Harvesting on a State-Dependent Time Delay Model

Ruijun XIE; Xin ZHANG; Wei ZHANG

doi:10.21078/JSSI-2020-082-15

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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

\begin{aligned} \begin{array}{lll} {\begin{cases} u_{1}^{'} (t) = α_{1} e^{- γ_{1} τ (u_{1})} u_{1} (t - τ (u_{1})) - β_{1} u_{1}^{2} + μ_{1} u_{1} u_{2}, \\ u_{2}^{'} (t) = α_{2} e^{- γ_{2} τ (u_{2})} u_{2} (t - τ (u_{2})) - β_{2} u_{2}^{2} + μ_{2} u_{1} u_{2}, \\ u_{1} (θ) = φ_{1} (θ) \geq 0, u_{2} (θ) = φ_{2} (θ) \geq 0, θ \in [- τ_{M}, 0] . \end{cases} \end{array} \end{aligned}

(1)

The variables

u_{i} (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

α_{i}

and

γ_{i}

(

i = 1, 2

) denote the birth rate of mature species

i

and the death rate of immature species

i

respectively, and

β_{i}

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

{\begin{cases} x^{'} (t) = r x (t) (1 - \frac{x (t)}{k}) - a y (t) x (t) - h_{1} x (t), \\ y^{'} (t) = b e^{- (γ + h_{3}) τ (u)} y (t - τ (u)) x (t - τ (u)) - d y (t) - h_{2} y (t), \\ y_{i}^{'} (t) = b y (t) x (t) - γ y_{i} (t) - b e^{- (γ + h_{3}) τ (u)} y (t - τ (u)) x (t - τ (u)) - h_{3} y_{i} (t), \end{cases}

where

u = y_{i} + y

and

x (t)

and

y (t)

represent prey and mature predator densities, respectively.

y_{i} (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

\begin{aligned} \begin{array}{lll} {\begin{cases} v_{1}^{'} (t) = α_{1} u_{1} - γ_{1} v_{1} - α_{1} e^{- (γ_{1} + {\tilde{h}}_{1}) τ (u_{1})} u_{1} (t - τ (u_{1})) - {\tilde{h}}_{1} v_{1}, \\ u_{1}^{'} (t) = α_{1} e^{- (γ_{1} + {\tilde{h}}_{1}) τ (u_{1})} u_{1} (t - τ (u_{1})) - β_{1} u_{1}^{2} + μ_{1} u_{1} u_{2} - h_{1} u_{1}, \\ v_{2}^{'} (t) = α_{2} u_{2} - γ_{2} v_{2} - α_{2} e^{- (γ_{2} + {\tilde{h}}_{2}) τ (u_{2})} u_{1} (t - τ (u_{2})) - {\tilde{h}}_{2} v_{2}, \\ u_{2}^{'} (t) = α_{2} e^{- (γ_{2} + {\tilde{h}}_{2}) τ (u_{2})} u_{2} (t - τ (u_{2})) - β_{2} u_{2}^{2} + μ_{2} u_{1} u_{2} - h_{2} u_{2} . \end{cases} \end{array} \end{aligned}

(2)

We take the following initial conditions in model (2)

\begin{aligned} \begin{array}{lll} u_{1} (θ) = φ_{1} (θ) \geq 0, u_{2} (θ) = φ_{2} (θ) \geq 0, θ \in [- τ_{M}, 0], t \in R^{+}, \end{array} \end{aligned}

(3)

and

v_{i 0} = \int_{- τ (u_{i} (0))}^{0} α_{i} u_{i} e^{(γ_{i} + {\tilde{h}}_{i}) s} d s, i = 1, 2.

The variables

v_{i} (t)

and

u_{i} (t)

(

i = 1, 2

) denote the densities of cooperative immature and mature species at time

t

and at position

x

, respectively. The parameters

α_{i}

and

γ_{i}

(

i = 1, 2

) denote the birth rate of mature species

i

and the death rate of immature species

i

respectively, and

β_{i}

measures the competitive effect within mature species

i

. The term

α_{i} e^{- γ_{i} τ (u_{i})} u_{i} (t - τ (u_{i}))

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

μ_{1} > 0

and

μ_{2} > 0

represent interspecific cooperative effects of

u_{2}

u_{1}

and

u_{1}

u_{2}

, respectively. The terms

h_{i}, \tilde{h_{i}}

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

\begin{aligned} {\begin{cases} u_{1}^{'} (t) = α_{1} e^{- (γ_{1} + {\tilde{h}}_{1}) τ (u_{1})} u_{1} (t - τ (u_{1})) - β_{1} u_{1}^{2} + μ_{1} u_{1} u_{2} - h_{1} u_{1}, \\ u_{2}^{'} (t) = α_{2} e^{- (γ_{2} + {\tilde{h}}_{2}) τ (u_{2})} u_{2} (t - τ (u_{2})) - β_{2} u_{2}^{2} + μ_{2} u_{1} u_{2} - h_{2} u_{2}, \\ u_{1} (θ) = φ_{1} (θ) \geq 0, u_{2} (θ) = φ_{2} (θ) \geq 0, θ \in [- τ_{M}, 0] . \end{cases} \end{aligned}

(4)

So we give the following assumptions for model (4):

(A1)

τ^{'} (u_{i}) \geq 0

and

0 < τ_{m} \leq τ (u_{i}) \leq τ_{M}

with

τ_{m} = τ (0)

and

τ_{M} = τ (\infty)

(A2) As discussed in [6], for our model to make sense, that is,

t - τ (u_{i} (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 $φ_{i} (θ) > 0$ , $i = 1, 2,$ for $θ \in [- τ_{M}, 0]$ , Then

$(a)$ $u_{i} (t) > 0$ for $t > 0$ ;

$(b)$ If $h_{i} < α_{i} e^{- (r_{i} + {\tilde{h}}_{i}) τ_{M}}$ , there exists $δ_{i} = δ_{i} (φ_{i}) > 0$ such that $u_{i} (t) > δ_{i}$ for all $t \geq 0$ , where

\begin{aligned} δ_{i} (φ_{i}) = \frac{1}{2} min {inf_{- τ_{M} \leq θ \leq 0} φ_{i} (θ), β_{i}^{- 1} (α_{i} e^{- (r_{i} + {\tilde{h}}_{i}) τ_{M}} - h_{i})} . \end{aligned}

(5)

Proof (a) We show the positivity of

u_{i}

. Otherwise, there would be

t_{1}

so that

u_{i} (t_{1}) = 0

. Let

t^{*} := inf {t : t > 0, u_{i} (t) = 0}

. Then we have

\begin{array}{lll} u_{i} (t) > 0 f o r t \in [- τ_{M}, t_{*}) a n d u_{i}^{'} (t^{*}) \leq 0. \end{array}

From the

i^{t h}

Equaiton (4), we have

u_{i}^{'} (t^{*}) = α_{i} e^{- (γ_{i} + {\tilde{h}}_{i}) τ_{m}} u_{i} (t^{*} - τ_{m}) .

From assumption

t^{*} - τ_{m} \in [- τ_{M}, t_{*}]

, then

u_{i}^{'} (t^{*}) > 0

. This is a contradiction. So we obtain the results (a).

(b) We show that

u_{i}

is uniformly bounded away from zero for a given positive initial function. Set

δ_{i} (φ_{i}) = \frac{1}{2} min {inf_{- τ_{M} \leq θ \leq 0} φ_{i} (θ), β_{i}^{- 1} (α_{i} e^{- (r_{i} + {\tilde{h}}_{i}) τ_{M}} - h_{i})} .

Otherwise, there exists a

s_{1}

such that

s_{1} = inf {t : t \geq 0, u_{i} (t) = δ_{i}}

and

u_{i}^{'} (s_{1}) \leq 0

. By the definition of

δ_{i}

, we have

u_{i} (0) = φ_{i} (0) \geq 2 δ_{i}

. It follows from the continuity that

s_{1} > 0

. Thus,

\begin{array}{lll} u_{i}^{'} (s_{1}) & = α_{i} e^{- (r_{i} + {\tilde{h}}_{i}) τ (u_{i})} u_{i} (s_{1} - τ (u_{i})) - β_{i} u_{i}^{2} (s_{1}) + μ_{i} u_{1} (s_{1}) u_{2} (s_{1}) - h_{i} u_{i} (s_{1}) \\ \geq α_{i} e^{- (r_{i} + \tilde{h_{i}}) τ_{M}} δ_{i} - β_{i} δ_{i}^{2} - h_{i} δ_{i} \geq α_{i} e^{- (r_{i} + \tilde{h_{i}}) τ_{M}} δ_{i} \\ - \frac{1}{2} (α_{i} e^{- γ_{i} (r_{i} + {\tilde{h}}_{i}) τ_{M}} - h_{i}) δ_{i} - h_{i} δ_{i} \\ = \frac{1}{2} (α_{i} e^{- (r_{i} + {\tilde{h}}_{i}) τ_{M}} - h_{i}) δ_{i} > 0. \end{array}

We have a contradiction. So, such

s_{1}

does not exist and

u_{i} (t) > δ_{i}

for all

t > 0

Theorem 2 Assume that $(A 1) \sim (A 2)$ holds. If $β_{1} β_{2} > μ_{1} μ_{2}$ , then every solution $(u_{1} (t), u_{2} (t))$ of system $(4)$ is uniformly bounded.

Proof Our proof is divided into two steps.

(i) Suppose that

u_{i}^{'} (t) \geq 0

for all

t > T

for some

T \geq 0

. Then for

t > T + τ_{M}

\begin{array}{lll} 0 \leq u_{i}^{'} (t) & = α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (u_{i})} u_{i} (t - τ (u_{i})) - β_{i} u_{i}^{2} (t) + μ_{i} u_{1} (t) u_{2} (t) - h_{i} u_{i} (t) \\ \leq α_{i} u_{i} (t) - β_{i} u_{i}^{2} (t) + μ_{i} u_{1} (t) u_{2} (t) - h_{i} u_{i} (t), \end{array}

since

u_{i} (t - τ (u_{i})) \leq u_{i} (t)

. This means that

β_{1} u_{1} - μ_{1} u_{2} \leq α_{1}, β_{2} u_{2} - μ_{2} u_{1} \leq α_{2}, t > T + τ_{M} .

Thus, it follows from

u_{i} (t) > 0

and assumption that

u_{1} (t) \leq \frac{β_{2} α_{1} + μ_{1} α_{2}}{β_{1} β_{2} - μ_{1} μ_{2}}, u_{2} (t) \leq \frac{β_{1} α_{2} + μ_{2} α_{1}}{β_{1} β_{2} - μ_{1} μ_{2}},

so our desired results are obtained.

(ii) Assume that

u (t)

is not monotone in the future. Then there are two sequences

{t_{n}}_{n = 1}^{\infty}

and

{s_{m}}_{m = 1}^{\infty}

such that

u_{1}^{'} (t_{n}) = 0

u_{2}^{'} (s_{m}) = 0

, and

u_{1} (t_{n})

u_{2} (s_{m})

is a local maximum, where

u_{1} (t) \leq u_{1} (t_{n})

0 < t < t_{n}

for all

n

, and

u_{2} (t) \leq u_{2} (s_{m})

0 < t < s_{m}

for all

m

, then by a similar analysis at

t = t_{n}

and

t = s_{m}

, it follows that

\begin{aligned} β_{1} u_{1} (t_{n}) - μ_{1} u_{2} (t_{n}) \leq α_{1}, β_{2} u_{2} (s_{m}) - μ_{2} u_{1} (s_{m}) \leq α_{2} . \end{aligned}

(6)

For any given

t_{n}

, we take

s_{n} = max {s_{m} : s_{m} \leq t_{n}}

. If

s_{n} = t_{n}

, it follows from (i) that the solutions

u_{1} (t)

and

u_{2} (t)

are bounded above by a bound. If

s_{n} < t_{n}

and

u_{2} (s_{n}) < u_{2} (t_{n})

, then

u_{2}^{'} (t_{n}) > 0

and

u_{2} (t) \leq u_{2} (t_{n})

for all

t \leq t_{n}

. Otherwise, there is a

s_{n} < t < t_{n}

such that

u_{2}^{'} (t) = 0

, which contradicts the definition of

s_{n}

. Thus, we have

\begin{array}{lll} 0 < u_{2}^{'} (t_{n}) & = α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} u_{2} (t_{n} - τ (u_{2})) - β_{2} u_{2}^{2} (t_{n}) + μ_{2} u_{1} (t_{n}) u_{2} (t_{n}) - h_{2} u_{2} (t_{n}) \\ \leq α_{2} u_{2} (t_{n}) - β_{2} u_{2}^{2} (t_{n}) + μ_{2} u_{1} (t_{n}) u_{2} (t_{n}) - h_{2} u_{2} (t_{n}) . \end{array}

In a similar way of (i), we get the same results that

u_{1} (t)

and

u_{2} (t)

are bounded above by a bound. If

s_{n} < t_{n}

and

u_{2} (t_{n}) < u_{2} (s_{n})

, then it follows from the first inequality of (6) that

β_{1} u_{1} (s_{n}) - μ_{1} u_{2} (s_{n}) \leq β_{1} u_{1} (t_{n}) - μ_{1} u_{2} (t_{n}) \leq α_{1} .

Combined with the second inequality of (6), we obtain the same results that

u_{1} (t)

and

u_{2} (t)

are bounded above by a bound. thus, every solution

(u_{1} (t), u_{2} (t))

of system (4) is uniformly bounded.

Remark 1 It shows that the eventually population densities are only subject to interspecific cooperative effects

μ_{i}

and competitive effect

β_{i}

and independents on harvesting effects

h_{i}

and

h_{i}^{'}

. If

β_{1} β_{1} > μ_{1} μ_{2}

, the eventually population densities is uniformly bounded.

Next, we discuss steady states of system (4). We first examine the nullclines of the system

\begin{aligned} {\begin{cases} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} - β_{1} u_{1} + μ_{1} u_{2} - h_{1} = 0, \\ α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} - β_{2} u_{2} + μ_{2} u_{1} - h_{2} = 0. \end{cases} \end{aligned}

(7)

It is easy to show that system (4) has the extinction equilibrium

E_{0} = (0, 0)

, where

h_{i}

and

h_{i}^{'}

are no any restrictions, the boundary equilibria

E_{1} = ({\hat{u}}_{1}, 0)

and

E_{2} = (0, {\hat{u}}_{2})

where

{\hat{u}}_{i}

satisfies the following equation

α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (u_{i})} - β_{i} u_{i} - h_{i} = 0, i = 1, 2.

We observe that

α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (u_{i})}

is a decreasing function with respect to

u_{i}

, and

α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (0)} = α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} > 0, lim_{u_{i} \to \infty} α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (u_{i})} = α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{M}} .

Furthermore,

lim_{u_{i} \to \infty} β_{i} u_{i} + h_{i} = \infty

. If

h_{i} \geq α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

, there is no boundary equilibria, this has only the unique extinction equilibrium

E_{0}

. If

h_{i} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

, there exists a boundary equilibria

E_{1} = ({\hat{u}}_{1}, 0)

E_{2} = (0, {\hat{u}}_{2})

. Therefore, if the boundary equilibria exists, then

0 < {\hat{u}}_{i} = β_{i}^{- 1} (α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (\hat{u_{i}})} - h_{i}) < β_{i}^{- 1} (α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} - h_{i})

is unique.

For the well-extended logistic model (4), we denote by

{\hat{u}}_{i}

the carrying capacity of the species

i

and imply that harvesting rate

h_{i}

and

h_{i}^{'}

have a impact on the the carrying capacity of the species

i

, that means that the harvesting

h_{i}

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

\begin{aligned} {\begin{cases} - β_{1} u_{1} + μ_{1} u_{2} = h_{1} - α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})}, \\ μ_{2} u_{1} - β_{2} u_{2} = h_{2} - α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} . \end{cases} \end{aligned}

(8)

β_{1} β_{1} > μ_{1} μ_{2}

, then we can get that the corresponding determinant

| \begin{array}{lll} - β_{1} & μ_{1} \\ μ_{2} & - β_{2} \end{array} | = β_{1} β_{2} - μ_{1} μ_{2} > 0,

system (8) is equivalent to the system

\begin{aligned} {\begin{cases} u_{1} = \frac{β_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} - h_{1}) + μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} - h_{2})}{β_{1} β_{2} - μ_{1} μ_{2}}, \\ u_{2} = \frac{β_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} - h_{2}) + μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} - h_{1})}{β_{1} β_{2} - μ_{1} μ_{2}} . \end{cases} \end{aligned}

(9)

Denote

g (u_{i}) = (β_{1} β_{2} - μ_{1} μ_{2}) u_{i}, i = 1, 2

and

\begin{array}{lll} f (u_{1}) = β_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} - h_{1}) + μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} - h_{2}), \\ f (u_{2}) = β_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} - h_{2}) + μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} - h_{1}), \end{array}

then we know that

g (u_{i})

is a increasing function and

f (u_{i})

is a decreasing function with respect to

u_{i}

, then the uniqueness of coexistence equilibrium is obtained providing

\begin{aligned} {\begin{cases} lim_{u_{1} \to 0} f (u_{1}) = β_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - h_{1}) + μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} - h_{2}) > 0, \\ lim_{u_{2} \to 0} f (u_{2}) = β_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - h_{2}) + μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} - h_{1}) > 0. \end{cases} \end{aligned}

(10)

Inequality (10) is established providing

\begin{aligned} {\begin{cases} β_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - h_{1}) + μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}} - h_{2}) > 0, \\ β_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - h_{2}) + μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{M}} - h_{1}) > 0 \end{cases} \end{aligned}

(11)

\begin{aligned} \Leftarrow {\begin{cases} β_{2} h_{1} + μ_{1} h_{2} < β_{2} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} + μ_{1} α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}, \\ μ_{2} h_{1} + β_{1} h_{2} < β_{1} α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} + μ_{2} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{M}} . \end{cases} \end{aligned}

(12)

Inequality (12) is established providing

β_{1} β_{2} > μ_{1} μ_{2}

and

h_{i} \leq h_{i}^{*}, i = 1, 2

, where

{\begin{cases} h_{1}^{*} = \frac{β_{1} β_{2} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{M}} - β_{1} μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}{β_{1} β_{2} - μ_{1} μ_{2}}, \\ h_{2}^{*} = \frac{β_{1} β_{2} α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}} - β_{2} μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} - μ_{1} μ_{2}} . \end{cases}

In order to make

h_{i}^{*} > 0, i = 1, 2,

it provides that

\begin{aligned} {\begin{cases} β_{1} β_{2} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{M}} > β_{1} μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}), \\ β_{1} β_{2} α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}} > β_{2} μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}) . \end{cases} \end{aligned}

(13)

Inequality (13) is established providing

\begin{aligned} \frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} . \end{aligned}

(14)

Here, it is not difficulty to prove that

\frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} > \frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} > 0,

so inequality (14) is meaningful.

Solving

(9)

, we get the uniqueness of coexistence equilibrium

(u_{1}^{*}, u_{2}^{*})

, where

u_{1}^{*} = \frac{β_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1}^{*})} - h_{1}) + μ_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2}^{*})} - h_{2})}{β_{1} β_{2} - μ_{1} μ_{2}},

\begin{aligned} u_{2}^{*} = \frac{β_{1} (α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2}^{*})} - h_{2}) + μ_{2} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1}^{*})} - h_{1})}{β_{1} β_{2} - μ_{1} μ_{2}}, \end{aligned}

(15)

provided

u_{1}^{*} > 0, u_{2}^{*} > 0

if and only if

β_{1} β_{2} > μ_{1} μ_{2},

h_{i} \leq h_{i}^{*}, i = 1, 2,

and

\frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} .

Particularly, if

\frac{β_{1}}{μ_{2}} > \frac{α_{1} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{α_{2} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}, \frac{β_{2}}{μ_{1}} > \frac{α_{2} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}{α_{1} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})},

we can get that

β_{1} β_{1} > μ_{1} μ_{2} a n d α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{M}} < h_{i}^{*} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} .

Therefore, we have the following Theorem.

Theorem 3 Let $h_{i} \geq 0, i = 1, 2.$ Then system $(4)$ has

$(a)$ a unique extinction equilibrium $E_{0} = (0, 0)$ ;

$(b)$ positive boundary equilibrium $E_{1} = ({\hat{u}}_{1}, 0)$ and $E_{2} = (0, {\hat{u}}_{2})$ if $h_{i} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}$ ;

$(c)$ coexistence equilibrium $E^{*} = (u_{1}^{*}, u_{2}^{*})$ if $β_{1} β_{2} > μ_{1} μ_{2}$ and $h_{i} \leq h_{i}^{*}, i = 1, 2,$ and

\begin{array}{lll} \frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} . \end{array}

$(d)$ coexistence equilibrium $E^{*} = (u_{1}^{*}, u_{2}^{*})$ if $h_{i} \leq h_{i}^{*}, i = 1, 2,$ and

\frac{β_{1}}{μ_{2}} > \frac{α_{1} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{α_{2} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}, \frac{β_{2}}{μ_{1}} > \frac{α_{2} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}{α_{1} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})},

where $α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{M}} < h_{i}^{*} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} .$

Remark 2 Here, (d) is specific csae of (c). From above analysis,

h_{i}

h_{i}^{'}

\frac{α_{2}}{α_{1}}

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

h_{i}

h_{i}^{'}

\frac{α_{2}}{α_{1}}

to control the population. We can conclude that if the coexistence equilibria exists, then

u_{1}^{*} > {\hat{u}}_{1}

and

u_{2}^{*} > {\hat{u}}_{2}

, which implies that the mutualism effects raises the equilibrium levels of each species. That is, the coexistence equilibrium values

u_{i}^{*}

is greater than the level

{\hat{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 = (u_{1}^{0}, u_{2}^{0})

is any equilibrium, the system (4) can be linearized to next equation about

E

\begin{array}{l} (\begin{array}{c} u_{1}^{'} (t) \\ u_{2}^{'} (t) \end{array}) = (\begin{array}{c} - 2 β_{1} u_{1}^{0} - ξ_{1}^{*} + μ_{1} u_{2}^{0} - h_{1} & μ_{1} u_{1}^{0} \\ μ_{2} u_{2}^{0} & - 2 β_{2} u_{2}^{0} - ξ_{2}^{*} + μ_{2} u_{1}^{0} - h_{2} \end{array}) (\begin{array}{c} u_{1} (t) \\ u_{2} (t) \end{array}) \\ + (\begin{array}{c} α_{1} e^{- (γ_{1} + {\tilde{h}}_{1}) τ (u_{1}^{0})} & 0 \\ 0 & α_{2} e^{- (γ_{2} + {\tilde{h}}_{2}) τ (u_{2}^{0})} \end{array}) (\begin{matrix} u_{1} (t - τ (u_{1}^{0})) \\ u_{2} (t - τ (u_{2}^{0})) \end{matrix}), \end{array}

(16)

where

\begin{aligned} ξ_{i}^{*} = α_{i} γ_{i} u_{i}^{0} e^{- (γ_{i} + {\tilde{h}}_{i}) τ (u_{i}^{0})} τ^{'} (u_{i}^{0}), i = 1, 2, \end{aligned}

(17)

and trial solutions proportional to

(c_{1}, c_{2}) \exp (λ t)

leads to the characteristic equation

\begin{aligned} | \begin{array}{lll} a_{11} - ξ_{1}^{*} + μ_{1} u_{2}^{0} - λ - h_{1} & μ_{1} u_{1}^{0} \\ μ_{2} u_{2}^{0} & a_{22} - ξ_{2}^{*} + μ_{2} u_{1}^{0} - λ - h_{2} \end{array} | = 0, \end{aligned}

(18)

where

a_{i i} = α_{i} e^{- τ (u_{i}^{0}) (γ_{i} + h_{i}^{'} + λ)} - 2 β_{i} u_{i}^{0}

For the extinction equilibrium

E_{0} = (0, 0)

, the Equation (18) reduces to

(α_{1} e^{- τ_{m} (γ_{1} + h_{1}^{'} + λ)} - λ - h_{1}) (α_{2} e^{- τ_{m} (γ_{2} + h_{2}^{'} + λ)} - λ - h_{2}) = 0.

All eigenvalues are given by solutions of

λ_{i} = α_{i} e^{- τ_{m} (γ_{i} + h_{i}^{'} + λ_{i})} - h_{i}, i = 1, 2.

h_{i} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

, assume that

λ_{i} \leq 0

, then

λ_{i} = α_{i} e^{- τ_{m} (γ_{i} + h_{i}^{'} + λ_{i})} - h_{i} \geq α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} - h_{i} > 0

, this is a contradiction, then

λ_{i} > 0

, so

E_{0}

is an unstable point. If

h_{i} > α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

, in a similar way, we get

λ_{i} < 0

, then

E_{0}

is linear stable. If

h_{i} = α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

, we have

λ_{i} = α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} (e^{- λ_{i} τ_{m}} - 1)

, we get

λ_{i} = 0

E_{0}

is an degenerate point. This implies that species may get extinct ultimately if

h_{i} > α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

For the boundary equilibrium

E_{1} = ({\hat{u}}_{1}, 0)

, the eigenvalues are the roots of equation

(α_{1} e^{- τ (\hat{u_{1}}) (γ_{1} + h_{1}^{'} + λ)} - 2 β_{1} \hat{u_{1}} - ξ_{1}^{*} - λ - h_{1}) (α_{2} e^{- τ_{m} (γ_{2} + h_{2}^{'} + λ)} + μ_{2} \hat{u_{1}} - λ - h_{2}) = 0,

so that some of the eigenvalues are given by the equation

λ + 2 β_{1} \hat{u_{1}} + ξ_{1}^{*} + h_{1} = α_{1} e^{- τ (\hat{u_{1}}) (γ_{1} + h_{1}^{'} + λ)} .

Now we show that all eigenvalues have negative real parts. Suppose that Re

λ \geq 0,

since

h_{i} < α_{i} e^{- (γ_{1} + h_{1}^{'}) τ_{m}}

and

{\hat{u}}_{i} = β_{i}^{- 1} (α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (\hat{u_{i}})} - h_{i}),

then from the above equation and Equation (17), we have

\begin{array}{lll} R e λ + 2 β_{1} \hat{u_{1}} + (β_{1} μ_{1} + h_{1}) (γ_{1} \hat{u_{1}} τ^{'} (\hat{u_{1}})) + h_{1} \\ = & α_{1} e^{- γ_{1} τ (\hat{u_{1}})} e^{- τ (\hat{u_{1}}) R e λ} \cos (- τ (\hat{u_{1}} I m λ) \\ = & (β_{1} μ_{1} + h_{1}) e^{- τ (\hat{u_{1}}) R e λ} \cos (- τ (\hat{u_{1}} I m λ) \leq (β_{1} μ_{1} + h_{1}), \end{array}

then we can get that

R e λ \leq - β_{1} \hat{u_{1}} - (β_{1} μ_{1} + h_{1}) (γ_{1} \hat{u_{1}} τ^{'} (\hat{u_{1}})) < 0,

a contradiction proving the claim.

The others are given by the equation

λ - μ_{2} \hat{u_{1}} = α_{2} e^{- τ_{m} (γ_{2} + h_{2}^{'} + λ)} - h_{2} .

Since

h_{i} < α_{i} e^{- (γ_{1} + h_{1}^{'}) τ_{m}},

it is easy to prove that the all eigenvalues have one positive real parts by the counter example. Thus, the boundary equilibrium

E_{1} = (\hat{u_{1}}, 0)

is linearly unstable.

In a similar way, we can show that

E_{2} = (0, \hat{u_{2}})

is linear unstable.

Now we show that the coexistence equilibrium state

E^{*} = (u_{1}^{*}, u_{2}^{*})

is linear stable if

β_{1} β_{2} > μ_{1} μ_{2}, h_{i} < h_{i}^{*}

and

\frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} .

From (18), we get the characteristic equation

\begin{aligned} (λ - a_{11} + ξ_{1}^{*} - μ_{1} u_{2}^{*} + h_{1}) (λ - a_{22} + ξ_{2}^{*} - μ_{2} u_{1}^{*} + h_{2}) - μ_{1} μ_{2} u_{1}^{*} u_{2}^{*} = 0. \end{aligned}

(19)

Let

\begin{array}{lll} D_{1} = a + β_{1} u_{1}^{*} + a_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1}^{*})} (1 + γ_{1} u_{1}^{*} τ^{'} (u_{1}^{*}) - e^{- a τ (u_{1}^{*})} \cos (b τ (u_{1}^{*}))), \\ D_{2} = a + β_{2} u_{2}^{*} + a_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2}^{*})} (1 + γ_{2} u_{2}^{*} τ^{'} (u_{2} *) - e^{- a τ (u_{2}^{*})} \cos (b τ (u_{2}^{*}))), \\ E_{1} = b + a_{1} e^{- τ (u_{1}^{*}) (γ_{1} + h_{1}^{'} + a)} \sin (b τ (u_{1}^{*})), \\ E_{2} = b + a_{2} e^{- τ (u_{2}^{*}) (γ_{2} + h_{2}^{'} + a)} \sin (b τ (u_{2}^{*})) . \end{array}

Let

λ = a + b i,

where

a

and

b

are real numbers. Since

β_{1} u_{1}^{*} + h_{1} - μ_{1} u_{2}^{*} = a_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1}^{*})}, β_{2} u_{2}^{*} + h_{2} - μ_{2} u_{1}^{*} = a_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2}^{*})},

Substituting

λ = a + b i

into Equation (19), we have

D_{1} D_{2} - E_{1} E_{2} = μ_{1} μ_{2} u_{1}^{*} u_{2}^{*} a n d D_{1} E_{2} + D_{2} E_{1} = 0.

Then

\begin{array}{lll} (μ_{1} μ_{2} u_{1}^{*} u_{2}^{*})^{2} & = & (D_{1} D_{2})^{2} + (E_{1} E_{2})^{2} - 2 D_{1} D_{2} E_{1} E_{2} \\ = & (D_{1} D_{2})^{2} + (E_{1} E_{2})^{2} + 2 (D_{1} E_{2})^{2} > (D_{1} D_{2})^{2} . \end{array}

Now, we assume that Re

λ = a \geq 0.

1 + γ_{1} u_{1}^{*} τ^{'} (u_{1}^{*}) - e^{- a τ (u_{1}^{*})} \cos (b τ (u_{1}^{*})) \geq 0,

we have

D_{i} \geq β_{i} u_{i}^{*} .

Since

β_{1} β_{2} \geq μ_{1} μ_{2},

we can get that

D_{1} D_{2} \geq μ_{1} μ_{2} u_{1}^{*} u_{2}^{*},

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 $E_{0}, E^{*}$**

According to above analysis, we know that only

E_{0}

and

E^{*}

are linear stable under given conditions. From biological prospective, the globally asymptotically stable of the equilibrium

E_{0} = (0, 0)

and the the coexistence equilibrium state

E^{*} = (u_{1}^{*}, u_{2}^{*})

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 $α > β > 0, τ \geq 0$ are constants and $x (t)$ is nonnegative function on $[t_{0} - τ, + \infty)$ . If

\begin{array}{lll} \dot{x} (t) \leq - α x (t) + β sup_{t - τ < θ < t} x (θ) f o r a l l t \geq t_{0}, \end{array}

then,

\begin{aligned} x (t) \leq sup_{t_{0} - τ < θ < t_{0}} x (θ) e^{- λ (t - t_{0})}, \end{aligned}

(20)

where $λ$ is the unique positive solution of $λ = α - β e^{λ τ} .$

Now, we show that state-dependent delay has an impact on the system (4) by comparing Theorem 4 with Theorem 5. Let

\tilde{h} = min {h_{1}, h_{2}}, \tilde{P} = max {α_{1} e^{- (γ_{1} + h_{1}^{'}) τ}, α_{2} e^{- (γ_{2} + h_{2}^{'}) τ}} .

Theorem 4 Let $τ (u_{i}) = τ$ , where $τ$ is constant. If $\tilde{h} > \tilde{P}$ and $β_{i} > \frac{μ_{1} + μ_{2}}{2}, i = 1, 2$ , then the extinction equilibrium $E_{0}^{'} = (0, 0)$ of system $(4)$ is exponentially asymptotically stable.

Proof Let

τ (u_{i}) = τ,

then system

(4)

can be reduced to the following equation

\begin{aligned} {\begin{cases} u_{1}^{'} (t) = α_{1} e^{- (γ_{1} + h_{1}^{'}) τ} u_{1} (t - τ) - β_{1} u_{1}^{2} + μ_{1} u_{1} u_{2} - h_{1} u_{1}, \\ u_{2}^{'} (t) = α_{2} e^{- (γ_{2} + h_{2}^{'}) τ} u_{2} (t - τ) - β_{2} u_{2}^{2} + μ_{2} u_{1} u_{2} - h_{2} u_{2} . \end{cases} \end{aligned}

(21)

Let

u (t) = u_{1} (t) + u_{2} (t)

, it follows from system (4), we have

\begin{array}{lll} u^{'} (t) = α_{1} e^{- (γ_{1} + h_{1}^{'}) τ} u_{1} (t - τ) + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ} u_{2} (t - τ) - β_{1} u_{1}^{2} (t) - β_{2} u_{2}^{2} (t) \\ + (μ_{1} + μ_{2}) u_{1} (t) u_{2} (t) - h_{1} u_{1} (t) - h_{2} u_{2} (t) \\ \leq α_{1} e^{- (γ_{1} + h_{1}^{'}) τ} u_{1} (t - τ) + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ} u_{2} (t - τ) - β_{1} u_{1}^{2} (t) - β_{2} u_{2}^{2} (t) \\ + (μ_{1} + μ_{2}) \frac{u_{1}^{2} + u_{2}^{2}}{2} - h_{1} u_{1} (t) - h_{2} u_{2} (t) \\ \leq \tilde{P} u (t - τ) - \tilde{h} u (t) \\ \leq \tilde{P} sup_{t - τ \leq θ \leq t} u (θ) - \tilde{h} u (t), f o r t \geq 0, \end{array}

due to the conditions

β_{i} > \frac{μ_{1} + μ_{2}}{2}, i = 1, 2.

Since

\tilde{h} > \tilde{P},

it follows from the Lemma 1, we can get

u_{i} (t) \leq u (t) \leq sup_{- τ \leq θ \leq 0} u (θ) e^{- \tilde{λ} t}, i = 1, 2,

where

\tilde{λ}

is the unique positive solution of

\tilde{λ} = h - \tilde{P} e^{\tilde{λ} τ} .

Theorem 5 Let $τ^{'} (u_{i}) \neq 0$ . If $\tilde{h} > 2 \bar{P}$ and $β_{i} > \frac{μ_{1} + μ_{2}}{2}, i = 1, 2$ , then the extinction equilibrium $E_{0}$ of system $(4)$ is exponentially asymptotically stable.

Proof Let

u (t) = u_{1} (t) + u_{2} (t)

and

\bar{P} = max {α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}}, α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}}}

, it follows from system (4), we can get

\begin{array}{lll} u^{'} (t) & = & α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} u_{1} (t - τ (u_{1})) + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ (u_{2})} u_{2} (t - τ (u_{2})) \\ - β_{1} u_{1}^{2} (t) - β_{2} u_{2}^{2} (t) + (μ_{1} + μ_{2}) u_{1} (t) u_{2} (t) - h_{1} u_{1} (t) - h_{2} u_{2} (t) \\ \leq & α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} u_{1} (t - τ (u_{1})) + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} u_{2} (t - τ (u_{2})) - β_{1} u_{1}^{2} (t) \\ - β_{2} u_{2}^{2} (t) + (μ_{1} + μ_{2}) \frac{u_{1}^{2} + u_{2}^{2}}{2} - h_{1} u_{1} (t) - h_{2} u_{2} (t) \\ \leq & \tilde{P} (u_{1} (t - τ (u_{1})) + u_{2} (t - τ (u_{2}))) - (β_{1} - \frac{μ_{1} + μ_{2}}{2}) u_{1}^{2} (t) \\ - (β_{2} - \frac{μ_{1} + μ_{2}}{2}) u_{2}^{2} (t) - \tilde{h} u (t) \\ \leq & 2 \bar{P} sup_{t - τ_{M} \leq θ \leq t} u (θ) - \tilde{h} u (t), f o r t \geq 0, \end{array}

due to the conditions

β_{i} > \frac{μ_{1} + μ_{2}}{2}, i = 1, 2.

Since

\tilde{h} > 2 \bar{P},

it follows from the Lemma 1, we can get

u_{i} (t) \leq u (t) \leq sup_{- τ_{M} \leq θ \leq 0} u (θ) e^{- \tilde{λ} t}, i = 1, 2,

where

\tilde{λ}

is the unique positive solution of

\tilde{λ} = h - 2 \tilde{P} e^{\tilde{λ} τ_{M}} .

According to above analysis, we know that the extinction equilibrium

E_{0}

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

E_{0} .

When the state-dependent delay

τ (u)

introduced into system (21), in order to make the extinction equilibrium

E_{0}

of system (4) exponentially asymptotically stable, we must make harvesting

h_{i}

get larger, this implies the state-dependent delay

τ (u)

affects harvesting

h_{i} .

Lemma 2^[14] Let $u_{1} (t)$ be the solution of

u_{1}^{'} (t) = α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} u_{1} (t - τ (u_{1})) - β_{1} u_{1}^{2} + λ u_{1}, t > 0,

and $u_{2} (t)$ some function satisfying

\begin{aligned} u_{2}^{'} (t) \geq α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{2})} u_{2} (t - τ (u_{2})) - β_{1} u_{2}^{2} + λ u_{2}, t > 0. \end{aligned}

(22)

Assume also that $u_{2} (θ) \geq u_{1} (θ)$ for all $θ \in [- τ_{M}, 0]$ . Then $u_{2} (t) \geq u_{1} (t)$ for all $t > 0$ .

For the general case, let $ε > 0$ and $u_{ε} (t)$ be the solution of

u_{1}^{'} (t) = α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1})} u_{1} (t - τ (u_{1})) - β_{1} u_{1}^{2} + λ u_{1} - ε

with initial data $u_{ε} (θ) = u_{1} (θ) - ε$ , $θ \in [- τ_{M}, 0)$ . we can get $u_{1} (t) = lim_{ε \to 0} u_{ε} (t) \leq u_{2} (t)$ .

Theorem 6 Let $u_{1} (t)$ be the solution of

\begin{aligned} \frac{d u_{1} (t)}{d t} = α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1} (t))} u_{1} (t - τ (u_{1} (t))) - β_{1} u_{1}^{2} (t) - h u_{1} (t), \end{aligned}

(23)

with the initial data $u_{1} (θ) = φ_{1} (θ) > 0,$ for $θ \in [- τ_{M}, 0]$ .

h < α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}}

, then

lim_{t \to \infty} u_{1} (t) = {\overset{ˇ}{u}}_{1},

where

{\overset{ˇ}{u}}_{1} = β_{1}^{- 1} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ ({\overset{ˇ}{u}}_{1})} - h) .

Proof It is not difficult to prove that

u_{1} (t)

is positive and bounded, here, it can be omitted. The proof can be divided into two steps. (i) Assume that

u_{1} (t)

is eventually monotonic. In this case, we know that

lim_{t \to \infty} u_{1} (t)

exists and the system has two equilibrium

E_{0} = 0

and

E_{1} = {\overset{ˇ}{u}}_{1}

, where

{\overset{ˇ}{u}}_{1} = β_{1}^{- 1} (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ ({\overset{ˇ}{u}}_{1})} - h) .

h < α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{m}}

, then

E_{0} = 0

is not stable by using a standard linearized analysis and

lim_{t \to \infty} u_{1} (t) = {\overset{ˇ}{u}}_{1} .

(ii) Assume that

u_{1} (t)

is oscillatory. There has two cases. (a)

u_{1} (t)

is eventually oscillatory below the

{\overset{ˇ}{u}}_{1}

. Then

u_{1} (t)

has an infinite sequence of local minima and define the sequence

t_{j}

, which satisfies

u_{1}^{'} (t_{j}) = 0,

u_{1}^{″} (t_{j}) > 0,

and

u_{1} (t_{j}) < {\overset{ˇ}{u}}_{1},

for

j = 1, 2, \dots

. Now, we prove that

inf_{t \geq t_{1}} u_{1} (t) = u_{1} (t_{k})

for some integer

k

. Otherwise, after every local minimum

u_{1} (t_{j})

, there is another that is lower, and a subsequence of

t_{j}

(still relabeled

t_{j}

), then we can choose sequence

t_{j}

, which satisfies

t_{1} \leq t_{j} < t_{j + 1}, t_{j} \to \infty

and

u_{1} (t_{j}) > u_{1} (t_{j + 1})

, Since

u_{1} (t_{j} - τ (u_{1} (t_{j}))) > u_{1} (t_{j})

, we have

\begin{array}{lll} 0 = u_{1}^{'} (t_{j}) = α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1} (t_{j}))} u_{1} (t_{j} - τ (u_{1} (t_{j}))) - β_{1} u_{1}^{2} (t_{j}) - h u_{1} (t_{j}) \\ > u_{1} (t_{j}) (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1} (t_{j}))} - β_{1} u_{1} (t_{j}) - h_{1}) \\ > u_{1} (t_{j}) (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ ({\overset{ˇ}{u}}_{1})} - β_{1} {\overset{ˇ}{u}}_{1} - h_{1}) \\ = 0, \end{array}

this is a contradiction. So

inf_{t \geq t_{1}} u_{1} (t) = u_{1} (t_{k})

for some integer

k

. Let

t_{k} = s_{1}

. For

t \geq t_{k + 1}

, in a similar way, there is

t_{l}

such that

inf_{t \geq t_{k + 1}} u_{1} (t) = u_{1} (t_{l})

for some integer

l > k .

Let

t_{l} = s_{2} .

Continuing this process, we can get an infinite sequence

{s_{j}}

, which satisfies

s_{1} \leq s_{j} < s_{j + 1}, s_{j} \to \infty

and

u_{1} (s_{j}) < u_{1} (s_{j + 1})

with

u_{1} (s_{j}) < {\overset{ˇ}{u}}_{1},

it is not difficult to get that

u_{1} (s_{j}) \to {\overset{ˇ}{u}}_{1},

s_{j} \to \infty

, this also implies the amplitude converge zero eventually.

(b)

u_{1} (t)

is not eventually oscillatory below the

{\overset{ˇ}{u}}_{1}

. Then

u_{1} (t)

has an infinite sequence of local maxima, without loss of generality, define the sequence

t_{j}

, which satisfies

u_{1}^{'} (t_{j}) = 0

and

u_{1}^{″} (t_{j}) < 0,

with the case of the local maximum

u_{1} (t_{j}) > {\overset{ˇ}{u}}_{1},

for

j = 1, 2, \dots

. Similar to (a), there is an infinite sequence

{s_{j}^{'}}

, which satisfies

s_{1}^{'} \leq s_{j}^{'} < s_{j + 1}^{'}, s_{j}^{'} \to \infty

and

u_{1} (s_{j}^{'}) > u_{1} (s_{j + 1}^{'})

with

u_{1} (s_{j}^{'}) > {\overset{ˇ}{u}}_{1},

we have

u_{1} (s_{j}^{'}) \to {\overset{ˇ}{u}}_{1},

s_{j}^{'} \to \infty

, 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 $φ_{i} (θ) \geq u_{i}^{*}, i = 1, 2 - τ_{M} \leq θ \leq 0,$ Then $u_{i} (t) \geq u_{i}^{*}$ for all $t \geq 0.$

Proof Now we show that

u_{i} (t) \geq u_{i}^{*}

for all

t \geq 0.

Otherwise, without loss of generality, there exists a

s_{1}

such that

s_{1} = inf {t : t \geq 0, u_{1} (t) = u_{1}^{*}}

u_{1}^{'} (s_{1}) \leq 0

, and

u_{2} (t) \geq u_{2}^{*}

for

t \leq s_{1}

. It follows from the continuity that

s_{1} > 0

. Thus,

\begin{array}{lll} u_{1}^{'} (s_{1}) & = α_{1} e^{- (r_{1} + h_{1}^{'}) τ (u_{1}^{*})} u_{1} (s_{1} - τ (u_{1}^{*})) - β_{1} (u_{1}^{*})^{2} + μ_{i} u_{1}^{*} u_{2} (s_{1}) - h_{1} u_{1}^{*} \\ > α_{i} e^{- (r_{i} + h_{i}^{'}) τ (u_{1}^{*})} u_{1}^{*} - β_{1} (u_{1}^{*})^{2} + μ_{i} u_{1}^{*} u_{2}^{*} - h_{1} u_{1}^{*} \\ = 0. \end{array}

This is contradiction, the proof is complete.

Theorem 8 The coexistence equilibrium $E^{*}$ of system $(4)$ with initial condition $φ_{i} (θ) \geq u_{i}^{*}, i = 1, 2 - τ_{M} \leq θ \leq 0,$ is globally exponentially asymptotically stable if $min {β_{1}, β_{2}} > max {μ_{1}, μ_{2}}$ and $h_{i} \leq h_{i}^{*}, i = 1, 2,$ and

\frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} .

Proof From condition of Theorem 8, we have

β_{i} > \frac{μ_{1} + μ_{2}}{2} > \sqrt{μ_{1} μ_{2}}, i = 1, 2

and

β_{1} β_{2} > μ_{1} μ_{2} .

it follows from Theorem 3 and Theorem 7 that there exists the coexistence equilibrium

E^{*} = (u_{1}^{*}, u_{2}^{*})

in system (4) and

u_{i} (t) \geq u_{i}^{*}, i = 1, 2

. Let

y_{i} (t) = u_{i} (t) - u_{i}^{*} \geq 0

, where

(u_{1} (t), u_{2} (t))

is a solution of system (4). Let

y (t) = y_{1} (t) + y_{2} (t)

. Since

α_{i} e^{- (γ_{i} + h_{i}^{'}) τ (u_{i}^{*})} u_{i}^{*} - β_{i} (u_{i}^{*})^{2} + μ_{i} u_{1}^{*} u_{2}^{*} - h_{i} u_{i}^{*} = 0, i = 1, 2.

From (4), we have

\begin{array}{lll} y^{'} (t) & = & y_{1}^{'} (t) + y_{2}^{'} (t) \\ = & α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (y_{1} + u_{1}^{*})} [y_{1} (t - τ (y_{1} + u_{1}^{*})) + u_{1}^{*}] - β_{1} (y_{1} + u_{1}^{*})^{2} \\ + μ_{1} (y_{1} + u_{1}^{*}) (y_{2} + u_{2}^{*}) - h_{1} (y_{1} + u_{1}^{*}) \\ + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{(} y_{2} + u_{2}^{*})} [y_{2} (t - τ (y_{2} + u_{2}^{*})) + u_{2}^{*}] \\ - β_{2} (y_{2} + u_{2}^{*})^{2} + μ_{2} (y_{1} + u_{1}^{*}) (y_{2} + u_{2}^{*}) - h_{2} (y_{2} + u_{2}^{*}) \\ \leq & (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1}^{*})} + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{(} u_{2}^{*})}) sup_{t - τ_{M} \leq θ \leq t} y (θ) \\ + (\frac{μ_{1} + μ_{2}}{2} - β_{1}) y_{1}^{2} + (\frac{μ_{1} + μ_{2}}{2} - β_{2}) y_{2}^{2} + (μ_{1} u_{2}^{*} + μ_{2} u_{2}^{*} - 2 β_{1} u_{1}^{*} - h_{1}) y_{1} \\ + (μ_{2} u_{1}^{*} + μ_{1} u_{1}^{*} - 2 β_{2} u_{2}^{*} - h_{2}) y_{2} \\ \leq & (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ (u_{1}^{*})} + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{(} u_{2}^{*})}) sup_{t - τ_{M} \leq θ \leq t} y (θ) \\ - [(h_{1} + h_{2} + 2 β_{1} u_{1}^{*} + 2 β_{2} u_{2}^{*} - (μ_{1} + μ_{2}) (u_{1}^{*} + u_{2}^{*})] y (t) . \end{array}

By Lemma 1, if

h_{1} + h_{2} + 2 β_{1} u_{1}^{*} + 2 β_{2} u_{2}^{*} - (μ_{1} + μ_{2}) (u_{1}^{*} + u_{2}^{*}) > α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{(} u_{1}^{*})} + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{(} u_{2}^{*})},

that is

(β_{1} - μ_{1}) u_{1}^{*} + (β_{2} - μ_{2}) u_{2}^{*} > 0,

which is forever admitted due to the condition

min {β_{1}, β_{2}} > max {μ_{1}, μ_{2}}

. we have

y (t) \leq sup_{- τ_{M} < θ < 0} y (θ) e^{- λ t},

where

λ

is the unique positive solution of

λ = h_{1} + h_{2} + 2 β_{1} u_{1}^{*} + 2 β_{2} u_{2}^{*} - (μ_{1} + μ_{2}) (u_{1}^{*} + u_{2}^{*}) - (α_{1} e^{- (γ_{1} + h_{1}^{'}) τ_{(} u_{1}^{*})} + α_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{(} u_{2}^{*})}) e^{λ τ_{M}} .

Then we have

u_{i} (t) \leq u_{i}^{*} + sup_{- τ_{M} < θ < 0} y (θ) e^{- λ t}, 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^{*} = (u_{1}^{*}, u_{2}^{*})

is obtained under given conditions in Theorem 3, which implies that harvesting rate

h_{i}

h_{i}^{'}

, the ratio of birth rate

\frac{α_{2}}{α_{1}}

, the ratio of intraspecific competition effect and mutualistic effect

\frac{β_{i}}{μ_{j}}, i \neq j, 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.,

E_{0}

E_{1}

E_{2}

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

E_{0}

is unstable if

h_{i} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}}

and it is a stable if

h_{i} > α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} .

For the boundary equilibrium

E_{i}

is linearly unstable if

h_{i} < α_{i} e^{- (γ_{i} + h_{i}^{'}) τ_{m}} .

For the coexistence equilibrium state

E^{*} = (u_{1}^{*}, u_{2}^{*})

is linearly stable if

β_{1} β_{1} > μ_{1} μ_{2}

and

h_{i} \leq h_{i}^{*}, i = 1, 2,

and

\frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} .

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

h_{i} \leq h_{i}^{*}, i = 1, 2,

and

\frac{β_{1}}{μ_{2}} > \frac{α_{1} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{α_{2} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}, \frac{β_{2}}{μ_{1}} > \frac{α_{2} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})}{α_{1} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})} .

Furthermore, If

\tilde{h} > 2 \bar{P}

and

β_{i} > \frac{μ_{1} + μ_{2}}{2}, i = 1, 2

, then the extinction equilibrium

E_{0}

of system

(4)

is exponentially asymptotically stable. The coexistence equilibrium

E^{*}

of system (4) with initial condition

φ_{i} (θ) \geq u_{i}^{*}, i = 1, 2 - τ_{M} \leq θ \leq 0,

is globally exponentially asymptotically stable if

min {β_{1}, β_{2}} > max {μ_{1}, μ_{2}}

and

h_{i} \leq h_{i}^{*}, i = 1, 2,

and

\frac{β_{2} μ_{2} (e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - e^{- (γ_{1} + h_{1}^{'}) τ_{M}})}{β_{1} β_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{2} + h_{2}^{'}) τ_{M}}} < \frac{α_{2}}{α_{1}} < \frac{β_{1} β_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{m}} - μ_{1} μ_{2} e^{- (γ_{1} + h_{1}^{'}) τ_{M}}}{β_{1} μ_{1} (e^{- (γ_{2} + h_{2}^{'}) τ_{m}} - e^{- (γ_{2} + h_{2}^{'}) τ_{M}})} .

Compared Theorem 4 with Theorem 5, it can be founded that the state-dependent delay

τ (u)

affects the stability of the extinction equilibrium

E_{0} .

When the state-dependent delay

τ (u)

introduced into system (4), in order to make the extinction equilibrium

E_{0}

of system (4) exponentially asymptotically stable, we must make harvesting

h_{i}

get larger, this implies the state-dependent delay

τ (u)

affects harvesting

h_{i} .

References

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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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1 Introduction
2 Existence and Linearized Stability of Equilibria
3 Globally Exponentially Asymptotically Stability of $E_{0}, E^{*}$
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**3 Globally Exponentially Asymptotically Stability of $E_{0}, E^{*}$**