Bifurcation of nontrivial periodic solutions for a biochemical model with impulsive perturbations

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Abstract

In this paper, a biochemical model with the impulsive perturbations is considered. By using the Floquet theorem, we find the boundary-periodic solution is asymptotically stable if the impulsive period is larger than a critical value. On the contrary, it is unstable if the impulsive period is less than the critical value. The problem of finding nontrivial periodic solutions is reduced to showing the existence of the nontrivial fixed points for the associated stroboscopic mapping of time snapshot equal to the common period of input. It is then shown that once a threshold condition is reached, a stable nontrivial periodic solution emerges via a supercritical bifurcation. Furthermore, influences of the impulsive input on the inherent oscillations are studied numerically, which shows the rich dynamics in the positive octant.

Introduction

Most biochemical reactions can present rich phenomena in vessels, such as chemical oscillation [1], [2], [3], [4], [5], periodic doubling, chemical waves [6], [7], and chaos [8], [9]. Analysis of forced nonlinear oscillations plays an important role in understanding their dynamic phenomena of electronic generators, mechanical chemical and biological systems. Broad classes of chemical reaction systems exhibit nonlinear dynamic behavior and are typically modelled by systems of nonlinear differential equations [10], [11], [12]. It has turned out that numerous models in chemistry, biochemistry, biology and population ecology obey a set of differential equations of the following form:dxdt=g(x)-u(x,y),dydt=εv(x,y)-r(y),where g(x) denotes the input and possible output of one agent, r(y) denotes the output (i.e removal) and possible input of a second agent. u(x,y) and v(x,y) describe some mechanism of conversion of the first agent to the second agent (e.g., in an enzyme reaction). ε is a scaling parameter.

In fact, many perturbations are not continuous, which brings to sudden changes to the system. For example, we put the reactants into the reaction vessel in between some time during many chemical reactions. Systems with such sudden perturbations are involving in impulsive differential equations which has been studied in the literature [13], [14], [15], [16], [17]. Impulsive perturbations make systems more intractable except that in some instance, the models can be rewritten as simple discrete-time mapping or difference equations when the corresponding continuous models can be solved explicitly. That is why most of the investigations related to impulsive systems are focused on the basic theory of impulsive equations and seldom give applications on the enzyme kinetics.

In this paper, we consider the simplest autocatalytic chemical reactions set involving two internal reactants X,Y whose concentrations vary in time, which takes the following form:X0k0X,XkP,X+Yb2Y,YqQFig. 1 the basic biochemical reaction, where k0,k and b are the reaction rate constants, respectively. q is the constant of the maximum output.

According to Fig. 1, we introduce a function qc+y of the saturation output into the following model:x˙(t)=a-kx(t)-bx(t)y(t),y˙(t)=bx(t)y(t)-qy(t)c+y(t),tnT,Δx(t)=p,Δy(t)=0,,t=nT,where x(t) and y(t) denote the concentrations of X and Y, respectively. Δx(t)=x(t+)-x(t),Δy(t)=y(t+)-y(t),a=k0X0,n={1,2,3,}. p(p>0) represents that input concentration of the reactant X at time t=nT. T is the impulsive period. Other parameters are all positive constants.

The purpose of this paper introduces the impulsive perturbations into the biochemical reaction and investigates the effect of the impulsive input reactant on the system. By using Floquet theorem and small-amplitude perturbation skills, we obtain the boundary-periodic solution (x(t),0) is locally stable if T<T. On the contrary, the boundary-periodic solution is unstable if T<T. Hence T practises a bifurcation threshold, as far as the stability of the trivial periodic solution is concerned. Using a bifurcation theorem, we obtain a positive periodic solution. By numerical simulation, we can show system presents rich dynamics including quasi-periodic oscillation, periodic doubling cascade, periodic halving cascade, attractor crisis and chaotic bands with periodic windows.

Section snippets

The stability of the boundary-periodic solution

Firstly, we give some basic properties about the following subsystem of system (1.3)x˙(t)=a-kx(t),tnT,x(t+)=x(t)+p,t=nT,x(0+)=x0.Clearly, x˜(t)=ak+p1-e-kTe-k(t-nT)t(nT,(n+1)T],x˜(0+)=ak+p1-e-kT is a positive periodic solution of system (2.1). Since x(t)=(x(0+)-p1-e-kT-ak)e-kt+x˜(t) is the solution of system (2.1) with initial value x00, where t(nT,(n+1)T].

Lemma 2.1

For a positive periodic solution x˜(t) of system (2.1) and every solution x(t) of system (2.1) with x(0+)=x00, we have |x(t)-x˜(t)|0

Existence of positive periodic solution via bifurcation

We now proceed to study bifurcation, which occurs at T=T using the bifurcation theory of Lakmeche and Arino [19]. To this purpose, we shall employ a fixed point argument. We denote by Φ(t,U0) the solution of the (unperturbed) system consisting of the first two equations of (1.3) for the initial data U0=u01,u02; also, Φ=(Φ1,Φ2). We define the mapping I1,I2:R2R2 byI1(x1,x2)=x1+p,I2(x1,x2)=x2and the mapping F1,F2:R2R2 byF(x1,x2)=a-kx1-bx1x2,F2(x1,x2)=bx1x2-qx2c+x2.Furthermore, let us define Ψ:[0

Effect of impulsive input on the system (1.3)

In this section, we will study the influence of periodic forcing and impulsive perturbations on inherent oscillation. We will see that periodic forcing and impulsive perturbations cause complicated dynamics for system (1.3).

The influence of bifurcation parameters (p and T) may be documented by stroboscopically sampling one of the variables over a range of bifurcation parameters (p and T).

In Fig. 1, we have displayed the effect of bifurcation diagram p on the system (1.3) with a=3.4,b=3.4,c=1,k=

Discussion

In this paper, we investigate the autocatalytic chemical reactions set involving two internal reactants X,Y whose concentrations vary in time. Now we have to added to a remark upon the significance of the threshold condition T=T, that is 0Tbx˜(t)-qcdt=0. Then, as the conversion rate of the first agent X to the second Y is of the form bx(t)y(t), the integral 0Tbx˜(t)dt approximates the normalized conversion of the first agent X to the send agent Y for a period due to the chemical reaction,

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This work is supported by the National Natural Science Foundation of China (No. 10771179) and Henan Science and Technology Department (No. 082102140025).

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