Elsevier

Mathematical Biosciences

Volume 278, August 2016, Pages 30-36
Mathematical Biosciences

Computer simulation of two chemostat models for one nutrient resource

https://doi.org/10.1016/j.mbs.2016.05.004Get rights and content

Highlights

  • We provide an analysis of two Michaelis–Menten chemostat dynamic models.

  • We obtain analytical solutions of these models for some parametric relations.

  • We construct software modules that allow modeling the chemostat cultivation.

  • We provide a visualization of the dynamics of the development of each microorganism.

Abstract

We consider Michaelis–Menten chemostat dynamic models, describing the process of continuous cultivation of bacteria with one organic substrate and two types of microorganisms in a case where the Michaelis–Menten constants for the two competing species of microorganisms are equal. For such a system we obtain solutions with the finite initial conditions assuming only positive values. As it is shown the problem is reduced to the solution of the nonlinear differential equation of the first order. For some parametric relations the solutions of the differential system are found in the analytical form. Using numerical procedures we construct software modules that allow modeling the chemostat cultivation for the changing parameters and visualizing the dynamics of the development process for each microorganism. A comparative analysis of some numerical methods that are used to integrate the resulting nonlinear differential equation is given.

Introduction

We consider a basic, resource-based model of growth in the chemostat. Such models have applications in ecology to a simple lake model and in biotechnology to the commercial bio-reactor model. Experimental verification of the match between theory and experiment in the chemostat can be found in [1]. Motivation of the such research is in [2]. The goal of the study is to attract the attention of the scientific community to the opportunities for the use of this chemostat model in solving a wide range of problems in natural sciences (biotechnology, ecology, etc.). The classical chemostat model was developed in the 1970s. Since then the evolution of mathematical methods and the design and development of computer algebra systems allowed to carry out the mathematical modeling and simulations in the biological sciences, in particular in the chemostat theory. Noteworthy to mention that during the computer simulation we can visualize and do animations of the solutions of the chemostat model with the changing parameters.

Mathematical modeling of the dynamics of the two types of microorganisms which consume a given substrate is an important task, often arising in the medical and food industries production, microbiological industry, ecology. Nowadays the two general ways for organizing the biochemical process exist. These are the periodical and the continuous cultivation. During the periodical process of the limited population growth the initial substrate and biomass of microorganisms are placed in a closed vessel (fermenter). Thus the substrate is being consumed only during the growth of microorganisms. After completing the process the fermenter is emptied, the product is purified and the cycle repeats. Population dynamics in a batch cultivation is described by the following system of equations known as the Mono model [3], [4]. dXdt=X·μmaxsks+s,dsdt=1YdXdt.where X is a biomass concentration, s is a concentration of the limiting substrate, t is time, μmax  – maximum specific growth rate, ks –constant of saturation, Y – economic coefficient. The system (1) has an analytic solution of the form ksY+s0Y+x0s0Y+x0ln(xx0)ksYs0Y+x0ln(s0Y+x0Xs0Y)=μmaxt,where x0 and s0 are initial concentrations of microbial biomass and substrate respectively. The disadvantages of a batch cultivation are the impossibility of identifying the degree of influence of external factors on the processes which change the concentrations of microorganisms and the lack of management method. During the continuous process of cultivation of microorganisms the supply of nutrient substrate takes place without interruption and is performed at the same time as the removal of excess biomass. Due to this, the size of the breeding population remains unchanged. A system in which the feed rate of the substrate and the removal of biomass are constant, is called the chemostat [2], [3]. For the description of the limited population growth in the chemostat the following system of equations is used [2], [4] dXdt=X·(μmaxsks+sD),dsdt=D(s0s)XYμmaxsks+s.The notation here is the same as in system (1), the parameter D is called a flow and it is numerically equal to the feed rate of nutrient substrate in the fermenter. The hemostat cultivation method shows that it is possible to fully control the action of any environmental factor.

In the simplest models of the chemostat [2] the competition of several species of microorganisms that feed on a limited nutrient called substrate is studied. If the competition between two or more populations is “exploitative manner” at one limited type substrate, just one of the populations survives and the rest die. This situation occurs for the majority of the given constant values of the parameters – run-off rates and the input concentration of the nutrient substrate. The study of these processes is given in [2], [5]. It was also proved that it is theoretically possible for two or more populations eating a limited substrate to coexist short-term. In nature there are examples that demonstrate the coexistence of several populations quite for a long time. More detailed consideration of the duration of time processes (short-term and long term) in the chemostat is given in [2, pp. 20 and 21].

Under real conditions the parameters that describe the chemostat are not a constant. Availability of natural seasonal changes makes it necessary to clarify a simple chemostat model, namely, the consideration of models with periodically varying coefficients. There are two main ways to describe such a model: the first – to make the feed rate of the input concentration of the nutrient substrate periodic, the second – to consider periodic flushing rate of the substrate. The first of these modifications has been studied in [5]. Such approach is natural from the point of view of ecology, since it can be expected that the levels of nutrients in many ecosystems depend on day and night time, or have a long-term seasonal dependence. The system of differential equations describing this model will have the form {s˙(t)=f(t)s(t)x1(t)μ1(s(t))x2(t)μ2(s(t)),x˙1(t)=(μ1(s(t))1)x1(t),x˙2(t)=(μ2(s(t))1)x2(t),where μi(s(t))=mis(t)ai+s(t)(i=1,2), s(t) is a density of the nutrient substrate, x1(t), x2(t)  are densities of microorganisms changing in time t, other parameters m1, a1, m2, a2 of the model (2) are the given positive numbers, a periodic function f(t) is the rate of filing of nutrient substrate in a chemostat.

The second modification was investigated in [6], [7], [8] and the corresponding experiment is a pump speed control that changes the speed of run-off (wastewater treatment plant model). The system of differential equations describing this modification has the following form {s˙(t)=(1s(t))D(t)x1(t)μ1(s(t))x2(t)μ2(s(t)),x˙1(t)=(μ1(s(t))D(t))x1(t),x˙2(t)=(μ2(s(t))D(t))x2(t),where D(t) is a positive periodic function defining the run-off rate of the substrate. For both considered models the results are similar and lead to the conclusion that the long-term coexistence of populations is possible.

Systems of the form (2) and (3) have been investigated with numerical, asymptotic, topological methods and methods of the qualitative theory of the ordinary differential equations (including the Poincare–Bendixson theory), based on which the conclusions of the effectiveness of the chemostat periodic control modes were made, that allow to provide a useful biomass concentration levels unattainable at the constant control actions [6], [7], [9], [10]. The new method and software were proposed to obtain two-parametric families solutions with the given accuracy for chemostat models in the paper [11].

The developed methods were applied not only to the models that describe the competition between two populations of a single substrate limited. For example, in [12] the model of the chemostat with n competing populations was considered. In the works [2], [13], combination of the N bacterial species or N phytoplankton species with one limiting substrate at constant feed rate control substrate were studied. In the paper [14] the research in competition between three different types of microorganisms (containing n spices) was further developed.

Section snippets

The model

In this paper we consider dynamic models of the Michaelis–Menten chemostat, describing the process of continuous cultivation of bacteria with a given organic substrate and two species of microorganisms when the Michaelis–Menten constants for the two competing populations of microorganisms are equal. That is, the solutions of system (2) satisfying the initial conditions s(0)=s00,x1(0)=x100,x2(0)=x200at a sufficiently large interval of time (including possible infinitely large time interval)

Computational study of chemostat model with equal parameters of the Michaelis–Menten

We consider the case of the system (1), when the Michaelis–Menten coefficients satisfy the condition a1=a2. We assume without losing the generality that m2=ρm1, where ρ is a positive real number different from 1 [18], [19]. For this coefficient condition the system (1) takes the form s(t)=f(t)s(t)m1x1(t)s(t)a1+s(t)ρm1x2(t)s(t)a1+s(t),x1(t)=(m1s(t)a1+s(t)1)x1(t),x2(t)=(ρm1s(t)a1+s(t)1)x2(t).For the system (5) we look for solutions that take positive values and satisfy the initial

Example

As an example we consider the numerical solution of Eq. (8) for the parameter values m1=8,a1=0,3,ρ=0,4,C1=1,C2=1,initial condition x1(0)=0,0001 and a constant feed rate of the input substrate f(t) ≡ 1. Here is a comparison of the six solutions of the Eq. (8) for given values of the parameters (9), found by using the following numerical methods:

  • (1)

    the numerical method selected automatically by the Mathematica system;

  • (2)

    the implicit Runge–Kutta method of the eighth order of accuracy;

  • (3)

    the implicit

Module with visualization

We now carry out a numerical investigation of the solutions of a differential system (5). With the help of the following software module we simulate possible states of the dynamic system (5) for f(t)=1 and different values of its parameters.

Fig. 2 shows the graphs of the three unknown functions included in the system (5), and in the lower right corner, a phase curve in the plane (x1, x2) is built. For the parameter values m1=2,a1=0,01,x10=0,001,ρ=0,85 it is shown that each population increases

Analytical solutions of the models

In the system (3) there are solutions in the analytical form. Indeed, Theorems 2 holds true [18].

Theorem 2

Leta1=a2 andD(t)=m1=2. Then the system (3) takes the form s(t)=2a1(s(t)1)+s(t)(m2x2(t)+2(s(t)x1(t)1))a1+s(t),x1(t)=2a1x1(t)a1+s(t),x2(t)=(m2s(t)a1+s(t)2)x2(t),and its two-parameter family of solutions can be written in the form s=1x1C1e(m22)tx1m2/2,x2=C1e(m22)tx1m2/2,4a1(m22)2(2a1C1exp(m222a1(a1(lnx1+2t)x1+lnx1))(m22)(x1)2m22a1+1Em222a1(m222a1x1)=C2),where C1, C2 are

Conclusions

This work focuses on the application of computer simulation tools for solving tasks of chemostat describing dynamic model of the Michaelis–Menten when using numerical-analytical methods. We found a new analytical solution of the system (12), which has the form (13). Note that one of the purposes of this paper is to illustrate the theoretical possibility for the emergence of complex modes of fluctuations in the number of microorganisms.

Besides, the considered mathematical models have real-world

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