1 Introduction

Cooperative system is an important system in the field of biology, and the importance of the system is the same as for prey–predator and competitive systems. Many scholars have done research on the cooperative ecosystem (see [1–12]). May [1] described a cooperative system with the following equations:

$$ \begin{gathered}\frac{dx_{1}}{dt}=r_{1}x_{1} \biggl(1-\frac {x_{1}}{a_{1}+b_{1}x_{2}}-c_{1}x_{1} \biggr), \\ \frac{dx_{2}}{dt}=r_{2}x_{2} \biggl(1- \frac {x_{2}}{a_{2}+b_{2}x_{1}}-c_{2}x_{2} \biggr), \end{gathered} $$
(1.1)

where \({x_{1}}\), \({x_{2}}\) are the densities of the species \({x_{1}}\), \({x_{2}}\) at time t, respectively, \({r_{i}} \) refers to the intrinsic rate of population \({x_{i}}\), \(i=1,2\), and \({b_{i}}\), \(i=1,2\), refers to the coefficients of cooperation, \({r_{i}}\), \({a_{i}}\), \({b_{i}}\), \({c_{i}}\), \(i=1,2\) are positive constants. His research shows that the cooperative system has a unique positive equilibrium point and it is globally asymptotically stable.

Cui and Chen [2] think that a non-autonomous form is more reasonable. They put forward the following cooperation system:

$$ \begin{gathered}\frac{dx_{1}}{dt}=r_{1}(t)x_{1} \biggl(1-\frac {x_{1}}{a_{1}(t)+b_{1}(t)x_{2}}-c_{1}(t)x_{1} \biggr), \\ \frac{dx_{2}}{dt}=r_{2}(t)x_{2} \biggl(1- \frac {x_{2}}{a_{2}(t)+b_{2}(t)x_{1}}-c_{2}(t)x_{2} \biggr), \end{gathered} $$
(1.2)

where the function \({r_{i}(t)}\), \({a_{i}(t)}\), \({b_{i}(t)}\), \({c_{i}(t)}\), \(i=1,2\) are continuous functions and bounded above and below by positive constants. Under the premise of \({r_{i}(t)}\), \({a_{i}(t)}\), \({b_{i}(t)}\), \({c_{i}(t)}\), \(i=1,2\) are periodic function, they get the sufficient conditions which guarantee the global asymptotic stability of positive periodic solutions of this system.

In view of the influence of time delay, species interactions and feedback, Chen, Liao and Huang [3] proposed the following n-species cooperation system:

$$ \begin{gathered}\begin{aligned} \frac{dx_{i}(t)}{dt}={}&r_{i}(t)x_{i}(t) \biggl[1- \frac {x_{i}(t)}{a_{i}(t)+{\sum_{j=1,j\neq i}^{n}}b_{ij}(t)\int _{-T_{ij}}^{0}K_{ij}(s)x_{j}(t+s)\,ds}-c_{i}(t)x_{i}(t) \biggr] \\ &-d_{i}(t)u_{i}(t)x_{i}(t)-e_{i}(t)x_{i}(t) \int_{-\tau _{i}}^{0}H_{i}(s)u_{i}(t+s)\,ds, \end{aligned} \\ \frac{du_{i}(t)}{dt}=-\alpha_{i}(t)u_{i}(t)+\beta _{i}(t)x_{i}(t)+r_{i}(t) \int_{-\eta_{i}}^{0}G_{i}(s)x_{i}(t+s)\,ds, \end{gathered} $$
(1.3)

where \(x_{i}(t)\), \(i=1,\ldots,n\) is the density of cooperation species \(X_{i}\), \(u_{i}(t)\), \(i=1,\ldots,n\), is the feedback control variable. The authors obtained the sufficient conditions which guarantee the permanence by using differential inequality theory. For more work as regards the system, we can refer to [4–6].

In the real world, individual organisms are associated with a strong and weak differential. Mohammadi [13] proposed a Leslie–Gower predator–prey model:

$$ \begin{gathered}\frac{dH_{1}}{dt}=(r_{1}-bH_{1}- \alpha H_{2})H_{1}, \\ \frac{dH_{2}}{dt}=(\alpha H_{1}-c_{1}-c_{2}P)H_{2}, \\ \frac{dP}{dt}= \biggl(r_{2}-\frac{a_{2}P}{H_{2}} \biggr)P, \end{gathered} $$
(1.4)

where \(r_{1}\), \(b_{1}\), α, \(c_{1}\), \(c_{2}\), \(r_{2}\), \(a_{2}\) are positive constants, the predators can distinguish between strong and weak prey and predator eats only weak prey, when a prey becomes weak, it does not become strong again; by constructing a suitable Lyapunov function, it is shown that the unique equilibrium point is stable in the positive octant.

Conversely, in many cooperative ecosystems, partners like strong partners, because the strong partners are more conducive to their survival. This shows that the cooperative object should only be part instead of the whole.

There are two populations:

The partner H, whose total density is H, is divided into two categories \({H_{1}}\), \({H_{2}}\). \({H_{1}}\) denotes the strong partner density and \({H_{2}}\) denotes the weak. Of the other partner, the total density is P.

The May cooperative model (1.1) is our basic model and we consider the following assumptions to improve the model:

\((A_{1})\) :

The partner P can distinguish between strong partner \({H_{1}}\) and weak partner \({H_{2}}\) and the partner P cooperates only with strong partner \({H_{1}}\).

\((A_{2})\) :

When provided with food resources, the weak partner \({H_{2}}\) has no negative influence on the stronger partner, that is to say, the weak partners can only eat the food after the strong ones have used enough.

\((A_{3})\) :

Due to the lack of sufficient food resources, once it becomes weak, the weak partner \({H_{2}}\) and their descendants will no longer be strong.

\((A_{4})\) :

The rate of becoming weak is described by the simple mass action \(\alpha{H_{1}}{H_{2}}\).

By the above assumptions, we propose a model as follows:

$$ \begin{gathered}\frac{dH_{1}}{dt}=r_{1}H_{1} \biggl(1-\frac {H_{1}}{a_{1}+b_{1}P}-c_{1}H_{1}-\frac{\alpha{H_{2}}}{r_{1}} \biggr), \\ \frac{dH_{2}}{dt}=H_{2} (\alpha{H_{1}}+d-e{H_{2}} ), \\ \frac{dP}{dt}=r_{2}P \biggl(1-\frac{P}{a_{2}+b_{2}H_{1}}-c_{2}P \biggr), \end{gathered} $$
(1.5)

where \({r_{i}}\), \({a_{i}}\), \({b_{i}}\), \({c_{i}}\), d, \(i=1,2\) are positive constants.

The structure of this article as follows. In Sect. 2 we will introduce several useful lemmas and prove permanence and non-permanence. In Sect. 3 we will discuss the existence of the equilibrium point. In Sect. 4 global stability of equilibrium points is studied. In Sect. 5 two examples are given to show the feasibility of our results. We end this paper by a brief discussion.

2 Permanence and non-permanence

In view of the actual ecological implications of system (1.5), we assume that the initial value \(H_{i}(0)>0\), \(i=1, 2\), \(P(0)>0\) in system (1.5). Obviously, any solution of system (1.5) remains positive for all \(t\geq0\).

Lemma 2.1

(see [14])

Let \(a>0\), \(b>0\).

  1. (I)

    If \(\frac{dx}{dt}\geq x(b-ax)\), then \(\liminf_{t\rightarrow+\infty}{x(t)}\geq\frac{b}{a}\) for \(t \geq0\) and \(x(0)>0\).

  2. (II)

    If \(\frac{dx}{dt}\leq x(b-ax)\), then \(\limsup_{t\rightarrow+\infty}{x(t)}\leq\frac{b}{a}\) for \(t\geq0\) and \(x(0)>0\).

Lemma 2.2

(see [15])

Let \(a>0\), \(b>0\).

If \(\frac{dx}{dt}\leq x(-b-ax)\), then \(\lim_{t\rightarrow+\infty}{x(t)}=0\) for \(t\geq0\) and \(x(0)>0\).

Theorem 2.1

If the assumptions \((B_{1})\) and \((B_{2})\) hold,

\((B_{1})\) :

\(M=1-\frac{\alpha d}{r_{1}e}>0\),

\((B_{2})\) :

\(1>\frac{{\alpha }^{2}(a_{1}c_{2}+b_{1})}{r_{1}e(a_{1}c_{1}c_{2}+b_{1}c_{1}+c_{2})}\),

then system (1.5) is permanent.

Proof

Let \((H_{1}(t),H_{2}(t),P(t))^{T}\) be any positive solution of system (1.5), from the second equation of system (1.5), it follows that

$$\frac{dH_{2}}{dt}\geq H_{2}(d-e{H_{2}}). $$

According to Lemma 2.1, we have

$$ \liminf_{t\rightarrow+\infty }{H_{2}(t)}\geq \frac{d}{e}\stackrel{\text{def}}{=} H_{2}^{i}>0.$$
(2.1)

For any positive constant ε small enough, it follows from (2.1) that there exists a large enough \(T_{1}>0\) such that

$$ H_{2}(t)> H_{2}^{i}-\varepsilon,\quad t \geq T_{1}.$$
(2.2)

From the third equation, we have

$$\frac{dP}{dt}\leq r_{2}P(1-c_{2}P). $$

According to Lemma 2.1, we have

$$ \limsup_{t\rightarrow+\infty }{P(t)}\leq\frac{1}{c_{2}} \stackrel{\text{def}}{=} {P}^{s}>0.$$
(2.3)

For any positive constant ε small enough, it follows from (2.3) that there exists a large enough \(T_{2}>T_{1}\) such that

$$ P(t)\leq{P}^{s}+\varepsilon,\quad t\geq T_{2}.$$
(2.4)

By applying (2.2) and (2.4), from the first equation of system (1.5), we have

$$\frac{dH_{1}}{dt}\leq r_{1}H_{1} \biggl(1- \frac{H_{1}}{a_{1}+b_{1}({P}^{s}+\varepsilon )}-c_{1}H_{1}-\frac{\alpha(H_{2}^{i}-\varepsilon)}{r_{1}} \biggr),\quad t \geq T_{2}. $$

According to Lemma 2.1, we have

$$\limsup_{t\rightarrow+\infty}{H_{1}(t)}\leq \biggl(1- \frac{\alpha (H_{2}^{i}-\varepsilon)}{r_{1}} \biggr)\frac{a_{1}+b_{1} ({P}^{s}+\varepsilon)}{1+a_{1}c_{1}+b_{1}c_{1}({P}^{s}+\varepsilon)}. $$

Letting \(\varepsilon\rightarrow0\) and by applying (2.1) and (2.3)

$$ \limsup_{t\rightarrow+\infty }{H_{1}(t)}\leq \frac{a_{1}c_{2}+b_{1}}{a_{1}c_{1}c_{2}+b_{1}c_{1}+c_{2}}M\stackrel {\text{def}}{=} H_{1}^{s}>0.$$
(2.5)

For any positive constant ε small enough, it follows from (2.5) that there exists a large enough \(T_{3}>T_{2}\) such that

$$ H_{1}(t)\leq H_{1}^{s}+ \varepsilon,\quad t\geq T_{3}.$$
(2.6)

Then the second equation of (1.5) leads to

$$\frac{dH_{2}}{dt}\leq H_{2} \bigl(\alpha \bigl(H_{1}^{s}+ \varepsilon \bigr)+d-e{H_{2}} \bigr),\quad t\geq T_{3}. $$

According to Lemma 2.1, we have

$$\limsup_{t\rightarrow+\infty}{H_{2}(t)}\leq\frac{\alpha (H_{1}^{s}+\varepsilon)+{d}}{e}. $$

Letting \(\varepsilon\rightarrow0\) in the above inequality leads to

$$ \limsup_{t\rightarrow+\infty }{H_{2}(t)}\leq \frac{\alpha{H_{1}^{s}}+{d}}{e}\stackrel{\text{def}}{=} H_{2}^{s}.$$
(2.7)

For any positive constant ε small enough, it follows from (2.7) that there exists a large enough \(T_{4}>T_{3}\) such that

$$ H_{2}(t)\leq H_{2}^{s}+\varepsilon,\quad t\geq T_{4}.$$
(2.8)

Then substituting (2.8) into the first equation of (1.5), we have

$$\frac{dH_{1}}{dt}\geq r_{1}H_{1} \biggl(1- \frac {H_{1}}{a_{1}}-c_{1}H_{1}-\frac{\alpha(H_{2}^{s}+\varepsilon)}{r_{1}} \biggr),\quad t \geq T_{4}. $$

According to Lemma 2.1, we have

$$\liminf_{t\rightarrow+\infty}{H_{1}(t)}\geq \biggl(1- \frac{\alpha (H_{2}^{s}+\varepsilon)}{r_{1}} \biggr)\frac{a_{1}}{a_{1}c_{1}+1}. $$

Letting \(\varepsilon\rightarrow0\) and by applying (2.5) and (2.7)

$$ \liminf_{t\rightarrow+\infty }{H_{1}(t)}\geq M- \frac{\alpha^{2}H_{1}^{s}}{r_{1}e}\stackrel{\text{def}}{=} H_{1}^{i}>0.$$
(2.9)

From the third equation of system(1.5), it follows that

$$\frac{dP}{dt}\geq r_{2}P \biggl(1-\frac{P}{a_{2}}-c_{2}P \biggr). $$

According to Lemma 2.1, we have

$$ \liminf_{t\rightarrow+\infty }{P(t)}\geq\frac{a_{2}}{a_{2}c_{2}+1} \stackrel{\text{def}}{=} {P}^{i}.$$
(2.10)

(2.1), (2.3), (2.5), (2.7), (2.9) and (2.10) show that if the assumptions \((B_{1})\), \((B_{2})\) hold, then system (1.5) is permanent. □

Theorem 2.2

If the assumption \((B_{3})\) holds,

\((B_{3})\) :

\(M=1-\frac{\alpha d}{r_{1}e}<0\),

then the weak partners \(H_{2}\) and partners P are permanent, the strong partners \(H_{1}\) are non-permanent.

Proof

Let \((H_{1}(t),H_{2}(t),P)^{T}\) be any positive solutions of system (1.5) for \(t\geq0\).

From the proof Theorem 2.1, we know

$$\frac{dH_{1}}{dt}\leq r_{1}H_{1} \biggl(1- \frac{H_{1}}{a_{1}+b_{1}({P}^{s}+\varepsilon )}-c_{1}H_{1}-\frac{\alpha(H_{2}^{i}-\varepsilon)}{r_{1}} \biggr),\quad t \geq T_{2}. $$

Noting that condition \((B_{3})\) implies that \(1-\frac{\alpha(H_{2}^{i}-\varepsilon)}{r_{1}}<0\).

According to Lemma 2.2, we have

$$ \lim_{t\rightarrow+\infty}{H_{1}(t)}=0.$$
(2.11)

By applying (2.11), from the second equation of system (1.5), it is easy to prove that

$$ \lim_{t\rightarrow+\infty }{H_{2}(t)}= \frac{d}{e}.$$
(2.12)

(2.3), (2.10), (2.11) and (2.12) show that if the assumptions \((B_{3})\) hold, then the weak partners \(H_{2}\) and partners P are permanent, the strong partners \(H_{1}\) are non-permanent. □

3 Existence of equilibrium point

Theorem 3.1

If the assumption \((B_{1})\) holds, then system (1.5) have a unique positive equilibrium point.

Proof

We determine the positive equilibrium of the system (1.5) through solving the following equations:

$$ \left \{ \textstyle\begin{array}{l} 1-\frac{H_{1}}{a_{1}+b_{1}P}-c_{1}H_{1}-\frac{\alpha{H_{2}}}{r_{1}}=0,\\ \alpha{H_{1}}+d-e{H_{2}}=0,\\ 1-\frac{P}{a_{2}+b_{2}H_{1}}-c_{2}P=0. \end{array}\displaystyle \right .$$
(3.1)

Here we transform Eqs. (3.1) into the following form:

$$ \left \{ \textstyle\begin{array}{l} 1-\frac{H_{1}}{a_{11}+b_{11}P}-c_{11}H_{1}=0,\\1-\frac{P}{a_{2}+b_{2}H_{1}}-c_{2}P=0,\\ \alpha{H_{1}}+d-e{H_{2}}=0, \end{array}\displaystyle \right .$$
(3.2)

where \(a_{11}=Ma_{1}\), \(b_{11}=Mb_{1}\), \(c_{11}=(c_{1}+\frac{\alpha d}{r_{1}e})/M\), from the first and second equations of (3.2), we have

$$ DH_{1}^{2}+E{H_{1}}+F=0,$$
(3.3)

where

$$\begin{gathered} D=b_{2}(a_{11}c_{11}c_{2}+b_{11}c_{11}+c_{2}),\qquad F= -a_{11}(a_{2}c_{2}+b_{2}+1), \\ E= \bigl[(a_{2}c_{2}+1)+c_{11}(a_{11}+a_{11}a_{2}c_{2}+a_{2}b_{11})-b_{2}(a_{11}c_{2}+b_{11}) \bigr].\end{gathered} $$

From the terms D and F of (3.3), we know that there is a unique positive solutions \(H_{1}^{*}\). Substitute \(H_{1}^{*}\) into the second and third equations of (3.1). Then system (1.5) has a unique positive equilibrium point \(E_{1}(H_{1}^{*}, H_{2}^{*}, {P}^{*})\). □

Theorem 3.2

Clearly, the system (1.5) has an equilibrium point \(E_{2}(0, {H_{2}}_{*}, {P}_{*})\).

4 Global stability

Theorem 4.1

If the assumptions \((B_{1})\) and \((B_{4})\) hold,

\((B_{4})\) :

\({\alpha}^{2}<\frac{a_{2}r_{1}c_{1}e}{a_{2}+b_{2}H_{1}^{s}}\),

then the positive equilibrium of system (1.5) is globally asymptotically stable.

Proof

Inspired by the idea of Li, Han and Chen [7] and Leon [8], the following Lyapunov function is presented:

We define \(L: \{(H_{1},H_{2},P)\in R_{+}^{3}:H_{1}>0,H_{2}>0,P>0 \}\rightarrow R\) by

$$\begin{aligned} L(H_{1},H_{2},P)={}&\eta_{1} \int_{H_{1}^{*}}^{H_{1}}\frac{(\theta -H_{1}^{*})}{(a_{2}+b_{2}\theta)\theta}\,d\theta+ \eta_{2} \int_{{ P}^{*}}^{P}\frac{(\theta-{P}^{*})}{(a_{1}+b_{1}\theta)\theta}\,d\theta\\ & + \eta_{3} \biggl({H_{2}}-H_{2}^{*} -H_{2}^{*}\ln\frac{H_{2}}{H_{2}^{*}} \biggr),\end{aligned} $$

where \(\eta_{1}=1\), \(\eta _{2}=r_{1}b_{1}H_{1}^{*}(a_{2}+b_{2}H_{1}^{*})/r_{2}b_{2}{P}^{*}(a_{1}+b_{1}{P}^{*})\), \(\eta_{3}=1/a_{2}\). The function \(L(H_{1},H_{2},P)\) is defined, continuous and positive definite for all \(H_{1},H_{2},P>0\). and the minimum \(L(H_{1},H_{2},P)=0\) occurs at the equilibrium point \((H_{1}^{*},H_{2}^{*},{P}^{*})^{T}\). Calculating the derivative of L along the solution \((H_{1}(t),H_{2}(t),P(t))^{T}\) of the system (1.5), we have

$$\begin{aligned}\frac{dL}{dt}={}&\frac{\eta _{1}r_{1}(H_{1}-H_{1}^{*})}{a_{2}+b_{2}H_{1}} \biggl(1-\frac {H_{1}}{a_{1}+b_{1}P}-c_{1}H_{1}-\frac{\alpha{H_{2}}}{r_{1}} \biggr) \\ &+\eta_{3} \bigl(H_{2}-H_{2}^{*} \bigr) ( \alpha{H_{1}}+d-e{H_{2}} )+\frac{\eta _{2}r_{2}(P-{P}^{*})}{a_{1}+b_{1}P} \biggl(1- \frac {P}{a_{2}+b_{2}H_{1}}-c_{2}P \biggr) \\ ={}&\frac{\eta_{1}r_{1}(H_{1}-H_{1}^{*})}{a_{2}+b_{2}H_{1}} \biggl(-\frac {\alpha(H_{2}-H_{2}^{*})}{r_{1}}-c_{1} \bigl(H_{1}-H_{1}^{*} \bigr)+\frac {H_{1}^{*}}{a_{1}+b_{1}{P}^{*}}- \frac{H_{1}}{a_{1}+b_{1}P} \biggr) \\ &+\eta_{3} \bigl(H_{2}-H_{2}^{*} \bigr) \bigl( \alpha \bigl(H_{1}-H_{1}^{*} \bigr)-e \bigl(H_{2}-H_{2}^{*} \bigr) \bigr) \\ &+\frac{\eta_{2}r_{2}(P-{P}^{*})}{a_{1}+b_{1}P} \biggl(-c_{2} \bigl(P-{P}^{*} \bigr)+ \frac{{P}^{*}}{a_{2}+b_{2}H_{1}^{*}}-\frac {P}{a_{2}+b_{2}H_{1}} \biggr) \\ ={}&\frac{\eta_{1}r_{1}(H_{1}-H_{1}^{*})}{a_{2}+b_{2}H_{1}} \biggl(-\frac {\alpha(H_{2}-H_{2}^{*})}{r_{1}}-c_{1} \bigl(H_{1}-H_{1}^{*} \bigr)-\frac {(H_{1}-H_{1}^{*})}{a_{1}+b_{1}{P}} \\ &+\frac {b_{1}H_{1}^{*}(P-{P}^{*})}{(a_{1}+b_{1}P)(a_{1}+b_{1}{P}^{*})} \biggr)+\eta _{3} \bigl(H_{2}-H_{2}^{*} \bigr) \bigl(\alpha \bigl(H_{1}-H_{1}^{*} \bigr)-e \bigl(H_{2}-H_{2}^{*} \bigr) \bigr) \\ &+\frac{\eta_{2}r_{2}(P-{P}^{*})}{a_{1}+b_{1}P} \biggl(-c_{2} \bigl(P-{P}^{*} \bigr)- \frac{(P-{P}^{*})}{a_{2}+b_{2}H_{1}}+\frac {b_{2}{P}^{*}(H_{1}-H_{1}^{*})}{(a_{2}+b_{2}H_{1}^{*})(a_{2}+b_{2}{H_{1}})} \biggr) \\ \leq{}& A(t)+B(t), \end{aligned} $$

where

$$\begin{gathered}\begin{aligned}A(t)={}& \frac {1}{(a_{1}+b_{1}P)(a_{2}+b_{2}{H_{1}})} \biggl(-\eta _{1}r_{1} \bigl(H_{1}-H_{1}^{*} \bigr)^{2}- \eta_{2}r_{2} \bigl(P-{P}^{*} \bigr)^{2} \\ &+ \biggl(\frac{\eta_{1}r_{1}b_{1}H_{1}^{*}}{a_{1}+b_{1}{P}^{*}}+\frac {\eta_{2}r_{2}b_{2}{P}^{*}}{a_{2}+b_{2}H_{1}^{*}} \biggr) \bigl(H_{1}-H_{1}^{*} \bigr) \bigl(P-{P}^{*} \bigr) \biggr), \end{aligned} \\ \begin{aligned}B(t)={}&{-}\frac{\alpha }{a_{2}+b_{2}H_{1}} \bigl(H_{1}-H_{1}^{*} \bigr) \bigl(H_{2}-H_{2}^{*} \bigr)-\frac {c_{1}r_{1}}{a_{2}+b_{2}H_{1}} \bigl(H_{1}-H_{1}^{*} \bigr)^{2} \\ &+\eta_{3}\alpha \bigl(H_{1}-H_{1}^{*} \bigr) \bigl(H_{2}-H_{2}^{*} \bigr)-\eta_{3}e \bigl(H_{2}-H_{2}^{*} \bigr)^{2}. \end{aligned} \end{gathered} $$

Now, we prove \(A(t)\), \(B(t)\) are negative definite.

Let \(A(t)\stackrel{\text{def}}{=}\frac {1}{(a_{1}+b_{1}P)(a_{2}+b_{2}{H_{1}})}Y^{T}AY\) where \(Y=((H_{1}-H_{1}^{*}),(P-{P}^{*}))^{T}\) and

$$ A=\left ( \textstyle\begin{array}{c@{\quad}c} -\eta_{1}r_{1}&\frac{\eta _{1}r_{1}b_{1}H_{1}^{*}}{2(a_{1}+b_{1}{P}^{*})}+\frac{\eta _{2}r_{2}b_{2}{P}^{*}}{2(a_{2}+b_{2}H_{1}^{*})}\\\frac{\eta_{1}r_{1}b_{1}H_{1}^{*}}{2(a_{1}+b_{1}{P}^{*})}+\frac{\eta _{2}r_{2}b_{2}{P}^{*}}{2(a_{2}+b_{2}H_{1}^{*})}& -\eta_{2}r_{2} \end{array}\displaystyle \right ). $$
(4.1)

Note first that both of the off-diagonal elements of matrix A are negative and

$$\begin{gathered}\eta_{1}\eta_{2}r_{1}r_{2}- \biggl(\frac{\eta _{1}r_{1}b_{1}H_{1}^{*}}{2(a_{1}+b_{1}{P}^{*})}+\frac{\eta _{2}r_{2}b_{2}{P}^{*}}{2(a_{2}+b_{2}H_{1}^{*})} \biggr)^{2} \\ \quad=\eta_{1}\eta_{2}r_{1}r_{2} \biggl(1-\frac {b_{1}H_{1}^{*}}{a_{1}+b_{1}{P}^{*}}\frac {b_{2}{P}^{*}}{a_{2}+b_{2}H_{1}^{*}} \biggr)+ \biggl(\frac{\eta_{1}r_{1}b_{1}H_{1}^{*}}{ 2(a_{1}+b_{1}{P}^{*})}- \frac{\eta _{2}r_{2}b_{2}{P}^{*}}{2(a_{2}+b_{2}H_{1}^{*})} \biggr)^{2} \\ \quad=\eta_{1}\eta_{2}r_{1}r_{2} \biggl(1-\frac {b_{1}H_{1}^{*}}{a_{1}+b_{1}{P}^{*}}\frac {b_{2}{P}^{*}}{a_{2}+b_{2}H_{1}^{*}} \biggr)>0, \end{gathered} $$

thus \(A(t)\leq0\).

Noting that \(ab\leq\frac{\theta a^{2}}{2}+\frac{b^{2}}{2\theta}\), \(\theta>0\), it follows

$$\begin{aligned}B(t)\leq{}&\frac{\alpha}{a_{2}+b_{2}H_{1}} \biggl( \frac{1}{2\theta _{1}} \bigl(H_{1}-H_{1}^{*} \bigr)^{2}+ \frac{\theta_{1}}{2\theta _{1}} \bigl(H_{2}-H_{2}^{*} \bigr)^{2}-r_{1}c_{1} \bigl(H_{1}-H_{1}^{*} \bigr)^{2} \biggr) \\ &+\frac{\eta_{3}\alpha}{2\theta_{2}} \bigl(H_{1}-H_{1}^{*} \bigr)^{2}+\frac{\eta _{3}\alpha\theta_{2}}{2} \bigl(H_{2}-H_{2}^{*} \bigr)^{2}-\eta_{3}e \bigl(H_{2}-H_{2}^{*} \bigr)^{2} \\ \leq{}&{-} \biggl(\frac{r_{1}c_{1}}{a_{2}+b_{2}H_{1}^{s}}-\frac {\alpha}{2a_{2}\theta_{1}}-\frac{\eta_{3}\alpha}{2\theta_{2}} \biggr) \bigl(H_{1}-H_{1}^{*} \bigr)^{2} \\ &- \biggl(\eta_{3}e-\frac{\eta_{3}\alpha\theta_{2}}{2}-\frac {\alpha\theta_{1}}{2a_{2}} \biggr) \bigl(H_{2}-H_{2}^{*} \bigr)^{2}. \end{aligned} $$

Denote \(\delta_{1}=\frac{r_{1}c_{1}}{a_{2}+b_{2}H_{1}^{s}}-\frac{\alpha }{2a_{2}\theta_{1}}-\frac{\eta_{3}\alpha}{2\theta_{2}}\) and \(\delta_{2}=\eta_{3}c_{2}-\frac{\eta_{3}\alpha\theta_{2}}{2}-\frac {\alpha\theta_{1}}{2a_{2}}\). Then taking

$$\eta_{3}=\frac{1}{a_{2}}, \qquad\theta_{1}= \theta_{2}=\frac{2\alpha ea_{2}+2\alpha eb_{2}H_{1}^{s}}{a_{2}r_{1}c_{1}c_{2}+a_{2}{\alpha }^{2}+b_{2}H_{1}^{s}{\alpha}^{2}} $$

gives

$$\delta_{1}=\frac{e (a_{2}r_{1}c_{1}e-a_{2}{\alpha }^{2}-b_{2}H_{1}^{s}{\alpha}^{2} )}{a_{2} (a_{2}r_{1}c_{1}e+a_{2}{\alpha}^{2}+b_{2}H_{1}^{s}{\alpha}^{2} )},\qquad \delta_{2}= \frac{a_{2}r_{1}c_{1}e-a_{2}{\alpha }^{2}-b_{2}H_{1}^{s}{\alpha}^{2}}{2a_{2}e (a_{2}+b_{2}H_{1}^{s} )}. $$

From \((B_{4})\), we know that \(\delta_{i}>0\), \(i=1,2\). It is easy to see that \(B(t)\leq0\).

Obvious, \(\frac{dL}{dt}<0\) for all \(H_{1}>0\), \(H_{2}>0\), \(P>0\) except the equilibrium point \((H_{1}^{*}, H_{2}^{*}, {P}^{*})\) where \(\frac{dL}{dt}=0\). According to the Lyapunov asymptotic stability theorem [9], the equilibrium point \((H_{1}^{*}, H_{2}^{*}, {P}^{*})\) is globally asymptotically stable in the interior of \(R_{+}^{3}\). This completes the proof. □

Theorem 4.2

If the assumptions \((B_{3})\) and \((B_{5})\) hold,

\((B_{5})\) :

\({\alpha}^{2}< r_{1}c_{1}e\),

then the equilibrium point \(E_{2}(0, {H_{2}}_{*}, {P}_{*})\) system is globally asymptotically stable.

Proof

We define \(L: \{(H_{1},H_{2},P)\in R_{+}^{3}:H_{1}>0,H_{2}>0,P>0 \}\rightarrow R\) by

$$\begin{aligned} L(H_{1},H_{2},P)={}&\eta_{1} \int_{0}^{H_{1}}\frac{1}{(a_{2}+b_{2}\theta )}\,d\theta+ \eta_{2} \int_{{ P}_{*}}^{P}\frac{(\theta-{P}_{*})}{(a_{1}+b_{1}\theta)\theta}\,d\theta \\ &+ \eta_{3} \biggl({H_{2}}-{H_{2}}_{*} -{H_{2}}_{*}\ln\frac{H_{2}}{{H_{2}}_{*}} \biggr),\end{aligned} $$

where \(\eta_{1}=1\), \(\eta_{2}=r_{1}a_{2}^{2}/r_{2}b_{2}^{2}P_{*}^{2}\), \(\eta_{3}=1/a_{2}\). The function \(L(H_{1},H_{2},P)\) is defined, continuous and positive definite for all \(H_{1},H_{2},P>0\). and the minimum \(L(H_{1},H_{2},P)=0\) occurs at the equilibrium point \((H_{1}^{*},H_{2}^{*},{P}^{*})^{T}\). Calculating the derivative of L along the solution \((H_{1}(t),H_{2}(t),P(t))^{T}\) of the system (1.5), we have

$$\begin{aligned}\frac{dL}{dt}={}&\frac{\eta_{1}r_{1}H_{1}}{a_{2}+b_{2}H_{1}} \biggl(1-\frac{H_{1}}{a_{1}+b_{1}P}-c_{1}H_{1}-\frac{\alpha{H_{2}}}{r_{1}} \biggr)+\eta_{3}(H_{2}-{H_{2}}_{*}) ( \alpha{H_{1}}+d-e{H_{2}} ) \\ &+\frac{\eta_{2}r_{2}(P-{P}_{*})}{a_{1}+b_{1}P} \biggl(1-\frac {P}{a_{2}+b_{2}H_{1}}-c_{2}P \biggr) \\ ={}&\frac{\eta_{1}r_{1}H_{1}}{a_{2}+b_{2}H_{1}} \biggl(-\frac{\alpha (H_{2}-{H_{2}}_{*})}{r_{1}}-c_{1}H_{1}- \frac{H_{1}}{a_{1}+b_{1}P} \biggr) \\ &+\eta_{3}(H_{2}-{H_{2}}_{*}) \bigl(\alpha H_{1}-e(H_{2}-{H_{2}}_{*}) \bigr) \\ &+\frac{\eta_{2}r_{2}(P-{P}_{*})}{a_{1}+b_{1}P} \biggl(-c_{2}(P-{P}_{*})+\frac{{P}_{*}}{a_{2}}- \frac{P}{a_{2}+b_{2}H_{1}} \biggr) \\ ={}&\frac{\eta_{1}r_{1}H_{1}}{a_{2}+b_{2}H_{1}} \biggl(-\frac{\alpha (H_{2}-{H_{2}}_{*})}{r_{1}}-c_{1}H_{1}- \frac{H_{1}}{a_{1}+b_{1}P} \biggr) \\ &+\eta_{3}(H_{2}-{H_{2}}_{*}) \bigl(\alpha H_{1}-e(H_{2}-{H_{2}}_{*}) \bigr) \\ &+\frac{\eta_{2}r_{2}(P-{P}_{*})}{a_{1}+b_{1}P} \biggl(-c_{2}(P-{P}_{*})-\frac{(P-{P}_{*})}{a_{2}+b_{2}H_{1}}+ \frac {b_{2}{P}_{*}H_{1}}{a_{2}(a_{2}+b_{2}{H_{1}})} \biggr) \\ \leq{}& C(t)+D(t), \end{aligned} $$

where

$$\begin{gathered}C(t)=\frac{1}{(a_{1}+b_{1}P)(a_{2}+b_{2}{H_{1}})} \biggl(-\eta _{1}r_{1}H_{1}^{2}- \eta_{2}r_{2}(P-{P}_{*})^{2} +\frac{\eta_{2}r_{2}b_{2}{P}_{*}}{a_{2}}H_{1}(P-{P}_{*}) \biggr), \\ D(t)= \biggl(\eta_{3}\alpha-\frac{\alpha}{a_{2}+b_{2}H_{1}} \biggr)H_{1}(H_{2}-{H_{2}}_{*})-\frac{c_{1}r_{1}}{a_{2}+b_{2}H_{1}}H_{1}^{2}- \eta_{3}e(H_{2}-{H_{2}}_{*})^{2}. \end{gathered} $$

Now, we prove \(C(t)\), \(D(t)\) is negative definite.

Let \(C(t)\stackrel{\text{def}}{=}\frac {1}{(a_{1}+b_{1}P)(a_{2}+b_{2}{H_{1}})}Y^{T}CY\) where \(Y=((H_{1}),(P-{P}_{*}))^{T}\)

$$ C=\left ( \textstyle\begin{array}{c@{\quad}c} -\eta_{1}r_{1}& \frac{\eta_{2}r_{2}b_{2}{P}^{*}}{2a_{2}}\\\frac{\eta_{2}r_{2}b_{2}{P}^{*}}{2a_{2}}& -\eta_{2}r_{2} \end{array}\displaystyle \right ). $$
(4.2)

Note first that both of the off-diagonal elements of matrix A are negative and

$$\begin{aligned}\eta_{1}\eta_{2}r_{1}r_{2}- \biggl(\frac{\eta _{2}r_{2}b_{2}{P}^{*}}{2a_{2}} \biggr)^{2} =\eta_{2}r_{2} \biggl(\eta_{1}r_{1}-\frac{\eta _{2}r_{2}b_{2}^{2}P_{*}^{2}}{4a_{2}^{2}} \biggr) = \frac{3\eta_{2}r_{1}r_{2}}{4}>0, \end{aligned} $$

thus \(C(t)\leq0\).

Noting that \(ab\leq\frac{\theta a^{2}}{2}+\frac{b^{2}}{2\theta}\), \(\theta>0\), from (2.11), for sufficiently small constant \(\varepsilon_{0}>0\), there is an integer \(T>0\) such that if \(t>T\), \(b_{2}H_{1}(t)<\varepsilon_{0}\), it follows that

$$\begin{aligned}B(t)\leq{}&\frac{\alpha}{a_{2}+b_{2}H_{1}} \biggl( \frac{1}{2\theta _{1}}H_{1}^{2}+\frac{\theta_{1}}{2}(H_{2}-{H_{2}}_{*})^{2}-r_{1}c_{1}H_{1}^{2} \biggr) \\ &+\frac{\eta_{3}\alpha}{2\theta_{2}}H_{1}^{2}+\frac{\eta_{3}\alpha \theta_{2}}{2}(H_{2}-{H_{2}}_{*})^{2}- \eta_{3}e(H_{2}-{H_{2}}_{*})^{2} \\ \leq{}&{-} \biggl(\frac{r_{1}c_{1}}{a_{2}+\varepsilon_{0}}-\frac {\alpha}{2a_{2}\theta_{1}}-\frac{\eta_{3}\alpha}{2\theta_{2}} \biggr)H_{1}^{2} \\ &- \biggl(\eta_{3}e-\frac{\eta_{3}\alpha\theta_{2}}{2}-\frac {\alpha\theta_{1}}{2a_{2}} \biggr) \bigl(H_{2}-H_{2}^{*} \bigr)^{2}. \end{aligned} $$

Denote \(\delta_{1}=\frac{r_{1}c_{1}}{a_{2}+\varepsilon_{0}}-\frac{\alpha }{2a_{2}\theta_{1}}-\frac{\eta_{3}\alpha}{2\theta_{2}}\) and \(\delta_{2}=\eta_{3}c_{2}-\frac{\eta_{3}\alpha\theta_{2}}{2}-\frac {\alpha\theta_{1}}{2a_{2}}\). Then taking

$$\eta_{3}=\frac{1}{a_{2}}, \qquad\theta_{1}= \theta_{2}=\frac{2\alpha ea_{2}+2\alpha e\varepsilon_{0}}{a_{2}r_{1}c_{1}c_{2}+a_{2}+{\alpha}^{2}\varepsilon_{0}}, $$

gives

$$\delta_{1}=\frac{e (a_{2}r_{1}c_{1}e-a_{2}{\alpha}^{2}-{\alpha }^{2}\varepsilon_{0} )}{a_{2} (a_{2}r_{1}c_{1}e+a_{2}{\alpha }^{2}+{\alpha}^{2}\varepsilon_{0} )}, \qquad\delta_{2}= \frac{a_{2}r_{1}c_{1}e-a_{2}{\alpha}^{2}-{\alpha }^{2}\varepsilon_{0}}{2a_{2}e (a_{2}+\varepsilon_{0} )}. $$

From \((B_{5})\), we know that, for sufficiently small constant \(\varepsilon_{0}>0\), \(\delta_{i}>0\), \(i=1,2\). It is easy to see that \(D(t)\leq0\).

Obviously, \(\frac{dL}{dt}<0\) for all \(H_{1}>0\), \(H_{2}>0\), \(P>0\) except the equilibrium point \((0, {H_{2}}_{*}, {P}_{*})\) where \(\frac{dL}{dt}=0\). According to the Lyapunov asymptotic stability theorem [9], the equilibrium point \((0, {H_{2}}_{*}, {P}_{*})\) is globally asymptotically stable in the interior of \(R_{+}^{3}\). This completes the proof. □

5 Example and numeric simulation

Consider the following system:

$$ \begin{gathered}\frac{dH_{1}}{dt}=3H_{1} \biggl(1-\frac{H_{1}}{2+2P}-2H_{1}-\frac {{H_{2}}}{10} \biggr), \\ \frac{dH_{2}}{dt}=H_{2}(0.3{H_{1}}+2-2{H_{2}}), \\ \frac{dP}{dt}=2P \biggl(1-\frac{P}{2+0.8H_{1}}-1.5P \biggr). \end{gathered} $$
(5.1)

By calculation, we have \(M=1-\frac{\alpha d}{r_{1}e}=0.9>0\), \({\alpha}^{2}=0.09<8<\frac{a_{2}r_{1}c_{1}e}{a_{2}+b_{2}H_{1}^{s}}\), and it is easy to see that the conditions \((B_{1})\) and \((B_{4})\) are verified. It follows from Theorem 4.1 that there is a unique positive equilibrium point \((H_{1}^{*},H_{2}^{*},{P}^{*})=(0.383868, 1.057580, 0.517211)\) of system (5.1) and it is globally asymptotically stable. Our numerical simulation supports our result (see Fig. 1).

Figure 1
figure 1

Dynamical behavior of system (5.1) with initial values \((0.4,1.2,0.7)^{T}\), \((0.1,0.8,0.2)^{T}\) and \((0.25,0.6,0.3)^{T}\)

Full size image

Consider the following system:

$$ \begin{gathered}\frac{dH_{1}}{dt}=3H_{1} \biggl(1-\frac{H_{1}}{2+2P}-2H_{1}-\frac {3.5{H_{2}}}{3} \biggr), \\ \frac{dH_{2}}{dt}=H_{2}(3.5{H_{1}}+2-2{H_{2}}), \\ \frac{dP}{dt}=2P \biggl(1-\frac{P}{2+0.8H_{1}}-1.5P \biggr). \end{gathered} $$
(5.2)

By calculation, we have \(M=1-\frac{\alpha d}{r_{1}e}=-0.1<0\), \({\alpha}^{2}=10.89<\frac{a_{2}r_{1}c_{1}e}{a_{2}+b_{2}H_{1}^{s}}=12\), it is easy to see that the conditions \((B_{3})\) and \((B_{5})\) are verified. It follows from Theorem 3.1 that there is a unique positive equilibrium point \(({H_{1}}_{*},{H_{2}}_{*},{P}_{*})=(0, 1, 0.5)\) of system (5.2) and it is globally asymptotically stable. Our numerical simulation supports our result (see Fig. 2).

Figure 2
figure 2

Dynamical behavior of system (5.2) with initial values \((0.4,1.4,0.8)^{T}\), \((0.1,0.9,0.3)^{T}\) and \((1.2,0.5,0.1)^{T}\)

Full size image

6 Discussion

In this paper, a May cooperative system with strong and weak cooperative partners is studied. We obtained the sufficient conditions that guarantee the permanence, non-permanence and the global stability of the equilibrium points. By comparing the conditions of \((B_{1})\) and \((B_{3})\), we found that as α becomes larger and larger, the strong partner changes from persistent to extinct. The ecological explanation is that more and more the strong partner become a weak partner, thus we have extinction of the strong. The above numerical simulations also supports this conclusion.

At the end of this paper, we point out that conditions \((B_{1})\) and \((B_{3})\) can be weakened or even are unnecessary. This problem is very interesting and worthy of further study in the future.