Table Of ContentUNSTABLE PATTERNS IN AUTOCATALYTIC
REACTION-DIFFUSION-ODE SYSTEMS
ANNA MARCINIAK-CZOCHRA, GRZEGORZ KARCH, AND KANAKO SUZUKI
3
1 Abstract. The aim of this paper is to contribute to understanding of the pattern for-
0 mation phenomenon in reaction-diffusion equations coupled with ordinary differential
2 equations. Such systems of equations arise, for example, from the modeling of inter-
n actions between proliferating cells and diffusing growth factors. We focus on stability
a of solutions to an initial-boundary value problem for a system consisting of a single
J
reaction-diffusionequationcoupledwithanordinarydifferentialequation. Weshowthat
9 suchsystemsexhibitdiffusion-driveninstabilities(Turinginstability)underthecondition
of autocatalysis of non-diffusing component. Nevertheless, there exist no stable Turing
]
P patterns, i.e. all continuous spatially heterogenous stationary solutions are unstable. In
A addition, we formulate instability conditions for discontinuous patterns for a class of
. nonlinearities.
h
t
a
m
[ 1. Introduction
1
Diffusion-driven instability (DDI), called also Turing-type instability, is a phenomenon
v
2 in mathematical biology, which has been often used to explain de novo pattern forma-
0 tion. Let us recall that DDI is a bifurcation that arises in a reaction-diffusion system,
0
when there exists a spatially homogeneous solution, which is asymptotically stable with
2
. respect to spatially homogeneous perturbations, but unstable to spatially heterogeneous
1
0 perturbations. Models with DDI describe then a process of a destabilization of stationary
3 spatially homogeneous steady states and evolution of spatially heterogeneous structures
1
towards spatially heterogeneous steady states. The ideas on DDI have inspired develop-
:
v
ment of a vast number of mathematical models since the seminal paper of Turing [36],
i
X
providing some explanations on symmetry breaking and de novo pattern formation dur-
r ing development, explaining shape of animal coat markings, and predicting oscillating
a
chemical reactions, see e.g. [22] and references therein.
In particular, the following system of reaction-diffusion equations
u = ε∆u+f(u,v), v = D∆v +g(u,v),
t t
considered in a bounded domain and supplemented with boundary and initial conditions,
has been proposed as a mathematical model describing the diffusion-driven instability.
Here, the unknown function u = u(x,t), sometimes called the activator, diffuses with a
very small diffusion coefficient ε > 0. On the the hand, v = v(x,t) describes the density of
the inhibitorandthe corresponding equation contains a largeconstant diffusion coefficient
D > 0.
In several models, however, the diffusion of u is either so small that it is practically
negligible or u does not diffuse at all, which leads to a system of one ordinary differential
equationcoupledwithonereaction-diffusionequation. Suchsystemsofequationsarisee.g.
from modeling of interactions between cellular processes and diffusing growth factors, see
[18, 13] for derivation of such mathematical models, [9, 12, 15, 16, 17] for some biological
Date: January 11, 2013.
1
2 ANNAMARCINIAK-CZOCHRA,GRZEGORZ KARCH,ANDKANAKOSUZUKI
applications and [14] for their mathematical analysis. Such models are different from
classical Turing-type models and the spatial structure of the pattern emerging from the
destabilization of the spatially homogeneous steady state cannot be determined based on
linear stability analysis. Therefore, the existence and stability of spatially heterogeneous
patterns arising in models exhibiting diffusion-driven instability, but consisting of only
one reaction-diffusion equation is an interesting issue. As shown in [14] for a particular
example of a reaction-diffusion-ODE system, it may happen that there exist no stable
Turing-typepatternsandtheemergingspatiallyheterogenousstructureshaveadynamical
character. In numerical simulations, solutions having the form of periodic or irregular
spikes have been observed.
Theaimofthisworkistoinvestigate towhichextent theseresults, obtainedin[14], con-
cerning the instability of all stationary structures can be generalized to reaction-diffusion-
ODE models with a single diffusion operator.
In the remainder of this paper, we focus on the following two-equation system
(1.1) u = f(u,v), for x Ω, t > 0,
t
∈
(1.2) v = ∆v +g(u,v) for x Ω, t > 0
t
∈
in a bounded domain Ω RN for N 1, with a sufficiently regular boundary ∂Ω,
⊂ ≥
supplemented with the Neumann boundary condition
(1.3) ∂ v = 0 for x ∂Ω, t > 0,
ν
∈
where ∂ = ∂/∂ν and ν denotes the unit outer normal vector to ∂Ω, and initial data
ν
(1.4) u(x,0) = u (x), v(x,0) = v (x).
0 0
The nonlinearities f = f(u,v) and g = g(u,v) are arbitrary C2-functions that satisfy
certain natural (biologically motivated) assumptions. Notice also that the diffusion term
in equation (1.2) could have the form D∆v with a constant diffusion coefficient D > 0,
however, after rescaling the time variable and the functions f and g, we can assume that
D = 1.
By a standard theory (see e.g. [29] and the beginning of Section 4), the boundary value
problem (1.1)-(1.4) has a unique local-in-time solution e.g. for every u ,v L∞(Ω). The
0 0
∈
aim of this work is to investigate the stability properties of stationary solutions.
The paper is organized as follows. In Section 2, main results of this paper are formu-
lated. Section 3 provides examples of reaction-diffusion-ODE systems considered in math-
ematical biology, which fit the framework of the theory developed in this work. Proofs
of the results are presented in Sections 4 and 5. Section 4 is devoted to showing insta-
bility of continuous stationary solutions under so-called autocatalysis and compensation
conditions. Moreover, for the instability of discontinuous solutions additional conditions
on the model nonlinearities are required. In Section 5, constant and non-constant sta-
tionary solutions of considered models are characterized and it is shown that they satisfy
the conditions formulated in Section 4. Appendix contains additional information on the
model of early carcinogenesis which was the main motivation for the research reported in
this work.
2. Results and comments
2.1. Instability of steady states. The main goal of this paper is to formulate assump-
tions on stationary solutions of problem (1.1)-(1.4), which lead to their instability. Then,
UNSTABLE PATTERNS IN REACTION-DIFFUSION-ODE SYSTEMS 3
we show that such assumptions are satisfied in a natural way in reaction-diffusion-ODE
systems modeling the Turing instability.
First, we consider regular stationary solutions (U,V) of problem (1.1)-(1.3), namely, we
assumethatthereexistsasolution(notnecessarilyunique)oftheequationf U(x),V(x) =
0 that is given by the relation U(x) = k(V(x)) for all x Ω with a C1-funct(cid:0)ion k = k(V(cid:1)).
∈
Thus, every regular stationary solution (U,V) of the boundary value problem
(2.1) f(U,V) = 0 for x Ω,
∈
(2.2) ∆V +g(U,V) = 0 for x Ω,
∈
(2.3) ∂ V = 0 for x ∂Ω
ν
∈
satisfies the elliptic problem
(2.4) ∆V +h(V) = 0 for x Ω,
∈
(2.5) ∂ V = 0 for x ∂Ω,
ν
∈
where
(2.6) h(V) = g k(V),V and U(x) = k(V(x)).
(cid:0) (cid:1)
Regular stationary solutions appear to be unstable solutions to problem (1.1)-(1.4)
under a simple assumption imposed on the first equation.
Theorem 2.1 (Instabilityofregularsolutions). Let (U,V) be a regular solution of problem
(2.1)-(2.3) satisfying the following “autocatalysis condition”:
(2.7) f U(x),V(x) > 0 for all x Ω.
u
∈
(cid:0) (cid:1)
Then, (U,V) is an unstable solution the initial-boundary value problem (1.1)-(1.4).
Inequality (2.7) can be interpreted as an autocatalysis in the dynamics of u at the
steady state (U,V). Notice that, in this work, the stability of a stationary solution
is understood in the Lyapunov sense. Moreover, we prove the nonlinear instability of
stationary solutions to problem (1.1)-(1.3) and not only their linear instability, i.e. the
instability of zero solution of the corresponding linearized problem, see Section 4 for more
explanations.
Each constant solution (u¯,v¯) R2 of problem (2.1)-(2.3) is a particular case of regular
∈
solutions. Thus, Theorem2.1providesasimplecriterionforthediffusion-driveninstability
of (u¯,v¯).
Corollary 2.2. If a constant solution (u¯,v¯) of problem (1.1)-(1.4) (namely, f(u¯,v¯) =
and g(u¯,v¯) = 0) satisfies the inequalities
f (u¯,v¯) f (u¯,v¯)
(2.8) f (u¯,v¯) > 0, f (u¯,v¯)+g (u¯,v¯) < 0, det u v > 0,
u u v (cid:18) g (u¯,v¯) g (u¯,v¯) (cid:19)
u v
then it has the DDI property.
This corollary results immediately from Theorem 2.1, because the second and the third
inequality in (2.8) imply that (u¯,v¯) is stable under homogeneous perturbations, see Re-
mark 5.2 below for more details.
Remark 2.3. The instability results from Theorem 2.1 and Corollary 2.2 can be summa-
rized in the following way: This is a classical idea that, in a system of reaction-diffusion
equations with a constant solution having the DDI property, one expects stable patterns
to appear around that constant steady state. For the initial-boundary value problem
4 ANNAMARCINIAK-CZOCHRA,GRZEGORZ KARCH,ANDKANAKOSUZUKI
for reaction-diffusion-ODE system with a single diffusion equation (1.1)-(1.3), station-
ary solutions can be constructed in the case of several interesting models (see Section
3). However, the same mechanism which destabilizes constant solutions of such models,
destabilizes also non-constant solutions.
We prove Theorem 2.1 in Section 4 by applying ideas invented for analysis of the Euler
equation and other fluid dynamics models. In that approach, it suffices to show that the
spectrum of the linearized operator
u 0 f (U,V) f (U,V) u
+ u v
L(cid:18) v (cid:19) ≡ (cid:18) ∆v (cid:19) (cid:18) g (U,V) g (U,V) (cid:19)(cid:18) v (cid:19)
u v
e e
with the Neumann boundary condition ∂ v = 0, has so-called spectral gap, namely, there
e e ν e
exists a subset of the spectrum σ( ), which has a positive real part, separated from zero.
Here, we prove that σ( ) C coLnsists ofethe interval f (U(x),V(x)) : x Ω and of
u
L ⊂ { ∈ }
isolated eigenvalues of , see Section 4 and, in particular, Fig. 4.1 for more details.
L
More precise description of σ( ) is possible in the case of our model examples gath-
L
ered in Section 3, where the autocatalysis condition (2.7) is satisfied, and moreover, the
quantity f (U(x),V(x)) is independent of x for every regular stationary solution.
u
Corollary 2.4. Let (U,V) be a regular solution of problem (2.1)-(2.3). Assume that there
exists a constant λ > 0 such that the following autocatalysis condition is satisfied:
0
(2.9) 0 < λ = f U(x),V(x) for all x Ω.
0 u
∈
(cid:0) (cid:1)
Suppose also the following “compensation condition” at the stationary solution (U,V)
(2.10) g U(x),V(x) f U(x),V(x) < 0 for all x Ω.
u v
∈
(cid:0) (cid:1) (cid:0) (cid:1)
Then, the spectrum σ( ) contains the number λ , which is an element of the continuous
0
L
spectrum of and a sequence of real eigenvalues λ ∞ of converging towards λ .
L { n}n=1 L 0
In Section 3, we recall models from mathematical biology where both conditions: auto-
catalysis(2.7)andcompensation(2.10)arefulfilled, whichcanbechecked inasurprisingly
simple way.
Remark 2.5. If one replaces the compensation condition (2.10) by the following one
(2.11) g U(x),V(x) f U(x),V(x) > 0 for all x Ω,
u v
∈
(cid:0) (cid:1) (cid:0) (cid:1)
then the statement of Theorem 2.4 remains true, what can be proved by an obvious
modification of the proof given in Section 4. However, below in Corollaries 2.14 and
2.15, we show that the compensation condition (2.10) has to be satisfied for reaction-
diffusion-ODE problems which have constant solutions with the DDI property as well as
non-constant stationary solutions.
The initial-boundary value problem (1.1)-(1.4) may also have non-regular steady states
in the case when the equation f(U,V) = 0 is not uniquely solvable. Choosing different
branches of solutions of the equation f U(x),V(x) = 0, we obtain the relation U(x) =
k V(x) with a discontinuous, piecewi(cid:0)se C1-functi(cid:1)on k. Here, we recall that a couple
U(cid:0),V (cid:1) L∞(Ω) W1,2(Ω) is a weak solution of problem (2.1)-(2.3) if the equation
∈ ×
f(cid:0) U(x(cid:1)),V(x) = 0 is satisfied for almost all x Ω and if
∈
(cid:0) (cid:1)
V(x) ϕ(x) dx+ g U(x),V(x) ϕ(x) dx = 0
−Z ∇ ·∇ Z
Ω Ω (cid:0) (cid:1)
for all test functions ϕ W1,2(Ω).
∈
UNSTABLE PATTERNS IN REACTION-DIFFUSION-ODE SYSTEMS 5
In this work, we do not prove the existence of such discontinuous solutions and we refer
the reader to classical works [2, 19, 30] as well as to our recent paper [14, Thm. 2.9] for
explanations how to construct such solutions to one dimensional problems using the phase
portrait analysis. Our goal is to formulate a counterpart of the autocatalysis condition
(2.7) which leads also to instability ofweak (including discontinuous) stationarysolutions.
The following two corollaries can be proved in the same way as Theorem 2.1.
Corollary 2.6. Assume that (U,V) is a weak bounded solution of problem (2.1)-(2.3)
satisfying the following counterpart of the autocatalysis condition
(2.12) Rangef (U,V) f (U(x),V(x)) : x Ω [λ ,Λ ]
u u 0 0
≡ { ∈ } ⊂
for some constants 0 < λ Λ < . Suppose, moreover, that there exists x Ω such
0 0 0
≤ ∞ ∈
that f (U,V) is continuous in a neighborhood of x . Then, (U,V) is an unstable solution
u 0
the initial-boundary value problem (1.1)-(1.4).
In fact, Theorem 2.1 is a particular case of Corollary 2.6, however, we prefer to state
those two results separately to emphasize the role of the autocatalysis condition (2.7) in
the proof of instability of stationary solutions.
The following corollary covers the case when the generalized autocatalysis condition
(2.12) fails to hold.
Corollary 2.7 (Instabilityofweaksolutions). Assume that the nonlinearterm in equation
(1.1) satisfies f(0,v) = 0 for all v R. Suppose that (U,V) is a weak bounded solution
∈
of problem (2.1)-(2.3) with the following property: there exist constants 0 < λ < Λ <
0 0
∞
such that
(2.13) λ f U(x),V(x) Λ for all x Ω, where U(x) = 0.
0 u 0
≤ ≤ ∈ 6
(cid:0) (cid:1)
Suppose, moreover, that there exists x Ω such that U(x ) = 0 and that the functions
0 0
∈ 6
U = U(x) as well as f (U,V) are continuous in a neighborhood of x . Then, (U,V) is an
u 0
unstable solution the initial-boundary value problem (1.1)-(1.4).
Remark 2.8. A typical nonlinearity satisfying the assumptions of Corollary 2.7 has the
form f(u,v) = r(u,v)u and appears in the models where the unknown variable u evolves
according to the Malthusian law with a growth rate r depending on u and other variables
of the model.
We defer the proofs of Theorems 2.1 and of Corollaries 2.4, 2.6, and 2.7 to Section 4.
2.2. Turing-type mechanism of pattern formation. Let us recall that, in initial-
boundary value problems for reaction-diffusion equations modeling biological pattern for-
mation,
(2.14) u = ε∆u+f(u,v), v = D∆v +g(u,v),
t t
the nonlinearities are typically satisfying the inequalities f > 0 and g < 0. Then, the
u v
DDI phenomenon requires the condition f g f g > 0, and therefore, f and g satisfy
u v v u v u
−
also the compensation condition f g < 0. Consequently, the signs of f and g should
v u v u
be opposite and there are only two possibilities
f f + + +
u v = either − or .
(cid:18) g g (cid:19) (cid:18) + (cid:19) (cid:18) (cid:19)
u v
− − −
The former on the right-hand side above is called an activator-inhibitor system, and the
latter is a resource-consumer system. We refer the reader to the Murray books [21, 22]
6 ANNAMARCINIAK-CZOCHRA,GRZEGORZ KARCH,ANDKANAKOSUZUKI
and to the review article [33] as well as to references therein for more information on DDI
in the case of two component reaction-diffusion systems and to the paper [31] in the case
of several component systems. A detailed discussion of the DDI phenomena in the case
of three component systems with some diffusion coefficients equal to zero can be found in
the recent work [1].
Our next goal is to show that the autocatalysis inequality f > 0 and the compensation
u
inequality f g < 0 at a stationary solution (cf. the assumptions of Corollary 2.4) appear
v u
indeed in system (2.14) with ε = 0 exhibiting DDI if there exist spatially heterogeneous
stationary solutions.
Since it is natural to expect the existence of such steady states in models based on the
Turing idea, let us first state a simple but fundamental property of stationary solutions
to (1.1)-(1.4).
Proposition 2.9. Assume that (U,V) is a non-constant regular solution of stationary
problem (2.1)-(2.3). Then, there exists x Ω, such that vector (u¯,v¯) U(x ),V(x )
0 0 0
∈ ≡
is a constant solution of problem (2.1)-(2.3). (cid:0) (cid:1)
The proof of Proposition 2.9 is short. It suffices to integrate equation (2.2) over Ω and
to use the Neumann boundary condition (2.3) to obtain g U(x),V(x) dx = 0. Hence,
Ω
there exists x Ω such that g U(x ),V(x ) = 0, beRcaus(cid:0)e U and V(cid:1)are continuous.
0 0 0
∈
Thus, by equation (2.1), we also (cid:0)have f U(x ),(cid:1)V(x ) = 0.
0 0
Next, we formulate our standing assu(cid:0)mption on co(cid:1)nstant solutions. In this work, we
consider only non-degenerate constant stationary solution (u¯,v¯) of the reaction-diffusion-
ODE system (1.1)-(1.3), vectors i.e. (u¯,v¯) satisfying f(u¯,v¯) = 0 and g(u¯,v¯) = 0. It
means
f (u¯,v¯) f (u¯,v¯)
(2.15) f (u¯,v¯)+g (u¯,v¯) = 0, det u v = 0, and f (u¯,v¯) = 0.
u v 6 (cid:18) g (u¯,v¯) g (u¯,v¯) (cid:19) 6 u 6
u v
Notice that the first two conditions in (2.15) allow us to study the asymptotic stability of
(u¯,v¯) as a solution to the corresponding kinetic system of ordinary differential equations
du dv
(2.16) = f(u,v), = g(u,v),
dt dt
by analyzing eigenvalues of the corresponding linearization matrix, see Remark 5.2 below
for more details. On the other hand, by last condition in (2.15), the equation f(U,V) = 0
can be uniquely solved with respect to U in the neighborhood of (u¯,v¯).
In the case described by Proposition 2.9, we say that a non-constant solution (U,V)
touches a constant solution (u¯,v¯). Now, we prove an important property of the con-
stant solutions which are touched by non-constant solutions, and, for simplicity of the
exposition, we limit our study to regular solutions.
Theorem 2.10. Let U(x),V(x) be a regular stationary solution of problem (1.1)–(1.3)
and assume that all c(cid:0)onstant stat(cid:1)ionary solutions which are touched by (U,V), are non-
degenerate (cf. (2.15)). Then, at least at one of those constant solutions, denoted here by
(u¯,v¯), the following inequality holds
1 f (u¯,v¯) f (u¯,v¯)
(2.17) det u v > 0.
f (u¯,v¯) (cid:18) g (u¯,v¯) g (u¯,v¯) (cid:19)
u u v
The proof of Theorem 2.10 is based on properties of solutions to the elliptic Neumann
problem (2.4)–(2.5) (see Theorem 5.1, below) and we present it in Section 5.
UNSTABLE PATTERNS IN REACTION-DIFFUSION-ODE SYSTEMS 7
Now, let us derive properties of a constant solution (u¯,v¯) which satisfies inequality
(2.17).
Remark 2.11. One should notice in the following corollaries that we do not impose the
autocatalysis condition (2.7) to prove the instability of a constant steady state. Here, this
condition is a consequence of the fact that the constant solution (u¯,v¯) is touched by a
non-constant solution and because (u¯,v¯) satisfies inequality (2.17).
Corollary 2.12. Denote by (u¯,v¯) a non-degenerate constant solution of problem (1.1)-
(1.4), which satisfies inequality (2.17). Then, we have the following alternative:
either (u¯,v¯) is the constant solution of the reaction-diffusion-ODE system (1.1)-
•
(1.4) with the DDI property,
or (u¯,v¯) is an unstable solution of the kinetic system (2.16).
•
Intheexample discussed inSubsection 3.2, we show thatbothcases fromCorollary2.12
can indeed occur in one model, depending on its parameters.
Now, we have to emphasize the following necessary condition for the DDI property,
which is obtained immediately from Corollary 2.12.
Corollary 2.13. If problem (1.1)-(1.4) has only one constant solution (u¯,v¯) which is
asymptotically stable as a solution to kinetic system (2.16) and if there exists a noncon-
stant regular stationary solution which touches (u¯,v¯), than the constant steady state (u¯,v¯)
is an unstable solutions of (1.1)-(1.4) (thus, (u¯,v¯) has the DDI property).
Next, we discuss the compensation condition (2.10) to link Theorem 2.10 with the the
assumptions of Corollary 2.4.
Corollary 2.14. Let (u¯,v¯) be a constant solution of problem (1.1)-(1.4) satisfying in-
equality (2.17). If (u¯,v¯) is an asymptotically stable solution to kinetic system (2.16), then
it satisfies the compensation inequality f (u¯,v¯)g (u¯,v¯) < 0.
v u
Thus, we obtain the following local version of the compensation assumption (2.10) for
a regular non-constant stationary solution.
Corollary 2.15. Let (U,V) be a non-constant regular stationary solution of problem
(1.1)-(1.4). Denote by (u¯,v¯) a constant solution which is touched by (U,V) and which
satisfies inequality (2.17). Assume that (u¯,v¯) is an asymptoticallystable solution of kinetic
system (2.16). Then, there exists x Ω such that f U(x ),V(x ) g U(x ),V(x ) < 0.
0 v 0 0 u 0 0
∈
(cid:0) (cid:1) (cid:0) (cid:1)
In other words, every non-constant solution U(x),V(x) which touches a constant
solution with the DDI property has to satisfy co(cid:0)mpensation(cid:1)property (2.10) at least at
the point x and, by continuity, in a neighborhood of x . Notice, however, that we require
0 0
this condition in Corollary 2.4 to be satisfied for all x Ω.
∈
3. Model examples
In this section, our results are illustrated by applying them to nonlinearities from math-
ematical biology.
3.1. Gray-Scott model. Instabilityphenomena describedinSection2appear, forexam-
ple, inthefollowinginitial-boundaryvalueproblemwithnonlinearitiesasinthecelebrated
Gray-Scott system describing pattern formation in chemical reactions [7]
(3.1) u = (B +k)u+u2v for x Ω, t > 0,
t
− ∈
(3.2) v = ∆v u2v +B(1 v) for x Ω, t > 0,
t
− − ∈
8 ANNAMARCINIAK-CZOCHRA,GRZEGORZ KARCH,ANDKANAKOSUZUKI
where B and k are positive constants, with the zero-flux boundary condition for v and
withnonnegative initial conditions. Here, every regularpositive stationarysolution(U,V)
of the Neumann boundary-initial value problem for equations (3.1)-(3.2) has to satisfy
the relation U = (B +k)/V, hence,
(B +k)2
(3.3) ∆V BV +B = 0 for x Ω,
− − V ∈
(3.4) ∂ V = 0 for x ∂Ω.
ν
∈
All continuous positive solutions of such boundary value problem in one dimensional case
have been constructed in our recent paper [14, Sec. 5]. A construction of discontinuous
stationary solutions to the reaction-diffusion-ODE problem for (3.1)-(3.2) can be also
found in [14, Thm. 2.9].
Our instability results, formulated in Section 2.1, implicate that all stationary solutions
(constant, regular as well as discontinuous) of the reaction-diffusion-ODE problem for
equations (3.1)-(3.2) are unstable under heterogeneous perturbations. For the proof, it
suffices to notice that the autocatalysis assumption (2.7) holds true because, for U =
(B +k)/V, the function f U(x),V(x) is independent of x and satisfies
u
(cid:0) (cid:1)
λ = f U(x),V(x) = (B +k)+2U(x)V(x) = B +k > 0 for all x Ω.
0 u
− ∈
(cid:0) (cid:1)
This is the case of nonlinearities to which one can apply Corollary 2.1, because the com-
pensation assumption (2.10) at a positive solution (U,V) is also valid due to the following
calculation f (U,V) g (U,V) = 2U3V < 0 for all positive stationary solutions.
v u
· −
Now, it suffices to apply either Theorem 2.1 or Corollary 2.4.
3.2. Activator-inhibitor system. Now, we consider the following activator-inhibitor
system with no diffusion of activator
up
(3.5) u = u+ for x Ω, t > 0,
t − vq ∈
ur
(3.6) τv = ∆v v + for x Ω, t > 0,
t
− vs ∈
supplemented with positive initial data and with the zero-flux boundary condition for
v = v(x,t). We assume that τ > 0 and exponents satisfy p > 1, q,r > 0 and s 0. Such
≥
nonlinearities were proposed by Gierer and Meinhardt [6] to describe a system consisting
of a slowly diffusing activator and a rapidly diffusing inhibitor. In their model, a high
concentration of activator is supposed to induce morphogenetic changes in a biological
tissue.
First, we observe that every positive regular stationary solution satisfies the relation
Up−1 = Vq, thus, the function V = V(x) is a solution of the boundary value problem
(3.7) ∆V V +VQ = 0 for x Ω,
− ∈
(3.8) ∂ V = 0 for x ∂Ω,
ν
∈
where Q = qr/(p 1) s. This problem with Q > 1 was considered in a series of papers,
− −
e.g. [35, 11, 25, 26]. In particular, when 1 < Q < N+2 for N 3, and 1 < Q < for
N−2 ≥ ∞
N = 1,2, it was shown that it has a positive solution V for a sufficiently large domain Ω.
We refer the reader to the review article [23] for more comments and references on this
problem.
UNSTABLE PATTERNS IN REACTION-DIFFUSION-ODE SYSTEMS 9
Here, the autocatalysis condition (2.7) holds true for regular stationary solutions of
system (3.5)-(3.6), because
Up−1
f (U,V) = 1+p = 1+p > 0 for Up−1 = Vq.
u − Vq −
Notice that λ = f U(x),V(x) is independent of x, as required in Corollary 2.4. Next,
0 u
we check directly th(cid:0)e compensat(cid:1)ion assumption (2.10)
Up Ur−1 Up+r−1
f U,V g U,V = q r = rq < 0 for all x Ω,
v · u (cid:18)− Vq+1(cid:19)(cid:18) Vs (cid:19) − Vs+q+1 ∈
(cid:0) (cid:1) (cid:0) (cid:1)
for every positive stationary solution (U,V). As a consequence either of Theorem 2.1 or of
Corollary 2.4, we obtain that all positive regular stationary solutions to system (3.5)-(3.6)
with the Neumann boundary condition for v are unstable.
By Proposition 2.9, every regular stationary solution of the Neumann boundary-initial
value problem for system (3.5)-(3.6) has to touch the only constant positive solution
(u¯,v¯) = (1,1) of system (3.5)-(3.6). In the case of the kinetic system corresponding to
(3.5)–(3.6), it is shown in [24] that this constant solution is
(1) asymptotically stable if the parameter τ > 0 is small;
(2) unstable if the parameter τ > 0 is large.
Thus, byCorollary2.12,theconstantsolution(u¯,v¯) = (1,1)ofthereaction-diffusion-ODE
system (3.5)-(3.6) with small τ > 0, has the DDI property.
3.3. Model of early carcinogenesis. The main motivation for the research reported in
this work comes fromour studies of the reaction-diffusion system of three ordinary/partial
differential equations modeling the diffusion-regulated growth of a cell population of the
following form
av
(3.9) u = d u for x Ω, t > 0,
t c
u+v − ∈
(cid:16) (cid:17)
(3.10) v = d v +u2w dv for x Ω, t > 0,
t b
− − ∈
(3.11) w = D∆w d w u2w+dv +κ for x Ω, t > 0,
t g 0
− − ∈
supplemented withzero-fluxboundaryconditions forthefunctionw andwithnonnegative
initial conditions, [14]. Here, the letters a,d ,d ,d ,d,D,κ denote positive constants.
c b g 0
In the present paper, we focus on a reduced two-equation version of model (3.9)-(3.11),
obtained by quasi-steady state approximation of the dynamics of v. Applying the quasi-
steady state approximation in equation (3.10) (namely, assuming that v 0), we obtain
t
≡
the relation v = u2w/(d +d), which after substituting into remaining equations (3.9) and
b
(3.11) yields the following initial-boundary value problem for the reaction-diffusion-ODE
system
auw
(3.12) u = d u for x Ω, t > 0,
t c
1+uw − ∈
(cid:16) (cid:17)
d
(3.13) w = D∆w d w b u2w +κ for x Ω, t > 0.
t g 0
− − d +d ∈
b
Here, the autocatalysis assumption (2.9) and the compensation assumption (2.10) are
satisfied by simple calculations, which are analogous to those in previous examples. As
a consequence, all nonnegative stationary solutions of system (3.12)-(3.13) are unstable
due to Theorem 2.1 and Corollary 2.7. This gives a counterpart of our results on three
equation model (3.9)-(3.11) proved in [14].
10 ANNAMARCINIAK-CZOCHRA,GRZEGORZ KARCH,ANDKANAKOSUZUKI
Numerical simulations suggest that the two-equation model exhibits qualitatively the
same dynamics as (3.9)-(3.11). A rigorous derivation of the two equation model (3.12)-
(3.13) from the model (3.9)-(3.11) as well as other properties of solutions to (3.12)-(3.13)
are presented in Appendix A of this work. Our detailed studies of stability properties
of space homogeneous solutions of the two equation model (3.12)-(3.13) are reported
in Appendix B. In particular, by Corollary 2.12, constant steady states of (3.12)-(3.13)
are either unstable solutions of the corresponding kinetic system or they have the DDI
property.
4. Instability of stationary solutions
4.1. Existence of solutions. Webegin ourstudy ofproperties of solutions tothe initial-
boundary value problem (1.1)–(1.4) by recalling the results on local-in-time existence and
uniqueness of solutions for all bounded initial conditions.
Theorem 4.1 (Local-in-time solution). Assume that u ,v L∞(Ω). Then, there exists
0 0
∈
T = T( u , v ) > 0 such that the initial-boundary value problem (1.1)–(1.4) has a
0 ∞ 0 ∞
k k k k
unique local-in-time mild solution u,v L∞ [0,T], L∞(Ω) .
∈
(cid:0) (cid:1)
We recall that a mild solution of problem (1.1)–(1.4) is a couple of measurable functions
u,v : [0,T] Ω R satisfying the following system of integral equations
× 7→
t
(4.1) u(x,t) = u (x)+ f u(x,s),v(x,s) ds,
0
Z
0 (cid:0) (cid:1)
t
(4.2) v(x,t) = et∆v (x)+ e(t−s)∆g u(x,s),v(x,s) ds,
0
Z
0 (cid:0) (cid:1)
where et∆ is the semigroup of linear operators generated by Laplacian with the Neumann
boundary condition. Since our nonlinearities f = f(u,v) and g = g(u,v) are locally
Lipschitz continuous, to construct a local-in-time unique solution of system (4.1)–(4.2), it
suffices toapplytheBanachfixedpointtheorem. Detailsofsuch areasoningandtheproof
of Theorem 4.1 in a case of much more general systems of reaction-diffusion equations can
be found in [29, Thm. 1, p. 111], see also our recent work [14, Ch. 3] for a construction
of nonnegative solutions of particular reaction-diffusion-ODE problems.
Remark 4.2. If u and v are more regular, i.e. if for some α (0,1) we have u Cα(Ω),
0 0 0
∈ ∈
v C2+α(Ω) and ∂ v = 0 on∂Ω, then the mildsolution of problem (1.1)–(1.4) issmooth
0 ν 0
∈
and satisfies u C1,α [0,T] Ω and v C1+α/2,2+α [0,T] Ω . We refer the reader to
∈ × ∈ ×
[29, Thm. 1, p. 112] a(cid:0)s well as to(cid:1)[5] for studies of gene(cid:0)ral reactio(cid:1)n-diffusion-ODE systems
in the H¨older spaces.
4.2. General instability result. The next goal in this section isto show that, under ap-
propriate conditions, linear instability of stationary solutions of a reaction-diffusion-ODE
problem implies their nonlinear instability. Such a theorem is well-known for ordinary
differential equations. Furthermore, in the case of reaction-diffusion equations, one can
apply the abstract result from the book by Henry [8, Thm. 5.1.3]. However, in the
case of reaction-diffusion-ODE problems, we often encounter discontinuous stationary so-
lutions and the corresponding linearization operator may have a non-empty continuous
spectrum. Hence, checking the assumptions of the general result from [8] does not seem
to be straightforward.
Therefore, we propose here a different approach which was developed to study stability
of stationarysolutions offluid dynamic equations [10, 4], using the, so-called, linearization
