1 Introduction

During the last decade, many scholars [1–12] investigated the dynamic behaviors of the amensalism model; here, amensalism means that a species inflicts harm to other species without any costs or benefits received by the other. Such topics as the stability of the equilibrium [1, 3–8], the existence of the positive periodic solution [2, 9, 11], the extinction of the species [8, 10], the influence of the cover [8, 12], the influence of the functional response [10], etc. have been extensively studied. Recently, Xiong et al. [1] proposed the following amensalism model:

$$ \begin{aligned} &\frac{dN_{1}}{dt}=r_{1}N_{1} \biggl(1-\frac{N_{1}}{P_{1}}-u\frac{N_{2}}{P_{1}} \biggr), \\ &\frac{dN_{2}}{dt}=r_{2}N_{2} \biggl(1-\frac{N_{2}}{P_{2}} \biggr), \end{aligned} $$
(1.1)

where \(r_{i}, P_{i}, u, i=1, 2\), are all positive constants. They investigated the local stability property of the equilibria of system (1.1).

On the other hand, in 1973, in their study on the validity of the competition modeling, Ayala et al. [13] found that the following nonlinear competition model accounts best for the experimental results:

$$ \begin{aligned} &\dot{x}_{1}(t)=r_{1}x_{1}(t) \biggl(1-\biggl(\frac{x_{1}(t)}{K_{1}}\biggr)^{\theta_{1}}-\alpha _{12} \frac{x_{2}(t)}{K_{2}} \biggr), \\ &\dot{x}_{2}(t)=r_{2}x_{2}(t) \biggl(1-\biggl( \frac{x_{2}(t)}{K_{2}}\biggr)^{\theta_{2}}-\alpha _{21}\frac{x_{1}(t)}{K_{1}} \biggr). \end{aligned} $$
(1.2)

Since then, the dynamic behaviors of the nonlinear competition system and nonlinear competition-predator-prey system have been extensively studied by many scholars [13–30]. Such topics as the persistence [14, 17, 20, 23], extinction [15, 17–19, 26, 28, 30], the stability of the equilibrium [19–22], the existence and stability of the periodic solution [18, 22, 24, 25, 29], etc. have been extensively investigated, and many excellent results have been obtained.

It brings to our attention that to this day, there is still no scholar to propose and investigate the nonlinear amensalism model. Stimulated by the works of Xiong et al. [1], Chen et al. [14, 15], Lu et al. [23], Lu [24], and Wang [26], in this paper, we propose the following nonlinear amensalism model:

$$ \begin{aligned} &\frac{dN_{1}}{dt}=r_{1}N_{1} \biggl(1- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl( \frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}{dt}=r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), \end{aligned} $$
(1.3)

where \(r_{i}, P_{i}, u, i=1, 2, \alpha_{j}, j=1, 2, 3\), are all positive constants.

As far as system (1.3) is concerned, one interesting issue is:

Find out the influence of the parameter \(\alpha_{i}\), \(i=1, 2, 3\), which reflects the influence of the nonlinearity.

The paper is arranged as follows. We investigate the existence and local stability property of the equilibrium solutions of system (1.3) in the next section. In Sect. 3, by applying the differential inequality theory, we investigate the global stability property of the equilibria. The influence of the parameter \(\alpha_{i}, i=1, 2, 3\), is then discussed in Sect. 4. Some examples together with their numeric simulations are presented in Sect. 5 to show the feasibility of the main results. We end this paper with a brief discussion.

2 Local stability

The system always admits three boundary equilibria \(A(0,0)\), \(B(P_{1}, 0)\), \(C(0, P_{2})\). Also, if \(1> u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}\), the system admits a unique positive equilibrium

$$D\bigl(N_{1}^{*},N_{2}^{*}\bigr)= \biggl(P_{1} \biggl(1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}}, P_{2} \biggr). $$

We shall now investigate the local stability property of the above equilibria.

The variational matrix of the system of Eq. (1.3) is

$$ J(N_{1},N_{2})= \begin{pmatrix} \Delta_{1} & -\frac{r_{1}uN_{1}\alpha_{2}N_{2}^{\alpha _{2}-1}}{P_{1}^{\alpha _{2}}}\\ 0& \Delta_{2} \end{pmatrix} , $$
(2.1)

where

$$\begin{aligned} &\Delta_{1}=r_{1} \biggl(1- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha_{1}} -u \biggl(\frac {N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr) -r_{1} \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}} \alpha_{1}, \\ &\Delta_{2}= r_{2} \biggl(1 - \biggl(\frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr)- r_{2} \biggl(\frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}}\alpha_{3}. \end{aligned}$$

Theorem 2.1

Assume that \(\alpha_{2}\geq1\), then

  1. (1)

    \(A(0,0)\) is unstable;

  2. (2)

    \(B(P_{1}, 0) \) is a saddle point, thus, is unstable;

  3. (3)

    if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), \(C(0, P_{2})\) is a saddle point and consequently unstable; if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\), \(C(0, P_{2})\) is a stable node;

  4. (4)

    if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), \(D(N_{1}^{*},N_{2}^{*})\) is a stable node.

Remark 2.1

If \(\alpha_{i}=1\), \(i=1, 2, 3\), then Theorem 2.1 degenerates to the main result of Xiong et al. [1], hence, we generalize the main result of [1]. Note that the boundary equilibria are independent of \(\alpha_{i}, i=1, 2, 3\), hence, \(\alpha_{i}\), \(i=1, 3\), has no influence on the existence and stability of the boundary equilibria.

Remark 2.2

From (2.1), the second term in \(J(N_{1}, N_{2})\) is \(-\frac{r_{1}uN_{1}\alpha_{2}N_{2}^{\alpha_{2}-1}}{P_{1}^{\alpha _{2}}}\), which means that if \(\alpha_{2}<1\), then at \(N_{2}=0\), the value of this term could not be computed. Hence, for \(0<\alpha_{2}<1\) case, the local stability of the equilibrium \(A(0,0)\) and \(B(P_{1}, 0)\) could not be determined by analyzing the Jacobian matrix.

Proof of Theorem 2.1

(1) From (2.1) we could see that the Jacobian of the system about the equilibrium point \(A(0,0)\) is given by

$$ \begin{pmatrix} r_{1} & 0\\ 0& r_{2} \end{pmatrix}. $$
(2.2)

The eigenvalues of the matrix are \(\lambda_{1}=r_{1}>0, \lambda_{2}=r_{2}>0\). Hence, \(A(0,0)\) is unstable;

(2) The Jacobian of the system about the equilibrium point \(B(P_{1},0)\) is given by

$$ \begin{pmatrix} -r_{1}\alpha_{1}& 0\\ &0&r_{2} \end{pmatrix}. $$
(2.3)

The eigenvalues of the matrix are \(\lambda_{1}=-r_{1}\alpha_{1}<0, \lambda_{2}= r_{2}>0\). So \(B(P_{1}, 0)\) is a saddle point, it is unstable;

(3) The Jacobian of the system about the equilibrium point \(C(0,P_{2})\) is given by

$$ \begin{pmatrix} r_{1} (1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} ) &0\\ 0&-r_{2}\alpha_{3} \end{pmatrix}. $$
(2.4)

The two eigenvalues of the matrix satisfy \(\lambda_{1}=r_{1} (1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} ), \lambda_{2}=-r_{2}\alpha_{3}<0\). Obviously, if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), then \(\lambda _{1}>0\); and consequently, \(C(0, P_{2})\) is a saddle point, it is unstable; if \(1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}}<0\), then \(\lambda_{1}<0\), \(C(0, P_{2})\) is locally stable, it is a stable node.

(4) Note that the positive equilibrium \(D(N_{1}^{*}, N_{2}^{*})\) satisfies

$$ \begin{aligned} &r_{1} \biggl(1- \biggl( \frac{N_{1}^{*}}{P_{1}} \biggr)^{\alpha_{1}}-u \biggl(\frac {N_{2}^{*}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)=0, \\ &r_{2} \biggl(1- \biggl(\frac{N_{2}^{*}}{P_{2}} \biggr)^{\alpha_{3}} \biggr)=0. \end{aligned} $$
(2.5)

Combining with (2.1) and (2.5), we could see that the Jacobian of the system about the equilibrium point \(D(N_{1}^{*},N_{2}^{*})\) is given by

$$ \begin{pmatrix} -r_{1} (\frac{N_{1}^{*}}{P_{1}} )^{\alpha_{1}}\alpha_{1}& -\frac{r_{1}uN_{1}^{*}\alpha_{2}(N_{2}^{*})^{\alpha_{2}-1}}{P_{1}^{\alpha_{2}}}\\ 0&- r_{2} (\frac{N_{2}^{*}}{P_{2}} )^{\alpha_{3}}\alpha_{3} \end{pmatrix} $$
(2.6)

The eigenvalues of the variational matrix (2.6) is the roots \(\lambda_{1}=-r_{1} (\frac{N_{1}^{*}}{P_{1}} )^{\alpha_{1}}\alpha_{1}<0\), \(\lambda_{2}=- r_{2} (\frac{N_{2}^{*}}{P_{2}} )^{\alpha_{3}}\alpha_{3}<0\). Thus, \(D(N_{1}^{*}, N_{2}^{*})\) is locally stable.

The proof of Theorem 2.1 is finished. □

3 Global stability

As was pointed out in the previous section, for the case \(\alpha_{2}<1\), the local stability property of the boundary equilibrium \(A(0,0)\) and \(B(P_{1},0)\) could not be determined by using the Jacobian matrix (2.1). The aim of this section is to try to solve this problem and to further investigate the global stability property of the equilibria of system (1.3).

Lemma 3.1

Consider the system

$$ \frac{dN_{2}}{dt}=r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), $$
(3.1)

where \(r_{2}, P_{2}, \alpha_{3}\) are all positive constants. The unique positive equilibrium \(N_{2}^{*}=P_{2}\) of system (3.1) is globally stable.

Proof

Set

$$ F(N_{2})=r_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), $$
(3.2)

then

$$\begin{aligned} & F(0)=r_{2}>0,\\ & F(+\infty)=-\infty, \end{aligned}$$

hence \(F(N_{2})=0\) has at least one positive solution on the interval \((0,+\infty)\). Also, for \(N_{2}\geq0\), from (3.2)

$$\frac{dF(N_{2})}{dN_{2}}=-\frac{r_{2}a_{3}N_{2}^{\alpha_{3}-1}}{P_{2}^{\alpha_{3}}} < 0. $$

Hence, \(F(N_{2})\) is strictly decreasing on the interval \((0,+\infty)\), therefore, \(F(N_{2})=0\) has at most one positive solution on the interval \((0,+\infty)\). One could easily see that \(N_{2}=P_{2}\) is the solution of (3.2). The above analysis shows that \(N_{2}^{*}=P_{2}\) is the unique positive equilibrium of system (3.1).

On the other hand, from the above analysis, we also have

  1. (i)

    For all \(N_{2}^{*} > N_{2} > 0, F(N_{2}) > 0\).

  2. (ii)

    For all \(N_{2} > N_{2}^{*} > 0, F(N_{2}) < 0\).

Hence, it immediately follows from Theorem 2.1 of [27] that the unique positive equilibrium \(N_{2}^{*}\) of system (3.1) is globally stable. □

The next lemma is a direct corollary of Lemma 2.2 of Chen [16].

Lemma 3.2

If \(a>0, b>0\), and \(\dot{x}\geq x(b-ax^{\alpha})\), where α is some positive constant, when \(t\geq{0}\) and \(x(0)>0\), we have

$$\liminf_{t\rightarrow+\infty} x(t)\geq \biggl(\frac {b}{a} \biggr)^{1/\alpha}. $$

If \(a>0\), \(b>0\), and \(\dot{x}\leq x(b-ax^{\alpha})\), where α is some positive constant, when \(t\geq{0}\) and \(x(0)>0\), we have

$$\limsup_{t\rightarrow+\infty} x(t)\leq \biggl(\frac{b}{a} \biggr)^{1/\alpha}. $$

Theorem 3.1

  1. (a)

    Assume that \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\), then \(C(0, P_{2})\) is globally attractive;

  2. (b)

    Assume that \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\), then \(D(N_{1}^{*},N_{2}^{*})\) is globally attractive.

Proof

(a) Condition \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\) implies that for enough small positive constant \(\varepsilon>0\) (\(\varepsilon<\frac {1}{2}P_{2}\)), the following inequality holds:

$$ 1-u \biggl(\frac{P_{2}-\varepsilon}{P_{1}} \biggr)^{\alpha_{2}}< 0. $$
(3.3)

Let \((N_{1}(t), N_{2}(t))\) be any positive solution of system (1.3). It follows from Lemma 3.1 that

$$ \lim_{t\rightarrow+\infty}N_{2}(t)=N_{2}^{*}=P_{2}. $$
(3.4)

For \(\varepsilon>0\) which satisfies (3.3), from (3.4) there exists an enough large \(T_{1}\) such that

$$ N_{2}(t)\geq P_{2}-\varepsilon\quad \mbox{for all }t\geq T_{1}. $$
(3.5)

Since function \(y=x^{\alpha_{2}}\) is strictly increasing for all \(x>0\), it follows from (3.5) that

$$ \biggl(\frac{N_{2}(t)}{P_{1}} \biggr)^{\alpha_{2}}\geq \biggl( \frac{P_{2}-\varepsilon }{P_{1}} \biggr)^{\alpha_{2}}\quad \mbox{for all } t\geq T_{1}. $$
(3.6)

For \(t\geq T_{1}\), from the right-hand side of (3.5) and the first equation of system (1.3), one has

$$\begin{aligned} \frac{dN_{1}}{dt} \leq&r_{1}N_{1} \biggl(1- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{P_{2}-\varepsilon}{P_{1}} \biggr)^{\alpha_{2}} \biggr) \\ \leq& r_{1} \biggl(1 -u \biggl(\frac{P_{2}-\varepsilon}{P_{1}} \biggr)^{\alpha_{2}} \biggr)N_{1}. \end{aligned}$$
(3.7)

Equation (3.7) together with (3.1) leads to

$$N_{1}(t)\leq N_{1}(T_{1})\exp \biggl\{ r_{1} \biggl(1 -u \biggl(\frac{P_{2}-\varepsilon }{P_{1}} \biggr)^{\alpha_{2}} \biggr) (t-T_{1}) \biggr\} \rightarrow0\quad \mbox{as }t\geq T_{1}. $$

(b) Condition \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}>0\) implies that for enough small positive constant \(\varepsilon>0\) (\(\varepsilon<\frac {1}{2}P_{2}\)), the following inequality holds:

$$ 1-u \biggl(\frac{P_{2}+\varepsilon}{P_{1}} \biggr)^{\alpha_{2}}>0. $$
(3.8)

Let \((N_{1}(t), N_{2}(t))\) be any positive solution of system (1.3), it follows from Lemma 3.1 that

$$ \lim_{t\rightarrow+\infty}N_{2}(t)=N_{2}^{*}=P_{2}. $$
(3.9)

For \(\varepsilon>0\) which satisfies (3.8), from (3.9) there exists an enough large \(T_{2}\) such that

$$ P_{2}-\varepsilon< N_{2}(t)< P_{2}+ \varepsilon\quad \mbox{for all } t\geq T_{2}. $$
(3.10)

For \(t\geq T_{2}\), from the left-hand side of (3.10) and the first equation of system (1.3), one has

$$\begin{aligned} \frac{dN_{1}}{dt} = &r_{1}N_{1} \biggl(1- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr) \\ \leq& r_{1} \biggl(1 -u \biggl(\frac{P_{2}-\varepsilon}{P_{1}} \biggr)^{\alpha _{2}}- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha_{1}} \biggr)N_{1} \\ =& \biggl( r_{1} \biggl(1 -u \biggl(\frac{P_{2}-\varepsilon}{P_{1}} \biggr)^{\alpha _{2}} \biggr)-\frac{r_{1}}{P_{1}^{\alpha_{1}}}N_{1}^{\alpha_{1}} \biggr)N_{1}. \end{aligned}$$
(3.11)

Applying Lemma 3.2 to (3.11), we could obtain

$$ \limsup_{t\rightarrow+\infty}N_{1}(t)\leq P_{1} \biggl(1-u \biggl(\frac {P_{2}-\varepsilon}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}}. $$
(3.12)

Since ε is an arbitrary small positive constant, setting \(\varepsilon\rightarrow0\) leads to

$$ \limsup_{t\rightarrow+\infty}N_{1}(t)\leq P_{1} \biggl(1-u \biggl(\frac {P_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}}. $$
(3.13)

For \(t\geq T_{2}\), from the right-hand side of (3.10) and the first equation of system (1.3), one has

$$\begin{aligned} \frac{dN_{1}}{dt} = &r_{1}N_{1} \biggl(1- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr) \\ \geq& r_{1} \biggl(1 -u \biggl(\frac{P_{2}+\varepsilon}{P_{1}} \biggr)^{\alpha _{2}}- \biggl(\frac{N_{1}}{P_{1}} \biggr)^{\alpha_{1}} \biggr)N_{1} \\ =& \biggl( r_{1} \biggl(1 -u \biggl(\frac{P_{2}+\varepsilon}{P_{1}} \biggr)^{\alpha _{2}} \biggr)-\frac{r_{1}}{P_{1}^{\alpha_{1}}}N_{1}^{\alpha_{1}} \biggr)N_{1}. \end{aligned}$$
(3.14)

Applying Lemma 3.2 to (3.14), we could obtain

$$ \liminf_{t\rightarrow+\infty}N_{1}(t)\geq P_{1} \biggl(1-u \biggl(\frac {P_{2}+\varepsilon}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}}. $$
(3.15)

Since ε is an arbitrary small positive constant, setting \(\varepsilon\rightarrow0\) leads to

$$ \liminf_{t\rightarrow+\infty}N_{1}(t)\geq P_{1} \biggl(1-u \biggl(\frac {P_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}}. $$
(3.16)

Equation (3.13) together with (3.16) leads to

$$ \lim_{t\rightarrow+\infty}N_{1}(t)= P_{1} \biggl(1-u \biggl(\frac {P_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}}=N_{1}^{*}. $$
(3.17)

This ends the proof of Theorem 3.1. □

Remark 3.1

Theorems 2.1 and 3.1 show that if system (1.3) admits the unique positive equilibrium, then the positive equilibrium is globally attractive.

Remark 3.2

Noting that if \(C(0,P_{2})\) or \(D(N_{1}^{*}, N_{2}^{*})\) is globally attractive, then all the solutions with positive initial conditions will finally asymptotically to the equilibrium, which means that the solutions with positive solution could not be asymptotically to \(A(0,0)\) and \(B(P_{1},0)\), thus, \(A(0,0)\) and \(B(P_{1},0)\) is unstable. Theorem 3.1 shows that for almost all the cases (only \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}=0\) could not be determined), \(A(0,0)\) and \(B(P_{1},0)\) are unstable.

Remark 3.3

Xiong et al. [1] proposed system (1.1) and investigated the local stability property of the equilibria. However, they did not give any information about the global stability property of the equilibrium. Thus, Theorem 3.1 can be seen as the supplement of the main result of [1].

4 The influence of the parameter \(\alpha_{i}\)

Now let us consider the influence of the parameter \(\alpha _{i}, i=1, 2, 3\), on the finial density of the two species. We will focus our attention on the positive equilibrium. Noting that

$$D\bigl(N_{1}^{*},N_{2}^{*}\bigr)= \biggl(P_{1} \biggl(1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac{1}{\alpha_{1}}},P_{2} \biggr), $$

\(N_{2}^{*}=P_{2}\) is independent of the parameter \(\alpha_{i}\), hence, \(\alpha _{i}\) has no influence on the final density of the second species. However, \(N_{1}^{*}\) is the function of the parameters \(\alpha_{1} \) and \(\alpha_{2}\), hence, it is necessary to investigate the relationship of \(N_{1}^{*}\) and \(\alpha_{i}\), \(i= 1, 2\). Noting that

$$ \frac{\partial N_{1}^{*}}{\partial\alpha_{1}}=- \frac{P_{1} \ln (1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} ) (1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} )^{\frac{1}{\alpha_{1}}} }{\alpha_{1}^{2}}>0, $$
(4.1)

hence, \(N_{1}^{*}\) is the increasing function of \(\alpha_{1}\).

$$ \frac{\partial N_{1}^{*}}{\partial\alpha_{2}}=- \frac{P_{1} u\ln(\frac {P_{2}}{P_{1}}) (\frac{P_{2}}{P_{1}})^{\alpha_{2}} (1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}} )^{\frac{1}{\alpha_{1}}} }{\alpha_{1} (1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} ) }, $$
(4.2)

thus

  1. (1)

    If \(P_{2} >P_{1} \), then \(\frac{\partial N_{1}^{*}}{\partial\alpha_{2}}<0\), and \(N_{1}^{*}\) is the strictly decreasing function of \(\alpha_{2}\);

  2. (2)

    If \(P_{2} < P_{1}\), then \(\frac{\partial N_{1}^{*}}{\partial\alpha_{2}}>0\), and \(N_{1}^{*}\) is the strictly increasing function of \(\alpha_{2}\).

  3. (3)

    If \(P_{2}=P_{1}\), then \(\frac{\partial N_{1}^{*}}{\partial\alpha_{2}}=0\), and \(N_{1}^{*}\) is independent of \(\alpha_{2}\).

5 Numeric simulations

Example 5.1

Consider the following amensalism system:

$$ \begin{aligned} &\frac{dN_{1}}{dt}=N_{1} \biggl(1- \frac{N_{1}}{2} - \biggl(\frac {N_{2}}{2} \biggr)^{2} \biggr), \\ &\frac{dN_{2}}{dt}=N_{2} (1- N_{2} ). \end{aligned} $$
(5.1)

Here, corresponding to system (1.3), we take \(r_{1}=r_{2}=P_{2}=1\), \(\alpha_{2}=2\), \(P_{1}=2\), \(\alpha_{3}=\alpha_{1}=1\), \(u=1\). One could easily check that

$$1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}}=1-\frac{1}{4}= \frac{3}{4}>0. $$

So, from Theorem 3.1, the unique positive equilibrium \(D(1.5,1)\) is globally attractive. Figure 1 also supports this assertion.

Figure 1
figure 1

Numeric simulations of system (5.1), the initial conditions \((x(0), y(0))=(0.4,5), (1,5), (1, 0.02), (1.5,5), (5,5), (5,0.2)\), and \((5,3)\), respectively

Full size image

Example 5.2

Consider the following amensalism system:

$$ \begin{aligned} &\frac{dN_{1}}{dt}= N_{1} \bigl(1- N_{1} - (N_{2} )^{2} \bigr), \\ &\frac{dN_{2}}{dt}= N_{2} \biggl(1-\frac{N_{2}}{2} \biggr). \end{aligned} $$
(5.2)

Here, corresponding to system (1.3), we take \(r_{1}=r_{2}=P_{1}=1\), \(\alpha_{2}=2\), \(P_{2}=2\), \(\alpha_{3}=\alpha_{1}=1\), \(u=1\). One could easily check that

$$1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}}=1-4=-3< 0. $$

So, from Theorem 3.1, the boundary equilibrium \(C(0,2)\) is globally attractive. Figure 2 also supports this assertion.

Figure 2
figure 2

Numeric simulations of system (5.2), the initial conditions \((x(0), y(0))=(0.4,5), (5,1), (5, 3), (5,5)\), and \((5,0.2)\), respectively

Full size image

Example 5.3

Consider the following amensalism system:

$$ \begin{aligned} &\frac{dN_{1}}{dt}=N_{1} \bigl(1- N_{1}(t) - \bigl(N_{2}(t) \bigr)^{2} \bigr), \\ &\frac{dN_{2}}{dt}=N_{2} \bigl(1-N_{2}(t) \bigr). \end{aligned} $$
(5.3)

Here, corresponding to system (1.3), we take \(r_{1}=r_{2}=P_{1}=1\), \(\alpha_{2}=2\), \(P_{2}=1\), \(\alpha_{3}=\alpha_{1}=1\), \(u=1\). One could easily check that

$$1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}}=1-1=0. $$

Numeric simulation (Fig. 3) shows that in this case the boundary equilibrium \(C(0,1)\) is globally asymptotically stable.

Figure 3
figure 3

Numeric simulations of system (5.3), the initial conditions \((x(0), y(0))=(1,5), (0.1,0.002), (1.5, 5), (5,5), (5,3)\), and \((5,0.2)\), respectively

Full size image

Example 5.4

Consider the function

$$N_{1}^{*}= P_{1} \biggl(1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr)^{\frac {1}{\alpha_{1}}}. $$
  1. (1)

    Let us take \(P_{1}=1, P_{2}=2, \alpha_{2}=1, u=\frac{1}{4}\). From Sect. 4 we know that \(N_{1}^{*}\) is the strictly increasing function of \(\alpha_{1}\). Figure 4 also supports this assertion;

    Figure 4
    figure 4

    The relationship of \(N_{1}^{*}\) and \(\alpha_{1}\), here we choose \(P_{1}=1\), \(P2=2\), \(\alpha_{2}=1\)

    Full size image
  2. (2)

    Let us take \(P_{1}=1, P_{2}=2, \alpha_{1}=1, u=\frac{1}{4}\). From Sect. 4 we know that \(N_{1}^{*}\) is the strictly decreasing function of \(\alpha_{2}\). Figure 5 also supports this assertion;

    Figure 5
    figure 5

    The relationship of \(N_{1}^{*}\) and \(\alpha_{2}\), here we choose \(P_{1}=1\), \(P_{2}=2\), \(\alpha_{1}=1\)

    Full size image
  3. (3)

    Let us take \(P_{1}=1, P_{2}=\frac{1}{2}, \alpha_{1}=1, u=\frac{1}{4}\). From Sect. 4 we know that \(N_{1}^{*}\) is the strictly increasing function of \(\alpha_{2}\). Figure 6 also supports this assertion;

    Figure 6
    figure 6

    The relationship of \(N_{1}^{*}\) and \(\alpha_{2}\), here we choose \(P_{1}=1\), \(P_{2}=\frac{1}{2}\)

    Full size image
  4. (4)

    Let us take \(P_{1}=1, P_{2}=1, \alpha_{1}=1, u=\frac{1}{4}\). From Sect. 4 we know that \(N_{1}^{*}\) is independent of \(\alpha_{2}\). Figure 7 also supports this assertion.

    Figure 7
    figure 7

    The relationship of \(N_{1}^{*}\) and \(\alpha_{2}\), here we choose \(P_{1}=P_{2}=1\)

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6 Discussion

Stimulated by the works of Xiong et al. [1] and Chen et al. [14–19], in this paper, we propose the nonlinear amensalism model (1.3).

We first investigated the local stability property of the equilibria, and we found that, by introducing the nonlinear term, the situation became complicated, and only under the assumption \(\alpha _{2}\geq1\) could we give the local stability result of \(A(0,0)\) and \(B(P_{1},0)\). To overcome this difficulty, we used the differential inequality theory, and finally proved the following results: if the positive equilibrium exists, then it is globally attractive.

We also investigated the relationship among \(N_{1}^{*}\) and \(\alpha _{i}\), and found that \(\alpha_{2}\) and the ratio of \(\frac{P_{2}}{P_{1}}\) play most important roles on the final density of the first species.

We mention here that we did not discuss the degenerate case

$$1-u \biggl(\frac{P_{2}}{P_{1}} \biggr)^{\alpha_{2}}=0. $$

However, numeric simulation (Fig. 3) shows that in this case, the boundary equilibrium \(C(0, P2)\) is globally asymptotically stable. For fixed \(\alpha_{i}, i=1, 2, 3\), we could put forward some progress, such as numeric simulation, on this direction. However, we still have difficulty obtaining the analysis result, we leave this for the future investigation.