Hopf bifurcation analysis of a turbidostat model with discrete delay

Abstract

In this contribution, the dynamic behaviors of a turbidostat model with discrete delay are investigated. With the delay of digestion chosen to be a bifurcation parameter, we find that Hopf bifurcations occur when the delay crosses some critical values. Also, by employing the normal form theory and the center manifold theorem, we determine the type and stability of the bifurcating periodic solutions. Finally, numerical simulations are offered to support our results.

Introduction

It is well known that mathematical models of continuous culture of microorganism are very important biomathematical models. The study of these mathematical models has been one of the hottest subjects concerned by mathematical and theoretical biologists. Mathematical models of continuous culture of microorganisms commonly include mathematical models of the chemostat and the turbidostat. These mathematical models not only have good ecological significance, but also play an important role in the study of chemical sciences and bio-engineering. Also, they are used widely in studying the law of interaction between microbial populations, and are also used widely in the production of micro-organisms, food processing, management and forecast of ecosystems, especially aquatic ecosystems, and control of environmental pollution.

Researchers have recognized that time delays often yield a complex impact to the dynamic behaviors of ecological systems in reality (see Gopalsamy [1], Kuang [2], Smith [3]). Delays occur naturally in the chemostat by two obvious sources of delays: delays due to the cell cycle and delays due to the possibility that the organism digests (stores) the nutrient. Digestion delays (delays due to the possibility that the organism digests the nutrient) are first introduced to a chemostat model by Caperon [4]. Bush and Cook [5] corrected Caperon’s model and then discussed the local stability of solutions and observed the oscillation of solutions. Mathematical models of the chemostat have received great attention of mathematical and theoretical biologists, and many valuable results have been obtained. For example, Smith and Waltman [6] introduced in detail the theory on ecological models and the mathematical results on the chemostat. Ruan [7] introduced the theory and the results before 1997 about different types of chemostat models governed by ordinary differential equations, partial differential equations and delay differential equations. A number of good results concerning with chemostat models with discrete delay (see, for example, Caperon [4], MacDonald [8], Wolkowicz and Xia [9]), distributed delay (see, for example,MacDonald [10], Ruan and Wolkowicz [11], Wolkowicz et al. [12], Li et al. [13]), nutrient recycling (Yuan and Zhang [14], Lin et al. [15]), biological control (Dimitrova and Krastanov [16]) as well as unstirred chemostat models (So and Waltman [17], Dung and Smith [18], Hsu et al. [19], Guo et al. [20], Guo and Zheng [21], Grover et al. [22]) were also acquired.

Recently, researchers are attracted by mathematical models of the turbidostat. Walz et al. [23] analyzed and demonstrated the stability of mathematical models of the turbidostat on the basis of experimental methods. Flegr [24] showed the coexistence of two species in the turbidostat by using numerical analysis, and De Leenheer and Smith [25] also verified Flegr’s results by employing theoretical analysis. Cammarota and Miccio [26] established a mathematical model for the turbidostat described by ordinary differential equations with switched system, with corresponding conditions for the stability of the equilibrium obtained. Li [27] established a mathematical model of competition in a turbidostat for an inhibitory nutrient, and achieved the sufficient conditions for the stability of the equilibrium and the existence of limit cycles. Li and Chen [28] studied a mathematical model of the turbidostat with impulsive state feedback control to obtain sufficient conditions for the existence and the stability of periodic solutions of order one, and verified the main results by numerical simulation. Tagashira and Hara [29] showed that the stability of the equilibrium point of a turbidostat model is changed by ”time-delay” caused in controlling the dilution rate after measuring the concentrations of two organisms. By qualitative analysis, Yuan et al. [30] investigated a mathematical model of the turbidostat with delayed feedback control and found that the system generates Hopf bifurcation when the delay reaches a certain value.

Because of the difficulty caused by the dilution rate, results on turbidostat models with delay are still rare to date, so far as we are aware, Tagashira et al. [29] and Yuan et al. [30] investigated a turbidostat model of the dilution rate with delayed feedback control. Further noting that the chemostat models with discrete delay due to the possibility that the organism digests (stores) the nutrient are natural and reasonable (see Bush and Cook [5], Freedman et al. [31]), and basing on motivation from the work of Bush and Cook [5] , De Leenheer and Smith [25], we in this study focus on the dynamics of a turbidostat model with time delay of digestion which follows as

$$\{\begin{array}{c}{\displaystyle \frac{dS\left(t\right)}{dt}=(d+kx\left(t\right))(S^0-S\left(t\right))-\frac{x\left(t\right)}{\gamma }f\left(S\left(t\right)\right),}\hfill \\ {\displaystyle \frac{dx\left(t\right)}{dt}=x\left(t\right)(f\left(S(t-\tau )\right)-d-kx\left(t\right)),}\hfill \end{array}$$

where S(t) and x(t) present the nutrient concentration and the density of the organism at time t, respectively, S⁰ > 0 stands for the input concentration of the nutrient, γ > 0 is yield constant, ${\displaystyle f\left(S\left(t\right)\right)=\frac{mS\left(t\right)}{a+S\left(t\right)}}$ (m > 0, a > 0) is uptakes function, τ > 0 is time delay of digestion, and d + kx(t) (d > 0, k > 0) is the dilution rate of the turbidostat.

The objective of this paper is to establish the existence of Hopf bifurcation and determine the dynamics in system (1). This paper is organized as follows. In Section 2, we analyze local stability and Hopf bifurcation of system (1), and establish sufficient conditions for local stability and Hopf bifurcation of the unique positive equilibrium. Section 3 mostly focuses on the stability and type of the bifurcating periodic solutions to system (1). We in Section 4 further illustrate our main results by numerical simulations.