Periodic solution and dynamical analysis for a delayed food chain model with general functional response and discontinuous harvesting

Abstract

In this paper, a delayed food chain model with general functional response and discontinuous harvesting is considered. Under some reasonable assumptions, one proves the positivity and boundedness of the solutions. Moreover, the sufficient conditions for the existence of the periodic solution are found by using differential inclusion theory and topological degree theory. Most interestingly, the globally asymptotically stable of the periodic solution is studied by using a selected Lyapunov function and the sufficient conditions for it are given. Finally, one gives the numerical simulations to confirm the theoretical results.

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Correspondence to Zhen Wang.

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This work was supported by the National Natural Science Foundation of China (Nos. 61573008, 61973199, 61973200), and Taishan Scholar Project of Shandong Province of China.

Appendix

Appendix

\({\varvec{Proof A_1}}\): If \({\varvec{H_1}}-{\varvec{H_3}}\) are satisfied, it is clear that \((t,x)\rightarrow F(t,x)=(F_1(t,x),F_2(t,x),\cdots ,F_n(t,x))^T\) is a U.S.C. set-valued map with nonempty compact convex values. Then the local existence of a solution x(t) to system (1) on \([-\tau ,T)\) for some \(T\in (0,+\infty )\) is a straightforward consequence of ( [36], p. 77, Th. 1).

When \(x_i(t)=0\), \(\bar{co}[h(x_i(t))]x_i(t)=0\) and \(h(x_i(t))x_i(t)\) is continuous at \(x_i=0\). Then, \(\exists \delta _i>0\) such that when \(|x_i|<\delta _i\), \(h(x_i(t))x_i(t)\) is continuous. System (1) can be rewritten as the following right hand continuous function

$$\begin{aligned} \left\{ \begin{aligned}&\frac{dx_1(t)}{dt}=x_1(t)[r_1(t)-u_1(t)x_1(t)-c_1(t)h_1(x_1(t))-a_1(t)\varphi _1(x_1(t))x_2(t)],&\\&\frac{dx_i(t)}{dt}=x_i(t)[-r_i(t)+b_{i-1}(t)\phi _{i-1}(x_{i-1}(t-\tau _{i-1}))\\ {}&-u_{i}(t)x_i(t)-c_i(t)h_i(x_i(t))-a_i(t)\varphi _{i}(x_i(t))x_{i+1}(t)],&\\&\frac{dx_n(t)}{dt}=x_n(t)[-r_n(t)+b_{n-1}(t)\phi _{n-1}(x_{n-1}(t-\tau _{n-1}))\\ {}&-u_{n}(t)x_n(t)-c_n(t)h_n(x_n(t))].&\end{aligned} \right. \end{aligned}$$
(10)

One asserts that \(x_i(t) > 0\) for all \(t\in [-\tau ,T)\). Otherwise, let \(t_i=\inf \{t|x_i(t)=0\}\). Owing to \(x_i(t)\) is continuous on \([-\tau ,T)\), then there exists a positive constant \(\vartheta _i\) such that \(t_i-\vartheta _i>0\) and \(0<x_i(t)<\delta _i\) on \([t_i-\vartheta _i,t_i]\). One can obtain

$$\begin{aligned} 0=x_i(t_i)=x_i(t_i-\vartheta _i)exp\left\{ \int _{t_i-\vartheta _i}^{t_i}\frac{F(s,x(s))}{x(s)}ds\right\} >0. \end{aligned}$$
(11)

It is a contradiction. Thus, \(x_i(t) > 0\) for all \(t\in [-\tau ,T)\). That is to say \(x_i(t)\) is positive.

\({\varvec{Proof A_2}}\): By (1) one can see that

$$\begin{aligned} \frac{dx_1(t)}{dt}<x_1(t)[r_1^M-u_1^Lx_1(t)]. \end{aligned}$$
(12)

Substituting \(x_1(t)=\frac{r_1^M}{u_1^L}=N_1\) into (12), one gets

$$\begin{aligned} \frac{dx_1(t)}{dt}<N_1[r_1^M-u_1^LN_1]=0. \end{aligned}$$
(13)

So if \(0<x_1(0)\le N_1\), for all \(t>0\), \(x_1(t)\le N_1\); else if \(x_1(0)> N_1\), according to (13), there exists a \(t^*_1>0\) such that when \(t>t^*_1\), \(x_1(t)\le N_1\).

From (1), one can get that

$$\begin{aligned} \frac{dx_i(t)}{dt}<x_i(t)[b_{i-1}^M\Psi _i N_{i-1}-u_{i}(t)x_i(t)], \end{aligned}$$
(14)

where \(N_i= \frac{b_{i-1}^M\Psi _i N_{i-1}}{u_i^L}\) for \(i=2,3,\ldots ,n\). By using the same method, one can see that when \(0<x_i(0)\le N_i\), \(x_i(t)\le N_i\) for all \(t>0\). When \(x_i(0)> N_i\), there exists a \(t^*_i>0\) such that when \(t>t^*_i\), \(x_1(t)\le N_i\). Let \(t^*=\max _{i=1,2,\ldots ,n}\{t^*_i\}\), then when \(t>t^*\)\(x_i(t)\le N_i\) for \(i=1,2,\ldots ,n\). By theorem 1, \(x_i(t)>0\). Thus, \(x_i(t)\) is ultimately bounded.

\({\varvec{Proof A_3}}\): By Theorem 1, it is easy to see that the solution of system (1) remains positive for all \(t>0\). Let \(\mu _i(t)=ln(x_i(t))\). Substituting them into differential inclusion (3), one derives that

$$\begin{aligned} \left\{ \begin{aligned} \frac{d\mu _1(t)}{dt}\in&r_1(t)-u_1(t)e^{\mu _1(t)}-c_1(t)\bar{co}[h_1(e^{\mu _1(t)})]-a_1(t)\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)}=F_1(t,\mu ),&\\ \frac{d\mu _i(t)}{dt}\in&-r_i(t)+b_{i-1}(t)\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})-u_{i}(t)e^{\mu _i(t)}\\ {}&-c_i(t)\bar{co}[h_i(e^{\mu _i(t)})]-a_i(t)\varphi _{i}(e^{\mu _i(t)})e^{\mu _{i+1}(t)}\\&=F_i(t,\mu ),&\\ \frac{d\mu _n(t)}{dt}\in&-r_n(t)+b_{n-1}(t)\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})-u_{n}(t)e^{\mu _n(t)}\\ {}&-c_n(t)\bar{co}[h_n(e^{\mu _n(t)})]=F_n(t,\mu ).&\end{aligned} \right. \end{aligned}$$
(15)

It is easy to see that any positive solution x(t) of system (1) is absolutely continuous on any compact interval of \([-\tau ,T)\), and \(\mu _i(t)=ln(x_i(t))\) are absolutely continuous on any compact interval of \([0,+\infty )\) with respect to t. Obviously, if differential inclusion (15) has one \(\omega \)-periodic solution \(\mu ^*(t)=(\mu _1^*(t),\mu _2^*(t),\ldots ,\mu _n^*(t))^T\), \(x^*(t)=(x_1^*(t),x_2^*(t),\ldots ,x_n^*(t))\) is a positive \(\omega \)-periodic solution of differential inclusion (1). Define

$$\begin{aligned} C_\omega =\{\mu (t)\in C(R,R^n):\mu (t+\omega )=\mu (t)\},\qquad \Vert \mu (t)\Vert _{C_\omega }=\sum _{i=1}^n\max _{t\in [0,\omega ]}|\mu _i(t)|.\nonumber \\ \end{aligned}$$
(16)

where \(C(R,R^n)\) donates the continuous function on \((R,R^n)\). Let \(F(t,\mu )=(F_1(t,\mu ),F_2(t,\mu ),\ldots ,F_n(t,\mu ))^T\) for \(\mu (t)\in C_\omega \). It is clear that \(F : R \times R^n \longrightarrow R^n\) is a U.S.C. set-valued map with nonempty compact convex values. Next one will search for appropriate open, bounded subset \(\Omega \). Corresponding to the differential inclusion \(\frac{d\mu }{dt}=\lambda F(t,\mu ),\lambda \in (0,1)\),

$$\begin{aligned} \left\{ \begin{aligned} \frac{d\mu _1(t)}{dt}\in&\lambda \{r_1(t)-u_1(t)e^{\mu _1(t)}-c_1(t)\bar{co}[h_1(e^{\mu _1(t)})]-a_1(t)\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)}\},&\\ \frac{d\mu _i(t)}{dt}\in&\lambda \{-r_i(t)+b_{i-1}(t)\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})-u_{i}(t)e^{\mu _i(t)}\\ {}&-c_i(t)\bar{co}[h_i(e^{\mu _i(t)})]-a_i(t)\varphi _{i}(e^{\mu _i(t)})e^{\mu _{i+1}(t)}\},\\ \frac{d\mu _n(t)}{dt}\in&\lambda \{-r_n(t)+b_{n-1}(t)\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})-u_{n}(t)e^{\mu _n(t)}\\ {}&-c_n(t)\bar{co}[h_n(e^{\mu _n(t)})]\},&\end{aligned} \right. \end{aligned}$$
(17)

By the measurable selection theorem [37], it is easy to find a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)\) such that \(\gamma _i(t)\in \bar{co}[h_1(e^{u_1(t)})]\)\((i=1,2,\ldots ,n)\) for a.e. \(t\in [0,T)\). Then, one has

$$\begin{aligned} \left\{ \begin{aligned} \frac{d\mu _1(t)}{dt}=&\lambda \{r_1(t)-u_1(t)e^{\mu _1(t)}-c_1(t)\gamma _1(t)-a_1(t)\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)}\},&\\ \frac{d\mu _i(t)}{dt}=&\lambda \{-r_i(t)+b_{i-1}(t)\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})-u_{i}(t)e^{\mu _i(t)}-c_i(t)\gamma _i(t)\\ {}&-a_i(t)\varphi _{i}(e^{\mu _i(t)})e^{\mu _{i+1}(t)}\},\\ \frac{d\mu _n(t)}{dt}=&\lambda \{-r_n(t)+b_{n-1}(t)\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})-u_{n}(t)e^{\mu _n(t)}-c_n(t)\gamma _n(t)\}.&\end{aligned} \right. \end{aligned}$$
(18)

Integrating (18) over the interval \([0,\omega ]\), one obtains

$$\begin{aligned} \left\{ \begin{aligned}&\int _0^\omega r_1(t)dt=\int _0^\omega u_1(t)e^{\mu _1(t)}dt+\int _0^\omega c_1(t)\gamma _1(t)dt+\int _0^\omega a_1(t)\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)}dt,&\\&\int _0^\omega b_{i-1}(t)\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})dt=\int _0^\omega u_{i}(t)e^{\mu _i(t)}dt\\ {}&+\int _0^\omega c_i(t)\gamma _i(t)dt+\int _0^\omega a_i(t)\varphi _{i}(e^{\mu _i(t)})e^{\mu _{i+1}(t)}dt&\\&\qquad +\int _0^\omega r_i(t)dt,\\&\int _0^\omega b_{n-1}(t)\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})dt=\int _0^\omega u_{n}(t)e^{\mu _n(t)}dt\\ {}&+\int _0^\omega c_n(t)\gamma _n(t)dt+\int _0^\omega r_n(t)dt,&\end{aligned} \right. \end{aligned}$$
(19)

Then, one has

$$\begin{aligned} \int _0^\omega |\dot{\mu }_1|dt\le & {} \int _0^\omega r_1(t)dt+\int _0^\omega u_1(t)e^{\mu _1(t)}dt+\int _0^\omega c_1(t)\gamma _1(t)dt\nonumber \\&\quad +\int _0^\omega a_1(t)\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)}dt=2\bar{r}\omega ,\nonumber \\ \int _0^\omega |\dot{\mu }_i|dt\le & {} \int _0^\omega b_{i-1}(t)\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})dt+\int _0^\omega r_i(t)dt\nonumber \\&\quad +\int _0^\omega u_{i}(t)e^{\mu _i(t)}dt+\int _0^\omega c_i(t)\gamma _i(t)dt\nonumber \\&\quad +\int _0^\omega a_i(t)\phi _{i}(e^{\mu _i(t)})e^{\mu _{i+1}(t)}dt\nonumber \\&\le 2\int _0^\omega b_{i-1}(t)\Psi _{i-1}e^{\mu _{i-1}(t-\tau _{i-1})}dt,\nonumber \\ \int _0^\omega |\dot{\mu }_n|dt\le & {} \int _0^\omega b_{n-1}(t)\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})dt+\int _0^\omega u_{n}(t)e^{\mu _n(t)}dt\nonumber \\&\quad +\int _0^\omega c_n(t)\gamma _n(t)dt+\int _0^\omega r_n(t)dt\nonumber \\&\le 2\int _0^\omega b_{n-1}(t)\Psi _{n-1}e^{\mu _{n-1}(t-\tau _{n-1})}dt. \end{aligned}$$
(20)

Because of \(\mu (t)\in C_\omega \), then there exist \(\xi _i,\eta _i\in [0,\omega ]\) such that

$$\begin{aligned} \mu _i(\xi _i)=\min _{t\in [0,\omega ]}\mu _i(t),\mu _i(\eta _i)=\max _{t\in [0,\omega ]}\mu _i(t). \end{aligned}$$
(21)

By (19), it is easy to see that \(\bar{r}_1\omega >\int _0^\omega u_1(t)e^{\mu _1(t)}dt\) and \(\int _0^\omega b_{i-1}(t)\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})dt>\int _0^\omega u_{i}(t)e^{\mu _i(t)}dt\) for \(i=2,3,\ldots ,n\).

$$\begin{aligned} \begin{aligned}&\mu _1(\xi _1)\le \ln \frac{\bar{r}_1}{\bar{u}_1}, \qquad \mu _i(\xi _i)\le \ln \frac{\bar{b}_{i-1}\Psi _{i-1}}{\bar{u}_i}. \end{aligned} \end{aligned}$$
(22)

Then,

$$\begin{aligned} \begin{aligned}&\mu _1(t)\le \mu _1(\xi _1)+\int _0^\omega |\dot{\mu }_1|dt\le \ln \frac{\bar{r}_1}{\bar{u}_1}+2\bar{r}\omega =K_1,\\&\mu _i(t)\le \mu _i(\xi _i)+\int _0^\omega |\dot{\mu }_i|dt\le \ln \frac{\bar{b}_{i-1}\Psi _{i-1}}{\bar{u}_i}+ 2\omega \bar{b}_{i-1}(t)\Psi _{i-1}e^{K_{i-1}}=K_i. \end{aligned} \end{aligned}$$
(23)

By (20) and (23), one can see that

$$\begin{aligned} \begin{aligned} \int _0^\omega |\dot{\mu }_1|dt&\le 2\bar{r}_1\omega =A_1,\\ \int _0^\omega |\dot{\mu }_i|dt&\le 2\omega \bar{b}_{i-1}(t)\Psi _{i-1}e^{K_{i-1}}=A_i, \end{aligned} \end{aligned}$$
(24)

for \(i=2,3,\ldots ,n\). By (19), one can see that

$$\begin{aligned} \int _0^\omega r(t)dt\le \int _0^\omega u_1(t)e^{\mu _1(\xi _i)}dt+\int _0^\omega c_1(t)Mdt +\int _0^\omega a_1(t)\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)}dt.\nonumber \\ \end{aligned}$$
(25)

One has

$$\begin{aligned} \begin{aligned}&\omega \bar{r}_1\le \omega \bar{u}_1e^{\mu _1(\eta _1)}+\omega M\bar{c}_1+\omega \bar{a}_1\Psi _1 e^{K_2},\\&\mu _1(\eta _1)\ge \ln \frac{\bar{r}_1-M_1\bar{c}_1-\bar{a}_1\Psi _1 e^{K_2}}{\bar{u}_1}. \end{aligned} \end{aligned}$$
(26)

Then,

$$\begin{aligned} \mu _1(t)\ge \mu _1(\eta _1)-\int _0^\omega |\dot{\mu }_1|dt\ge \ln \frac{\bar{r}_1-M_1\bar{c}_1-\bar{a}_1\Psi _1 e^{K_2}}{\bar{u}_1}-2\bar{r}\omega =P_1. \end{aligned}$$
(27)

By (19) and (24), one obtains

$$\begin{aligned} \begin{aligned}&\omega \bar{b}_{i-1}\phi ^*_{i-1}\le \omega \bar{r}_{i}+\omega \bar{u}_{i}e^{\mu _i(\eta _i)}+\omega \bar{c}_i M_i+\omega a_i\Psi _{i}e^{K_{i+1}},\\&\omega \bar{b}_{n-1}\phi ^*_{n-1}\le \omega \bar{r}_{n}+\omega \bar{u}_{n}e^{\mu _n(\eta _n)}+\omega \bar{c}_n M_n,\\&\mu _i(\eta _i)\ge \ln \frac{\bar{b}_{i-1}\phi ^*_{i-1}- \bar{c}_iM_i- \bar{a}_i\Psi _{i}e^{K_{i+1}}-\bar{r}_{i}}{\bar{u}_{i}}\\&\mu _i(\eta _n)\ge \ln \frac{\bar{b}_{n-1}\phi ^*_{n-1}- \bar{c}_nM_n-\bar{r}_{n}}{\bar{u}_n},\\&\mu _i(t)\ge \mu _i(\eta _i)-\int _0^\omega |\dot{\mu }_i|dt\ge \ln \frac{\bar{b}_{i-1}\phi ^*_{i-1}- \bar{c}_iM_i- \bar{a}_i\Psi _{i}e^{K_{i+1}}-\bar{r}_{i}}{\bar{u}_{i}}-A_i=P_i,\\&\mu _n(t)\ge \mu _n(\eta _n)-\int _0^\omega |\dot{\mu }_n|dt\ge \ln \frac{\bar{b}_{n-1}\phi ^*_{n-1}- \bar{c}_nM_n-\bar{r}_{n}}{\bar{u}_{n}}-A_n=P_n. \end{aligned} \end{aligned}$$
(28)

where \(\phi ^*_{i-1}=\min \limits _{\mu _{i-1}(t)\in [P_{i-1},K_{i-1}]}\phi _i(e^{\mu _{i-1}(t)})\), \(i=2,3,\ldots ,n-1\). So

$$\begin{aligned} \begin{aligned}&\max \limits _{t\in [0,\omega ]}|\mu _1(t)|<\max \left\{ |\ln \frac{\bar{r}_1}{\bar{u}_1}|+A_1,|\ln \frac{\bar{r}_1-M\bar{c}_1-\bar{a}_1\Psi _1 e^{K_2}}{\bar{u}_1}|+A_1\right\} =R_1\\&\max \limits _{t\in [0,\omega ]}|\mu _i(t)|<\max \left\{ |\ln \frac{\bar{b}_{i-1}\Psi _{i-1}}{\bar{u}_i}|\right. \\&\quad \left. + A_i,|\ln \frac{\bar{b}_{i-1}\phi ^*_{i-1}- \bar{c}_iM_i- \bar{a}_i\Psi _{i}e^{K_{i+1}}-\bar{r}_{i}}{\bar{u}_{i}}|+ A_i\right\} =R_i\\&\max \limits _{t\in [0,\omega ]}|\mu _n(t)|<\max \left\{ |\ln \frac{\bar{b}_{n-1}\Psi _{n-1}}{\bar{u}_n}|+ A_n,|\ln \frac{\bar{b}_{n-1}\phi ^*_{n-1}- \bar{c}_nM_n-\bar{r}_{n}}{\bar{u}_{n}}|+ A_n\right\} =R_n \end{aligned} \end{aligned}$$
(29)

Obviously, \(R_i\) in (29) are independent of \(\lambda \). Consider the following system of algebraic inclusion:

$$\begin{aligned} \left\{ \begin{aligned} 0\in&\bar{r}_1-\bar{u}_1e^{\mu _1(t)}-\bar{c}_1\bar{co}[h_1(e^{\mu _1(t)})]-\bar{a}_1\varphi _1(e^{\mu _1(t)})e^{\mu _2(t)},&\\ 0\in&-\bar{r}_{i}+\bar{b}_{i-1}\phi _{i-1}(e^{\mu _{i-1}(t-\tau _{i-1})})-\bar{u}_{i}e^{\mu _i(t)}-\bar{c}_i\bar{co}[h_i(e^{\mu _i(t)})]-\bar{a}_i\varphi _{i}(e^{\mu _i(t)})e^{\mu _{i+1}(t)},\\ 0\in&-\bar{r}_{n}+\bar{b}_{n-1}\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})-\bar{u}_{n}e^{\mu _n(t)}-\bar{c}_n\bar{co}[h_n(e^{\mu _n(t)})],&\end{aligned} \right. \nonumber \\ \end{aligned}$$
(30)

It is clear that the set of all solutions for (30) are bounded if there exists. Denote \(R=\sum _{i=0}^nR_i\) where \(R_0\) is taken sufficiently large such that each solution \(u^*\in \mathbb {R}^n\) of the algebraic inclusion (30) satisfies \(\sum _{i=1}^n\mu ^*_i<R\). Let \(\Omega =\{(\mu _1,\mu _2,\ldots ,\mu _n)^T\in C_\omega :\Vert (\mu _1,\mu _2,\ldots ,\mu _n)^T\Vert _{C_\omega }<R,\forall t\in \mathbb {R}\}\). Clearly, \(\Omega \) is an open bounded set of \(C_\omega \) and \(u\notin \partial \Omega \) for any \(\lambda \in (0,1)\).

Suppose that there exists a solution \(\mu =(\mu _1,\mu _2,\ldots ,\mu _n)^T\in \partial \Omega \bigcap \mathbb {R}^n\) of the inclusion \(0\in \frac{1}{\omega }\int _0^\omega F(t,\mu )dt\), then \(\Sigma _{i=1}^{n}\mu _i=R\). Since each solution \(u^*\in R^n\) of the algebraic inclusion (30) satisfies \(\sum _{i=1}^n\mu ^*_i<R\), one has that

$$\begin{aligned}&0\notin \frac{1}{\omega }\int _0^\omega F(t,\mu )dt\nonumber \\&\quad =g_0(\mu ) =\left( {\begin{array}{*{10}c} \bar{r}_1-\bar{u}_1e^{\mu _1}-\bar{c}_1\bar{co}[h_1(e^{\mu _1})]-\bar{a}_1\varphi _1(e^{\mu _1})e^{\mu _2}\\ -\bar{r}_{i}+\bar{b}_{i-1}\phi _{i-1}(e^{\mu _{i-1}})-\bar{u}_{i}e^{\mu _i}-\bar{c}_i\bar{co}[h_i(e^{\mu _i})]-\bar{a}_i\varphi _{i}(e^{\mu _i})e^{\mu _{i+1}}\\ -\bar{r}_{n}+\bar{b}_{n-1}\phi _{n-1}(e^{\mu _{n-1}})-\bar{u}_{n}e^{\mu _n}-\bar{c}_n\bar{co}[h_n(e^{\mu _n})] \end{array}}\right) ,\nonumber \\ \end{aligned}$$
(31)

for \(i=2,3,\ldots ,n-1\). This is a contradiction.

Define a homotopic set-valued map

$$\begin{aligned} G(\mu ,\nu )= & {} \left( {\begin{array}{*{10}c} \bar{r}_1-\bar{u}_1e^{\mu _1} \\ \bar{b}_{i-1}\phi _{i-1}(e^{\mu _{i-1}})-\bar{u}_{i}e^{\mu _i} \\ \bar{b}_{n-1}\phi _{n-1}(e^{\mu _{n-1}})-\bar{u}_{n}e^{\mu _n} \end{array}} \right) \\&+ \nu \left( {\begin{array}{*{10}c} -\bar{c}_1\bar{co}[h_1(e^{\mu _1})]-\bar{a}_1\varphi _1(e^{\mu _1})e^{\mu _2}\\ -\bar{r}_{i}-\bar{c}_i\bar{co}[h_i(e^{\mu _i})]-\bar{a}_i\varphi _{i}(e^{\mu _i})e^{\mu _{i+1}}\\ -\bar{r}_{n}-\bar{c}_n\bar{co}[h_n(e^{\mu _n})] \end{array}} \right) ,\nu \in [0,1]. \end{aligned}$$

If \(\mu =(\mu _1,\mu _2,\ldots ,\mu _n)^T\in \partial \Omega \bigcap \mathbb {R}^n\), then \(\mu \) is a constant vector in \(\partial \Omega \bigcap \mathbb {R}^n\) with \(\sum _{i=1}^n|\mu _i|=R\). If

$$\begin{aligned} \left\{ \begin{aligned} 0\in&\bar{r}_1-\bar{u}_1e^{\mu _1}+\nu (\bar{c}_1\bar{co}[h_1(e^{\mu _1})]-\bar{a}_1\varphi _1(e^{\mu _1})e^{\mu _2}),&\\ 0\in&\bar{b}_{i-1}\phi _{i-1}(e^{\mu _{i-1}})-\bar{u}_{i}e^{\mu _i}+\nu (-\bar{r}_{i}-\bar{c}_i\bar{co}[h_i(e^{\mu _i})]-\bar{a}_i\varphi _{i}(e^{\mu _i})e^{\mu _{i+1}}),\\ 0\in&\bar{b}_{n-1}\phi _{n-1}(e^{\mu _{n-1}(t-\tau _{n-1})})-\bar{u}_{n}e^{\mu _n(t)}+\nu (-\bar{r}_{n}-\bar{c}_n\bar{co}[h_n(e^{\mu _n(t)})]).&\end{aligned} \right. \end{aligned}$$
(32)

One has that

$$\begin{aligned} \begin{aligned}&|\mu _1(t)|<\max \left\{ |\ln \frac{\bar{r}_1}{\bar{u}_1}|,|\ln \frac{\bar{r}_1-M\bar{c}_1-\bar{a}_1\Psi _1 e^{K_2}}{\bar{u}_1}|\right\}<R_1,\\&|\mu _i(t)|<\max \left\{ |\ln \frac{\bar{b}_{i-1}\Psi _{i-1}}{\bar{u}_i}|,|\ln \frac{\bar{b}_{i-1}\phi ^*_{i-1}- \bar{c}_i(t)M_i- \bar{a}_i\Psi _{i}e^{K_{i+1}}-\bar{r}_{i}}{\bar{u}_{i}}|\right\}<R_i,\\&|\mu _n(t)|<\max \left\{ |\ln \frac{\bar{b}_{n-1}\Psi _{n-1}}{\bar{u}_n}|,|\ln \frac{\bar{b}_{n-1}\phi ^*_{n-1}- \bar{c}_n(t)M_n-\bar{r}_{n}}{\bar{u}_{n}}|\right\} <R_n. \end{aligned}\nonumber \\\ \end{aligned}$$
(33)

So \(0\notin G(\mu ,\nu )\). It is easy to see that \(G(\mu ,0)=0\) has a unique solution. Assume \(\mu ^*=(\mu _1^*,\mu _2^*,\ldots ,\mu _n^*)\) is the solution of \(G(\mu ,0)=0\). Then

$$\begin{aligned} \begin{aligned} \deg \{g_0,\Omega \bigcap \mathbb {R}^n,0\}&=\deg \{G(\mu ,1),\Omega \bigcap \mathbb {R}^n,0\}=\deg \{G(\mu ,0),\Omega \bigcap \mathbb {R}^n,0\}\\&=sign \left| {\begin{array}{*{10}ccccc} -\bar{u}_1e^{\mu _1^*} &{}0 &{}0 &{}\cdots &{}0\\ * &{}-\bar{u}_{2}e^{\mu _2^*} &{}0 &{}\cdots &{}0\\ 0 &{}* &{}-\bar{u}_{3}e^{\mu _3^*} &{}\cdots &{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}0\\ 0 &{}0 &{}0 &{}\cdots &{}-\bar{u}_{n}e^{\mu ^*_n} \end{array}} \right| \ne 0 \end{aligned} \end{aligned}$$
(34)

where \(deg(\cdot ,\cdot ,\cdot )\) denotes the topological degree for upper semi-continuous set-valued maps with compact convex values.

Above all, \(\Omega \) satisfies all the requirements in Lemma 1, then the differential inclusion (15) has at least one \(\omega \)-periodic solution. As a consequence, system (1) has at least one positive \(\omega \)-periodic solution.

\({\varvec{Proof A_4}}\): Suppose that \(\gamma (t)=(\gamma _1(t),\gamma _2(t),\ldots ,\gamma _n(t))\) is the harvesting solution associated with the solution \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))\), and \(\gamma ^*(t)=(\gamma _1^*(t),\gamma ^*_2(t),\ldots ,\gamma ^*_n(t))\) is the harvesting solution associated with the positive \(\omega \)-periodic solution \(x^*(t)=(x_1^*(t),x_2^*(t),\ldots ,x_n^*(t))\). Define

$$\begin{aligned} \begin{aligned}&z_i(t)=\ln x_i(t)-\ln x_i^*(t),&V_i(z_i(t))=|z_i(t)|, \end{aligned} \end{aligned}$$
(35)

for \(i=1,2,\ldots ,n\). Then

$$\begin{aligned} \frac{dV_i(t)}{dt}=\kappa _i\left( \frac{\dot{x}_i(t)}{x_i(t)}-\frac{\dot{x}^*_i(t)}{x_i^*(t)}\right) , \end{aligned}$$
(36)

where \(\kappa _i\) can be chosen as follow

$$\begin{aligned} \kappa _i=\left\{ \begin{array}{l@{\quad }l} 0, &{}x_i(t)-x_i^*(t)=0,\gamma _i(t)-\gamma _i^*(t)=0, \\ sign(\gamma _i(t)-\gamma _i^*(t)), &{}x_i(t)-x_i^*(t)=0,\gamma _i(t)-\gamma _i^*(t)\ne 0, \\ sign(x_i(t)-x_i^*(t)) &{}x_i(t)-x_i^*(t)\ne 0. \end{array}\right. \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} \kappa _i(x_i(t)-x_i^*(t))=|x_i(t)-x_i^*(t)|,\\ \kappa _i(\gamma _i(t)-\gamma _i^*(t))=|\gamma _i(t)-\gamma _i^*(t)|. \end{aligned} \end{aligned}$$
(37)

So one has that

$$\begin{aligned}&\begin{aligned} \frac{dV_1(t)}{dt}&=\kappa _1\left( \frac{\dot{x}_1(t)}{x_1(t)}-\frac{\dot{x}^*_1(t)}{x_1^*(t)}\right) \\&=\kappa _1[-u_1(t)(x_1(t)-x_1^*(t))-c_1(t)(\gamma _1(t)-\gamma ^*_1(t))-a_1(t)(\varphi _1(x_1(t))x_2(t)\\&\quad -\varphi _1(x^*_1(t))x^*_2(t))]\\&\le -u_1(t)|x_1(t)-x_1^*(t)|-c_1(t)|\gamma _1(t)-\gamma _1^*(t)|+a_1(t)\varphi _1(x_1(t))|x_2(t)-x^*_2(t)|\\&\quad +a_1(t)x^*_2(t)|\varphi _1(x_1(t))-\varphi _1(x^*_1(t))| \end{aligned} \end{aligned}$$
(38)
$$\begin{aligned}&\begin{aligned} \frac{dV_i(t)}{dt}&=\kappa _i\left( \frac{\dot{x}_i(t)}{x_i(t)}-\frac{\dot{x}^*_i(t)}{x_i^*(t)}\right) \\&=\kappa _i[b_{i-1}(t)(\phi _{i-1}(x_{i-1}(t-\tau _{i-1}))\\&\quad -\phi _{i-1}(x^*_{i-1}(t-\tau _{i-1})))-u_{i}(t)(x_i(t)-x_i^*(t))\\&\quad -c_i(t)(\gamma _i(t)-\gamma ^*_i(t))-a_i(t)(\varphi _{i}(x_i(t))x_{i+1}(t)-\varphi _{i}(x^*_i(t))x^*_{i+1}(t))]\\&\le b_{i-1}(t)|\phi _{i-1}(x_{i-1}(t-\tau _{i-1}))-\phi _{i-1}(x^*_{i-1}(t-\tau _{i-1}))|-u_{i}(t)|x_i(t)-x_i^*(t)|\\&\quad -c_i(t)|\gamma _i(t)-\gamma ^*_i(t)|+a_i(t)|\varphi _{i}(x_i(t))x_{i+1}(t)-\varphi _{i}(x^*_i(t))x^*_{i+1}(t)| \end{aligned} \end{aligned}$$
(39)
$$\begin{aligned}&\begin{aligned} \frac{dV_n(t)}{dt}&=\kappa _n\left( \frac{\dot{x}_n(t)}{x_n(t)}-\frac{\dot{x}^*_n(t)}{x_n^*(t)}\right) \\&=\kappa _n[b_{n-1}(t)(\phi _{n-1}(x_{n-1}(t-\tau _{n-1}))\\&\quad -\phi _{n-1}(x^*_{n-1}(t-\tau _{n-1})))-u_{n}(t)(x_n(t)-x_n^*(t))\\&\quad -c_n(t)(\gamma _n(t)-\gamma ^*_n(t))]\\&\le b_{n-1}(t)|\phi _{n-1}(x_{n-1}(t-\tau _{n-1}))-\phi _{n-1}(x^*_{n-1}(t-\tau _{n-1}))|-u_{n}(t)|x_n(t)-x_n^*(t)|\\&\quad -c_n(t)|\gamma _n(t)-\gamma ^*_n(t)| \end{aligned} \end{aligned}$$
(40)

Since \(\varphi _i(x_i(t))\) and \(\phi _i(x_i(t))\) have continuous derivatives for \(x_i\in [0,+\infty )\), \(\varphi ^{'}_i(x_i(t))\) and \(\phi _i(x_i(t))\) have maximum value on \([0,N_i]\). Assume that \(\max \limits _{x_i\in [0,N_i]}|\varphi ^{'}_i(x_i(t))|=D_i\) and \(\max \limits _{x_i\in [0,N_i]}|\phi ^{'}_i(x_i(t))|=E_i\), then \(|\varphi _i(x_i(t))-\varphi _i(x^*_i(t))|\le D_i|x_i(t)-x_i^*(t)|\) and \(|\phi _i(x_i(t))-\phi _i(x^*_i(t))|\le E_i|x_i(t)-x_i^*(t)|\). So

$$\begin{aligned}&\frac{dV_1(z_1(t))}{dt}\le -(u_1(t)-a_1(t)x^*_2(t)D_1)|x_1(t)-x_1^*(t)|\nonumber \\&\qquad -\,c_1(t)|\gamma _1(t)-\gamma _1(t)|+a_1(t)\Phi _1|x_2(t)-x^*_2(t)|. \end{aligned}$$
(41)
$$\begin{aligned}&\frac{dV_i(z_i(t))}{dt}\le -(u_i(t)-a_i(t)x^*_{i+1}(t)D_i)|x_i(t)-x_i^*(t)|\nonumber \\&\qquad -\,c_i(t)|\gamma _i(t)-\gamma _i(t)|+a_i(t)\Phi _i|x_{i+1}(t)-x^*_{i+1}(t)|\nonumber \\&\qquad +\,b_{i-1}(t)E_{i-1}|x_{i-1}(t-\tau _{i-1})-x_{i-1}^*(t-\tau _{i-1})|. \end{aligned}$$
(42)
$$\begin{aligned}&\frac{dV_n(z_n(t))}{dt}\le b_{n-1}(t)E_{n-1}|x_{n-1}(t-\tau _{n-1})-x_{n-1}^*(t-\tau _{n-1})|\nonumber \\&\qquad -\,c_n(t)|\gamma _n(t)-\gamma ^*_n(t)|-u_{n}(t)|x_n(t)-x_n^*(t)|. \end{aligned}$$
(43)

Define \(y(t)=(y_1(t),y_2(t),\ldots ,y_n(t))\) and \(y_i(t)=x_i(t)-x_i^*(t)\). The selected Lyapunov function is defined as follow

$$\begin{aligned} \begin{aligned} V(z(t),y(t))=&\sum _{i=1}^nV_i(z_i(t))+\sum _{i=1}^{n-1}b_{i}(t)E_{i}\int _{t-\tau _{i}}^t|y_i(t)|dt \end{aligned} \end{aligned}$$
(44)

It is easy to see that V(z(t), y(t)) is C-regular. By (41), (42) and (43), one has that

$$\begin{aligned} \begin{aligned} \frac{dV(z(t),y(t))}{dt}\le&-\sum _{i=2}^{n-1}(u_i(t)-a_i(t)x^*_{i+1}(t)D_i-b_{i}(t)E_{i}\\&-a_{i-1}(t)\Phi _{i-1})|x_i(t)-x_i^*(t)|-u_{n}(t)|x_n(t)-x_n^*(t)|\\&-(u_1(t)-a_1(t)x^*_{2}(t)D_1-b_{1}(t)E_{1})|x_1(t)-x_1^*(t)|\\&-\sum _{i=1}^nc_i(t)|\gamma _i(t)-\gamma _i(t)|\\ \le&-\sum _{i=2}^{n-1}(u^L_i-a^M_iN_{i+1}D_i-b^M_{i}E_{i}-a^M_{i-1}\Phi _{i-1})|x_i(t)-x_i^*(t)|\\&-\sum _{i=1}^nc^L_i|\gamma _i(t)-\gamma _i(t)|\\&-u_{n}^L|x_n(t)-x_n^*(t)|-(u_1^L-a_1^M N_{2}D_1-b^M_{1}E_{1})|x_1(t)-x_1^*(t)|. \end{aligned} \end{aligned}$$
(45)

By the conditions \(u^L_i-a^M_iN_{i+1}D_i-b^M_{i}E_{i}-a^M_{i-1}\Phi _{i-1}>0\) for \(i=2,3,\ldots ,n-1\) and \(u_1^L-a_1^M N_{2}D_1-b^M_{1}E_{1}>0\), there exist a positive constant \(\beta =\min \limits _{i=2,3,\ldots ,n-1}\{u_1^L-a_1^M N_{2}D_1-b^M_{1}E_{1},u^L_i-a^M_iN_{i+1}D_i-b^M_{i}E_{i}-a^M_{i-1}\Phi _{i-1},u_{n}^L\}\) such that

$$\begin{aligned} \frac{dV(z(t),y(t))}{dt}<-\beta \sum _{i=1}^n|x_i(t)-x_i^*(t)|, \end{aligned}$$
(46)

for a.e. \(t>t^*\) where \(t^*\) is defined in Theorem 2. Integrating both sides of inequality, one has

$$\begin{aligned} \begin{aligned} V(z(t),y(t))+\beta \int _{t^*}^t\sum _{i=1}^{n}|x_i(s)-x_i^*(s)|ds\le V(z(t^*),y(t^*)). \end{aligned} \end{aligned}$$
(47)

So V(t) is bounded on \([0,+\infty )\).

$$\begin{aligned} \int _{t^*}^t\sum _{i=1}^{n}|x_i(s)-x_i^*(s)|ds<+\infty . \end{aligned}$$
(48)

By using Lemma 2, one can conclude that

$$\begin{aligned} lim_{t\rightarrow +\infty }\left( \sum _{i=1}^n|x_i(t)-x_i^*(t)|\right) =0. \end{aligned}$$
(49)

That is to say, the positive \(\omega \)-periodic solution of system (1) is globally asymptotically stable and the positive \(\omega \)-periodic solution \(x^*\) of system (1) is unique.

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Xie, Y., Wang, Z. Periodic solution and dynamical analysis for a delayed food chain model with general functional response and discontinuous harvesting. J. Appl. Math. Comput. (2020). https://doi.org/10.1007/s12190-020-01389-6

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Keywords

  • Periodic solutions
  • General functional response
  • Food chain model
  • Discontinuous harvesting

Mathematics Subject Classification

  • 34C25
  • 03C45