## 具有无穷分布时滞和反馈控制的周期阶段结构单种群模型

1新疆大学数学与系统科学学院 乌鲁木齐 830017

2新疆维吾尔自治区应用数学重点实验室 乌鲁木齐 830017

## A Periodic Stage Structure Single-Population Model with Infinite Delay and Feedback Control

Yin Ruixia1, Wang Zedong1, Zhang Long,1,2,*

1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017

2The Key Laboratory of Applied Mathematics of Xinjiang Uygur of Xinjiang, Urumqi 830017

 基金资助: 新疆维吾尔自治区应用数学重点实验室开放课题(2021D04014)新疆维吾尔自治区天山英才领军人才项目(2023TSYCLJ0054)国家自然科学基金(12261087)国家自然科学基金(12262035)国家自然科学基金(12201540)国家自然科学基金(11861065)新疆维吾尔自治区自然科学基金(2022D01E41)新疆维吾尔自治区高校科研项目(XJEDU2021I002)

 Fund supported: Open Project of Key Laboratory of Applied Mathematics of XinjiangUygur Autonomous Region(2021D04014)Leading Talents of Tianshan Mountains Project in Xinjiang Uygur Autonomous Region(2023TSYCLJ0054)NSFC(12261087)NSFC(12262035)NSFC(12201540)NSFC(11861065)Natural Science Foundation of Xinjiang Province(2022D01E41)Scientific Research Programmes of Colleges in Xinjiang(XJEDU2021I002)

Abstract

In this paper, we investigate a periodic stage structure single-population model with infinitely distributed delay and feedback control. Firstly, a sufficient criterion for the existence of a unique globally asymptotically stable periodic solution for the auxiliary system-periodic stage structure single-population model is derived. Secondly, we establish sufficiency criteria on the permanence of the model and the existence of a positive periodic solution in the form of integral. Finally, we illustrate the results with numerical simulations.

Keywords： Permanence; Periodic solution; Infinite delay; Feedback control; Stage structure

Yin Ruixia, Wang Zedong, Zhang Long. A Periodic Stage Structure Single-Population Model with Infinite Delay and Feedback Control[J]. Acta Mathematica Scientia, 2024, 44(4): 994-1011

## 1 引言

\left\{\begin{aligned}&\dot x{_{1}}(t)=\alpha x_{2}(t)-\gamma_1 x_{1}(t)-\beta x_{1}(t)-\eta_{1} x^2_{1}(t),\\&\dot x{_{2}}(t)=\beta x_{1}(t)-\eta_{2} x^2_{2}(t),\\\end{aligned}\right.

\left\{\begin{aligned}&\dot x{_{1}}(t)=\alpha(t) x_{2}(t)-\gamma_{1}(t)x_{1}(t)-\beta(t) x_{1}(t)-\eta_{1}(t)x^2_{1}(t),\\&\dot x{_{2}}(t)=\beta(t) x_{1}(t)-\gamma_{2}(t)x_{2}(t)-\eta_{2}(t) x^2_{2}(t),\\\end{aligned}\right.

$$$\left\{\begin{array}{l}\dot N_i(t) =N_{i}(t)[b_{i}(t)-h_{i}(t) N_{i}(t)- \displaystyle\sum_{j=1,i\neq j}^{n} a_{i j}(t) N_{j}(t)-c_{i}(t) N_{i}\left(t-\tau_{i}\right)-d_{i}(t) P_{i}(t)], \\\dot P_i(t) =P_{i}(t)\left[r_{i}(t) N_{i}(t)-D_{i}(t) P_{i}(t)\right],(i=1,2, \cdots, n),\end{array}\right.$$$

## 2 预备知识

\begin{equation*}\begin{aligned}A_w(g)=w^{-1}\int_{0}^{w} g(t){\rm d}t, \quad g^M=\max_{t\in \mathbb{R}_+} g(t), \quad g^L=\min_{t\in \mathbb{R}_+} g(t).\end{aligned}\end{equation*}

(H1) $\alpha^L \beta^L> \gamma_2^M(\gamma_1^M+\beta^M)$;

(H2) $\alpha^L \geq 0, \beta^L>0, A_w(\gamma_1)>0,A_w(\eta_1)>0, e^L \geq 0,f^L>0, d^L>0$, 对所有 $t \in [w]$ 成立.

$C_w(\mathbb{R})$ 表示 $\mathbb{R}$ 上连续向量函数 $\phi(\theta)=\{\phi_1(\theta),\phi_2(\theta),\phi_3(\theta)\}$ 构成的 Banach 空间, 且范数定义为 $\|\phi\|=\sup |\phi(\theta) |,\ \theta \in \mathbb{R}$.C_w^+(\mathbb{R})=\{\phi=(\phi_1,\phi_2,\phi_3) \in C_w(\mathbb{R}): \phi_i(0)>0, i=1,2,3\}. 基于系统 (1.4) 的生物背景, 其任意解 x(t)=(x_1(t),x_2(t),u(t)) 满足以下初始条件 \begin{align*} \begin{aligned}& x_i(s)=\phi_i(s), s\in (-\infty,0],\ i=1,2.\\& u(s)=\phi_3(s), s\in (-\infty,0].\end{aligned}\end{align*} 这里 \phi=(\phi_1,\phi_2,\phi_3) \in C^+_w(\mathbb{R}). 接着引入一些定义以及引理便于后续证明. 定义 2.1[5] 称系统 (1.4) 是持久的, 若对系统 (1.4) 满足初始条件 (2.1) 的任意正解 x(t)=(x_1(t),x_2(t),u(t)), 存在正常数 m,M, 满足 \begin{equation*}\begin{aligned}& m \leq \liminf_{t\rightarrow +\infty} x_i(t) \leq \limsup_{t\rightarrow +\infty} x_i(t) \leq M,\ (i=1,2),\\& m \leq \liminf_{t\rightarrow +\infty} u(t) \leq \limsup_{t\rightarrow +\infty} u(t) \leq M.\end{aligned}\end{equation*} 考虑下列泛函微分方程 \begin{aligned} \dot x=f(t,x_t),\end{aligned} 其中 f(t,\phi)=(f_1(t,\phi),f_2(t,\phi),f_3(t,\phi)): \mathbb{R}\times C_w \rightarrow \mathbb{R}^3 是连续的且关于 \phi 满足 Lipschitz 条件, 根据泛函微分方程理论[16]对于任意 t_0 \in \mathbb{R}\phi \in C_w(\mathbb{R}), 系统 (2.2) 存在唯一解 $x(t,t_0,\phi)=(x_1(t,t_0,\phi),x_2(t,t_0,\phi),x_3(t,t_0,\phi))$. 如果下列条件成立, 则称系统 (2.2) 为合作的[16]

\begin{equation*}\begin{aligned}\text{对于任意} \phi,\ \varphi \in C_w(\mathbb{R}),\ \text{若} \phi < \varphi,\text{且} \phi_i(0)=\varphi_i(0),\ \text{那么} f_i(t,\phi)<f_i(t,\varphi).\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\dot y \geq (\text{或} \leq) f(t,y_t),\quad y_{t_0}\geq (\text{或} \leq) \phi, \quad t \in [t_0,T].\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}y_i(t) \geq (\text{或} \leq) x_i(t,t_0,\phi), \quad t \in [t_0,T], \quad i=1,2,3.\end{aligned}\end{equation*}

\begin{aligned}\dot x(t)=x(t)(a(t)-b(t)x(t)),\end{aligned}

\begin{equation*}\begin{aligned}m \leq \liminf_{t\rightarrow \infty} x(t) \leq \limsup_{t\rightarrow \infty}x(t) \leq M.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\liminf_{t \rightarrow \infty} x(t) & \leq \liminf _{t \rightarrow \infty} \int_{-\infty}^{0} k(s) x(t+s){\rm d}s \\& \leq \limsup _{t \rightarrow \infty} \int_{-\infty}^{0} k(s) x(t+s){\rm d}s\leq \limsup _{t \rightarrow \infty} x(t).\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned} \left\{\begin{aligned}&\dot x{_{1}}(t) \geq \alpha^L x_{2}(t)-\gamma_1^M x_{1}(t)-\beta^M x_{1}(t)-\eta_{1}^M x^2_{1}(t),\\&\dot x{_{2}}(t) \geq \beta^L x_{1}(t)-\gamma_2^M x_{2}(t)-\eta_{2}^M x^2_{2}(t).\end{aligned}\right.\end{aligned}\end{equation*}

Cui 等[4,引理 3] 已经证明了 $\alpha^L \beta^L> \gamma_2^M(\gamma_1^M+\beta^M)$ 成立时, 下列模型 (2.5) 有唯一全局渐近稳定的正平衡点 $(\hat{x}_1,\hat{x}_2)$

\begin{aligned} \left\{\begin{aligned}&\dot x{_{1}}(t)= \alpha^L x_{2}(t)-\gamma_1^M x_{1}(t)-\beta^M x_{1}(t)-\eta_{1}^M x^2_{1}(t),\\&\dot x{_{2}}(t) = \beta^L x_{1}(t)-\gamma_2^M x_{2}(t)-\eta_{2}^M x^2_{2}(t).\end{aligned}\right.\end{aligned}

\begin{equation*}\begin{aligned}\liminf_{t\rightarrow +\infty} \bar{x}_i(t) \geq \hat{x}_i\ \ (i=1,2).\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}& p_{1}<\min \{\hat{x}_1,\check{x}_1\}; p_{2}>\max \{\hat{x}_1,\check{x}_1\},\\& \frac{p_1(\eta_1^M p_{1}+\gamma_1^{M}+\beta^{M})}{a^{L}}<q_{1}<\min \left\{\frac{-\gamma^M+\sqrt{(\gamma^M)^2+4\eta_1^M\beta^Lp_1}}{2\eta_2^M}, \hat{x}_{2}, \check{x}_{2}\right\}, \\& \max \left\{\frac{-\gamma^L+\sqrt{(\gamma^L)^2+4\eta_2^L\beta^Mp_2 }}{2\eta_2^L},\hat{x}_{2}, \check{x}_{2}\right\} <q_{2} <\frac{p_2(\eta_1^L p_{2}+\gamma_1^{L}+\beta^{L})}{a^{M}}. \\\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\mathcal{A}X_0=X(w,X_0),\end{aligned}\end{equation*}

$\| (x_1,x_2) \|=\max\{|x_1|,|x_2|\},$

\begin{equation*}\begin{aligned}\left\{\begin{aligned}&\dot x{_{1}}(t)=\alpha(t) x_{2}(t)-\gamma_{1}(t)x_{1}(t)-\beta(t) x_{1}(t)-\eta_{1}(t)x^2_{1}(t),\\&\dot x{_{2}}(t) \leq \beta(t) x_{1}(t)-\gamma_{2}(t)x_{2}(t)-\eta_{2}(t) x^2_{2}(t).\\\end{aligned}\right.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\left\{\begin{aligned}&\dot x{_{1}}(t)=\alpha(t) x_{2}(t)-\gamma_{1}(t)x_{1}(t)-\beta(t) x_{1}(t)-\eta_{1}(t)x^2_{1}(t),\\&\dot x{_{2}}(t) = \beta(t) x_{1}(t)-\gamma_{2}(t)x_{2}(t)-\eta_{2}(t) x^2_{2}(t),\\\end{aligned}\right.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}u(t) < u(0) {\rm e}^{\int_{0}^{t} \left[f(u)\int_{-\infty}^{0}k_1(s)x_2(u+s){\rm d}s \right]{\rm d}u},\end{aligned}\end{equation*}

$x_i(t)\ (i=1,2)$ 的有界性, 易得 $u(t)$$t \in [0,T_\phi) 上有界. 显然这与假设矛盾, 因此 T_\phi=\infty, 即系统解的最大存在区间是 [0,\infty). M_1 >\max_{t \in R} |x^*(t)|, 其中 |x^*(t)|=\sum\limits_{i=1}^{2} x_i^*(t). 由于 t\rightarrow \infty$$\bar{x}_i(t) \rightarrow x_i^*(t)\ (i=1,2)$, 因此存在 $\tilde{T}>0$ 使得

\begin{equation*}\begin{aligned}\bar{x}_i(t) < M_1, \quad i=1,2, \quad t \geq \tilde{T},\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}x_i(t) < M_1, \quad i=1,2, \quad t \geq \tilde{T},\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\limsup_{t\rightarrow \infty} x_i(t) \leq M_1, \quad i=1,2.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\int_{-\infty }^{0}k_2(s)x_2(t+s){\rm d}s \leq M_1+\varepsilon,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\dot{u}(t) \leq u(t)[-e(t)u(t)+(M_1+\varepsilon) f(t)].\end{aligned}\end{equation*}

\begin{aligned}\dot{\tilde{u}}(t) = \tilde{u}(t)[-e(t)\tilde{u}(t)+(M_1+\varepsilon) f(t)],\end{aligned}

\begin{equation*}\begin{aligned}\limsup_{t\rightarrow \infty} \tilde{u}(t) \leq \bar{u},\end{aligned}\end{equation*}

\begin{aligned}u(t) \leq \tilde{u}(t), \quad \text{对所有 t \geq \bar{T} 成立}.\end{aligned}

\begin{equation*}\begin{aligned}\limsup_{t\rightarrow +\infty} x_i(t)\leq M(i=1,2), \quad \limsup_{t\rightarrow +\infty} u(t)\leq M,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}x_i(t)< M(i=1,2), \quad u(t) < M.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\int_{0}^{w} \left[-e(t) +f(t) \int_{-\infty}^{0} k_2(s)x_2^*(t+s){\rm d}s \right]{\rm d}t>0,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\liminf_{t\rightarrow +\infty} u(t) \geq \sigma_3.\end{aligned}\end{equation*}

\begin{aligned}\int_{0}^{w} \left[-e(t) +f(t) \int_{-\infty}^{0} k_2(s)x_2^*(t+s){\rm d}s -2f(t)\epsilon_0 \right]{\rm d}t> \epsilon_0.\end{aligned}

\begin{aligned}\left\{\begin{aligned}&\dot x{_{1}}(t)=\alpha(t) x_{2}(t)-\gamma_{1}(t)x_{1}(t)-\beta(t) x_{1}(t)-\eta_{1}(t)x^2_{1}(t),\\&\dot x{_{2}}(t)=\beta(t) x_{1}(t)-\gamma_{2}(t)x_{2}(t)-\eta_{2}(t) x^2_{2}(t)-2\xi d(t) x_2(t),\\\end{aligned}\right.\end{aligned}

\begin{equation*}\begin{aligned}\alpha^L \beta^L> (\gamma_2^M + 2\xi d^M) (\gamma_1^M+\beta^M),\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}| x^*_{i\xi}(t)- x_{i\xi}(t) | < \frac{\epsilon_0}{2}, \quad t>T_0, \quad i=1,2.\end{aligned}\end{equation*}

\begin{aligned}| x^*_{i\xi}(t)- x_i^*(t) | < \frac{\epsilon_0}{2}, \quad t \in R, \quad i=1,2.\end{aligned}

\begin{aligned}\limsup_{t \to \infty} u(t) \geq \xi,\end{aligned}

\begin{equation*}\begin{aligned}\limsup_{t \to \infty} u(t) < \xi,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}H \int_{-\infty }^{-t_1} k(s){\rm d}s<\xi,\end{aligned}\end{equation*}

\begin{equation*}\left\{\begin{aligned}\dot x_1(t) &=\alpha(t)x_2(t)-r_1(t)x_1(t)-\beta(t)x_1(t)-\eta_1(t)x^2_1(t),\\\dot x_2(t) & = \beta(t)x_1(t)-r_2(t)x_2(t)-\eta_2(t)x^2_2(t)- d(t)x_2(t) \left\{ \int_{-\infty }^{-t_1}+\int_{-t_1}^{0} \right\} k_1(s)u(t+s){\rm d}s,\\&\geq \beta(t)x_1(t)-r_2(t)x_2(t)-\eta_2(t)x^2_2(t)-2\xi d(t)x_2(t),\end{aligned}\right.\end{equation*}

\begin{aligned}x_i(t) \geq x_{i\xi}(t), \quad t \geq T_1+t_1,\end{aligned}

\begin{aligned}| x_{i\xi}(t)- x^*_{i\xi}(t) | < \frac{\epsilon_0}{2}, \quad t>T_2, \quad i=1,2.\end{aligned}

\begin{equation*}\begin{aligned}x_i(t) \geq x^*_i(t)-\epsilon_0, \quad t>T_2, \quad i=1,2.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\dot u(t)& \geq u(t)\bigg[-e(t)\epsilon_0 +f(t)\int_{-t_1}^{0}k_2(s)x_2(t+s){\rm d}s \bigg] \\& \geq u(t) \bigg[-e(t)+ f(t)\int_{-t_1}^{0}k_2(s)(x^*_2(t+s)-\epsilon_0){\rm d}s \bigg] \\& \geq u(t) \bigg[-e(t) + f(t)\int_{-t_1}^{0}k_2(s)x^*_2(t+s){\rm d}s-f(t)\epsilon_0\int_{-\infty}^{0}k_2(s){\rm d}s \bigg] \\& = u(t) \bigg[-e(t)+ f(t)\int_{-\infty}^{0}k_2(s)x^*_2(t+s){\rm d}s-f(t)\epsilon_0-f(t)\int_{-\infty}^{-t_1}k_2(s)x^*_2(t+s){\rm d}s \bigg] \\& \geq u(t) \bigg[-e(t)+ f(t)\int_{-\infty}^{0}k_2(s)x^*_2(t+s){\rm d}s-2f(t)\epsilon_0 \bigg],\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned} \liminf_{t \to \infty} u(t,\Phi _k)<\frac{\xi}{k+1}, \quad k=1,2,\cdots.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned} 0 < s_1^{(k)} < t_1^{(k)} < s_2^{(k)} < t_2^{(k)} < \cdots < s_q^{(k)} <t_q^{(k)} <\cdots,\end{aligned}\end{equation*}
\begin{equation*}\begin{aligned}s^{(k)}_q \rightarrow \infty, t^{(k)}_q \rightarrow \infty, \quad q \rightarrow \infty,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}u(s^{(k)}_q,\Phi_k)=\xi, \quad u(t^{(k)}_q,\Phi_k)=\frac{\xi}{k+1},\end{aligned}\end{equation*}
\begin{aligned}\frac{\xi}{k+1} < u(t,\Phi_k) < \xi <\epsilon_0, \quad t \in (s^{(k)}_q,t^{(k)}_q).\end{aligned}

\begin{aligned}H_0^{(k)} \int_{-\infty }^{-t^{(k)}_1} k(s){\rm d}s<\xi,\end{aligned}

\begin{equation*}\begin{aligned}\dot u(t,\Phi_k) &=u(t,\Phi_k) \bigg[-e(t)u(t,\Phi_k)+f(t)\int_{-\infty }^{0}k_2(s)x_2(t+s,\Phi_k){\rm d}s \bigg]\\& \geq u(t,\Phi_k)( -e(t)M),\\\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}u(t^{(k)}_q,\Phi_k)\geq u(s^{(k)}_q+L_1+t_1,\Phi_k){\rm e}^{\int_{s^{(k)}_q+L_1+t_1}^{t^{(k)}_q} \big[-e(t)+ f(t)\int_{-\infty }^{0}k_2(s)x_2(t+s,\Phi_k){\rm d}s- 2f(t)\epsilon_0 \big]{\rm d}t},\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\frac{\xi}{k+1} & \geq \frac{\xi}{k+1} {\rm e}^{\int_{s^{(k)}_q+L_1+t_1}^{t^{(k)}_q} \big[-e(t)+ f(t)\int_{-\infty }^{0}k_2(s)x_2(t+s,\Phi_k){\rm d}s- 2f(t)\epsilon_0 \big]{\rm d}t}& > \frac{\xi}{k+1},\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\liminf_{t\rightarrow +\infty} x_2(t) \geq \sigma_2.\end{aligned}\end{equation*}

\begin{aligned}x_i^*(t)-\delta > \delta (i=1,2),\end{aligned}
\begin{aligned}\int_{0}^{w} \bigg[-e(t)\sigma_3 +2 \delta f(t)\bigg]{\rm d}t < -\delta.\end{aligned}

\begin{aligned}\left\{\begin{aligned}&\dot x{_{1}}(t)=\alpha(t) x_{2}(t)-\gamma_{1}(t)x_{1}(t)-\beta(t) x_{1}(t)-\eta_{1}(t)x^2_{1}(t),\\&\dot x{_{2}}(t)=\beta(t) x_{1}(t)-\gamma_{2}(t)x_{2}(t)-\eta_{2}(t) x^2_{2}(t)-2 \tau d(t) x_2(t),\\\end{aligned}\right.\end{aligned}

\begin{equation*}\begin{aligned}\alpha^L \beta^L> (\gamma_2^M + 2 \tau d^M) (\gamma_1^M+\beta^M),\end{aligned}\end{equation*}

\begin{aligned}| x^*_{i \tau}(t)- x^*(t) | < \frac{\delta}{2}, \quad t \in \mathbb{R}, \quad i=1,2.\end{aligned}

\begin{aligned}\limsup_{t \to \infty} x_2(t) \geq \tau,\end{aligned}

\begin{equation*}\begin{aligned}\limsup_{t \to \infty} x_2(t) \leq \tau,\end{aligned}\end{equation*}

\begin{aligned}x_2(t)< \tau < \delta, \quad t>D_1.\end{aligned}

\begin{equation*}\begin{aligned} H_0 \int_{-\infty }^{-t_2} k(s){\rm d}s< \tau <\delta,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\dot u(t) &= u(t)\Bigg[-e(t)u(t)+f(t)\left\{\int_{-\infty }^{-t_2} + \int_{-t_2}^{0} \right\} k_2(s)x_2(t+s){\rm d}s \Bigg], \\&\leq u(t)[-e(t)\sigma_3 + 2 \delta f(t)],\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\dot x_2(t,\varphi_k) & > -r_2(t)x_2(t,\varphi_k)-\eta_2(t)x^2_2(t,\varphi_k)\\& -d(t)x_2(t,\varphi_k) \left \{ \int_{-\infty }^{-t^{(k)}_2}+\int_{-t^{(k)}_2}^{0} \right\} k_1(s)u(t+s,\varphi_k){\rm d}s \\&\geq -(r_2(t)+\eta_2(t)M+2d(t)M)x_2(t,\Phi_k)\\&\geq -\Psi^m x_2(t,\Phi_k),\end{aligned}\end{equation*}

\begin{aligned}V^{(k)}_q-S^{(k)}_q \geq \frac{\ln k}{\Psi^m},\quad \forall q \geq N_1^{(k)},k=1,2,\cdots.\end{aligned}

\begin{aligned}M {\rm e}^{\int_{t}^{t+a} \big[-e(u)\sigma_3+2 \delta f(u)\big]{\rm d}u} < \tau < \delta.\end{aligned}

\begin{aligned}H_0^{(k)}\int_{-\infty}^{D_3^{(k)}-S_q^{(k)}}k(s){\rm d}s <\frac{\delta}{2},\end{aligned}

\begin{aligned}M\int_{-\infty}^{-t_2}k(s){\rm d}s <\frac{ \delta }{2}.\end{aligned}

\begin{aligned}V^{(k)}_q-S^{(k)}_q>P_2+L_2, \quad k\geq N_0, q\geq N^{(k)}_2.\end{aligned}

\begin{equation*}\begin{aligned}\dot u(t,\varphi_k) &=u(t,\varphi_k)\Bigg[-e(t)u(t,\varphi_k)+f(t) \left\{\int_{-\infty }^{D_3^{(k)}}+\int_{D_3^{(k)}}^{S^{(k)}_q}+\int_{S^{(k)}_q}^{t} \right\} k_2(s-t)x_2(s,\varphi_k){\rm d}s \Bigg] \\& \leq u(t,\varphi_k) \Bigg[-e(t)\sigma_3+f(t) \left \{H_0^{(k)} \int_{-\infty }^{D_3^{(k)}-S^{(k)}_q}+ M \int_{-\infty}^{-t_2} + \delta \int_{-\infty}^{0} \right\}k_2(s){\rm d}s \Bigg]\\& \leq u(t,\varphi_k)[-e(t) \sigma_3 +2 \delta f(t)],\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}u(t,\varphi_k) \leq u(S^{(k)}_q+t_2) {\rm e}^{\int_{S^{(k)}_q+t_2}^{t} \big[-e(u)\sigma_3+2 \delta f(u)\big]{\rm d}u},\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}u(t,\varphi_k) \leq M {\rm e}^{\int_{S^{(k)}_q+t_2+p_2}^{t} \big[-e(u)\sigma_3+2 \delta f(u)\big]{\rm d}u} < \delta,\end{aligned}\end{equation*}

\begin{equation*}\left\{\begin{aligned}\dot x_1(t,\varphi_k) &=\alpha(t)x_2(t,\varphi_k)-r_1(t)x_1(t,\varphi_k)-\beta(t)x_1(t,\varphi_k)-\eta_1(t)x^2_1(t,\varphi_k),\\\dot x_2(t,\varphi_k) &= \beta(t)x_1(t,\varphi_k)-r_2(t)x_2(t,\varphi_k)-\eta_2(t)x^2_2(t,\varphi_k)\\& -d(t)x_2(t,\varphi_k) \left\{\int_{-\infty }^{D_3^{(k)}}+\int_{D_3^{(k)}}^{S^{(k)}_q+P_2}+\int_{{S^{(k)}_q+P_2}}^{t} \right\} k_1(s-t)u(s,\varphi_k){\rm d}s \\&\geq \beta(t)x_1(t,\varphi_k)-r_2(t)x_2(t,\varphi_k)-\eta_2(t)x^2_2(t,\varphi_k)\\& -d(t)x_2(t,\varphi_k) \left\{H_0^{(k)} \int_{-\infty }^{D_3^{(k)}-t} + M \int_{-\infty}^{S^{(k)}_q+P_2-t} + \delta \int_{-\infty}^{0}\right \}k_1(s){\rm d}s \\&\geq \beta(t)x_1(t,\varphi_k)-r_2(t)x_2(t,\varphi_k)-\eta_2(t)x^2_2(t,\varphi_k)-2 \tau d(t)x_2(t,\varphi_k).\end{aligned}\right.\end{equation*}

\begin{equation*}\begin{aligned}x_i(t,\varphi_k) \geq x_{i \tau}(t), \quad \forall t \in \big[S^{(k)}_q+t_2+P_2,V^{(k)}_q \big].\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}|x_{i \tau}(t,t_0,x_0)-x^*_{i \tau}(t)|< \frac{\delta}{2}, \quad t\geq S^{(k)}_q+P_2+L_2,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}x_i(t,\Phi_k) \geq x_{i \tau}(t,\Phi_k) \geq x^*(t,\Phi_k)-\delta > \delta,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\int_{0}^{w} \left[-e(t) +f(t) \int_{-\infty}^{0} k_2(s)x_2^*(t+s){\rm d}s \right]{\rm d}t>0,\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\liminf_{t\rightarrow +\infty} x_1(t) \geq \sigma_1.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}\dot x_1(t)&=\alpha(t)x_2(t)-\gamma_1(t)x_1(t)-\beta(t)x_1(t)-\eta_1(t)x^2_1(t) \\& \geq \alpha(t) \sigma_2 -(\gamma_1(t)+\beta(t)+\eta(t) M)x_1(t).\\\end{aligned}\end{equation*}

\begin{aligned}\dot {\tilde{x}}_1(t)& = \alpha(t) \sigma_2 -(\gamma_1(t)+\beta(t)+\eta(t) M)\tilde{x}_1(t),\end{aligned}

\begin{equation*}\begin{aligned}\liminf_{t\rightarrow \infty} \tilde{x}_1(t) \geq \sigma_1.\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}x_1(t) \geq \tilde{x}_1(t),\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}&\liminf_{t\rightarrow \infty} x_i(t) \geq m(i=1,2), \quad \liminf_{t\rightarrow \infty} u(t)>m.\end{aligned}\end{equation*}

## 4 数值模拟

$\int_{-\infty}^{0} k_1(s){\rm d}s=\int_{-\infty}^{0} k_2(s){\rm d}s=\int_{-\infty}^{0} {\rm e}^{s}{\rm d}s$(参见文献 [11]).

$g(t)=g_0[1+g_j \sin(\frac{2 \pi t}{w})],$

### 图1

\begin{equation*}\begin{aligned}&\int_{0}^{w} \Big[-e(t)+f(t) \int_{-\infty}^{0} k_2(s)x^*_2(t+s){\rm d}s \Big]{\rm d}t\\= \ & \int_{0}^{w} \Big[-0.16(1+0.1\sin(\frac{2\pi t}{365})+0.03(1+0.3\sin(\frac{2\pi t}{365}))*26.4 \Big]{\rm d}t\\> \ & 0.3944 >0,\end{aligned}\end{equation*}

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Vance R R, Coddington E A.

A nonautonomous model of population growth

J Math Biol, 1989, 27(5): 491-506

PMID:2794800

With x = population size, the nonautonomous equation x = xf(t, x) provides a very general description of population growth in which any of the many factors that influence the growth rate may vary through time. If there is some fixed length of time (usually long) such that during any interval of this length the population experiences environmental variability representative of the variation that occurs in all time, then definite conclusions about the population's long-term behavior apply. Specifically, conditions that produce population persistence can be distinguished from conditions that cause extinction, and the difference between any pair of solutions eventually converges to zero. These attributes resemble corresponding features of the related autonomous population growth model x = xf(x).

Chen F D, Yang J H, Chen L J.

Note on the persistent property of a feedback control system with delays

Nonlinear Anal: Real World Appl, 2010, 11(2): 1061-1066

Liu S Q, Chen L S, Agarwal R.

Recent progress on stage-structured population dynamics

Math Comput Model, 2002, 36: 1319-1460

Cui J A, Chen L S, Wang W D.

The effect of dispersal on population growth with stage-structure

Comput Math Appl, 2000, 39(1/2): 91-102

Wang C Y, Li L R, Zhang Q Y.

Dynamical behaviour of a Lotka-Volterra competitive-competitive-cooperative model with feedback controls and time delays

J Biol Dyn, 2019, 13(1): 43-68

Gopalsamy K, Weng P X.

Feedback regulation of logistic growth

Internat J Math Sci, 1993, 16(1): 177-192

Jana S, Kar T K.

A mathematical study of a prey-predator model in relevance to pest control

Nonlinear Dyn, 2013, 74: 667-683

Fu J B, Chen L S.

Positive periodic solution of multiple species comptition system with ecological environment and feedback controls

Acta Math Sci, 2017, 37A(3): 553-561

Basir F A, Noor M H.

A model for pest control using integrated approach: Impact of latent and gestation delays

Nonlinear Dyn, 2022, 108(2): 1805-1820

Teng Z D, Chen L S.

Permanence and extinction of periodic predator-prey systems in a patchy environment with delay

Nonlinear Anal: Real World Appl, 2003, 4(2): 335-364

Pu L Q, Adam B, Lin Z Q.

Extinction in a nonautonomous competitive system with toxic substance and feedback control

J Appl Anal Comput, 2019, 9(5): 1838-1854

Chen F D, Li Z, Huang Y J.

Note on the permanence of a competitive system with infinite delay and feedback controls

Nonlinear Anal: Real World Appl, 2007, 8(2): 680-687

Guo K, Song K Y, Ma W B.

Existence of positive periodic solutions of a delayed periodic Microcystins degradation model with nonlinear functional responses

Appl Math Lett, 2022, 131: 108056

Wang C Y, Li N, Jiang T, Yang Q.

On a Nonlinear non-autonomous ratio-dependent food chain model with delays and feedback controls

Acta Math Sci, 2022, 42A(1): 245-268

Shi L, Qi L X, Zhai S L.

Periodic and almost periodic solutions for a non-autonomous respiratory disease model with a lag effect

Acta Math Sci, 2022, 42B(1): 187-211

Zhang L, Teng Z D.

Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent

J Math Anal Appl, 2008, 338(1): 175-193

Hu H X, Teng Z D, Jiang H J.

Permanence of the nonautonomous competitive systems with infinite delay and feedback controls

Nonlinear Anal: Real World Appl, 2009, 10(4): 2420-2433

Liu Z Q, Kang S M.

Applications of Schauder's fixed-point theorem with respect to iterated functional equations

Appl Math Lett, 2001, 14(8): 955-962

Zhang G Q, Lin Y Q. Lecture Notes on Functional Analysis. Beijing: Peking University Press, 2004

Hale J K. Introduction to Functional Differential Equations. New York: Springer, 1993

Smith H L.

Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems

Providence, RI: American Mathematical Society, 1995

Teng Z D, Chen L S.

The positive periodic solutions of periodic Kolmogorov type systems with delays

Acta Math Appl Sin, 1999, 22(3): 446-456

Wu H, Chen W, Wang N, et al.

A delayed stage-structure brucellosis model with interaction among seasonality, time-varying incubation and density-dependent growth

Int J Biomath, 2023, 16(6): 2250114

Pandey S, Ghosh U, Das D, et al.

Rich dynamics of a delay-induced stage-structure prey-predator model with cooperative behaviour in both species and the impact of prey refuge

Math Comput Simul, 2024, 216: 49-76

Hu D P, Li Y Y, Liu M, et al.

Stability and Hopf bifurcation for a delayed predator-prey model with stage structure for prey and Ivlev-type functional response

Nonlinear Dynam, 2020, 99: 3323-3350

/

 〈 〉