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
The variables () denote the densities of cooperative mature species at time and at position . the delay is the time taken from birth to maturity. The parameters and () denote the birth rate of mature species and the death rate of immature species respectively, and measures the competitive effect within mature species .
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
where and and represent prey and mature predator densities, respectively. 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
We take the following initial conditions in model (2)
and
The variables and () denote the densities of cooperative immature and mature species at time and at position , respectively. The parameters and () denote the birth rate of mature species and the death rate of immature species respectively, and measures the competitive effect within mature species . The term appearing in both equations represents the rate at time and position at which individuals leave the immature and enter the mature class, having just reached maturity. Assume and represent interspecific cooperative effects of on and on , respectively. The terms are the harvesting efforts of the immature and mature species , 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
So we give the following assumptions for model (4):
(A1) and with and .
(A2) As discussed in [
6], for our model to make sense, that is,
is an increasing function of
. There exists
such that
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
, 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 . 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 , for , Then
for ;
If , there exists such that for all , where
Proof (a) We show the positivity of . Otherwise, there would be so that . Let . Then we have
From the Equaiton (4), we have
From assumption , then . This is a contradiction. So we obtain the results (a).
(b) We show that is uniformly bounded away from zero for a given positive initial function. Set
Otherwise, there exists a such that and . By the definition of , we have . It follows from the continuity that . Thus,
We have a contradiction. So, such does not exist and for all .
Theorem 2 Assume that holds. If , then every solution of system is uniformly bounded.
Proof Our proof is divided into two steps.
(i) Suppose that for all for some . Then for ,
since . This means that
Thus, it follows from and assumption that
so our desired results are obtained.
(ii) Assume that is not monotone in the future. Then there are two sequences and such that , , and , is a local maximum, where , for all , and , for all , then by a similar analysis at and , it follows that
For any given , we take . If , it follows from (i) that the solutions and are bounded above by a bound. If and , then and for all . Otherwise, there is a such that , which contradicts the definition of . Thus, we have
In a similar way of (i), we get the same results that and are bounded above by a bound. If and , then it follows from the first inequality of (6) that
Combined with the second inequality of (6), we obtain the same results that and are bounded above by a bound. thus, every solution of system (4) is uniformly bounded.
Remark 1 It shows that the eventually population densities are only subject to interspecific cooperative effects and competitive effect and independents on harvesting effects and . If , the eventually population densities is uniformly bounded.
Next, we discuss steady states of system (4). We first examine the nullclines of the system
It is easy to show that system (4) has the extinction equilibrium , where and are no any restrictions, the boundary equilibria and where satisfies the following equation
We observe that is a decreasing function with respect to , and
Furthermore, . If , there is no boundary equilibria, this has only the unique extinction equilibrium . If , there exists a boundary equilibria or . Therefore, if the boundary equilibria exists, then
is unique.
For the well-extended logistic model (4), we denote by the carrying capacity of the species and imply that harvesting rate and have a impact on the the carrying capacity of the species , that means that the harvesting is smaller, the the carrying capacity of the species is stronger. The explanation is accordance with the actual case.
The equations (7) can be rewritten as the following equations
If , then we can get that the corresponding determinant
system (8) is equivalent to the system
Denote and
then we know that is a increasing function and is a decreasing function with respect to , then the uniqueness of coexistence equilibrium is obtained providing
Inequality (10) is established providing
Inequality (12) is established providing and , where
In order to make it provides that
Inequality (13) is established providing
Here, it is not difficulty to prove that
so inequality (14) is meaningful.
Solving , we get the uniqueness of coexistence equilibrium , where
provided if and only if and
Particularly, if
we can get that Therefore, we have the following Theorem.
Theorem 3 Let Then system has
a unique extinction equilibrium ;
positive boundary equilibrium and if ;
coexistence equilibrium if and and
coexistence equilibrium if and
where
Remark 2 Here, (d) is specific csae of (c). From above analysis, , , and state-dependent delay 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 , or to control the population. We can conclude that if the coexistence equilibria exists, then and , which implies that the mutualism effects raises the equilibrium levels of each species. That is, the coexistence equilibrium values is greater than the level (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
is any equilibrium, the system (4) can be linearized to next equation about
where
and trial solutions proportional to leads to the characteristic equation
where .
For the extinction equilibrium , the Equation (18) reduces to
All eigenvalues are given by solutions of
If , assume that , then , this is a contradiction, then , so is an unstable point. If , in a similar way, we get , then is linear stable. If , we have , we get , is an degenerate point. This implies that species may get extinct ultimately if .
For the boundary equilibrium , the eigenvalues are the roots of equation
so that some of the eigenvalues are given by the equation
Now we show that all eigenvalues have negative real parts. Suppose that Re since and then from the above equation and Equation (17), we have
then we can get that
a contradiction proving the claim.
The others are given by the equation
Since it is easy to prove that the all eigenvalues have one positive real parts by the counter example. Thus, the boundary equilibrium is linearly unstable.
In a similar way, we can show that is linear unstable.
Now we show that the coexistence equilibrium state is linear stable if and
From (18), we get the characteristic equation
Let
Let where and are real numbers. Since
Substituting into Equation (19), we have
Then
Now, we assume that Re By
we have Since we can get that
which contradicts the assumption, then all eigenvalues of Equation (19) have the negative real parts. Therefore, the coexistence equilibrium state is linearly stable under the given conditions.
3 Globally Exponentially Asymptotically Stability of
According to above analysis, we know that only
and
are linear stable under given conditions. From biological prospective, the globally asymptotically stable of the equilibrium
and the the coexistence equilibrium state
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 are constants and is nonnegative function on . If
then,
where is the unique positive solution of
Now, we show that state-dependent delay has an impact on the system (4) by comparing Theorem 4 with Theorem 5. Let
Theorem 4 Let , where is constant. If and , then the extinction equilibrium of system is exponentially asymptotically stable.
Proof Let then system can be reduced to the following equation
Let , it follows from system (4), we have
due to the conditions Since it follows from the Lemma 1, we can get
where is the unique positive solution of
Theorem 5 Let . If and , then the extinction equilibrium of system is exponentially asymptotically stable.
Proof Let and , it follows from system (4), we can get
due to the conditions Since it follows from the Lemma 1, we can get
where is the unique positive solution of
According to above analysis, we know that the extinction equilibrium 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 affects the stability of the extinction equilibrium When the state-dependent delay introduced into system (21), in order to make the extinction equilibrium of system (4) exponentially asymptotically stable, we must make harvesting get larger, this implies the state-dependent delay affects harvesting
Lemma 2[14] Let be the solution of
and some function satisfying
Assume also that for all . Then for all .
For the general case, let and be the solution of
with initial data , . we can get .
Theorem 6 Let be the solution of
with the initial data for .
If , then where
Proof It is not difficult to prove that is positive and bounded, here, it can be omitted. The proof can be divided into two steps. (i) Assume that is eventually monotonic. In this case, we know that exists and the system has two equilibrium and, where If , then is not stable by using a standard linearized analysis and
(ii) Assume that is oscillatory. There has two cases. (a) is eventually oscillatory below the . Then has an infinite sequence of local minima and define the sequence , which satisfies and for . Now, we prove that for some integer . Otherwise, after every local minimum , there is another that is lower, and a subsequence of (still relabeled ), then we can choose sequence , which satisfies and , Since , we have
this is a contradiction. So for some integer . Let . For , in a similar way, there is such that for some integer Let Continuing this process, we can get an infinite sequence , which satisfies and with it is not difficult to get that as , this also implies the amplitude converge zero eventually.
(b) is not eventually oscillatory below the . Then has an infinite sequence of local maxima, without loss of generality, define the sequence , which satisfies and with the case of the local maximum for . Similar to (a), there is an infinite sequence , which satisfies and with we have as , 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 under given conditions.
Theorem 7 If Then for all
Proof Now we show that for all Otherwise, without loss of generality, there exists a such that , , and for . It follows from the continuity that . Thus,
This is contradiction, the proof is complete.
Theorem 8 The coexistence equilibrium of system with initial condition is globally exponentially asymptotically stable if and and
Proof From condition of Theorem 8, we have and it follows from Theorem 3 and Theorem 7 that there exists the coexistence equilibrium in system (4) and . Let , where is a solution of system (4). Let . Since
From (4), we have
By Lemma 1, if
that is
which is forever admitted due to the condition . we have
where is the unique positive solution of Then we have
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 is obtained under given conditions in Theorem 3, which implies that harvesting rate , , the ratio of birth rate , the ratio of intraspecific competition effect and mutualistic effect and state-dependent delay have an important influence on equilibriums of the system (4).
Then, we show the linear stability for all the equilibria, i.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
is unstable if
and it is a stable if
For the boundary equilibrium
is linearly unstable if
For the coexistence equilibrium state
is linearly stable if
and
and
Finally, the global exponentially asymptotical stability criteria of the coexistence equilibrium state is obtained by the improved Hanalay inequality. That is, the coexistence equilibrium is globally asymptotically stable if and
Furthermore, If and , then the extinction equilibrium of system is exponentially asymptotically stable. The coexistence equilibrium of system (4) with initial condition is globally exponentially asymptotically stable if and and
Compared Theorem 4 with Theorem 5, it can be founded that the state-dependent delay affects the stability of the extinction equilibrium When the state-dependent delay introduced into system (4), in order to make the extinction equilibrium of system (4) exponentially asymptotically stable, we must make harvesting get larger, this implies the state-dependent delay affects harvesting
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