compared to the literature [10], By controlling the harvesting efforts of the immature and mature species i, we get the positivity and boundedness property of the solution of (4). Furthermore, we obtain the exponentially asymptotically stable of extinction equilibrium and coexistence equilibrium by the improved Hanalay inequality.
The remaining parts of this paper are organized as follows. We show the existence and linearized stability of equilibria in Section 2. Section 3 is devoted to study the exponentially asymptotically stable of extinction equilibrium and coexistence equilibrium by controlling the harvesting efforts. Section 4 concludes.

2 Existence and Linearized Stability of Equilibria

We show the positivity and boundedness of the solution of (4) by controlling the harvesting efforts of the immature and mature species i. From the standpoint of biology, positivity means that the system persists, i.e., the populations may survive. Boundedness may be viewed as a natural restriction to growth as a result of limited resources in a closed environment.
Theorem 1 Let φi(θ)>0, i=1,2, for θ[τM,0], Then
(a) ui(t)>0 for t>0;
(b) If hi<αie(ri+h~i)τM, there exists δi=δi(φi)>0 such that ui(t)>δi for all t0, where
δi(φi)=12min{infτMθ0φi(θ),βi1(αie(ri+h~i)τMhi)}.
(5)
Proof (a) We show the positivity of ui. Otherwise, there would be t1 so that ui(t1)=0. Let t:=inf{t:t>0,ui(t)=0}. Then we have
ui(t)>0  fort[τM,t)  and  ui(t)0.
From the ith Equaiton (4), we have
ui(t)=αie(γi+h~i)τmui(tτm).
From assumption tτm[τM,t], then ui(t)>0. This is a contradiction. So we obtain the results (a).
(b) We show that ui is uniformly bounded away from zero for a given positive initial function. Set
δi(φi)=12min{infτMθ0φi(θ),βi1(αie(ri+h~i)τMhi)}.
Otherwise, there exists a s1 such that s1=inf{t: t0,ui(t)=δi} and ui(s1)0. By the definition of δi, we have ui(0)=φi(0)2δi. It follows from the continuity that s1>0. Thus,
ui(s1)=αie(ri+h~i)τ(ui)ui(s1τ(ui))βiui2(s1)+μiu1(s1)u2(s1)hiui(s1)αie(ri+hi~)τMδiβiδi2hiδiαie(ri+hi~)τMδi12(αieγi(ri+h~i)τMhi)δihiδi=12(αie(ri+h~i)τMhi)δi>0.
We have a contradiction. So, such s1 does not exist and ui(t)>δi for all t>0.
Theorem 2 Assume that (A1)(A2) holds. If β1β2>μ1μ2, then every solution (u1(t),u2(t)) of system (4) is uniformly bounded.
Proof Our proof is divided into two steps.
(i) Suppose that ui(t)0 for all t>T for some T0. Then for t>T+τM,
0ui(t)=αie(γi+hi)τ(ui)ui(tτ(ui))βiui2(t)+μiu1(t)u2(t)hiui(t)αiui(t)βiui2(t)+μiu1(t)u2(t)hiui(t),
since ui(tτ(ui))ui(t). This means that
β1u1μ1u2α1,β2u2μ2u1α2,t>T+τM.
Thus, it follows from ui(t)>0 and assumption that
u1(t)β2α1+μ1α2β1β2μ1μ2,u2(t)β1α2+μ2α1β1β2μ1μ2,
so our desired results are obtained.
(ii) Assume that u(t) is not monotone in the future. Then there are two sequences {tn}n=1 and {sm}m=1such that u1(tn)=0, u2(sm)=0, and u1(tn), u2(sm) is a local maximum, where u1(t)u1(tn), 0<t<tn for all n, and u2(t)u2(sm), 0<t<sm for all m, then by a similar analysis at t=tn and t=sm, it follows that
β1u1(tn)μ1u2(tn)α1,β2u2(sm)μ2u1(sm)α2.
(6)
For any given tn, we take sn=max{sm:smtn}. If sn=tn, it follows from (i) that the solutions u1(t) and u2(t) are bounded above by a bound. If sn<tn and u2(sn)<u2(tn), then u2(tn)>0 and u2(t)u2(tn) for all ttn. Otherwise, there is a sn<t<tn such that u2(t)=0, which contradicts the definition of sn. Thus, we have
0<u2(tn)=α2e(γ2+h2)τ(u2)u2(tnτ(u2))β2u22(tn)+μ2u1(tn)u2(tn)h2u2(tn)α2u2(tn)β2u22(tn)+μ2u1(tn)u2(tn)h2u2(tn).
In a similar way of (i), we get the same results that u1(t) and u2(t) are bounded above by a bound. If sn<tn and u2(tn)<u2(sn), then it follows from the first inequality of (6) that
β1u1(sn)μ1u2(sn)β1u1(tn)μ1u2(tn)α1.
Combined with the second inequality of (6), we obtain the same results that u1(t) and u2(t) are bounded above by a bound. thus, every solution (u1(t),u2(t)) of system (4) is uniformly bounded.
Remark 1 It shows that the eventually population densities are only subject to interspecific cooperative effects μi and competitive effectβi and independents on harvesting effects hi and hi. If β1β1>μ1μ2, the eventually population densities is uniformly bounded.
Next, we discuss steady states of system (4). We first examine the nullclines of the system
{α1e(γ1+h1)τ(u1)β1u1+μ1u2h1=0,α2e(γ2+h2)τ(u2)β2u2+μ2u1h2=0.
(7)
It is easy to show that system (4) has the extinction equilibrium E0=(0,0), where hi and hi are no any restrictions, the boundary equilibria E1=(u^1,0) and E2=(0,u^2) where u^i satisfies the following equation
αie(γi+hi)τ(ui)βiuihi=0,i=1,2.
We observe that αie(γi+hi)τ(ui) is a decreasing function with respect to ui, and
αie(γi+hi)τ(0)=αie(γi+hi)τm>0,limuiαie(γi+hi)τ(ui)=αie(γi+hi)τM.
Furthermore, limuiβiui+hi=. If hiαie(γi+hi)τm, there is no boundary equilibria, this has only the unique extinction equilibrium E0. If hi<αie(γi+hi)τm, there exists a boundary equilibria E1=(u^1,0) or E2=(0,u^2). Therefore, if the boundary equilibria exists, then
0<u^i=βi1(αie(γi+hi)τ(ui^)hi)<βi1(αie(γi+hi)τmhi)
is unique.
For the well-extended logistic model (4), we denote by u^i the carrying capacity of the species i and imply that harvesting rate hi and hi have a impact on the the carrying capacity of the species i, that means that the harvesting hi is smaller, the the carrying capacity of the species i is stronger. The explanation is accordance with the actual case.
The equations (7) can be rewritten as the following equations
{β1u1+μ1u2=h1α1e(γ1+h1)τ(u1),μ2u1β2u2=h2α2e(γ2+h2)τ(u2).
(8)
If β1β1>μ1μ2, then we can get that the corresponding determinant
|β1μ1μ2β2|=β1β2μ1μ2>0,
system (8) is equivalent to the system
{u1=β2(α1e(γ1+h1)τ(u1)h1)+μ1(α2e(γ2+h2)τ(u2)h2)β1β2μ1μ2,u2=β1(α2e(γ2+h2)τ(u2)h2)+μ2(α1e(γ1+h1)τ(u1)h1)β1β2μ1μ2.
(9)
Denote g(ui)=(β1β2μ1μ2)ui,i=1,2 and
f(u1)=β2(α1e(γ1+h1)τ(u1)h1)+μ1(α2e(γ2+h2)τ(u2)h2),f(u2)=β1(α2e(γ2+h2)τ(u2)h2)+μ2(α1e(γ1+h1)τ(u1)h1),
then we know that g(ui) is a increasing function and f(ui) is a decreasing function with respect to ui, then the uniqueness of coexistence equilibrium is obtained providing
{limu10f(u1)=β2(α1e(γ1+h1)τmh1)+μ1(α2e(γ2+h2)τ(u2)h2)>0,limu20f(u2)=β1(α2e(γ2+h2)τmh2)+μ2(α1e(γ1+h1)τ(u1)h1)>0.
(10)
Inequality (10) is established providing
{β2(α1e(γ1+h1)τmh1)+μ1(α2e(γ2+h2)τMh2)>0,β1(α2e(γ2+h2)τmh2)+μ2(α1e(γ1+h1)τMh1)>0
(11)
{β2h1+μ1h2<β2α1e(γ1+h1)τm+μ1α2e(γ2+h2)τM,μ2h1+β1h2<β1α2e(γ2+h2)τm+μ2α1e(γ1+h1)τM.
(12)
Inequality (12) is established providing β1β2>μ1μ2and hihi,i=1,2, where
{h1=β1β2α1e(γ1+h1)τmμ1μ2α1e(γ1+h1)τMβ1μ1(α2e(γ2+h2)τmα2e(γ2+h2)τM)β1β2μ1μ2,h2=β1β2α2e(γ2+h2)τmμ1μ2α2e(γ2+h2)τMβ2μ2(α1e(γ1+h1)τmα1e(γ1+h1)τM)β1β2μ1μ2.
In order to make hi>0,i=1,2, it provides that
{β1β2α1e(γ1+h1)τmμ1μ2α1e(γ1+h1)τM>β1μ1(α2e(γ2+h2)τmα2e(γ2+h2)τM),β1β2α2e(γ2+h2)τmμ1μ2α2e(γ2+h2)τM>β2μ2(α1e(γ1+h1)τmα1e(γ1+h1)τM).
(13)
Inequality (13) is established providing
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
(14)
Here, it is not difficulty to prove that
β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM)>β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM>0,
so inequality (14) is meaningful.
Solving (9), we get the uniqueness of coexistence equilibrium (u1,u2), where
u1=β2(α1e(γ1+h1)τ(u1)h1)+μ1(α2e(γ2+h2)τ(u2)h2)β1β2μ1μ2,
u2=β1(α2e(γ2+h2)τ(u2)h2)+μ2(α1e(γ1+h1)τ(u1)h1)β1β2μ1μ2,
(15)
provided u1>0,u2>0 if and only if β1β2>μ1μ2, hihi,i=1,2, and
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
Particularly, if
β1μ2>α1(e(γ1+h1)τme(γ1+h1)τM)α2(e(γ2+h2)τme(γ2+h2)τM),β2μ1>α2(e(γ2+h2)τme(γ2+h2)τM)α1(e(γ1+h1)τme(γ1+h1)τM),
we can get that β1β1>μ1μ2andαie(γi+hi)τM<hi<αie(γi+hi)τm. Therefore, we have the following Theorem.
Theorem 3 Let hi0,i=1,2. Then system (4) has
(a) a unique extinction equilibrium E0=(0,0);
(b) positive boundary equilibrium E1=(u^1,0) and E2=(0,u^2)if hi<αie(γi+hi)τm;
(c) coexistence equilibrium E=(u1,u2) if β1β2>μ1μ2 and hihi,i=1,2, and
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
(d) coexistence equilibrium E=(u1,u2) if hihi,i=1,2, and
β1μ2>α1(e(γ1+h1)τme(γ1+h1)τM)α2(e(γ2+h2)τme(γ2+h2)τM),β2μ1>α2(e(γ2+h2)τme(γ2+h2)τM)α1(e(γ1+h1)τme(γ1+h1)τM),
where αie(γi+hi)τM<hi<αie(γi+hi)τm.
Remark 2 Here, (d) is specific csae of (c). From above analysis, hi, hi, α2α1 and state-dependent delay τ(u) have an important influence on equilibriums of the system (4). As we know, equilibria usually represents the ultimate populations in the Biology. So we can regulate the harvesting rate hi, hi or α2α1 to control the population. We can conclude that if the coexistence equilibria exists, then u1>u^1 and u2>u^2, which implies that the mutualism effects raises the equilibrium levels of each species. That is, the coexistence equilibrium values ui is greater than the level u^i (the carrying capacities for each species) in the absence of cooperative interaction.
Next, we will discuss the linearized stability of equilibria (see [10, 17, 18] for more details about linearization and stability of state-dependent delay differential equations. let E=(u10,u20) is any equilibrium, the system (4) can be linearized to next equation about E
(u1(t)u2(t))=(2β1u10ξ1+μ1u20h1μ1u10μ2u202β2u20ξ2+μ2u10h2)(u1(t)u2(t))+(α1e(γ1+h~1)τ(u10)00α2e(γ2+h~2)τ(u20))(u1(tτ(u10))u2(tτ(u20))),
(16)
where
ξi=αiγiui0e(γi+h~i)τ(ui0)τ(ui0),i=1,2,
(17)
and trial solutions proportional to (c1,c2)exp(λt) leads to the characteristic equation
|a11ξ1+μ1u20λh1μ1u10μ2u20a22ξ2+μ2u10λh2|=0,
(18)
where aii=αieτ(ui0)(γi+hi+λ)2βiui0.
For the extinction equilibrium E0=(0,0), the Equation (18) reduces to
(α1eτm(γ1+h1+λ)λh1)(α2eτm(γ2+h2+λ)λh2)=0.
All eigenvalues are given by solutions of
λi=αieτm(γi+hi+λi)hi,i=1,2.
If hi<αie(γi+hi)τm, assume that λi0, then λi=αieτm(γi+hi+λi)hiαie(γi+hi)τmhi>0, this is a contradiction, then λi>0, so E0 is an unstable point. If hi>αie(γi+hi)τm, in a similar way, we get λi<0, then E0 is linear stable. If hi=αie(γi+hi)τm, we have λi=αie(γi+hi)τm(eλiτm1), we get λi=0, E0 is an degenerate point. This implies that species may get extinct ultimately if hi>αie(γi+hi)τm.
For the boundary equilibrium E1=(u^1,0), the eigenvalues are the roots of equation
(α1eτ(u1^)(γ1+h1+λ)2β1u1^ξ1λh1)(α2eτm(γ2+h2+λ)+μ2u1^λh2)=0,
so that some of the eigenvalues are given by the equation
λ+2β1u1^+ξ1+h1=α1eτ(u1^)(γ1+h1+λ).
Now we show that all eigenvalues have negative real parts. Suppose that Reλ0, since hi<αie(γ1+h1)τmandu^i=βi1(αie(γi+hi)τ(ui^)hi), then from the above equation and Equation (17), we have
Reλ+2β1u1^+(β1μ1+h1)(γ1u1^τ(u1^))+h1=α1eγ1τ(u1^)eτ(u1^)Reλcos(τ(u1^Imλ)=(β1μ1+h1)eτ(u1^)Reλcos(τ(u1^Imλ)(β1μ1+h1),
then we can get that
Reλβ1u1^(β1μ1+h1)(γ1u1^τ(u1^))<0,
a contradiction proving the claim.
The others are given by the equation
λμ2u1^=α2eτm(γ2+h2+λ)h2.
Since hi<αie(γ1+h1)τm, it is easy to prove that the all eigenvalues have one positive real parts by the counter example. Thus, the boundary equilibrium E1=(u1^,0) is linearly unstable.
In a similar way, we can show that E2=(0,u2^) is linear unstable.
Now we show that the coexistence equilibrium state E=(u1,u2) is linear stable if β1β2>μ1μ2,hi<hi and
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
From (18), we get the characteristic equation
(λa11+ξ1μ1u2+h1)(λa22+ξ2μ2u1+h2)μ1μ2u1u2=0.
(19)
Let
D1=a+β1u1+a1e(γ1+h1)τ(u1)(1+γ1u1τ(u1)eaτ(u1)cos(bτ(u1))),D2=a+β2u2+a2e(γ2+h2)τ(u2)(1+γ2u2τ(u2)eaτ(u2)cos(bτ(u2))),E1=b+a1eτ(u1)(γ1+h1+a)sin(bτ(u1)),E2=b+a2eτ(u2)(γ2+h2+a)sin(bτ(u2)).
Let λ=a+bi, where a and b are real numbers. Since
β1u1+h1μ1u2=a1e(γ1+h1)τ(u1),β2u2+h2μ2u1=a2e(γ2+h2)τ(u2),
Substituting λ=a+bi into Equation (19), we have
D1D2E1E2=μ1μ2u1u2   and   D1E2+D2E1=0.
Then
(μ1μ2u1u2)2=(D1D2)2+(E1E2)22D1D2E1E2=(D1D2)2+(E1E2)2+2(D1E2)2>(D1D2)2.
Now, we assume that Reλ=a0. By
1+γ1u1τ(u1)eaτ(u1)cos(bτ(u1))0,
we have Diβiui. Since β1β2μ1μ2, we can get that
D1D2μ1μ2u1u2,
which contradicts the assumption, then all eigenvalues of Equation (19) have the negative real parts. Therefore, the coexistence equilibrium state E is linearly stable under the given conditions.

3 Globally Exponentially Asymptotically Stability of E0,E

According to above analysis, we know that only E0 and E are linear stable under given conditions. From biological prospective, the globally asymptotically stable of the equilibrium E0=(0,0) and the the coexistence equilibrium state E=(u1,u2) represent the populations will get extinct and permanent respectively. So, by controlling the harvesting, we can control the population. The following lemma are useful, which can be found in [19].
Lemma 1 (Improved Hanalay Inequality) (see [19]) Assume that α>β>0,τ0 are constants and x(t) is nonnegative function on [t0τ,+). If
x˙(t)αx(t)+βsuptτ<θ<tx(θ)foralltt0,
then,
x(t)supt0τ<θ<t0x(θ)eλ(tt0),
(20)
where λ is the unique positive solution of λ=αβeλτ.
Now, we show that state-dependent delay has an impact on the system (4) by comparing Theorem 4 with Theorem 5. Let
h~=min{h1,h2},P~=max{α1e(γ1+h1)τ,α2e(γ2+h2)τ}.
Theorem 4 Let τ(ui)=τ, whereτ is constant. If h~>P~ and βi>μ1+μ22,i=1,2, then the extinction equilibrium E0=(0,0) of system (4) is exponentially asymptotically stable.
Proof Let τ(ui)=τ, then system (4) can be reduced to the following equation
{u1(t)=α1e(γ1+h1)τu1(tτ)β1u12+μ1u1u2h1u1,u2(t)=α2e(γ2+h2)τu2(tτ)β2u22+μ2u1u2h2u2.
(21)
Let u(t)=u1(t)+u2(t), it follows from system (4), we have
u(t)=α1e(γ1+h1)τu1(tτ)+α2e(γ2+h2)τu2(tτ)β1u12(t)β2u22(t)+(μ1+μ2)u1(t)u2(t)h1u1(t)h2u2(t)α1e(γ1+h1)τu1(tτ)+α2e(γ2+h2)τu2(tτ)β1u12(t)β2u22(t)+(μ1+μ2)u12+u222h1u1(t)h2u2(t)P~u(tτ)h~u(t)P~suptτθtu(θ)h~u(t),fort0,
due to the conditions βi>μ1+μ22,i=1,2. Sinceh~>P~, it follows from the Lemma 1, we can get
ui(t)u(t)supτθ0u(θ)eλ~t,i=1,2,
where λ~ is the unique positive solution of λ~=hP~eλ~τ.
Theorem 5 Let τ(ui)0. If h~>2P¯ and βi>μ1+μ22,i=1,2, then the extinction equilibrium E0 of system (4) is exponentially asymptotically stable.
Proof Let u(t)=u1(t)+u2(t) and P¯=max{α1e(γ1+h1)τm,α2e(γ2+h2)τm}, it follows from system (4), we can get
u(t)=α1e(γ1+h1)τ(u1)u1(tτ(u1))+α2e(γ2+h2)τ(u2)u2(tτ(u2))β1u12(t)β2u22(t)+(μ1+μ2)u1(t)u2(t)h1u1(t)h2u2(t)α1e(γ1+h1)τmu1(tτ(u1))+α2e(γ2+h2)τmu2(tτ(u2))β1u12(t)β2u22(t)+(μ1+μ2)u12+u222h1u1(t)h2u2(t)P~(u1(tτ(u1))+u2(tτ(u2)))(β1μ1+μ22)u12(t)(β2μ1+μ22)u22(t)h~u(t)2P¯suptτMθtu(θ)h~u(t),  fort0,
due to the conditions βi>μ1+μ22,i=1,2. Sinceh~>2P¯, it follows from the Lemma 1, we can get
ui(t)u(t)supτMθ0u(θ)eλ~t,i=1,2,
where λ~ is the unique positive solution of λ~=h2P~eλ~τM.
According to above analysis, we know that the extinction equilibrium E0 is exponentially asymptotically stable. This completes the proof.
Remark 3 From the Theorem 4, Theorem 5, it can be founded that the state-dependent delay τ(u) affects the stability of the extinction equilibrium E0. When the state-dependent delay τ(u) introduced into system (21), in order to make the extinction equilibrium E0 of system (4) exponentially asymptotically stable, we must make harvesting hi get larger, this implies the state-dependent delay τ(u) affects harvesting hi.
Lemma 2[14] Let u1(t) be the solution of
u1(t)=α1e(γ1+h1)τ(u1)u1(tτ(u1))β1u12+λu1,t>0,
and u2(t) some function satisfying
u2(t)α1e(γ1+h1)τ(u2)u2(tτ(u2))β1u22+λu2,t>0.
(22)
Assume also that u2(θ)u1(θ) for all θ[τM,0]. Then u2(t)u1(t) for all t>0.
For the general case, let ε>0 and uε(t) be the solution of
u1(t)=α1e(γ1+h1)τ(u1)u1(tτ(u1))β1u12+λu1ε
with initial data uε(θ)=u1(θ)ε, θ[τM,0). we can get u1(t)=limε0uε(t)u2(t).
Theorem 6 Let u1(t) be the solution of
du1(t)dt=α1e(γ1+h1)τ(u1(t))u1(tτ(u1(t)))β1u12(t)hu1(t),
(23)
with the initial data u1(θ)=φ1(θ)>0, for θ[τM,0].
If h<α1e(γ1+h1)τm, then limtu1(t)=uˇ1, where
uˇ1=β11(α1e(γ1+h1)τ(uˇ1)h).
Proof It is not difficult to prove that u1(t) is positive and bounded, here, it can be omitted. The proof can be divided into two steps. (i) Assume thatu1(t) is eventually monotonic. In this case, we know that limtu1(t)exists and the system has two equilibrium E0=0andE1=uˇ1, whereuˇ1=β11(α1e(γ1+h1)τ(uˇ1)h). If h<α1e(γ1+h1)τm, then E0=0 is not stable by using a standard linearized analysis and limtu1(t)=uˇ1.
(ii) Assume that u1(t) is oscillatory. There has two cases. (a) u1(t) is eventually oscillatory below the uˇ1. Then u1(t) has an infinite sequence of local minima and define the sequence tj, which satisfies u1(tj)=0, u1(tj)>0, andu1(tj)<uˇ1, for j=1,2,. Now, we prove that inftt1u1(t)=u1(tk) for some integer k. Otherwise, after every local minimum u1(tj), there is another that is lower, and a subsequence of tj (still relabeled tj), then we can choose sequence tj, which satisfies t1tj<tj+1,tj and u1(tj)>u1(tj+1), Since u1(tjτ(u1(tj)))>u1(tj), we have
0=u1(tj)=α1e(γ1+h1)τ(u1(tj))u1(tjτ(u1(tj)))β1u12(tj)hu1(tj)>u1(tj)(α1e(γ1+h1)τ(u1(tj))β1u1(tj)h1)>u1(tj)(α1e(γ1+h1)τ(uˇ1)β1uˇ1h1)=0,
this is a contradiction. So inftt1u1(t)=u1(tk) for some integer k. Let tk=s1. For ttk+1, in a similar way, there is tl such that infttk+1u1(t)=u1(tl) for some integer l>k. Let tl=s2. Continuing this process, we can get an infinite sequence {sj}, which satisfies s1sj<sj+1,sj and u1(sj)<u1(sj+1) with u1(sj)<uˇ1, it is not difficult to get that u1(sj)uˇ1, as sj, this also implies the amplitude converge zero eventually.
(b) u1(t) is not eventually oscillatory below the uˇ1. Then u1(t) has an infinite sequence of local maxima, without loss of generality, define the sequence tj, which satisfies u1(tj)=0 and u1(tj)<0, with the case of the local maximum u1(tj)>uˇ1, for j=1,2,. Similar to (a), there is an infinite sequence {sj}, which satisfies s1sj<sj+1,sj and u1(sj)>u1(sj+1) with u1(sj)>uˇ1, we have u1(sj)uˇ1, as sj, this implies the amplitude converge zero eventually. Considering all these things, we obtain the result.
Now we will show the globally exponentially asymptotically stable of the coexistence equilibrium E under given conditions.
Theorem 7 If φi(θ)ui,i=1,2τMθ0, Then ui(t)ui for all t0.
Proof Now we show that ui(t)ui for all t0. Otherwise, without loss of generality, there exists a s1 such that s1=inf{t: t0,u1(t)=u1}, u1(s1)0, andu2(t)u2 for ts1. It follows from the continuity that s1>0. Thus,
u1(s1)=α1e(r1+h1)τ(u1)u1(s1τ(u1))β1(u1)2+μiu1u2(s1)h1u1>αie(ri+hi)τ(u1)u1β1(u1)2+μiu1u2h1u1=0.
This is contradiction, the proof is complete.
Theorem 8 The coexistence equilibrium E of system (4) with initial condition φi(θ)ui,i=1,2τMθ0, is globally exponentially asymptotically stable if min{β1,β2}>max{μ1,μ2} and hihi,i=1,2, and
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
Proof From condition of Theorem 8, we have βi>μ1+μ22>μ1μ2,i=1,2 and β1β2>μ1μ2. it follows from Theorem 3 and Theorem 7 that there exists the coexistence equilibrium E=(u1,u2) in system (4) and ui(t)ui,i=1,2. Let yi(t)=ui(t)ui0, where (u1(t),u2(t)) is a solution of system (4). Let y(t)=y1(t)+y2(t). Since
αie(γi+hi)τ(ui)uiβi(ui)2+μiu1u2hiui=0,i=1,2.
From (4), we have
y(t)=y1(t)+y2(t)=α1e(γ1+h1)τ(y1+u1)[y1(tτ(y1+u1))+u1]β1(y1+u1)2+μ1(y1+u1)(y2+u2)h1(y1+u1)+α2e(γ2+h2)τ(y2+u2)[y2(tτ(y2+u2))+u2]β2(y2+u2)2+μ2(y1+u1)(y2+u2)h2(y2+u2)(α1e(γ1+h1)τ(u1)+α2e(γ2+h2)τ(u2))suptτMθty(θ)+(μ1+μ22β1)y12+(μ1+μ22β2)y22+(μ1u2+μ2u22β1u1h1)y1+(μ2u1+μ1u12β2u2h2)y2(α1e(γ1+h1)τ(u1)+α2e(γ2+h2)τ(u2))suptτMθty(θ)[(h1+h2+2β1u1+2β2u2(μ1+μ2)(u1+u2)]y(t).
By Lemma 1, if
h1+h2+2β1u1+2β2u2(μ1+μ2)(u1+u2)>α1e(γ1+h1)τ(u1)+α2e(γ2+h2)τ(u2),
that is
(β1μ1)u1+(β2μ2)u2>0,
which is forever admitted due to the condition min{β1,β2}>max{μ1,μ2}. we have
y(t)supτM<θ<0y(θ)eλt,
where λ is the unique positive solution of λ=h1+h2+2β1u1+2β2u2(μ1+μ2)(u1+u2)(α1e(γ1+h1)τ(u1)+α2e(γ2+h2)τ(u2))eλτM. Then we have
ui(t)ui+supτM<θ<0y(θ)eλt,i=1,2,
thus, the proof is complete.

4 Discussion

This paper deals with a cooperation model with harvesting and state-dependent delay which is the time taken from birth to maturity is directly related to the number of the species individuals and the delay is assumed to be an increasing function of the population density with lower and upper bound. the positivity, boundedness are firstly given by controlling the harvesting rate and the existence and the uniqueness of equilibria have been derived. Especially, coexistence equilibrium E=(u1,u2) is obtained under given conditions in Theorem 3, which implies that harvesting rate hi, hi, the ratio of birth rate α2α1, the ratio of intraspecific competition effect and mutualistic effect βiμj,ij,i,j=1,2 and state-dependent delay τ(u) have an important influence on equilibriums of the system (4).
Then, we show the linear stability for all the equilibria, i.e., E0, E1, E2, E are carried out by studying the sign of the real parts of eigenvalues of the associated characteristic equations (see [10, 17, 18] for more details about linearization and stability of state-dependent delay differential equations. The extinction equilibrium E0 is unstable if hi<αie(γi+hi)τm and it is a stable if hi>αie(γi+hi)τm. For the boundary equilibrium Ei is linearly unstable if hi<αie(γi+hi)τm. For the coexistence equilibrium state E=(u1,u2) is linearly stable if β1β1>μ1μ2 and hihi,i=1,2, and
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
Finally, the global exponentially asymptotical stability criteria of the coexistence equilibrium state E is obtained by the improved Hanalay inequality. That is, the coexistence equilibrium E is globally asymptotically stable if hihi,i=1,2, and
β1μ2>α1(e(γ1+h1)τme(γ1+h1)τM)α2(e(γ2+h2)τme(γ2+h2)τM),β2μ1>α2(e(γ2+h2)τme(γ2+h2)τM)α1(e(γ1+h1)τme(γ1+h1)τM).
Furthermore, If h~>2P¯ and βi>μ1+μ22,i=1,2, then the extinction equilibrium E0 of system (4) is exponentially asymptotically stable. The coexistence equilibrium E of system (4) with initial condition φi(θ)ui,i=1,2τMθ0, is globally exponentially asymptotically stable if min{β1,β2}>max{μ1,μ2} and hihi,i=1,2, and
β2μ2(e(γ1+h1)τme(γ1+h1)τM)β1β2e(γ2+h2)τmμ1μ2e(γ2+h2)τM<α2α1<β1β2e(γ1+h1)τmμ1μ2e(γ1+h1)τMβ1μ1(e(γ2+h2)τme(γ2+h2)τM).
Compared Theorem 4 with Theorem 5, it can be founded that the state-dependent delay τ(u) affects the stability of the extinction equilibrium E0. When the state-dependent delay τ(u) introduced into system (4), in order to make the extinction equilibrium E0 of system (4) exponentially asymptotically stable, we must make harvesting hi get larger, this implies the state-dependent delay τ(u) affects harvesting hi.

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Acknowledgements

The authors gratefully acknowledge the Editor and two anonymous referees for their insightful comments.

Funding

the Scientific Research Project of Beijing Municipal Education Commission(KM201811417013)
Foundation of Anhui University of Finance and Economics(ACKYC19051)
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