Permanence and existence of periodic solutions for a generalized system with feedback control

Abstract

Sufficient conditions are obtained for the permanence and the existence of positive periodic solutions of the delay differential system with feedback control

(0.1)$$\left\{\begin{array}{c}\frac{\mathit{dx}}{\mathit{dt}}=x(t)[F(t\text{,}x(t-{\tau }_1(t))\text{,}\dots \text{,}x(t-{\tau }_n(t)))-c(t)u(t-\delta (t))]\text{,}\hfill \\ \frac{\mathit{du}}{\mathit{dt}}=-\eta (t)u(t)+a(t)x(t-\sigma (t)).\hfill \end{array}\right.$$

The method involves the application of estimation for uniform upper and lower bounds of solutions. When these results are applied to some special population models with multiple delays, some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in Chen et al. [F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587] and Huo and Li [H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46].

Introduction

For dynamical systems, if the environment is not temporally constant (e.g., seasonal effects of weather, food supplies, mating habits, etc.), then the parameters become time dependent. It has been suggested by Nicholson [1] that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. Pianka [2] discussed the relevance of periodic environment to evolutionary theory. On the other hand, ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. The control variables discussed in most literatures are constants or time dependent (see [3], [4]).

Recently, logistic model with several delays and feedback control (see [5])

$$\left\{\begin{array}{c}\frac{\mathit{dx}(t)}{\mathit{dt}}=x(t)\left[r(t)-\sum _{i=1}^n{a}_i(t)x(t-{\tau }_i(t))-c(t)u(t-\delta (t))\right]\text{,}\hfill \\ \frac{\mathit{du}(t)}{\mathit{dt}}=-\eta (t)u(t)+a(t)x(t-\sigma (t))\text{,}\hfill \end{array}\right.$$

the mutualism delay model with feedback control (see [6])

$$\left\{\begin{array}{c}\frac{\mathit{dx}(t)}{\mathit{dt}}=r(t)x(t)\left[\frac{K(t)+{\sum }_{i=1}^n{a}_i(t)x(t-{\tau }_i(t))}{1+{\sum }_{i=1}^n{c}_i(t)x(t-{\tau }_i(t))}-x(t-\alpha (t))-c(t)u(t-\delta (t))\right]\text{,}\hfill \\ \frac{\mathit{du}(t)}{\mathit{dt}}=-\eta (t)u(t)+a(t)x(t-\sigma (t))\text{,}\hfill \end{array}\right.$$

the so-called Michaelis–Menton single-species growth model with feedback control (see [5])

$$\left\{\begin{array}{c}\frac{\mathit{dx}(t)}{\mathit{dt}}=r(t)x(t)\left[1-\sum _{i=1}^n\frac{a_i(t)x(t-{\tau }_i(t))}{1+c_i(t)x(t-{\tau }_i(t))}-c(t)u(t-\delta (t))\right]\text{,}\hfill \\ \frac{\mathit{du}(t)}{\mathit{dt}}=-\eta (t)u(t)+a(t)x(t-\sigma (t))\text{,}\hfill \end{array}\right.$$

have been studied by several authors. The coincidence degree theory [7] has been used to establish the existence of periodic solutions.

For more works related to the study of feedback control systems, we refer the reader to [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [19], [20] and the references cited therein.

Motivated by the above literature, in this paper, we consider the following general nonlinear nonautonomous delay differential system with feedback control

$$\left\{\begin{array}{c}\frac{\mathit{dx}}{\mathit{dt}}=x(t)\left[F(t\text{,}x(t-{\tau }_1(t))\text{,}\dots \text{,}x(t-{\tau }_n(t)))-c(t)u(t-\delta (t))\right]\text{,}\hfill \\ \frac{\mathit{du}}{\mathit{dt}}=-\eta (t)u(t)+a(t)x(t-\sigma (t))\text{,}\hfill \end{array}\right.$$

where $F(t\text{,}{z}_1\text{,}{z}_2\text{,}\dots \text{,}{z}_n\text{,}{z}_{n+1})\in C(R^{n+2}\text{,}[0\text{,}\infty ))\text{,}{\tau }_i(t)(i=1\text{,}2\text{,}\dots \text{,}n)\text{,}\delta (t)\text{,}\sigma (t)\text{,}c(t)\text{,}\eta (t)\text{,}a(t)\in C(R\text{,}(0\text{,}\infty ))$, we further assume that all of the above functions are $\omega$-periodic in t and $\omega >0$ is a constant. By employing the differential inequality developed by Chen et al. [6] and Fan and Wang [18], we obtain sufficient conditions of the permanence and the existence of periodic solutions for system (1.4). Then we apply the obtained criteria to some population models with multiple delays and feedback control, such as the periodic logistic equation with several delays (1.1), (1.2), (1.3). Some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in [6], [5].

For biological meaning, we only consider (1.4) with the following initial conditions

$$\begin{array}{cc}\hfill & x(t)=\phi (t)⩾0\text{,}t\in [-\tau \text{,}0]\text{,}\phi (0)>0\text{,}\hfill \\ \hfill & u(t)=\psi (t)⩾0\text{,}t\in [-\tau \text{,}0]\text{,}\psi (0)>0\text{,}\hfill \end{array}$$

where $\phi$ and $\psi$ are continuous on $[-\tau \text{,}0]$. Here

$$\tau =\max \left\{\underset{t\in [0\text{,}\omega ]}{\max }{\tau }_i(t)(i=1\text{,}2\text{,}\dots \text{,}n)\text{,}\underset{t\in [0\text{,}\omega ]}{\max }\delta (t)\text{,}\underset{t\in [0\text{,}\omega ]}{\max }\sigma (t)\right\}\text{.}$$

By the fundamental theory of functional differential equations [21], it is clear that the solution of (1.4) exists and remains nonnegative on its maximal existence interval.