# 13.6: Reaction-Diffusion Systems - Mathematics

Finally, I would like to introduce reaction-diffusion systems, a particular class of continuous ﬁeld models that have been studied extensively. They are continuous ﬁeld models whose equations are made of only reaction terms and diffusion terms, as shown below:

[dfrac{partial{f_1}}{partial{t}} =R_1(f_1,f_2,...,f_n) + D_1∇^{2}f_1 label{(13.53)} ]

[dfrac{partial{f_2}}{partial{t}} = R_2 (f_1,f_2,...,f_n) + D_2∇^{2}f_2 label{(13.54)} ]

[ vdots]

[dfrac{partial{f_n}}{partial{t}} =R_n(f_1,f_2,...,f_n) + D_n∇^{2}f_n label{(13.55)} ]

Reaction terms ((R_i(...))) describe only local dynamics, without any spatial derivatives involved. Diffusion terms ((D_i∇^{2}f_i)) are strictly limited to the Laplacian of the state variable itself. Therefore, any equations that involve non-diffusive spatial movement (e.g., chemotaxis) are not reaction-diffusion systems.

There are several reasons why reaction-diffusion systems have been a popular choice among mathematical modelers of spatio-temporal phenomena. First, their clear separation between non-spatial and spatial dynamics makes the modeling and simulation tasks really easy. Second, limiting the spatial movement to only diffusion makes it quite straightforward to expand any existing non-spatial dynamical models into spatially distributed ones. Third, the particular structure of reaction-diffusion equations provides an easy shortcut in the stability analysis (to be discussed in the next chapter). And ﬁnally, despite the simplicity of their mathematical form, reaction-diffusion systems can show strikingly rich, complex spatio-temporal dynamics. Because of these properties, reaction diffusion systems have been used extensively for modeling self-organization of spatial patterns. There are even specialized software applications available exactly to simulate reaction-diffusion systems3.

Exercise (PageIndex{1})

Extend the following non-spatial models into spatially distributed ones as reaction-diffusion systems by adding diffusion terms. Then simulate their behaviors in Python.

• Motion of a pendulum (Eq.(6.2.1)): This creates a spatial model of locally coupled nonlinear oscillators.
• Susceptible-Infected-Recovered (SIR) model (Exercise 7.1.3): This creates a spatial model of epidemiological dynamics.

In what follows, we will review a few well-known reaction-diffusion systems to get a glimpse of the rich, diverse world of their dynamics.

## Turing Pattern Formation

As mentioned at the very beginning of this chapter, Alan Turing’s PDE models were among the ﬁrst reaction-diffusion systems developed in the early 1950s [44]. A simple linear version of Turing’s equations is as follows:

[dfrac{partial{u}}{partial{t}} =a(u-h) +b(v-k) +D_{u}Delta^{2}u label{13.56}]

[dfrac{partial{u}}{partial{t}} =c(u-h) +d(v-k) +D_{u}Delta^{2}u label{13.57}]

The state variables (u) and (v) represent concentrations of two chemical species. (a, b, c,) and (d) are parameters that determine the behavior of the reaction terms, while (h) and (k) are constants. Finally, (D_u) and (D_v) are diffusion constants.

If the diffusion terms are ignored, it is easy to show that this system has only one equilibrium point, ((u_eq,v_eq) = (h,k)). This equilibrium point can be stable for many parameter values for (a, b, c,) and (d). What was most surprising in Turing’s ﬁndings is that, even for such stable equilibrium points, introducing spatial dimensions and diffusion terms to the equations may destabilize the equilibrium, and thus the system may spontaneously self-organize into a non-homogeneous pattern. This is called diffusion-induced instability or Turing instability. A sample simulation result is shown in Fig. 13.6.1.

The idea of diffusion-induced instability is quite counter-intuitive. Diffusion is usually considered a random force that destroys any structure into a homogenized mess, yet in this particular model, diffusion is the key to self-organization! What is going on? The trick is that this system has two different diffusion coefﬁcients, (D_u) and (D_v), and their difference plays a key role in determining the stability of the system’s state. This will be discussed in more detail in the next chapter.

There is one thing that needs particular attention when you are about to simulate Turing’s reaction-diffusion equations. The Turing pattern formation requires small random perturbations (noise) to be present in the initial conﬁguration of the system; otherwise there would be no way for the dynamics to break spatial symmetry to create non-homogeneous patterns. In the meantime, such initial perturbations should be small enough so that they won’t immediately cause numerical instabilities in the simulation. Here is a sample code for simulating Turing pattern formation with a suggested level of initial perturbations, using the parameter settings shown in Fig. 13.6.1:

This simulation starts from an initial conﬁguration ((u(x,y),v(x,y)) ≈ (1,1) = (h,k)), which is the system’s homogeneous equilibrium state that would be stable without diffusion terms. Run the simulation to see how patterns spontaneously self-organize!

Exercise (PageIndex{2})

Conduct simulations of the Turing pattern formation with several different parameter settings, and discuss how the parameter variations (especially for the diffusion constants) affect the resulting dynamics.

Exercise (PageIndex{3})

Discretize the Keller-Segel slime mold aggregation model (Eqs. (13.4.13) and (13.4.14)) (although this model is not a reaction-diffusion system, this is the perfect time for you to work on this exercise because you can utilize Code 13.8). Implement its simulation code in Python, and conduct simulations with (µ = 10^{−4}), (D = 10^{−4}), (f = 1), and (k = 1), while varying (χ) as a control parameter ranging from (0) to (10^{−3}). Use (a = 1) and (c = 0) as initial conditions everywhere, with small random perturbations added to them.

## Belousov-Zhabotinsky Reaction

The Belousov-Zhabotinsky reaction, or BZ reaction for short, is a family of oscillatory chemical reactions ﬁrst discovered by Russian chemist Boris Belousov in the 1950s and then later analyzed by Russian-American chemist Anatol Zhabotinsky in the 1960s. One of the common variations of this reaction is essentially an oxidation of malonic acid ((CH_{2}(COOH)_{2})) by an acidiﬁed bromate solution, yet this process shows nonlinear oscillatory behaviorf or a substantial length of time before eventually reaching chemical equilibrium. The actual chemical mechanism is quite complex, involving about 30 different chemicals. Moreover, if this chemical solution is put into a shallow petri dish, the chemical oscillation starts in different phases at different locations. Interplay between the reaction and the diffusion of the chemicals over the space will result in the self-organization of dynamic traveling waves (Figure (PageIndex{2})), just like those seen in the excitable media CA model in Section 11.5.

A simpliﬁed mathematical model called the “Oregonator” was among the ﬁrst to describe the dynamics of the BZ reaction in a simple form [50]. It was originally proposed as a non-spatial model with three state variables, but the model was later simpliﬁed to have just two variables and then extended to spatial domains [51]. Here are the simpliﬁed “Oregonator” equations:

[ egin{align} epsilon{dfrac{partial{u}}{partial{t}}} &= u(1-u)-dfrac{u-q}{u+q}fv + D_{u}Delta^{2}u label{(13.58)} [4pt] dfrac{∂v}{∂t} &=u-v + D_{u}Delta^{2}v label{(13.59)} end{align}]

Here, (u) and (v) represent the concentrations of two chemical species. If you carefully examine the reaction terms of these equations, you will noticethat the presenceof chemical (u) has a positive effect on both (u) and (v), while the presence of chemical (v) has a negative effect on both. Therefore, these chemicals are called the “activator” and the “inhibitor,” respectively. Similar interactions between the activator and the inhibitor were also seen in the Turing pattern formation, but the BZ reaction system shows nonlinear chemical oscillation. This causes the formation of traveling waves. Sometimes those waves can form spirals if spatial symmetry is broken by stochastic factors. A sample simulation result is shown in Fig. 13.6.3.

Exercise (PageIndex{5})

Implement a simulator code of the “Oregonator” model of the BZ reaction in Python. Then conduct simulations with several different parameter settings, and discuss what kind of conditions would be needed to produce traveling waves.

## Gray-Scott Pattern Formation

The ﬁnal example is the Gray-Scott model, another very well-known reaction-diffusion system studied and popularized by John Pearson in the 1990s [52], based on a chemical reaction model developed by Peter Gray and Steve Scott in the 1980s [53, 54, 55]. The model equations are as follows:

[ dfrac{∂u}{∂t} =F(1-u) -uv^{2} +D_{u}Delta^{2}u label{(13.60)} ]

[dfrac{∂v}{∂t} =-(F+k)v +uv^{2}+ D_{u}Delta^{2}v label{(13.61)} ]

The reaction terms of this model assumes the following autocatalytic reaction (i.e., chemical reaction for which the reactant itself serves as a catalyst):

[u+2v ightarrow 3v label{(13.62)}]

This reaction takes one molecule of (u) and turns it into one molecule of (v), with help of two other molecules of (v) (hence, autocatalysis). This is represented by the second term in each equation. In the meantime, (u) is continuously replenished from the external source up to 1 (the ﬁrst term of the ﬁrst equation) at feed rate (F), while (v) is continuously removed from the system at a rate slightly faster than (u) ’s replenishment ((F +k )seen in the ﬁrst term of the second equation). (F) and (k) are the key parameters of this model.

It is easy to show that, if the diffusion terms are ignored, this system always has an equilibrium point at ((u_{eq},v_{eq}) = (1,0)) (which is stable for any positive (F) and (k)). Surprisingly, however, this model may show very exotic, biological-looking dynamics if certain spatial patterns are placed into the above equilibrium. Its behaviors are astonishingly rich, even including growth, division, and death of “cells” if the parameter values and initial conditions are appropriately chosen. See Fig. 13.6.4 to see only a few samples of its wondrous dynamics!

Exercise (PageIndex{5})

Implement a simulator code of the Gray-Scott model in Python. Then conduct simulations with several different parameter settings and discuss how the parameters affect the resulting patterns.

Figure (PageIndex{4}): (Next page) Samples of patterns generated by the Gray-Scott model with (D_u = 2 × 10^{−5}) and (D_v = 10^{−5}). The concentration of chemical (u) is plotted in grayscale (brighter = greater, only in this ﬁgure). Time ﬂows from left to right. The parameter values of (F) and (k) are shown above each simulation result. The initial conditions are the homogeneous equilibrium ((u,v) = (1,0)) everywhere in the space, except at the center where the local state is reversed such that ((u,v) = (0,1)).

## Reaction-diffusion equation

where $u=u(x,t)=(u_1,ldots,u_n)$, $Delta$ is the Laplace operator in the spatial variables $x$, $D$ is a non-negative non-zero diagonal matrix, and $f$ is a function from a domain in $mathbf R^n$ into $mathbf R^n$. Many generalizations of these equations have also been studied, such as result when $f$ depends also on the first-order $x$-derivatives of $u$, when the operator $Delta$ is replaced by other, possibly non-linear, operators, or when the matrix $D$ is not diagonal. If extra first-order terms appear in the system as a model for convective transport effects, the system is sometimes termed reaction-advection-diffusion equation.

Such equations arise as models of diverse natural phenomena [a1], but their most natural roots lie in the study of chemical systems: the components of the vector $u$ may then represent concentrations of chemical species which are present, the term $DDelta u$ represents the diffusive transport of those species, possibly through a chemical solution, and $f(u)$ represents the production or destruction of species resulting from reactions among them (if the rates of all such reactions are known, as functions of $u$, then the explicit form of $f$ can be written down).

The variable $x$ is often confined to a domain $Omega$ with boundary $partialOmega$, and then solutions are sought which satisfy specific boundary conditions on $partialOmega$. These are generally of the form

where $partial/partial u$ is the derivative normal to $partialOmega$, $a_i$ and $b_i$ are not both zero (unless $u_i$ does not "diffuse" ), and $h_i$ is a given function. Again, generalizations, such as to non-linear boundary conditions, abound.

The specific problems of interest are: i) the initial value problem, in which $u(x,0)$ is given and $u(x,t)$ is sought for $tgeq0$ ii) the steady problem, in which solutions independent of $t$ are sought and iii) the travelling-wave problem, in which $Omega=mathbf R$ and solutions are sought of the special form $u(x,t)=U(x-ct)$.

Because of their strong connections with the applied sciences and the limited number of important properties common to all members of this unwieldy class of systems, the research impetus in this field comes more from viewing the systems as models of specific natural phenomena, rather than from interest in them for their own sake. A typical motivation, for example, may be to ask whether a certain system, in which specific natural effects are modelled, will have solutions which reflect some known natural phenomenon of interest whose causes are incompletely known. Then one looks for the existence and the stability of solutions of the system in question which have properties analogous to the phenomenon in question.

Regarding the initial value problem i), the theory of analytic semi-groups, which in this context relies on the operator $DDelta$ being sectorial, has developed as one of the most commonly used approaches to existence and uniqueness [a2]. The study of steady solutions ii) has used a variety of methods, such as recasting the problem as a fixed-point problem for a mapping in some suitable function space and using topological-degree methods. In the case when $n=1$ or the system has certain monotonicity properties, methods based on upper-and-lower solutions (cf. Upper-and-lower-functions method) provide easier alternatives (see [a1] and [a3], e.g.).

The travelling-wave problem iii) can be viewed as seeking a parameter $c$ for which there exists a connection between two critical points for the system of ordinary differential equations resulting from the substitution $u=U(x-ct)$. One of the principal tools in this connection has been the powerful Conley index [a4], [a3]. Below some other methods which have been devised more recently are mentioned.

The theory of reaction-diffusion systems can be viewed as incorporating all of the theory of autonomous ordinary differential systems $du/dt=f(u)$ (cf. Autonomous system), since when homogeneous Neumann boundary conditions are imposed, solutions of the latter system automatically constitute $x$-independent solutions of the corresponding reaction-diffusion systems. But, of course, solutions with striking spatial characteristics arise as well and,in fact, the possible spatial structure of solutions is one of the most often investigated aspects.

Some of the best-studied examples of reaction-diffusion systems are the following.

a) The scalar Fisher equation

[a5], [a6], where $f$ has exactly two zeros. This equation originally arose in connection with population genetics.

b) The scalar bistable diffusion equation, [a7], [a6], [a8], of the same form but where $f$ has exactly three simple zeros and is negative between the first two. This equation also has connections with population genetics, but knowledge of its properties is even more important in connection with the role it plays as a component part of more complicated systems.

c) The FitzHugh–Nagumo system

where $f$ has the properties given in b) (see the references in [a3] and [a9] for this and generalizations). It is a simplification of higher-order systems such as the Hodgkin–Huxley system, which arise as models of signal transmission on nerve axons and in cardiac tissue.

d) The thermal-diffusion model in chemical reactor theory and combustion [a10], [a11]. In this model, $u=(u_0,ldots,u_n)$, $u_0$ represents temperature, the other components of $u$ represent concentrations of chemical species, and the components of $f$ are given by

the summation being over all reactions occurring in the material (these reactions are indexed by $l$). Here, $m_l$ is a (mass action) monomial in $u_1,ldots,u_n$ appropriate to the $l$-th reaction, $b_l$ is the "reaction constant" for that reaction, and the numbers $a_$ are "stoichiometric parameters" , specifying the amount of species $j$ (or heat, in case $j=0$) produced or consumed in reaction $l$.

In all of the above examples, the existence and the stability of travelling-wave solutions is of paramount importance and in case d), other spatially or temporally ordered solutions are important as well.

In many applications, solutions with frontal or interfacial properties arise [a9]. For example, a moving surface in $3$-space may exist near which some components of $u$ experience dramatic changes. These changes form an interior layer at the surface in question. They have been studied in the context of phase-field equations (a reaction-diffusion system with non-diagonal $D$), in which they represent phase interfaces, of the bistable equation, and of the FitzHugh–Nagumo equations and their generalizations, in which they may represent phase changes, changes in the electrochemical properties of neural or cardiac tissue, or changes in the chemical state of a medium.

The stability of waves in one space dimension for systems with $n>1$ is a considerably more difficult field of investigation than is their existence. Much of the work here has been done for the FitzHugh–Nagumo equations [a12]. Recently a new technique, the stability index of J. Alexander, R. Gardner and C. Jones [a13], was developed and applied to a number of travelling-wave problems.

For travelling-wave and stationary solutions with interfaces (see above), a technique called the SLEP method has been developed to study stability questions (see [a4] and the references therein).

For FitzHugh–Nagumo and related systems, the most important patterned solutions in two space dimensions are rotating spirals, which are extremely prevalent and apparently extremely stable structures for this and many other models for excitable media. Despite the great interest in these rotating solutions and the large number of papers the concept has generated (see the references in [a9]), their mathematical foundation is still rudimentary. Analogous phenomena exist in three dimensions: structures which rotate about curves in space, called filaments, which themselves migrate according to certain approximate laws. An important challenge for the future is to better understand (and provide a firm mathematical foundation for) the connections between such dynamic spatial patterns and their laws of motion, on the one hand, and the underlying partial differential equations on the other.

## Alternative Mathematical Theory of Non-equilibrium Phenomena

### 6.3.2 DISSIPATION THEOREM

In Section 4.4.3 we discussed several reasons for splitting the energy flux density je into the two vectors q ∘ ∗ and w ∘ ∗ . Under the condition that even in continuum physics the notion of work rate w ∘ ∗ refers to any change of information content as a characteristic of matter, w ∘ ∗ may be measured by the change of shape encountered by each infinitesimal volume element dτ. Such an effect is induced by all interactions between dτ and its surroundings, but primarily by flow influences. In particular, the local pressure as the natural conjugate property of dτ is the driving force for shape variations.

In this context, it should be stressed that the pressure p* in question must not be taken for an equilibrium quantity of state. In addition, the volume element dτ is also subject to dissipative shearing stresses conventionally expressed by local velocity gradients.

There cannot be any doubt that irreversible phenomena control details of flow patterns. The same is true for the specially prescribed boundary conditions. Still, since it is proved even in the limiting case of Euler’s equation of motion, the main influence on flow field topology stems from the spatial distributions of local pressure and flow velocity as well as their mutual interaction.

For this reason it is opportune to decompose (6.44) into two parts

where, according to (6.42) , the work rate definition includes the non-equilibrium pressure p*. We may provisionally imagine the pressure p* as a quantity that does not obey the equation of state commonly used in practice.

The introduction of a viscous pressure tensor τ*, defined by reference to the momentum flux density JI and pressure p*, leads to an important conclusion from (6.40) : Both the quantities JI and p* appear in two additive terms in the third bracket of this equation term, which may be modified by means of the tensor identities (6.38) as follows:

This result, together with (6.41) and (6.44) , leads to the identity

This so-called dissipation theorem also extends beyond its application to multicomponent single-phase body-field systems, because the structure of (6.4) indicates that for other classes of systems only the pertaining production densities have to be built in.

This theorem summarizes in a characteristic manner all flux and production densities arising in the relevant balance equation. But the most striking result clearly concerns the fact that all these characteristics of non-equilibrium phenomena produce a zero sum. The theorem obviously refers to two kinds of dissipative effects. The first considers the occurring flux densities jz along with the corresponding gradients ∇ζz, where ζz stands for the intensive variables assigned to z and distinguished by the condition ζz ≥ 0. The second effect comprises all actually occurring production densities σz. From a mathematical point of view, it is reasonable to consider three notable arguments:

Avoiding any chemical reactions, diffusion, or dissipative flows, the system in question may be reduced in such a way that either all terms in (6.47) will vanish identically, or at least two terms distinguished by opposite signs will remain.

There is at least one element of (6.47) that can never be switched off by admissible procedures of process realization without simultaneously dropping the other existing elements.

In addition to the definitions of the two kinds of dissipative effects given above, it is beneficial to distinguish them by different signs.

With respect to the Gibbs main equation ρ D e ∗ ρ = D p ∗ − p f ⋅ v + ∑ j = 3 r ζ z , j ρ D z j , let us formalize argument (3) by the convention

which is assumed to be valid for all specific variables z under consideration. The subscript j refers to r variables and allows us to distinguish two types of elements concerning dissipation for each indicated z. Every first-type element of (6.48) needs the summation over the subscript j the second type does not. The values j = 1 and j = 2 of the subscript relate to the pressure p* and the specific field force f, which do not explicitly furnish any contribution to dissipation.

There is an important example for the first-type element of (6.48) 1, well-known as De Bonder’s inequality ( Prigogine and Defay, 1962 , p. 71). For the special case of a closed system realized by vanishing diffusional flux densities jj, it becomes

where V r is the velocity of the rth reaction. De Donder defined this quantity for the first time in reference to the progress variable λr for the rth reaction:

According to Equation (6.35) , the chemical production density Гj for the jth species agrees with its convection term ρDωj if the system is closed. By means of Equations (5.52) and (5.53) , the direct relation

may then be obtained between Гj for all j and the affinity density AV,r:= ρar for all r.

The total production, caused by simultaneously occurring chemical reactions per time unit, equals the sum over the products, which are each formed from the velocity of the rth reaction and the assigned affinity density. The same is true for the sum over all involved components consisting of the products, each of them now made up from the chemical production density of the jth component and the assigned chemical potential per mass unit. Reaction coupling along with feedback events may happen and are mathematically represented by the possibility that some of the sum terms are singled out by opposite signs as compared to (6.51) .

There is an alternative possibility in studying details of chemical reaction dynamics. Guldberg and Waage found by an empirical analysis that the concentrations of reacting species can be related to the corresponding reaction velocities. “Even when applied to complex reactions involving intermediate steps and transient species this empirical approach so excels in the elucidation of reaction mechanisms that Mass Action might be termed the First Law of Chemical Kinetics” ( Garfinkle, 1992 , p. 282).

Recently, a promising method, the so-called natural path approach, has been developed by Garfinkle, who studied the progress of a homogeneous stoichiometric chemical reaction in a closed isothermal system. He worked out chemical thermodynamic formalisms in a manner consistent with the laws of thermodynamics, and therefore, independently of mechanistic considerations. Reactions and their velocities are now successfully described in terms of the rate of change of a proper thermodynamic function, the affinity decay rate ÅTV at uniform temperature and fixed volume. In response to Garfinkle’s work a critical review was prepared by Hjelm-felt, Brauman, and Ross in 1990. They found that the affinity decay rate ÅTV is directly dependent on the respective reaction mechanisms, contrary to Garfinkle’s observation based on direct empirical analysis ( Garfinkle, 1992 , p. 283). Summarizing his extensive studies, Garfinkle confirmed his assertion that there exists a unique reaction path over the entire range of empirical observation independent of any reaction mechanisms. Moreover, he stated that “The excellent data correlation observed permits a thermodynamic reaction velocity to be computed that corresponds to the mechanistic reaction velocity determined by kinetics based on an understanding of reaction mechanism. Excellent agreement between these reaction velocities is achieved over the range of experimental observations” ( Garfinkle, 1992 , p. 299).

Statements like (6.48) or (6.51) are metaphysical inasmuch as they based on a universal theorem that can never be proven by a finite number of data. The same is true for the case where inequalities (6.48) are assumed to formalize the Second Law of Thermodynamics. In this context it is remarkable that the second part of (6.48) indicates the well-known relationship between a prescribed direction of any flux density jz and the subsequent direction of the assigned gradient ∇ζz. This is the deeper meaning of the general observation that heat fluxes can never flow in the direction of increasing temperature. Curiously, this observation is rarely mentioned in connection with the mathematical formulation of the Second Law.

Although the entropy production density σ obeys the same inequality (6.48) 1 as the other pertaining production densities, there is still an essential difference with respect to argument (2) listed above: In agreement with experience, the entropy production density a is presumed to be the one production density that must always exist in real processes. All others may be manipulated, at least in principle, in such a way that their respective values tend to zero. However, we must pay attention to various possibilities of creating dissipation through some hidden processes. Frozen chemical reactions are a prominent example.

To quantify the special role of the entropy production density σ, a further rule should supplement the dissipation theorem with its parts (6.47) and (6.48) :

This limiting law is indeed an essential part of the mathematical theory presented here and refers to all dissipation terms except the term T*σ itself. The primary significance of (6.52) is that we may couple states of equilibrium to those processes assumed to be dissipationless. This is particularly relevant for both the special case of kinetic equilibrium and the state at rest defined by (6.19) .

The two theorems (6.44) and (6.47) derived from the general axioms of the Alternative Theory deserve comment particularly with regard to the extended irreversible thermodynamics (EIT). The EIT is characterized by a rate equation commonly dedicated to Gibbs,

It is assumed to be valid for an irreversible multicomponent single-phase system ( Eu, 1986 , p. 217). The corresponding generic function U(S,V,Nii α ) refers to the internal energy U by definition, following an idea of Meixner (cf. Garcia-Colin and Uribe, 1991 , p. 111). The respective relation between the entropy production density σ and the dissipative contributions represented by the flux ϕi α of species i and its potential conjugate X α i = T ∂ S / ∂ Φ i α is given by the expression

where the tensorial properties Λi α are the representative dissipative terms of the system in question. The sum over the integers α (running from 1 to 4) relates to an ordering directed to the following characteristic gradients:

Although there is a formal similarity between the expression for α and Equation (6.47) , it is obvious that the conceptual differences are significant. The two theorems (6.44) and (6.47) are derived from Equation (4.54) applied to a multicomponent single-phase body-field system. This Pfaffian results from the Gibbs fundamental equation Г(E, P, r, S, V, Ni) = 0 and refers to the total energy E of the system under consideration. The internal energy U may be formally determined by Equation (6.16) , provided that the complete solution via Г(E, P, r, .S, V, Ni) = 0 is available. Thus, the Alternative Theory leads to a result that is inconsistent with the two relationships given above as representative statements of the EIT.

In the EIT formalism, the theoretical approach is not related to the total energy E. Under the condition that the EIT refers to Gibbsian ideas and notions, E is exclusively established by the complete set of extensive variables, which work as coordinates of the corresponding Gibbs space. The common practice of separating the contributions of kinetic and potential energies from E implies the same conclusion for the internal energy U as for E. Assuming U as an M–G function of the system, then U cannot depend on properties such as the moments ϕ i α , which do not belong to the extensive state variables by definition (cf. Eu, 1986 , p. 215).

Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, USA

Received April 2020 Revised July 2020 Published December 2020 Early access September 2020

This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is $0$. The existence assertion is established by exploring the fact that the total population density satisfies a porous media equation.

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## Analytical Solution of Nonlinear Dynamics of a Self-Igniting Reaction-Diffusion System Using Modified Adomian Decomposition Method

A mathematical model of the dynamics of the self-ignition of a reaction-diffusion system is studied in this paper. An approximate analytical method (modified Adomian decomposition method) is used to solve nonlinear differential equations under steady-state condition. Analytical expressions for concentrations of the gas reactant and the temperature have been derived for Lewis number (Le) and parameters

. Furthermore, in this work, the numerical simulation of the problem is also reported using MATLAB program. An agreement between analytical and numerical results is noted.

#### 1. Introduction

Nonlinear dynamical phenomena in combustion process are an active area of experimental and theoretical research. Mathematical models that describe this phenomenon can be considered as nonlinear dynamical systems. The dynamic characterization of such models was developed by Continillo et al. [1]. The detailed numerical simulation of autoignition of coal stockpiles leads to the observation of steady regimes. To better investigate this phenomenon, two simplified distributed-parameter models were discussed which incorporate heat conduction, mass diffusion, and one-step Arrhenius exothermic chemical reaction. Both model equations were solved with straight forward finite-difference schemes [2]. The problem of spontaneous ignition of coal stockpiles is challenging for safety implications and for its theoretical complexity: a spontaneous combustion reaction takes place in a bed of solid fuel, while flow, driven by natural convection generated by the onset of temperature gradients within the pile, occurs. Coal stockpiles self-ignite when reaction of coal with oxygen present in the atmosphere generates heat, that is, not efficiently removed toward the external ambient [3]. Continillo et al. [4, 5] have analyzed the self-combustion of coal piles in the absence of natural convection. The three main phenomena in the self-ignition of coal stockpile are convection, reaction, and diffusion. On the other hand, Continillo et al. [6] studied the dynamic behavior of a two-dimensional coal pile also by accounting for natural convection. As part of a comprehensive study of self-heating of coal stockpiles, a simple mathematical model has been developed. To our knowledge, no rigorous analytical expressions of gas reactant (

) have been derived for all possible values of parameters under steady-state conditions. The purpose of this paper is to derive approximate analytical expressions for gas reactant concentration and temperature using the modified Adomian decomposition method.

#### 2. Mathematical Formulation of the Boundary Value Problem

The nonlinear differential equations are those of a distributed-parameter dynamic model of heterogeneous reaction in a one-dimensional layer. The gaseous reactant diffuses through the reacting medium and a first order one-step exothermic chemical reaction takes place. The reaction rate depends on the temperature through the Arrhenius exponential. The Arrhenius rate equation is a mathematical expression which relates the rate constant of a chemical reaction to the exponential value of the temperature. The model nonlinear equations in dimensionless form are [1]

where is the concentration of the gas reactant, is the temperature, Le is the Lewis number (the ratio between mass and heat diffusivities), is the dimensionless heat of reaction, is the thermal Thiele modulus (the ratio of the time scale of the limiting transport mechanism to the time scale of intrinsic reaction kinetics [7]), and is the dimensionless activation energy (minimum amount of energy between reactant molecules for effective collisions between them). The boundary conditions are

Under steady-state condition, the equations become

#### 3. Analytical Solution of Nonlinear Dynamics of a Self-Igniting Reaction-Diffusion System under Steady-State Condition Using Modified Adomian Decomposition Method

In the recent years, much attention is devoted to the application of the Adomian decomposition method to the solution of various scientific models [8]. An efficient modification of the standard Adomian decomposition method for solving initial value problem in the second order partial differential equation yields the MADM. The MADM without linearization, perturbation, transformation, or discretization gives an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. The results show that the rate of convergence of modified Adomian decomposition method is higher than standard Adomian decomposition method [9–13]. Using this method (see Appendix A), we obtain an approximate analytical expression of concentration of gas reactant ( ) and temperature ( ) (see Appendix B) as follows:

#### 4. Numerical Simulation

Nonlinear diffusion equation (3) for the boundary condition (4) is also solved numerically. We have used the function pdex1 in MATLAB software to solve the initial-boundary value problems for the nonlinear differential equations numerically. This numerical solution is compared with our analytical results in Figures 1–4. Upon comparison, it gives a satisfactory agreement for all values of the dimensionless parameters , , and . The MATLAB program is also given in Algorithm 1.

 function pdex4 m = 0 x = linspace(0,1) t = linspace(0,100000) sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x,t) u1 = sol(. 1) u2 = sol(. 2) figure plot(x,u1(end,:)) title(‘u1(x,t)') xlabel(‘Distance x') ylabel(‘u1(x,2)') %–––––––––––––––––––––––––––––––––– figure plot(x,u2(end,:)) title(‘u2(x,t)') xlabel(‘Distance x') ylabel(‘u2(x,2)') %–––––––––––––––––––––––––––––––––– function

versus dimensionless spatial coordinate x using (5) for Le = 0.233,

= 13.6, and various values of

versus dimensionless spatial coordinate using (6) for Le = 0.233,

=13.6, and various values of

versus dimensionless spatial coordinate x using (5) for Le = 0.233,

= 1000, and various values of

versus dimensionless spatial coordinate x using (6) for Le = 0.233,

= 1000, and various values of

#### 5. Discussion

Equations (5) and (6) represent the simplest form of approximate analytical expressions for the concentration of gas reactant and temperature for all values of parameters

, , , and . The Thiele number (thermal) is the ratio of layer thickness (L) and thermal diffusivity ( ). Equation (5) represents the new approximate analytical expression of concentration of gas reactant. The numerical solution is compared with the analytical results in Figures 1–4. These figures represent the analytical and numerical concentration profiles of gas reactant and temperature for different values of parameters , , and . Figure 1 represents the dimensionless concentration versus dimensionless spatial coordinate for

. From the figure it is inferred that the value of decreases when the value of or layer thickness increases. Figure 2 illustrates the dimensionless concentration versus dimensionless spatial coordinate for and we infer that the dimensionless temperature increases with the increase in values of layer thickness. Figure 3 represents the dimensionless concentration versus dimensionless spatial coordinate for values of . From the figure it is inferred that the value of decreases when the value of dimensionless activation energy ( ) increases. Figure 4 illustrates the dimensionless temperature versus dimensionless spatial coordinate for the values of the dimensionless temperature increases with increase in values of dimensionless activation energy ( ). From Figures 2 and 4 it is evident that the maximum value for dimensionless concentration is 1 and that temperature attains its maximum value when the spatial coordinate

. Figure 5 confirms the results given by Figures 1 to 4.

(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)

versus dimensionless spatial coordinate

, (b) the normalized three-dimensional concentration of gas reactant

versus dimensionless spatial coordinate

and dimensionless activation energy

, (c) the normalized three-dimensional temperature

versus dimensionless spatial coordinate

, and (d) the normalized three-dimensional temperature

versus dimensionless spatial coordinate

and dimensionless activation energy

Our analytical results are compared with the numerical results for the dimensionless concentration in Table 1. The maximum relative error between our analytical results and simulation results for the concentration is 1.3%. Also in Table 2, our analytical results are compared with the numerical results for the dimensionless temperature . Satisfactory agreement is noted. The maximum relative error in this case is 0.4%.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

XP Cui is partially supported by National Science Foundation Grant ATD-1222718 and the University of California, Riverside AES-CE RSAP A01869 ZB Yang is partially supported by National Institute of General Medical Sciences Grant GM100130 JP Shi is partially supported by National Science Foundation Grant DMS-1715651 QY Shi is partially supported by China Scholarship Council.

## Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction-diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace-Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.

Keywords: Bifurcation analysis Branch tracing Cross-diffusion Large-scale systems Linear stability analysis Marginal stability analysis Multigrid approach Nonlinear PDEs Pattern formation Reaction diffusion Surface FEMs.

## The Stacks project

Given a triangulated category and a triangulated subcategory we can construct another triangulated category by taking the “quotient”. The construction uses a localization. This is similar to the quotient of an abelian category by a Serre subcategory, see Homology, Section 12.10. Before we do the actual construction we briefly discuss kernels of exact functors.

Definition 13.6.1 . Let $mathcal$ be a pre-triangulated category. We say a full pre-triangulated subcategory $mathcal'$ of $mathcal$ is saturated if whenever $X oplus Y$ is isomorphic to an object of $mathcal'$ then both $X$ and $Y$ are isomorphic to objects of $mathcal'$.

A saturated triangulated subcategory is sometimes called a thick triangulated subcategory. In some references, this is only used for strictly full triangulated subcategories (and sometimes the definition is written such that it implies strictness). There is another notion, that of an épaisse triangulated subcategory. The definition is that given a commutative diagram

where the second line is a distinguished triangle and $S$ and $T$ isomorphic to objects of $mathcal'$, then also $X$ and $Y$ are isomorphic to objects of $mathcal'$. It turns out that this is equivalent to being saturated (this is elementary and can be found in [Rickard-derived] ) and the notion of a saturated category is easier to work with.

Lemma 13.6.2 . Let $F : mathcal o mathcal'$ be an exact functor of pre-triangulated categories. Let $mathcal''$ be the full subcategory of $mathcal$ with objects

Then $mathcal''$ is a strictly full saturated pre-triangulated subcategory of $mathcal$. If $mathcal$ is a triangulated category, then $mathcal''$ is a triangulated subcategory.

Proof. It is clear that $mathcal''$ is preserved under $[1]$ and $[-1]$. If $(X, Y, Z, f, g, h)$ is a distinguished triangle of $mathcal$ and $F(X) = F(Y) = 0$, then also $F(Z) = 0$ as $(F(X), F(Y), F(Z), F(f), F(g), F(h))$ is distinguished. Hence we may apply Lemma 13.4.16 to see that $mathcal''$ is a pre-triangulated subcategory (respectively a triangulated subcategory if $mathcal$ is a triangulated category). The final assertion of being saturated follows from $F(X) oplus F(Y) = 0 Rightarrow F(X) = F(Y) = 0$. $square$

Then $mathcal'$ is a strictly full saturated pre-triangulated subcategory of $mathcal$. If $mathcal$ is a triangulated category, then $mathcal'$ is a triangulated subcategory.

Proof. It is clear that $mathcal'$ is preserved under $[1]$ and $[-1]$. If $(X, Y, Z, f, g, h)$ is a distinguished triangle of $mathcal$ and $H(X[n]) = H(Y[n]) = 0$ for all $n$, then also $H(Z[n]) = 0$ for all $n$ by the long exact sequence (13.3.5.1). Hence we may apply Lemma 13.4.16 to see that $mathcal'$ is a pre-triangulated subcategory (respectively a triangulated subcategory if $mathcal$ is a triangulated category). The assertion of being saturated follows from

for all $n in mathbf$. $square$

Each of these is a strictly full saturated pre-triangulated subcategory of $mathcal$. If $mathcal$ is a triangulated category, then each is a triangulated subcategory.

Proof. Let us prove this for $mathcal_ H^<+>$. It is clear that it is preserved under $[1]$ and $[-1]$. If $(X, Y, Z, f, g, h)$ is a distinguished triangle of $mathcal$ and $H(X[n]) = H(Y[n]) = 0$ for all $n ll 0$, then also $H(Z[n]) = 0$ for all $n ll 0$ by the long exact sequence (13.3.5.1). Hence we may apply Lemma 13.4.16 to see that $mathcal_ H^<+>$ is a pre-triangulated subcategory (respectively a triangulated subcategory if $mathcal$ is a triangulated category). The assertion of being saturated follows from

for all $n in mathbf$. $square$

Definition 13.6.5 . Let $mathcal$ be a (pre-)triangulated category.

Let $F : mathcal o mathcal'$ be an exact functor. The kernel of $F$ is the strictly full saturated (pre-)triangulated subcategory described in Lemma 13.6.2.

These are sometimes denoted $mathop>(F)$ or $mathop>(H)$.

The proof of the following lemma uses TR4.

Lemma 13.6.6 . Let $mathcal$ be a triangulated category. Let $mathcal' subset mathcal$ be a full triangulated subcategory. Set

Then $S$ is a multiplicative system compatible with the triangulated structure on $mathcal$. In this situation the following are equivalent

$S$ is a saturated multiplicative system,

$mathcal'$ is a saturated triangulated subcategory.

Proof. To prove the first assertion we have to prove that MS1, MS2, MS3 and MS5, MS6 hold.

Proof of MS1. It is clear that identities are in $S$ because $(X, X, 0, 1, 0, 0)$ is distinguished for every object $X$ of $mathcal$ and because $is an object of$mathcal'$. Let$f : X o Y$and$g : Y o Z$be composable morphisms contained in$S$. Choose distinguished triangles$(X, Y, Q_1, f, p_1, d_1)$,$(X, Z, Q_2, g circ f, p_2, d_2)$, and$(Y, Z, Q_3, g, p_3, d_3)$. By assumption we know that$Q_1$and$Q_3$are isomorphic to objects of$mathcal'$. By TR4 we know there exists a distinguished triangle$(Q_1, Q_2, Q_3, a, b, c)$. Since$mathcal'$is a triangulated subcategory we conclude that$Q_2$is isomorphic to an object of$mathcal'$. Hence$g circ f in S$. Proof of MS3. Let$a : X o Y$be a morphism and let$t : Z o X$be an element of$S$such that$a circ t = 0$. To prove LMS3 it suffices to find an$s in S$such that$s circ a = 0$, compare with the proof of Lemma 13.5.3. Choose a distinguished triangle$(Z, X, Q, t, g, h)$using TR1 and TR2. Since$a circ t = 0$we see by Lemma 13.4.2 there exists a morphism$i : Q o Y$such that$i circ g = a$. Finally, using TR1 again we can choose a triangle$(Q, Y, W, i, s, k)$. Here is a picture Since$t in S$we see that$Q$is isomorphic to an object of$mathcal'$. Hence$s in S$. Finally,$s circ a = s circ i circ g = 0$as$s circ i = 0$by Lemma 13.4.1. We conclude that LMS3 holds. The proof of RMS3 is dual. Proof of MS5. Follows as distinguished triangles and$mathcal'$are stable under translations Proof of MS6. Suppose given a commutative diagram with$s, s' in S$. By Proposition 13.4.23 we can extend this to a nine square diagram. As$s, s'$are elements of$S$we see that$X'', Y''$are isomorphic to objects of$mathcal'$. Since$mathcal'$is a full triangulated subcategory we see that$Z''$is also isomorphic to an object of$mathcal'$. Whence the morphism$Z o Z'$is an element of$S$. This proves MS6. MS2 is a formal consequence of MS1, MS5, and MS6, see Lemma 13.5.2. This finishes the proof of the first assertion of the lemma. Let's assume that$S$is saturated. (In the following we will use rotation of distinguished triangles without further mention.) Let$X oplus Y$be an object isomorphic to an object of$mathcal'$. Consider the morphism$f : 0 o X$. The composition o X o X oplus Y$ is an element of $S$ as $(0, X oplus Y, X oplus Y, 0, 1, 0)$ is a distinguished triangle. The composition $Y[-1] o 0 o X$ is an element of $S$ as $(X, X oplus Y, Y, (1, 0), (0, 1), 0)$ is a distinguished triangle, see Lemma 13.4.11. Hence o X$is an element of$S$(as$S$is saturated). Thus$X$is isomorphic to an object of$mathcal'$as desired. Finally, assume$mathcal'$is a saturated triangulated subcategory. Let be composable morphisms of$mathcal$such that$fg, gh in S$. We will build up a picture of objects as in the diagram below. First choose distinguished triangles$(W, X, Q_1)$,$(X, Y, Q_2)$,$(Y, Z, Q_3)(W, Y, Q_<12>)$, and$(X, Z, Q_<23>)$. Denote$s : Q_2 o Q_1[1]$the composition$Q_2 o X[1] o Q_1[1]$. Denote$t : Q_3 o Q_2[1]$the composition$Q_3 o Y[1] o Q_2[1]$. By TR4 applied to the composition$W o X o Y$and the composition$X o Y o Z$there exist distinguished triangles$(Q_1, Q_<12>, Q_2)$and$(Q_2, Q_<23>, Q_3)$which use the morphisms$s$and$t$. The objects$Q_<12>$and$Q_<23>$are isomorphic to objects of$mathcal'$as$W o Y$and$X o Z$are assumed in$S$. Hence also$s[1]t$is an element of$S$as$S$is closed under compositions and shifts. Note that$s[1]t = 0$as$Y[1] o Q_2[1] o X[2]$is zero, see Lemma 13.4.1. Hence$Q_3[1] oplus Q_1[2]$is isomorphic to an object of$mathcal'$, see Lemma 13.4.11. By assumption on$mathcal'$we conclude that$Q_3$and$Q_1$are isomorphic to objects of$mathcal'$. Looking at the distinguished triangle$(Q_1, Q_<12>, Q_2)$we conclude that$Q_2$is also isomorphic to an object of$mathcal'$. Looking at the distinguished triangle$(X, Y, Q_2)$we finally conclude that$g in S$. (It is also follows that$h, f in S$, but we don't need this.)$square$Definition 13.6.7 . Let$mathcal$be a triangulated category. Let$mathcal$be a full triangulated subcategory. We define the quotient category$mathcal/mathcal$by the formula$mathcal/mathcal = S^<-1>mathcal$, where$S$is the multiplicative system of$mathcal$associated to$mathcal$via Lemma 13.6.6. The localization functor$Q : mathcal o mathcal/mathcal$is called the quotient functor in this case. Note that the quotient functor$Q : mathcal o mathcal/mathcal$is an exact functor of triangulated categories, see Proposition 13.5.5. The universal property of this construction is the following. Lemma 13.6.8 . Let$mathcal$be a triangulated category. Let$mathcal$be a full triangulated subcategory of$mathcal$. Let$Q : mathcal o mathcal/mathcal$be the quotient functor. If$F : mathcal o mathcal'$is an exact functor into a pre-triangulated category$mathcal'$such that$mathcal subset mathop>(F)$then there exists a unique factorization$F' : mathcal/mathcal o mathcal'$such that$F = F' circ Q$and$F'$is an exact functor too. Proof. This lemma follows from Lemma 13.5.6. Namely, if$f : X o Y$is a morphism of$mathcal$such that for some distinguished triangle$(X, Y, Z, f, g, h)$the object$Z$is isomorphic to an object of$mathcal$, then$H(f)$, resp.$F(f)$is an isomorphism under the assumptions of (1), resp. (2). Details omitted.$square$The kernel of the quotient functor can be described as follows. Lemma 13.6.9 . Let$mathcal$be a triangulated category. Let$mathcal$be a full triangulated subcategory. The kernel of the quotient functor$Q : mathcal o mathcal/mathcal$is the strictly full subcategory of$mathcal$whose objects are In other words it is the smallest strictly full saturated triangulated subcategory of$mathcal$containing$mathcal$. Proof. First note that the kernel is automatically a strictly full triangulated subcategory containing summands of any of its objects, see Lemma 13.6.2. The description of its objects follows from the definitions and Lemma 13.5.7 part (4).$square$Let$mathcal$be a triangulated category. At this point we have constructions which induce order preserving maps between the partially ordered set of multiplicative systems$S$in$mathcal$compatible with the triangulated structure, and the partially ordered set of full triangulated subcategories$mathcal subset mathcal$. Namely, the constructions are given by$S mapsto mathcal(S) = mathop>(Q : mathcal o S^<-1>mathcal)$and$mathcal mapsto S(mathcal)$where$S(mathcal)$is the multiplicative set of (13.6.6.1), i.e., Note that it is not the case that these operations are mutually inverse. Lemma 13.6.10 . Let$mathcal$be a triangulated category. The operations described above have the following properties$S(mathcal(S))$is the “saturation” of$S$, i.e., it is the smallest saturated multiplicative system in$mathcal$containing$S$, and$mathcal(S(mathcal))$is the “saturation” of$mathcal$, i.e., it is the smallest strictly full saturated triangulated subcategory of$mathcal$containing$mathcal$. In particular, the constructions define mutually inverse maps between the (partially ordered) set of saturated multiplicative systems in$mathcal$compatible with the triangulated structure on$mathcal$and the (partially ordered) set of strictly full saturated triangulated subcategories of$mathcal$. Proof. First, let's start with a full triangulated subcategory$mathcal$. Then$mathcal(S(mathcal)) = mathop>(Q : mathcal o mathcal/mathcal)$and hence (2) is the content of Lemma 13.6.9. Next, suppose that$S$is multiplicative system in$mathcal$compatible with the triangulation on$mathcal$. Then$mathcal(S) = mathop>(Q : mathcal o S^<-1>mathcal)$. Hence (using Lemma 13.4.9 in the localized category) in the notation of Categories, Lemma 4.27.21. The final statement of that lemma finishes the proof.$square$Lemma 13.6.11 . Let$H : mathcal o mathcal$be a homological functor from a triangulated category$mathcal$to an abelian category$mathcal$, see Definition 13.3.5. The subcategory$mathop>(H)$of$mathcal$is a strictly full saturated triangulated subcategory of$mathcal$whose corresponding saturated multiplicative system (see Lemma 13.6.10) is the set The functor$H$factors through the quotient functor$Q : mathcal o mathcal/mathop>(H)$. Proof. The category$mathop>(H)$is a strictly full saturated triangulated subcategory of$mathcal$by Lemma 13.6.3. The set$S$is a saturated multiplicative system compatible with the triangulated structure by Lemma 13.5.4. Recall that the multiplicative system corresponding to$mathop>(H)$is the set By the long exact cohomology sequence, see (13.3.5.1), it is clear that$f$is an element of this set if and only if$f$is an element of$S$. Finally, the factorization of$H$through$Q$is a consequence of Lemma 13.6.8.$square\$

## A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in (mathbb^d) . For two-dimensional surfaces embedded in (mathbb^3) , these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at “scattered” locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

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## Specification

To describe a reaction-diffusion problem in NEURON, begin by loading the rxd library:

Then answer the three questions: where, who, and how.

### Where

We begin by identifying the domain i.e. where do the dynamics occur? For many electrophysiology simulations, the only relevant domains are the plasma membrane or the volumes immediately adjacent to it on either side, since these are the regions responsible for generating the action potential. Cell biology models, by contrast, have dynamics spanning a more varied set of locations. In addition to the previous three regions, the endoplasmic reticulum (ER), mitochondria, and nuclear envelop often play key roles.

The rxd.Region class is used to describe the domain:

In its simplest usage, rxd.Region simply takes a Python iterable (e.g. a list or a h.SectionList ) of h.Section objects. In this case, the domain is the interior of the sections, but the concentrations for any species created on such a domain will only be available through the rxd.Species object and not through HOC or NMODL.

Example: Region on all sections:

Example: Region on just a few sections:

If the region you are describing coincides with the domain on the immediate interior of the membrane, set nrn_region='i' , e.g.

Concentration in these regions increases when a molecule of the species of interest crosses from outside the membrane to the inside via an NMODL or kschan mechanism. If a species on such a region is named ca, then its concentrations can be read and set via the NEURON and NMODL range variable cai .

Also: for the region just outside the membrane (the Frankenhaeuser-Hodgkin space) use nrn_region='o' . For full 3D extracellular diffusion, define the region with rxd.Extracellular for more on extracellular diffusion in NEURON, see Newton et al., 2018 or the extracellular diffusion tutorial.

Many alternative geometries are available, including: rxd.membrane , rxd.inside (this is the default), rxd.Shell (used in the radial diffusion example coming soon), rxd.FractionalVolume (used in the calcium wave example), rxd.FixedCrossSection , and rxd.FixedPerimeter .

Who are the actors? Often they are chemical species (proteins, ions), sometimes they are State variables (such as a gating variable), and other times parameters that vary throughout the domain are key actors. All three of these can be described using the rxd.Species class:

although we also provide rxd.State and rxd.Parameter for the second and third case, respectively. For now these are exact synonyms of rxd.Species that exist for promoting the clarity of the model code, but this will likely change in the future (so that States can only be changed over time via rxd.Rate objects and that rxd.Parameter objects will no longer occupy space in the integration matrix.)

Note: charge must match the charges specified in NMODL files for the same ion, if any.

The regions parameter is mandatory and is either a single rxd.Region or an iterable of them, and specifies what region(s) contain the species.

Set d= to the diffusion coefficient for your species, if any.

Specify an option for the name= keyword argument to allow these state variables to map to the NEURON/NMODL range variables if the region's nrn_region is 'i' or 'o' .

Specify initial conditions via the initial= keyword argument. Set that to either a constant or a function that takes a node (defined below, and see also this example).

Example: to have concentrations set to 47 at h.finitialize(init_v) :

Note: For consistency with the rest of NEURON, the units of concentration are assumed to be in mM. Many cell biology models involve concentrations on the order of μM calcium is often even smaller. Besides the need to be consistent about units, this has implications for variable step integration which by default has an absolute tolerance of (10^<-3>) or 1 μM. To address this, use atolscale to indicate that the tolerance should be scaled for the corresponding variable e.g.

Note: initial is also a property of the Species/Parameter/State and may be changed at any time, as in:

Warning: Prior to NEURON 7.7, there was a bug in initial support: if initial=None and name=None , then concentration will not be changed at a subsequent h.finitialize() . The intended behavior is that this would reset the concentration to 0.

How do they interact? Species interact via one or more chemical reactions. The primary class used to specify reactions is rxd.Reaction:

Here lhs and rhs describe the reaction scheme and are expressed using arithmetic sums of integer multiples of a Species. For example, an irreversible reaction to form calcium chloride might be written:

where ca and cl are rxd.Species instances and kf is the reaction rate. Since custom_dynamics was not specified this is a mass-action reaction and given the reactants kf has units of 1 / (ms μM 2 ). This corresponds to the system of differential equations:

While we can sometimes ignore the reverse reactions due to them having a high energy barrier, the laws of physics imply that all reactions are in fact reversible. A more correct specification of a mass action reaction for calcium chloride would thus include a backward reaction rate kb here in units of 1/ms:

While mass-action works well for elementary reactions, this is often impractical for modeling intracellular dynamics where there are potentially many elementary reactions driving the observable reaction. In this case, we often turn to phenomenological models, such as Michelis-Menten kinetics or the Hill equation. To indicate these in NEURON, set custom_dynamics=True and specify the forward and backward rates as the corresponding formula, e.g.

Note that using Michaelis-Menten kinetics for enzymatic reactions is only appropriate under certain conditions, such as that the concentration of enzyme is low relative to the concentration of the substrate.

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