## Existence and Stability of Periodic Solution for a Lasota-Wazewska Model with Discontinuous Harvesting

Yang Chao,1, Li Runjie2

 基金资助: 国家自然科学基金.  11801042长沙市科技计划.  K1705081

 Fund supported: the NSFC.  11801042the Changsha Science and Technology Plan.  K1705081

Abstract

In this paper, we study a class of mixed time-varying delayed Lasota-Wazewska model with discontinuous harvesting, which is described by a periodic nonsmooth dynamical system. Base on nonsmooth analysis, Kakutani's fixed point method and the generalized Lyapunov method, easily verifiable delay-independent criteria are established to ensure the existence and exponential stability of positive periodic solutions. Finally, we give an example to further illustrate the effectiveness of our main results.

Keywords： Lasota-Wazewska model ; Mixed delayed system ; Discontinuous harvesting ; Positive periodic solution ; Exponential stability

Yang Chao, Li Runjie. Existence and Stability of Periodic Solution for a Lasota-Wazewska Model with Discontinuous Harvesting. Acta Mathematica Scientia[J], 2019, 39(4): 785-796 doi:

## 1 引言

$$$x' (t) = -ax(t)+be^{-cx(t-\tau)}, \ \ \ \ t\geq 0,$$$

(H1) $H\geq 0$是几乎处处有界的且单调非减, $H$除了可数个不连续点之外在${\Bbb R}$上是连续的,此外, $H$在任意${\Bbb R}$中的紧致区间上仅有有限个不连续点,并且$H(0) = H(0^{+}) = 0$.

${\Bbb R}^{n}(n\geq 1)$表示$n$维欧几里得空间,若$X\subseteq{\Bbb R}^{n}$,则定义

$$$\frac{{\rm d}x}{{\rm d}t} = f(t, x_{t}),$$$

(ⅱ)存在可测函数$\gamma:[0, T)\rightarrow{\Bbb R}^{n}$满足$\gamma(t)\in \overline{co}[H(x(t))]$对几乎处处$t\in [0, T)$

$\begin{eqnarray} x' (t)& = &-a(t)x(t)+\sum\limits_{i = 1}^n b_{i}(t)e^{-c_{i}(t)x(t-\tau_{i}(t))}\\ &&+\sum\limits_{i = 1}^n p_{i}(t)\int^{\infty}_{0}K_{i}(s)e^{-q_{i}(t)x(t-s)}{\rm d}s-d(t)\gamma(t). \end{eqnarray}$

$$$x_{t_{0}} = \varphi \ \ \mbox{且} \ \ \varphi(0)>0.$$$

$x_{t}(t_{0}, \varphi)(x(t;t_{0}, \varphi))$是模型(1.2)在$x_{t_{0}}(t_{0}, \varphi) = \varphi $$t_{0}\in {\Bbb R} 定义下的初值.令 [t_{0}, \eta(\varphi)) 是解 x_{t}(t_{0}, \varphi) 的一个最大存在区间. 引理1.1[5] 假设 \Omega 是Banach空间 X 的一个紧有界凸子,若存在一个集值映射 \phi:\Omega\rightarrow P_{kc}(\Omega) 是上半连续的凸紧映射,则 \phi$$ \Omega$中具有一个不动点,即存在$x\in \Omega $$x\in \phi(t) . 定义1.3[12] 称 V(x):{\Bbb R}^{n}\rightarrow{\Bbb R}^{n} 是C -正则函数,当且仅当满足 (ⅰ) V(x)$$ {\Bbb R}^{n}$是正则的;

(ⅱ) $V(x)$是正定的,即对$x\neq0 $$V(x)>0$$ V(0) = 0$;

(ⅲ) $V(x)$是径向无界,即当$\|x\|\rightarrow+\infty$时有$V(x)\rightarrow+\infty$.

## 2 周期解的存在性

(H2)存在非负且单调非减的函数$W$满足

假设$x(t)$是系统(1.2)的一个$\omega$ -周期解,根据(1.5)式可知有

$\begin{eqnarray} [x(t)e^{\int^{t}_{0}a(s){\rm d}s}]' \in e^{\int^{t}_{0}a(s){\rm d}s}{\mathfrak F}(s, x(s)), \end{eqnarray}$

$\begin{eqnarray} x(t) = \int^{t+\omega}_{t} G(t, s){\mathfrak F}(s, \gamma(s)){\rm d}s, \end{eqnarray}$

为方便起见,分别记$x(t;t_{0}, \varphi) $$\eta(t_{0}, \varphi)$$ x(t) $$\eta(\varphi) .首先证明 $$x(t)>0 , \ \ \ t\in(t_{0}, \eta({\varphi})).$$ 通过反证法,假设(2.3)式不成立,则必然存在 t^{*}\in(t_{0}, \eta(\varphi)) ,满足当 t\in[t_{0}, t^{*})$$ x(t)>0 $$x(t^{*}) = 0 ,则根据假设(H1)–(H2)有 矛盾.故有(2.3)式成立,所以当 t\in[t_{0}, \eta(\varphi)) 时可得 根据计算当 t\in[t_{0}, \eta(\varphi)) 时可得 \begin{eqnarray} x(t)\leq e^{-a^{-}(t-t_{0})}x(t_{0})+\frac{\sum\limits_{i = 1}^{n}(b^{+}_{i}+q_{i}^{+}\int^{\infty}_{0}K_{i}(s){\rm d}s)}{a^{-}}(1-e^{-a(t-t_{0})}). \end{eqnarray} 故根据(2.3)式和延拓定理可知 \eta(\varphi) = \infty . 根据 \eta(\varphi) = \infty 和(2.4)式可知 \lim\limits_{t\rightarrow\infty}\sup x(t)\leq (\sum\limits_{i = 1}^n (b^{+}_{i}+q_{i}^{+}\int^{\infty}_{0}K_{i}(s){\rm d}s))/a^{-} .得证. 命题2.3 如果假设条件(H1)–(H2)满足.再给出以下假设 (H3) 则对任意的 \theta\in (-\infty, 0]$$ C_{0} = \{\varphi|\varphi\in C, R_{2}<\varphi(\theta)<R_{1}\}$是系统(1.2)的一个正不变集,其中$R_{1} = (\sum\limits_{i = 1}^n b_{i}^{+}+\sum\limits_{i = 1}^n p_{i}^{+}\int_{0}^{\infty}K_{i}(s){\rm d}s)/a^{-}$,

令$\varphi\in C_{0}$.根据上述命题2.2可知,在区间$[t_{0}, \infty)$上, $x(t;t_{0}, \varphi)$是系统(1.2)满足$x(t;t_{0}, \varphi)>0(t\geq t_{0})$的解.为简便起见,记$x(t;t_{0}, \varphi) $$x(t) .对任意的 t\in[t_{0}, T) ,要证 \begin{eqnarray} 0<x(t)<R_{1}, \end{eqnarray} 假设(2.5)式不成立,即存在 t_{1}\in[t_{0}, T) 对所有的 t\in(-\infty, t_{1}) 满足 则由系统(1.2)和(2.1)可得 这就得到矛盾,则可知(2.5)式成立. 接下来证明对任意的 t\in[t_{0}, T) \begin{eqnarray} x(t)>R_{2}, \end{eqnarray} 若不然,则存在 t_{2}\in(t_{0}, \infty) 对任意的 t\in(-\infty, t_{2}) 满足 \begin{eqnarray} x(t_{2}) = R_{2}, \ \ \ \ x(t)>R_{2}, \end{eqnarray} 根据(2.1)和(2.7)计算可得 则可推出矛盾,故(2.7)式成立.综合(2.5)和(2.7)式,命题得证. 命题2.4 如果假设条件(H1)–(H3)满足,则对每一个 x\in C_{0} ,集值映射 \phi(x) 是凸集. 对任意 x\in C_{0} ,设 u\in \phi(x)$$ u^{*}\in \phi(x)$.则根据微分包含理论,存在可测函数$\gamma:[0, T)\rightarrow{\Bbb R}^{n}$满足$\gamma(t)\in \overline{co}[H(x(t))] $$|\gamma(t)|<W(t) ,同样存在可测函数 \gamma^{*}:[0, T)\rightarrow{\Bbb R}^{n} 满足 \gamma^{*}(t)\in \overline{co}[H(x(t))]$$ |\gamma^{*}(t)|<W(t)$,则可得下式

$0\leq\lambda\leq1$,注意到$\lambda\gamma(t)+(1-\lambda)\gamma^{*}(t)\in \overline{co}[H(x(t))]$,即有

根据Ascoli-Arzela定理,我们只需证明$\phi(C_{0})$是一致有界且等度连续的即可.

$t_{1}\rightarrow t_{2}$时可知不等式的右端趋于0.因此可得当$t_{1}\rightarrow t_{2} $$\|u(t_{1})-u(t_{2})\|\rightarrow 0 .这就证明了 \phi(C_{0}) 是等度连续的.命题得证. 命题2.6 假设条件(H1)–(H3)满足,则集值映射 \phi: C_{0}\rightarrow P_{kc}(C_{0}) 是上半连续(USC)映射. 只需证明 \phi 是闭图像即可.令 {\Bbb F}(t, x(t)) = -a(t)x(t)+{\mathfrak F}(t, x(t)) ,令 |||{\Bbb F}(t, x(t))||| = \sup\{|u|:u\in{\Bbb F}(t, x(t))\} , L^{1}([0, \omega], {\Bbb R}^{n}) 表示一个Banach空间,空间内所有函数 u:[0, \omega]\rightarrow{\Bbb R}^{n} 都是勒贝格可积的.定义算子 {\Bbb F}:C_{0}\rightarrow L^{1}([0, \omega], {\Bbb R}^{n}) 容易得到对每一个固定的 x\in C_{0} 来说, {\Bbb F}(x) 是非空的. 考虑线性连续算子 \hbar:L^{1}([0, \omega], {\Bbb R}^{n})\rightarrow C([0, \omega], {\Bbb R}^{n}) , \hbar u(t) = \int^{t+\omega}_{t}G(t, s)u(s){\rm d}s,$$ t\in[0, \omega].$因此根据定义1.4可知$\phi = L\circ{\Bbb F}$是一个闭图像算子.这就意味着$\phi$是一个上半连续(USC)算子.得证.

## 3 全局指数稳定性

(H4)

令$\varphi, \varphi^{*}\in C_{0}$,为方便起见,分别把$x(t;t_{0}, \varphi) $$x(t;t_{0}, \varphi^{*}) 记作 x(t)$$ x^{*}(t)$,根据上一小节的命题2.3可知对任意的$t\in[t_{0}, T)$

$t\geq t_{0}$,再考虑下述Lyapunov函数

## 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

Wazewska-Czyzewska , Maria , Lasota , et al.

Mathematical problems of the dynamics of a system of red blood cells

Mat Stos, 1976, 6 (3): 23- 40

Long Z .

Global asymptotic stability of pseudo almost periodic solutions to a Lasota-Wazewska model with distributed delays

Electronic Journal of Qualitative Theory of Differential Equations, 2015, 2015 (51): 1- 3

Duan L , Huang L , Chen Y .

Global exponential stability of periodic solutions to a delay Lasota-Wazewska model with discontinuous harvesting

Proceedings of the American Mathematical Society, 2016, 144 (2): 561- 573

Fink A . Almost Periodic Differential Equations. Berlin: Springer, 1974

Clark C .

The optimal management of renewable resources

Natural Resource Modelling, 1990, 43 (1): 31- 52

Martin A , Ruan S .

Predator-prey models with delay and prey harvesting

Journal of Mathematical Biology, 2001, 43 (3): 247- 267

Xiao D , Jennings L .

Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting

SIAM Journal on Applied Mathematics, 2005, 65 (3): 737- 753

Pei Y , Lv Y , Li C .

Evolutionary consequences of harvesting for a two-zooplankton one-phytoplankton system

Applied Mathematical Modelling, 2012, 36 (4): 1752- 1765

Costa M , Meza M .

Dynamical stabilization of grazing systems:An interplay among plant-water interaction, overgrazing and a threshold management policy

Mathematical Biosciences, 2006, 204 (2): 250- 259

Bischi G , Lamantia F , Tramontana F .

Sliding and oscillations in fisheries with on-off harvesting and different switching times

Communications in Nonlinear Science and Numerical Simulation, 2014, 19 (1): 216- 229

Huang L , Wu J .

The role of threshold in preventing delay-induced oscillations of frustrated neural networks with McCulloch-Pitts nonlinearity

International Journal of Mathematics, Game Theory and Algebra, 2001, 11 (6): 71- 100

Filippov A .

Differential equations with discontinuous right-hand sides and differential inclusions

Nonlinear Analysis and Nonlinear Differential Equations, 1988, 154 (2): 265- 288

Adhikari S , Yang C , Kim H , et al.

Memristor bridge synapse-based neural network and its learning

IEEE Transactions on Neural Networks and Learning Systems, 2012, 23 (9): 1426- 1435

Chua L .

Memristor-the missing circuit element

IEEE Transactions on Circuit Theory, 1971, 18 (5): 507- 519

Sprott J .

A new class of chaotic circuit

Physics Letters A, 2000, 266 (1): 19- 23

Chen G , Ueta T .

Yet another chaotic attractor

International Journal of Bifurcation and Chaos, 1999, 9 (7): 1465- 1466

Liang Z .

Asymptotically periodic solutions of a class of second order nonlinear differential equations

Proceedings of the American Mathematical Society, 1987, 99 (4): 693- 699

Li J , Wang Z .

Existence and global attractivity of positive periodic solutions of a survival model of red blood cells

Computers and Mathematics with Applications, 2005, 50 (1/2): 41- 47

Xu W , Li J .

Global attractivity of the model for the survival of red blood cells with several delays

Annals of Differential Equations, 1998, 14 (2): 259- 265

Wang J , Fec M , Zhou Y .

Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations

Communications in Nonlinear Science and Numerical Simulation, 2013, 18 (2): 246- 256

Zhang C . Almost Periodic Type Functions and Ergodicity. Berlin: Springer, 2009

Zhang C .

Pseudo almost periodic solutions of some differential equations

Journal of Mathematical Analysis and Applications, 1994, 181 (1): 62- 76

Filippov A . Differential Equations with Discontinuous Righthand Sides:Control Systems. New York: Springer Science and Business Media, 1988

Huang L , Guo Z , Wang J . Theory and Application of Right end Discontinuous Differential Equations. Beijing: Science Press, 2011

Zhang H , Zheng Z , Jiang W .

Existence of solutions for nonlinear fractional functional differential equations

Acta Mathematica Scientia, 2011, 31A (2): 289- 297

Lu H , Wang K .

Single population model with periodic coefficients and its optimal harvesting strategy

Acta Mathematica Scientia, 2005, 25A (6): 176- 182

Aubin J , Cellina A . Differential Inclusions. Berlin: Springer-Verlag, 1984: 8- 13

Forti M , Nistri P .

Global convergence of neural networks with discontinuous neuron activations

IEEE Transactions on Circuits and Systems Ⅰ:Fundamental Theory and Applications, 2003, 50 (11): 1421- 1435

Forti M , Grazzini M , Nistri P , et al.

Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations

Physica D:Nonlinear Phenomena, 2006, 214 (1): 88- 99

Yan J .

Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model

Journal of Mathematical Analysis and Applications, 2003, 279 (1): 111- 120

Wang D , Huang L .

Periodicity and multi-periodicity of generalized Cohen-Grossberg neural networks via functional differential inclusions

Nonlinear Dynamics, 2016, 1- 20

Hamaya Y .

Almost periodic dynamic of a discrete Wazewska-Lasota model

Communications in Mathematics and Applications, 2013, 4 (3): 189- 199

Liu G , Yan W , Yan J .

Periodic solutions for a class of neutral functional differential equations with infinite delay

Nonlinear Analysis:Theory, Methods and Applications, 2009, 71 (1): 604- 613

Saker S .

Oscillation and global attractivity in hematopoiesis model with periodic coefficients

Applied Mathematics and Computation, 2003, 142 (2): 477- 494

Liu X , Meng J .

The positive almost periodic solution for Nicholson-type delay systems with linear harvesting terms

Applied Mathematical Modelling, 2012, 36 (7): 3289- 3298

Fan M , Wang K .

Global existence of positive periodic solutions of periodic predator-prey system with infinite delays

Journal of Mathematical Analysis and Applications, 2001, 262 (1): 1- 11

Liu M , Wang K .

Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays

Communications in Nonlinear Science and Numerical Simulation, 2012, 17 (8): 3115- 3123

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