1 Introduction

In the ecological research, an isolated island is a good laboratory to study the evolution of ecosystems, because there is no external distortion factor. In [2], Basener and Ross developed one mathematical model which can describe the dynamics of the evolution of human population in Easter island. The model is the following two-dimensional nonlinear plane system

$$\begin{aligned} \left\{ \begin{array}{l} x'(t)=\nu x(t)\big (1-\frac{x(t)}{k}\big )-hy(t),\\ y'(t)=\theta y(t)\big (1-\frac{y(t)}{x(t)}\big ), \end{array}\right. \end{aligned}$$
(1.1)

where x(t) is the amount of resources of the island at time t, y(t) is the amount of the human population of the island at time t, \(\nu \) is the growth rate of the recourses, k is the capacity of the island to carry resources, h is the harvesting constant and \(\theta \) is the growth rate of the population. More details about the description of the above system can be found in [2], in which the numerical solutions of the system (1.1) were obtained and some qualitative analysis of the general behavior of the solutions both in finite and infinite time were given. In [20], Nucci and Sanchini applied the Lie group theory to the system (1.1) and proved that the system can be integrated by quadrature for some values of the ecological parameters. Moreover, it is obvious that the system (1.1) is of predator–prey type. Compared with other predator–prey systems, the system (1.1) presents all sorts of dynamical behaviours, such as the extinction at a finite time. Therefore, during the past few years, the system (1.1) has been acknowledged as a plausible model for evolution of population in ancient civilizations, and some generalizations and variants have been done and studied, see [3,4,5, 16] and the references therein.

But in the real world, due to the influences of many different factors such as the variations of the environmental conditions and human living habits, the ecological parameters in constant form is no longer valid. Therefore, many scholars have paid attention to the Basener–Ross system with time-dependent ecological parameters, see [1, 7, 13, 15] and the references therein. For example, by some integrability criteria which were developed in [6], Güngör and Torres in [15] studied the integrability of the system (1.1) with all coefficients are time-dependent. Recently, Cheng and Cui [7] used Leray–Schauder alternative principle, Manásevich–Mawhin continuation theorem and fixed point theorem in cones to study the existence of positive periodic solutions of the system (1.1) with the coefficients \(\nu \) and k are time-dependent.

As far as we know, there are no analytic results available about the stability of the Basener–Ross system up to now. Motivated by this, in this paper, we will study the stability problem for the following system

$$\begin{aligned} \left\{ \begin{array}{l} x'(t)=\nu (t)x(t)\big (1-\frac{x(t)}{k}\big )-hy(t),\\ y'(t)=\theta y(t)\big (1-\frac{y(t)}{x(t)}\big ), \end{array}\right. \end{aligned}$$
(1.2)

where \(\nu :\mathbb {R}\rightarrow (0,+\infty )\) is a T-periodic function, \(T>0\).

From the second equation of the system (1.2), we can find

$$\begin{aligned} x(t)=\frac{\theta y^{2}(t)}{\theta y(t)-y'(t)}. \end{aligned}$$
(1.3)

Then elimination of x in the first equation of the system (1.2) leads to the following damped equation

$$\begin{aligned} y''(t)+p(t)y'(t)-q(t)y(t)=\alpha \frac{(y'(t))^2}{y(t)}-\beta (t)y^{2}(t), \end{aligned}$$
(1.4)

where

$$\begin{aligned} p(t)=\theta +\nu (t)-2h,~~q(t)=\theta (\nu (t)-h), \\ \alpha =2-\frac{h}{\theta },~~ \beta (t)=\frac{\theta \nu (t)}{k}. \end{aligned}$$

Take the change of variable \(y=u^{\gamma }\), \(\gamma =\frac{1}{1-\alpha }\), we get that the Eq. (1.4) is equivalent to

$$\begin{aligned} u''(t)+p(t)u'(t)+\frac{\beta (t)}{\gamma }u^{\gamma +1}(t)-\frac{q(t)}{\gamma }u(t)=0. \end{aligned}$$
(1.5)

By the definitions of \(\alpha \) and \(\gamma \), it is easy to see that \(\gamma \ne 0\) and \(\gamma \ne -1\).

Notice that the positive periodic solutions of the Eq. (1.5) corresponds to the positive periodic solutions of the system (1.2). In this paper, we first study the existence of the positive periodic solutions for the Eq. (1.5) and obtain the explicit bounds. Then by using a stability criterion of damped equations which was obtained in [18] by using the third order approximation method [8], we will prove that the positive periodic solutions of the Eq. (1.5) are of twist type. Such twist periodic solutions are nonlinear stable in the sense of Lyapunov. Finally, we analyze the stability of the positive periodic solutions of the system (1.2).

The third order approximation method was developed by Ortega [21] and Zhang [27] for the Lagrangian equations. In recent years, some progress has been made on this topic, we just refer the reader to [9, 12, 19, 25, 28] for differential equations without a singularity and [10, 11, 17, 18, 24, 26] for singular differential equations. Obviously, (1.5) is a damped equation. In general case, the method of third order approximation is appropriate only for the conservative equations, and might not be applicable for the damped equations, which are usually dissipative. But, as shown in [8], if the average of the damped term in one period is zero, then such method can be applied to the damped differential equations. Therefore, for the Eq. (1.5), throughout this paper, we assume that

$$\begin{aligned} \bar{\nu }=\frac{1}{T}\int ^{T}_{0}\nu (t)\textrm{d}t=2h-\theta , \end{aligned}$$
(1.6)

which is equivalent to \(\bar{p}=0\). Moreover, according to the natural laws, the rate of human access to resources is not greater than the rate of growth of resources, otherwise there will be a catastrophe of human. Hence, throughout this paper, we also assume that \(\nu (t)>h\), \(t\in \mathbb {R}\). Then it follows from (1.6) that \(h>\theta .\) Therefore \(\alpha <1\) and \(\gamma >0\).

Finally, it is worth noting that the periodic solutions obtained in this paper are of twist type. As a consequence of Moser’s invariant curve theorem [22], a solution of the twist type is Lyapunov stable. Moreover, some complicated dynamics may appear around the twist type solutions, including the stability islands, the invariant tori, the chaotic regions and the existence of subharmonic solutions and quasiperiodic solutions, as a consequence of the Poincáre-Birkhoff and Moser theorems. Summing up, the neighborhood of a solution of twist type presents the typical KAM scenario. These may be our future research topics. Moreover, the stability results of the Basener–Ross system presented in this paper are the first ones available in the literature, and are a kind of global results.

The rest of this paper is organized as follows. In Sect. 2, we state some preliminary results. Our main results will be stated and proved in Sect. 3.

2 Preliminaries

Throughout this paper, for a given T-periodic function a, we denote

$$\begin{aligned}a^{*}=\sup _{t\in [0,T]}a(t),~~a_{_{*}}=\inf _{t\in [0,T]}a(t).\end{aligned}$$

Consider the following damped equation

$$\begin{aligned} u''+p(t)u'+g(t,u)=0, \end{aligned}$$
(2.1)

where \(p\in C(\mathbb {R}/T\mathbb {Z})\) with \(\bar{p}=0\), \(g\in C^{0,4}(\mathbb {R}/T\mathbb {Z}\times \mathbb {R},\mathbb {R})\). Then the third order approximation of (2.1) is

$$\begin{aligned} u''+p(t)u'+ a(t)u+ b(t) u^2 + c(t) u^3 + o(u^3)=0, \end{aligned}$$
(2.2)

where

$$\begin{aligned} a(t)=g_{u}(t,\phi (t)),\quad b(t)=\frac{1}{2}g_{uu}(t,\phi (t)),\quad c(t)=\frac{1}{6} g_{uuu}(t,\phi (t)), \end{aligned}$$

and \(\phi \) is a T-periodic solution of (2.1).

Based on the method of third order approximation for the damped equations [8], Liang and Liao in [18] obtained the following stability criterion.

Lemma 1

Assume that \(\phi \) is a T-periodic solution of (2.1) such that

  1. (I)  

    \(0<a_{_{*}}\le a^{*}\le \frac{1}{\sigma (2p)^{*}}(\frac{\pi }{2\tau (T)})^{2}\);

  2. (II) 

    \(c_{_{*}}>0\);

  3. (III)

    \(10a_{_{*}}^{\frac{3}{2}}|b|_{_{*}}^{2}(\sigma (2p)_{_{*}})^{\frac{7}{2}}> 9(a^{*})^{\frac{5}{2}}c^{*}(\sigma (2p)^{*})^{\frac{7}{2}}\).

Then \(\phi \) is of twist type.

3 Main results

Set

$$\begin{aligned} \sigma (p)(t)=e^{\int _{0}^{t}p(s)\textrm{d}s}. \end{aligned}$$

By (1.6), we know that \(\sigma (p)\) is T-periodic. Multiplying both sides of (1.5) by \(\sigma (p)(t)\), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}\big (\sigma (p)(t)u'(t)\big )+\frac{\sigma (p)(t)\beta (t)}{\gamma }u(t)\bigg (u^{\gamma }(t)- \frac{q(t)}{\beta (t)}\bigg )=0. \end{aligned}$$

Under the change of time

$$\begin{aligned} s=\tau (t)=\int _{0}^{t}\sigma (-p)(s)\textrm{d}s, \end{aligned}$$

the above equation is then transformed into

$$\begin{aligned} \ddot{\rho }(s)+\frac{\tilde{\sigma }(2p)(s)\tilde{\beta }(s)}{\gamma }\rho (s)\bigg (\rho ^{\gamma }(s)- \frac{\tilde{q}(s)}{\tilde{\beta }(s)}\bigg )=0, \end{aligned}$$
(3.1)

where

$$\begin{aligned} \rho (s)=u(\tau ^{-1}(s)), ~~\tilde{\sigma }(2p)(s)=\sigma (2p)(\tau ^{-1}(s)), \\ \tilde{\beta }(s)=\beta (\tau ^{-1}(s)),~~\tilde{q}(s)=q(\tau ^{-1}(s)). \end{aligned}$$

Note that \(\tilde{\sigma }(2p)(s)\), \(\tilde{\beta }(s)\) and \(\tilde{q}(s)\) are periodic with period \(\tilde{T}\) with

$$\begin{aligned} \tilde{T}=\tau (T)=\int _{0}^{T}\sigma (-p)(s)\textrm{d}s. \end{aligned}$$

Firstly, by using the method of upper and lower solutions on the reversed order (see [14, 23]), we obtain the following existence result.

Theorem 1

Assume that \(\nu ^{*}>\frac{\gamma +1}{\gamma }h\) and

$$\begin{aligned} \theta (\nu ^{*}-h)e^{2(\nu ^{*}-2h+\theta )T}\le \frac{\pi ^{2}}{\tau ^{2}(T)}. \end{aligned}$$
(3.2)

Then Eq. (1.5) has a positive T-periodic solution u such that

$$\begin{aligned} \left( \frac{k(\nu _{_{_{*}}}-h)}{\nu _{_{_{*}}}}\right) ^{\frac{1}{\gamma }}<u(t) <\left( \frac{k(\nu ^{*}-h)}{\nu ^{*}}\right) ^{\frac{1}{\gamma }},~~t\in [0,T]. \end{aligned}$$
(3.3)

Proof

Let

$$\begin{aligned} g(s,\rho )=\frac{\tilde{\sigma }(2p)(s)\tilde{\beta }(s)}{\gamma }\rho (s)\bigg (\rho ^{\gamma }(s)- \frac{\tilde{q}(s)}{\tilde{\beta }(s)}\bigg ), \\\varphi (s)=\bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}\bigg ]^{\frac{1}{\gamma }}\end{aligned}$$

and

$$\begin{aligned}\psi (s)=\bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )_{_{*}}\bigg ]^{\frac{1}{\gamma }}.\end{aligned}$$

It is noticeable that

$$\begin{aligned} \varphi ''+g(s,\varphi )=\,&\frac{\tilde{\sigma }(2p)(s)\tilde{\beta }(s)}{\gamma }\varphi \bigg (\varphi ^{\gamma }- \frac{\tilde{q}(s)}{\tilde{\beta }(s)}\bigg )\\ =\,&\frac{\tilde{\sigma }(2p)(s)\tilde{\beta }(s)}{\gamma } \bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}\bigg ]^{\frac{1}{\gamma }} \bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}- \frac{\tilde{q}(s)}{\tilde{\beta }(s)}\bigg ]\\ \ge&0 \end{aligned}$$

and

$$\begin{aligned} \psi ''+g(s,\psi )=\,&\frac{\tilde{\sigma }(2p)(s)\tilde{\beta }(s)}{\gamma }\psi \bigg (\psi ^{\gamma }- \frac{\tilde{q}(s)}{\tilde{\beta }(s)}\bigg )\\ =\,&\frac{\tilde{\sigma }(2p)(s)\tilde{\beta }(s)}{\gamma } \bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )_{_{*}}\bigg ]^{\frac{1}{\gamma }} \bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )_{_{*}}- \frac{\tilde{q}(s)}{\tilde{\beta }(s)}\bigg ]\\ \le \,&0, \end{aligned}$$

which implies that \(\varphi (s) \) and \(\psi (s) \) is a constant strict lower and upper solution of the periodic problem of the Eq. (1.5) on the reserved order \(\varphi (s)>\psi (s)\), respectively.

Define

$$\begin{aligned} \Omega =\{(s,\rho )\in [0,\tilde{T}]\times (0,+\infty )\big |~\psi \le \rho \le \varphi \}. \end{aligned}$$

By the method of upper and lower solutions on the reversed order [14, 23], we know that the Eq. (3.1) has at least one \(\tilde{T}\)-periodic solution \(\rho \) such that

$$\begin{aligned}\psi<\rho (s)<\varphi , \forall s\in \mathbb {R},\end{aligned}$$

if, for every \(\rho \in \Omega \), we have

$$\begin{aligned} g_{\rho }(s,\rho )=\,&\frac{\tilde{\sigma }(2p)(s)}{\gamma }\bigg ((\gamma +1)\tilde{\beta }(s) \rho ^{\gamma }(s)-\tilde{q}(s)\bigg )\nonumber \\ \le \,&\frac{\tilde{\sigma }(2p)(s)}{\gamma }\bigg ((\gamma +1)\tilde{\beta }(s) \bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}-\tilde{q}(s)\bigg )\nonumber \\ =\,&\frac{\theta \tilde{\sigma }(2p)(s)}{\gamma }\bigg [\bigg (\frac{(\gamma +1)}{k} \bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}-1\bigg )\tilde{\nu }(s)+h\bigg ]\nonumber \\ \le \,&\frac{\pi ^{2}}{\tilde{T}^{2}}. \end{aligned}$$
(3.4)

Obviously, the \(\tilde{T}\)-periodic solution \(\rho (s)\) of (3.1) corresponds to the T-periodic solution \(u(t) = \rho (\tau (t))\) of (1.5), and

$$\begin{aligned} \bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )_{_{*}}\bigg ]^{\frac{1}{\gamma }} =\psi<u(t) = \rho (\tau (t))<\varphi =\bigg [\bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}\bigg ]^{\frac{1}{\gamma }}.\end{aligned}$$
(3.5)

By the facts \(\tau ^{-1}(s)\) is increasing in s, \(\tau ^{-1}(0)=0\) and \(\tau ^{-1}(\tilde{T})=T\) that

and

Then, we have

$$\begin{aligned} \bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}=\bigg (\frac{q}{\beta }\bigg )^{*} =\bigg (\frac{k(\nu -h)}{\nu }\bigg )^{*}=\frac{k(\nu ^{*}-h)}{\nu ^{*}} \end{aligned}$$

and

$$\begin{aligned} \bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )_{*}=\bigg (\frac{q}{\beta }\bigg )_{*} =\bigg (\frac{k(\nu -h)}{\nu }\bigg )_{*}=\frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}, \end{aligned}$$

which leads to that (3.5) is equivalent to (3.3). Moreover, by \(\nu ^{*}>\frac{\gamma +1}{\gamma }h\), we have

$$\begin{aligned} \frac{(\gamma +1)(\nu ^{*}-h)}{\nu ^{*}} -1>0, \end{aligned}$$

then we have

$$\begin{aligned}&\frac{\theta \tilde{\sigma }(2p)(s)}{\gamma } \bigg [\bigg (\frac{(\gamma +1)}{k} \bigg (\frac{\tilde{q}}{\tilde{\beta }}\bigg )^{*}-1\bigg )\tilde{\nu }(s)+h\bigg ]\\&\quad =\frac{\theta \tilde{\sigma }(2p)(s)}{\gamma }\bigg [\bigg (\frac{(\gamma +1)(\nu ^{*}-h)}{\nu ^{*}} -1\bigg )\tilde{\nu }(s)+h\bigg ]\\&\quad \le \frac{\theta \tilde{\sigma }(2p)^{*}}{\gamma }\bigg [\bigg (\frac{(\gamma +1)(\nu ^{*}-h)}{\nu ^{*}} -1\bigg )\tilde{\nu }^*+h\bigg ]\\&\quad = \frac{\theta \sigma (2p)^{*}}{\gamma }\bigg [\bigg (\frac{(\gamma +1)(\nu ^{*}-h)}{\nu ^{*}} -1\bigg )\nu ^*+h\bigg ]\\&\quad \le \theta (\nu ^{*}-h)e^{2(\nu ^{*}-2h+\theta )T} , \end{aligned}$$

which implies that if (3.2) holds, then (3.4) is satisfied.\(\square \)

Now, we will apply Lemma 1 to prove that the T-periodic solution u of (1.5) obtained in Theorem 1 is of twist type.

Theorem 2

Assume that \(\nu _{_{*}}>\frac{\gamma +1}{\gamma }h\) and

$$\begin{aligned} \theta (\nu ^{*}-h)e^{2(\nu ^{*}-2h+\theta )T}\le \bigg (\frac{\pi }{2\tau (T)}\bigg )^{2}. \end{aligned}$$
(3.6)

Then we have the following conclusions:

  1. (1)

    Suppose further that \(1<\gamma \le 2\) and

    $$\begin{aligned} \frac{G_{1}(\nu _{_{*}})}{G_{1}(\nu ^{*})}>\frac{9(\gamma -1)}{10(\gamma +1)}, \end{aligned}$$
    (3.7)

    then the T-periodic solution u of (1.5) obtained in Theorem 1 is of twist type, where

    $$\begin{aligned} G_{1}(y)=y\big (y-h \big )^{\frac{5}{2}} e^{7Ty}; \end{aligned}$$
  2. (2)

    Suppose further that \(\gamma >2\) and

    $$\begin{aligned} \frac{G(\nu _{_{*}})}{G(\nu ^{*})}>\frac{9(\gamma -1)}{10(\gamma +1)}, \end{aligned}$$
    (3.8)

    then the T-periodic solution u of (1.5) obtained in Theorem 1 is of twist type, where

    $$\begin{aligned} G(x)=x^{\frac{2}{\gamma }} (x-h)^{\frac{7\gamma -4}{2\gamma }}e^{7Tx}. \end{aligned}$$

Proof

First, we calculate the coefficients in (2.2) for the Eq. (1.5) gives

$$\begin{aligned}{} & {} a(t)=\frac{1}{\gamma }\bigg [(\gamma +1)\beta (t)u^{\gamma }(t)-q(t)\bigg ], \\{} & {} b(t)=(\gamma +1)\beta (t)u^{\gamma -1}(t), \\{} & {} c(t)=(\gamma ^{2}-1)\beta (t)u^{\gamma -2}(t). \end{aligned}$$

Using the estimate (3.3), we get

$$\begin{aligned} a(t)\ge&\frac{1}{\gamma }\bigg [\beta (t)(\gamma +1)\frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}} -q(t)\bigg ]\nonumber \\ =\,&\frac{\theta }{\gamma }\bigg [\big (\frac{(\gamma +1)(\nu _{_{*}}-h)}{\nu _{_{*}}}-1\big )\nu (t) +h\bigg ], \end{aligned}$$
(3.9)
$$\begin{aligned} a(t)\le \,&\frac{1}{\gamma }\bigg [(\gamma +1)\beta (t)\frac{k(\nu ^{*}-h)}{\nu ^{*}}-q(t)\bigg ]\nonumber \\ =\,&\frac{\theta }{\gamma }\bigg [\big (\frac{(\gamma +1)(\nu ^{*}-h)}{\nu ^{*}}-1\big )\nu (t) +h\bigg ], \end{aligned}$$
(3.10)
$$\begin{aligned} b(t)&\ge (\gamma +1)\beta (t)\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -1}{\gamma }}. \end{aligned}$$
(3.11)

By (3.6), (3.9), (3.10) and \(\displaystyle \nu _{_{*}}>\frac{\gamma +1}{\gamma }h\), we have

$$\begin{aligned} a_{_{*}}\ge&\frac{\theta }{\gamma }\bigg [\big (\frac{(\gamma +1)(\nu _{_{*}}-h)}{\nu _{_{*}}}-1\big )\nu _{_{*}} +h\bigg ]\nonumber \\ =\,&\theta (\nu _{_{*}}-h)>0 \end{aligned}$$
(3.12)

and

$$\begin{aligned} a^{*}\le \,&\frac{\theta }{\gamma }\bigg [\big (\frac{(\gamma +1)(\nu ^{*}-h)}{\nu ^{*}}-1\big )\nu ^{*} +h\bigg ]\nonumber \\ =\,&\theta (\nu ^{*}-h)\\ \le \,&\frac{1}{\sigma (2p)^{*}}\bigg (\frac{\pi }{2\tau (T)}\bigg )^{2}\nonumber , \end{aligned}$$
(3.13)

so the condition (I) of Lemma 1 is satisfied.

By (3.11), we have

$$\begin{aligned} |b|_{_{*}}\ge&(\gamma +1)\beta _{_{*}}\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -1}{\gamma }}\nonumber \\ =\,&(\gamma +1)\frac{\theta \nu _{_{*}}}{k}\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -1}{\gamma }}\nonumber \\ =\,&\theta (\gamma +1)k^{-\frac{1}{\gamma }}\nu _{_{*}}^{\frac{1}{\gamma }}(\nu _{_{*}}-h)^{\frac{\gamma -1}{\gamma }}. \end{aligned}$$
(3.14)

By (3.12) and (3.14), we have

$$\begin{aligned} 10a_{_{*}}^{\frac{3}{2}}|b|_{_{*}}^{2}(\sigma (2p)_{_{*}})^{\frac{7}{2}} \ge&10(\gamma +1)^{2}\theta ^{\frac{7}{2}}k^{\frac{-2}{\gamma }}\nu _{_{*}}^{\frac{2}{\gamma }} \left( \nu _{_{*}}-h\right) ^{\frac{7\gamma -4}{2\gamma }} e^{7(\nu _{_{*}}-2h+\theta )T}\nonumber \\ =\,&10(\gamma +1)^{2}\theta ^{\frac{7}{2}}k^{\frac{-2}{\gamma }}e^{7(-2h+\theta )T}G(\nu _{_{*}}). \end{aligned}$$
(3.15)

If \(1<\gamma \le 2\), by (3.3), we have

$$\begin{aligned} c(t)\le (\gamma ^{2}-1)\beta (t)\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -2}{\gamma }} \end{aligned}$$

and

$$\begin{aligned} c(t)\ge (\gamma ^{2}-1)\beta (t)\left( \frac{k(\nu ^{*}-h)}{\nu ^{*}}\right) ^{\frac{\gamma -2}{\gamma }}. \end{aligned}$$

Then

$$\begin{aligned} c^{*}\le \,&(\gamma ^{2}-1)\beta ^{*}\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&(\gamma ^{2}-1)\frac{d\nu ^{*}}{k}\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&\theta (\gamma ^{2}-1)k^{-\frac{2}{\gamma }}\nu ^{*}\nu _{_{*}}^{\frac{2-\gamma }{\gamma }}(\nu _{_{*}}-h)^{\frac{\gamma -2}{\gamma }} \end{aligned}$$

and

$$\begin{aligned} c_{_{*}}\ge&(\gamma ^{2}-1)\beta _{_{*}}\left( \frac{k(\nu ^{*}-h)}{\nu {*}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&(\gamma ^{2}-1)\frac{\theta \nu _{_{*}}}{k}\left( \frac{k(\nu ^{*}-h)}{\nu ^{*}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&\theta (\gamma ^{2}-1)k^{-\frac{2}{\gamma }}\nu _{_{*}}(\nu ^{*})^{\frac{2-\gamma }{\gamma }}(\nu ^{*}-h)^{\frac{\gamma -2}{\gamma }}>0. \end{aligned}$$

Therefore, the condition (II) of Lemma 1 is satisfied. By direct computations, we have

$$\begin{aligned} 9(a^{*})^{\frac{5}{2}}&c^{*}(\sigma (2p)^{*})^{\frac{7}{2}}\nonumber \\ \le \,&9\theta ^{\frac{7}{2}}(\gamma ^{2}-1)k^{\frac{-2}{\gamma }}\left( \nu _{_{*}}\right) ^{\frac{2-\gamma }{\gamma }} \left( \nu _{_{*}}-h\right) ^{\frac{\gamma -2}{\gamma }}\nu ^{*}\big (\nu ^{*}-h \big )^{\frac{5}{2}}e^{7(\nu ^{*}-2h+\theta )T}\nonumber \\ =\,&9(\gamma ^{2}-1)\theta ^{\frac{7}{2}}k^{\frac{-2}{\gamma }}e^{7(-2h+\theta )T}G_{2}(\nu _{_{*}},\nu ^{*}), \end{aligned}$$

where

$$\begin{aligned} G_{2}(x,y)=x ^{\frac{2-\gamma }{\gamma }}\left( x-h\right) ^{\frac{\gamma -2}{\gamma }}G_{1}(y). \end{aligned}$$

Using (3.15) and the above facts, and by direct computations, we can get that the condition (III) of Lemma 1 is satisfied if (3.7) holds.

Up to now, all conditions of Lemma 1 are satisfied. Then by Lemma 1, we know that the T-periodic solution u of (1.5) obtained in Theorem 1 is of twist type if \(1<\gamma \le 2\) and (3.7) hold.

If \(\gamma >2\), we have

$$\begin{aligned} c(t)\ge (\gamma ^{2}-1)\beta (t)\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -2}{\gamma }} \end{aligned}$$

and

$$\begin{aligned} c(t)\le (\gamma ^{2}-1)\beta (t)\left( \frac{k(\nu ^{*}-h)}{\nu ^{*}}\right) ^{\frac{\gamma -2}{\gamma }}. \end{aligned}$$

Then we have

$$\begin{aligned} c_{_{*}}\ge&(\gamma ^{2}-1)\beta _{_{*}}\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&(\gamma ^{2}-1)\frac{\theta \nu _{_{*}}}{k}\left( \frac{k(\nu _{_{*}}-h)}{\nu _{_{*}}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&\theta (\gamma ^{2}-1)k^{-\frac{2}{\gamma }}\nu _{_{*}}^{\frac{2}{\gamma }}(\nu _{_{*}}-h)^{\frac{\gamma -2}{\gamma }}>0 \end{aligned}$$

and

$$\begin{aligned} c^{*}\le \,&(\gamma ^{2}-1)\beta ^{*}\left( \frac{k(\nu ^{*}-h)}{\nu ^{*}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&(\gamma ^{2}-1)\frac{\theta \nu ^{*}}{k}\left( \frac{k(\nu ^{*}-h)}{\nu ^{*}}\right) ^{\frac{\gamma -2}{\gamma }}\nonumber \\ =\,&\theta (\gamma ^{2}-1)k^{-\frac{2}{\gamma }}(\nu ^{*})^{\frac{2}{\gamma }}(\nu ^{*}-h)^{\frac{\gamma -2}{\gamma }}. \end{aligned}$$
(3.16)

Hence, the condition (II) of Lemma 1 is satisfied. By (3.13) and (3.16), we have

$$\begin{aligned} 9(a^{*})^{\frac{5}{2}}c^{*}(\sigma (2p)^{*})^{\frac{7}{2}}\le \,&9\theta ^{\frac{7}{2}}(\gamma ^{2}-1)k^{\frac{-2}{\gamma }}(\nu ^{*})^{\frac{2}{\gamma }} \left( \nu ^{*}-h\right) ^{\frac{7\gamma -4}{2\gamma }}e^{7(\nu ^{*}-2h+\theta )T}\nonumber \\ =\,&9\theta ^{\frac{7}{2}}(\gamma ^{2}-1)k^{\frac{-2}{\gamma }}e^{7(-2h+\theta )T}G(\nu ^{*}). \end{aligned}$$

Using (3.15) and the above facts, and by direct computations, we can get that the condition (III) of Lemma 1 is satisfied if (3.8) holds.

Analogously, by Lemma 1 again, we know that the T-periodic solution u of (1.5) obtained in Theorem 1 is of twist type if \(\gamma >2\) and (3.8) hold. \(\square \)

By calculating the limit, we have

$$\begin{aligned}{} & {} \lim \limits _{\nu ^{*}\rightarrow \nu _{_{*}}}\frac{G(\nu _{_{*}})}{G(\nu ^{*})}= \lim \limits _{\nu ^{*}\rightarrow \nu _{_{*}}}\frac{G_{1}(\nu _{_{*}})}{G_{1}(\nu ^{*})} \\{} & {} \quad =1>\frac{9(\gamma -1)}{10(\gamma +1)},~~\gamma >1. \end{aligned}$$

Therefore, by continuity, we can obtain the following result.

Corollary 1

Assume that \(\gamma >1\), \(\nu _{_{*}}>\frac{\gamma +1}{\gamma }h\) and (3.6) hold. Then there exists a constant \(\delta >0\) such that the T-periodic solution u of (1.5) obtained in Theorem 1 is of twist type if \(\nu ^{*}-\nu _{_{*}}<\delta \).

Finally, we prove that the system (1.2) has a stable positive T-periodic solution (x, y).

Theorem 3

Assume that (3.6) holds. Then the system (1.2) has a stable T-periodic solution if (3.7) holds for \(1<\gamma \le 2\) or (3.8) holds for \(\gamma >2\).

Proof

In Theorem 2, we have shown that the Eq. (1.5) has a stable positive T-periodic solution \(u(t)=u(t,u(0),u'(0))\). Therefore, u(t) and \(u'(t)\) are bounded. Suppose that \(u_{1}(t)=u_{1}(t,u_{1}(0),u_{1}'(0))\) be a new solution of the Eq. (1.5). According to the definition of Lyapunov stability, we know that for any given \(\varepsilon >0\), there exists a \(\delta _{1}>0\) such that

$$\begin{aligned} |u(0)-u_{1}(0)|+|u'(0)-u_{1}'(0)|<\delta _{1}, \end{aligned}$$
(3.17)

then

$$\begin{aligned} |u(t)-u_{1}(t)|+|u'(t)-u'_{1}(t)|<\frac{\varepsilon }{M}, \end{aligned}$$
(3.18)

here

$$\begin{aligned} M=\max \limits _{t\in \mathbb {R}}|F(u(t),u_{1}(t))| \end{aligned}$$

and F is a continuous function \(F\in {\mathbb C}(\mathbb {R}_{+}^{2},\mathbb {R})\) such that

$$\begin{aligned} |u^{\gamma }-v^{\gamma }|\le |F(u,v)||u-v|,~~\forall u,v\in \mathbb {R}_{+}. \end{aligned}$$

By Theorem 1, we know that the system (1.2) has a positive T-periodic solution (x, y) with \(y=u^{\gamma }\) and (1.3) hold. Since \(y(0)=u^{\gamma }(0)\), then it follows from (3.17) that there exists a \(\delta _{2}>0\) such that

$$\begin{aligned} |y(0)-y_{1}(0)|<\delta _{2}, \end{aligned}$$

and by (3.18), we have

$$\begin{aligned} |y(t)-y_{1}(t)|=\,&|u^{\gamma }(t)-u_{1}^{\gamma }(t)|\\ \le \,&|F(u(t),u_{1}(t))||u(t)-u_{1}(t)|\\ <&\varepsilon , \end{aligned}$$

which means that the positive T-periodic solution y is stable. Analogously, we can also prove that the positive T-periodic solution x is stable. \(\square \)