On a linear finite-difference model of a mixed-culture biological system arising in food safety studies
© Macías-Díaz; licensee Springer. 2013
Received: 22 July 2013
Accepted: 18 November 2013
Published: 5 December 2013
In this work, we introduce a linear finite-difference methodology to approximate non-negative and bounded solutions of a coupled system of nonlinear parabolic partial differential equations that describes the growth of two different microbial colonies on a substrate of nutrients. Some simpler versions of the model of interest possess qualitative results that guarantee the existence and uniqueness of non-negative and bounded solutions. However, the exact determination of analytical solutions of our system (and even of those simpler versions of our model) corresponding to physically meaningful initial conditions, may be a difficult task, whence the need to design computational methods to approximate the solutions of our mathematical model is pragmatically justified. Our numerical technique has the advantage of conditionally preserving the non-negative and bounded characters of initial-boundary data. The most important analytical results of this work are summarized as a theorem of existence and uniqueness of non-negative and bounded numerical solutions, whose proof relies on the non-singularity property of M-matrices, and the fact that the entries of the inverses of these matrices are positive real numbers. We provide some illustrative simulations to evince the fact that our method preserves the mathematical characteristics of the solutions that we mentioned above.
MSC:65Q10, 65M06, 39A14, 92D25.
Keywordscomputational modeling biological films positivity and boundedness linear finite-difference scheme food safety investigation
The problem of modeling mathematically the growth dynamics of biological films is an important topic of investigation in view of the many practical problems where these complex structures appear. For instance, microbial films find applications in the development of biological techniques to treat contaminated fluids [1–4], in the design of microbial fuel cells to produce electricity by means of biological or chemical procedures [5–8], in the production of new generations of sensors through bacterial signaling systems  and in the investigation of the corrosion of material surfaces for environmental engineering purposes [10, 11], among many other scientific and engineering problems of pragmatic relevance [12, 13]. Needless to mention that, according to a report by the Chinese National Institute of Health , biological films are likely to be directly or indirectly responsible for most of the bacterial infections in humans.
The mathematical modeling of biological films ideally offers the advantage of providing an inexpensive means to determine some analytical properties of the solutions of quantitative paradigms in general scenarios. However, even simple biological film structures possess a high level of complexity, which is difficult to be faithfully reflected in a single system of partial differential equations. As a consequence, most of the realistic, mathematical models used to describe the growth dynamics of microbial colonies are complex systems of equations for which the exact determination of meaningful analytical solutions is a difficult task. As expected, this complexity is substantially increased when one considers not only the interaction between a single colony of bacteria with the substrate, but also when different types of bacteria are present in the medium.
the existence of a sharp front of biomass at the fluid-solid transition,
the presence of a threshold of biomass density,
the fact that the spreading of biomass is significant only when the biomass is close to the threshold,
the application of reaction kinetics mechanisms in the production of biomass,
the compatibility of the biomass spreading mechanism with hydrodynamics and with nutrient transfer-consumption models,
among other features observed in the practice. The functions of interest in our model (concentration of nutrients and population sizes) are non-negative functions that have been normalized with respect to maximum allowed quantities, so that the sets of possible values that these variables may take on are subsets of .
In view of the difficulties to calculate analytical solutions of the model under study, we propose here a computational method to approximate them. Motivated by some previous and successful efforts in the design of numerical techniques to estimate the solutions of much simpler forms of the system of equations investigated in the present work [19, 20], we follow a linear approach [21–26] to provide a linear and implicit finite-difference discretization of the mathematical model of interest. After some algebraic manipulations, we will readily verify that our technique may be represented conveniently in vector form using a square matrix with real entries which, under suitable conditions, turns out to be an M-matrix, that is, a strictly diagonally dominant matrix with non-positive off-diagonal entries, and positive components in its main diagonal [27, 28].
Following some approaches of the specialized literature employed for much simpler models [29–32], the main properties of M-matrices will be used in order to establish the conservation of the non-negative and the bounded characters of approximations. More concretely, we will use the facts that every M-matrix is non-singular and that all the entries of their inverses are positive numbers . Using these tools, we will prove a theorem of existence and uniqueness of non-negative and bounded solutions of our technique. As expected, we will provide some simulations in order to evince the performance of our method and illustrate the facts that non-negativity and boundedness are preserved at each iteration when some suitable conditions on the model and the computational parameters are satisfied.
The present work is divided as follows. In Section 2, we introduce the mathematical model under investigation, the physical relevance of the model parameters, the mathematical nomenclature and some conventions. Some particular models, for which theorems on the existence and uniqueness of non-negative and bounded solutions are available, are described in that stage of our work. Section 3 introduces the numerical technique employed to approximate the solutions of our model. We introduce therein the discrete notation employed in this manuscript and provide a useful vector representation of our method. In Section 4, we derive some analytical properties of our finite-difference scheme (like its capability to conditionally preserve the non-negative and bounded characters of approximations), and we present some illustrative simulations obtained through a computational implementation of our technique. Finally, we close this work with a section of concluding remarks.
2.1 Mathematical model
for every . Here, the set Ω will represent a spatial domain which may physically be identified with a Petri dish for practical purposes, and the variable t denotes time. The spatial operators ∇, ∇⋅ and are, respectively, the gradient, the divergence and the Laplacian.
In our model, the constants (the substrate diffusion coefficient), and (the diffusion coefficients corresponding to and , respectively), and (the maximum specific consumption rate of each colony), and (the Monad half-saturation constant of each of the colonies), and (the maximum specific growth rate of anaerobic growth for and , respectively), and (the decay rate of the total biomass of the system for each colony), (the maximum specific growth rate of aerobic growth), α and β are all non-negative numbers with .
Before we close this section, it is worthwhile to notice that system (1) is a generalized form of the model investigated in . In fact, our model reduces to the one studied in that work if we let . In the present manuscript, the distinction between the decay rate of the total biomass for each colony is made in order to derive a particular model of interaction between a substrate of nutrients and a single microbial colony.
2.2 Substrate-biomass model
for suitable functions .
Our work is greatly motivated by the next analytical result.
and for every ,
for every , and .
for every , and .
Proof The proof is a direct consequence of Theorems 2.1 and 2.2 of . □
It is important to remark the fact that this result guarantees the existence and uniqueness of non-negative and essentially bounded solutions of the particular biofilm model (11). However, even for this simplified version of (1), the calculation of particular solutions for non-trivial initial conditions is a complicated task. These observations motivate the numerical approach reported in this manuscript.
2.3 Simple model
Under these circumstances, the following result is an immediate corollary of Proposition 1. It establishes sufficient conditions for the existence and uniqueness of non-negative and bounded solutions of Equation (15) subject to (16).
and for every ,
for every , and .
for every , and .
3 Computational model
For the rest of this work, we will restrict our attention to the case . Lower and higher dimensional scenarios are treated in a similar fashion.
3.2 Finite-difference scheme
for every , and . In addition, we will impose discrete, homogeneous Neumann boundary data for each of the three functions of interest.
The following remarks will be useful in Section 4.1 in order to establish the main properties of our method. They will also serve as an intermediate step in order to present our finite-difference scheme in vector form.
for , and every , and . Obviously, the coefficients and are negative, the coefficient is positive, and the inequality holds.
We analyze now the second recursive formula of (36).
whenever and .
Next, we examine the last iterative formula of (36).
3.3 Vector representation
Throughout, we define the vector exactly as the vector , except that the components corresponding to boundary points are equal to zero.
where the zeros are zero matrices of size .
Let be the diagonal matrix of size whose diagonal components are the constants arranged in lexicographic order for all those coordinates which do not correspond to the boundaries of Ω; otherwise, let those entries be equal to zero. Define the matrix in a similar fashion as , but using now the coefficients instead of .
and is the lexicographically ordered vector of initial approximations. Clearly, the zeros of this matrix are zero matrices of size .
In view of these remarks, the method introduced in this work is linear and implicit, and provides linear approximations to the solutions of (1). Computationally, our technique is coded using an implementation of the stabilized bi-conjugate gradient method, which is a method that has been employed in several works to solve sparse systems arising in biology [19, 20], among other disciplines of the natural science and engineering.
In this section, we establish firstly some results on the existence and uniqueness of non-negative and bounded solutions of the finite-difference method (36) using the vector representation (65). In a second stage, we provide some illustrative simulations that reflect the capability of the method to yield non-negative and bounded approximations for non-negative and bounded initial profiles.
4.1 Analytical results
the off-diagonal elements of A are non-positive numbers,
the diagonal entries are positive numbers, and
the matrix A is strictly diagonally dominant.
In general, the M-matrices are important in many areas of mathematics; in particular, they are useful tools in the field of numerical analysis, where the fact that M-matrices are non-singular is employed extensively. Moreover, all the entries of the inverse of any M-matrix are positive numbers . These facts are summarized conveniently in the following result.
Lemma 4.1 (Fujimoto )
Every M-matrix is invertible, and the entries of its inverse matrix are all positive numbers.
We say that a real vector x is non-negative (respectively, positive) if all of its components are non-negative (respectively, positive) numbers; this fact is represented by (respectively, ). We say that x is bounded from above (respectively, strictly bounded from above) by 1 if all the components of x are less than or equal to (respectively, strictly less than) 1, a fact that is denoted by (respectively, ). It is readily checked that holds if and only if , where e is the vector of the same size of x, all of whose components are equal to 1. We employ the notation (respectively, ) to signify that the vector x is non-negative and bounded from above (respectively, strictly bounded from above) by 1.
The following result establishes that the matrix in Equation (66) is an M-matrix under suitable conditions. We will use here the remarks of Section 3.2 and will follow the notation introduced previously. Throughout, k will be an element of the set .
Lemma 4.2 Assume that and . Then is an M-matrix if all the conditions (46), (47), (51) and (53) hold for every and .
Proof By Remarks 1, 2, 3 and the inequalities (46) and (51), the off-diagonal entries of are non-positive. In turn, the inequalities (47) and (53) assure the strict diagonal dominance of this matrix. We conclude then that is an M-matrix by definition. □
The following is the main analytical result of this work. It establishes sufficient conditions that guarantee the existence and uniqueness of non-negative and bounded new approximations at each iteration of the finite-difference scheme (36).
Theorem 4.3 (Existence and uniqueness)
are all satisfied for every and .
It is important to notice that the rows of matrix associated to the homogeneous Neumann boundary conditions contain a 1 and a −1 as their only nonzero entries, and that the corresponding components of are equal to zero. In view of that, those entries of are equal to zero. We analyze next the rest of the rows of , focusing on each of the three block rows of the presentation (66).
Row 1. Evidently, the corresponding component of is given by the expression for some and .
Row 2. In this case, the corresponding component of the vector is(71)
for suitable and .
Row 3. For each row of the third block row of the matrix , the corresponding component of assumes the form(72)
for some and .
We have verified that all the components of are non-negative in all cases. Thus, or, equivalently, that is bounded from above by 1, as desired. □
4.2 Numerical examples
for every which belongs to Ω. Here, L is a suitable positive integer, , and are positive numbers, and are points in the interior of Ω for each . For the sake of convenience, we define the vectors for each such l.
Our first example provides a qualitative comparison against some of the results reported in [19, 20]. To that end, observe that the partial differential equation (15) is obtained from system (1) by fixing the constants , , . In addition, we must consider the initial conditions and for every , and any fixed constant .
For our following example, we study the dynamics of growth of a biological film whose interaction with the (non-constant) substrate is described by (11). Observe that that system is obtained from (1) by letting , , and letting and being fixed positive constants, say, equal to 1. Also, we need to impose the initial condition for every .
We consider now the full system (1).
Next, we approximate the solutions of the problem proposed in Example 3 at longer periods of time.
In this work, we designed a finite-difference method to approximate the solutions of a mathematical model that appears in the investigation of food safety. The model is a coupled system of parabolic partial differential equations with nonlinear diffusion and reaction laws, which describes the interaction between a substrate function and two different microbial colonies. For simpler versions of the problem, the specialized literature provides suitable theorems on the existence and uniqueness of solutions which are non-negative and bounded from above at all times. However, even in the mathematically simpler versions of our system of equations, the exact determination of analytical solutions for significant initial-value problems is a difficult task.
The method we introduced in this manuscript is a convenient, linear discretization of the mathematical model of interest. After some calculations, we showed that the method is a linear and implicit technique which may be rewritten in vector form through the multiplication by a real square matrix that, under suitable conditions on the parameters of the analytical and computational models, turns out to be an M-matrix. The facts that every M-matrix is non-singular and that all the entries of their inverses are positive numbers are the most important tools to establish conditions that guarantee the existence and uniqueness of non-negative and bounded approximations for every set of non-negative and bounded initial conditions. The main result is summarized as a theorem of existence and uniqueness of numerical solutions, and the computer simulations provided show that the properties of non-negativity and boundedness are preserved at every iteration of the method when the conditions of our result are satisfied.
Before we close this work, we must mention that we have obtained more computational simulations of the evolution of solutions of model (1) with various sets of values of the model parameters. On the one hand, the results evince the complex dynamics of this biological system, but they also motivate future numerical studies toward the resolution of practical problems in the investigation of biological films.
The author wants to express his deepest gratitude to Dr. I. E. Medina-Ramírez of the Department of Chemistry at the Universidad Autónoma de Aguascalientes for all her invaluable comments and enlightening discussions during the preparation of this manuscript. Also, he wishes to thank the anonymous reviewers for their invaluable comments and criticisms.
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