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A Numerical Comparison Between Degenerate Parabolic and Quasilinear Hyperbolic Models of Cell Movements Under Chemotaxis

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A NUMERICAL COMPARISON BETWEEN DEGENERATEPARABOLIC AND QUASILINEAR HYPERBOLIC MODELSOF CELL MOVEMENTS UNDER CHEMOTAXIS
M. TWAROGOWSKA
1
, R. NATALINI
1
, AND M. RIBOT
2
Abstract.
We consider two models which were both designed todescribe the movement of eukaryotic cells responding to chemical signals.Besides a common standard parabolic equation for the diﬀusion of achemoattractant, like chemokines or growth factors, the two modelsdiﬀer for the equations describing the movement of cells. The ﬁrstmodel is based on a quasilinear hyperbolic system with damping, theother one on a degenerate parabolic equation. The two models havethe same stationary solutions, which may contain some regions withvacuum. We ﬁrst explain in details how to discretize the quasilinearhyperbolic system through an upwinding technique, which uses anadapted reconstruction, which is able to deal with the transitions tovacuum. Then we concentrate on the analysis of asymptotic preservingproperties of the scheme towards a discretization of the parabolicequation, obtained in the large time and large damping limit, in orderto present a numerical comparison between the asymptotic behavior of these two models. Finally we perform an accurate numerical comparisonof the two models in the time asymptotic regime, which shows that therespective solutions have a quite diﬀerent behavior for large times.
1.
Introduction
The movement of cells, bacteria or other microorganisms under theeﬀect of a chemical stimulus, represented by a chemoattractant, such aschemokines or growth factors, has been widely studied in mathematics in thelast two decades, see [25, 30, 31, 38], and various models involving partial
diﬀerential equations have been proposed to describe this evolution. Thebasic unknowns in these chemotactic models are the density of individualsand the concentrations of some chemical attractants. One of the mostconsidered models is the Patlak-Keller-Segel system [27, 36], where the
evolution of the density of cells is described by a parabolic equation, andthe concentration of a chemoattractant is generally given by a parabolic orelliptic equation, depending on the diﬀerent regimes to be described and on
keywords and phrases:
chemotaxis, quasilinear hyperbolic problems with source,degenerate parabolic problems, comparison between parabolic and hyperbolic models,stationary solutions with vacuum, well-balanced scheme, asymptotic behavior.
1
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionaledelle Ricerche, via dei Taurini 19, I-00185 Roma, Italy (
mtwarogowska@gmail.com
,
roberto.natalini@cnr.it
).
2
Laboratoire J. A. Dieudonn´e, UMR CNRS 7351, Universit´e de Nice-Sophia Antipolis,Parc Valrose, F-06108 Nice Cedex 02, France & Project Team COFFEE, INRIA SophiaAntipolis, France (
ribot@unice.fr
).
1
a r X i v : 1 4 0 2 . 2 8 3 1 v 2 [ m a t h . N A ] 1 5 F e b 2 0 1 4
2 M. TWAROGOWSKA, R. NATALINI, AND M. RIBOT
the authors’ taste. The behavior of this systems is quite well known now,at least for linear diﬀusions: in the one-dimensional case, the solution isalways global in time [32], while in two and more dimensions the solutionsexist globally in time or blow up according to the size of the initial data, see[9, 10] and references therein, and see the recent result of global existence
for large data in the parabolic-parabolic case [4]. However, a drawback of this model is that the diﬀusion leads alternatively to a fast dissipation oran explosive behavior, while in general, from a biological point of view, itis much more interesting to observe the creation of patterns and permanentstructures.In order to avoid these drawbacks and to improve the accuracy of the transient description, some modiﬁcations of the srcinal Keller-Segelformulation were introduced to prevent overcrowding, by taking into accountthe volume ﬁlling eﬀect; see [25, 38, 23, 35]. For instance, in [28, 8], a
nonlinear diﬀusion is considered. More precisely, denoting by
ρ
(
x,t
) thedensity of cells and by
φ
(
x,t
) the concentration of a generic chemoattractant,the Keller-Segel-like system with nonlinear diﬀusion reads
ρ
t
=
P
(
ρ
)
xx
−
χ
(
ρφ
x
)
x
,δφ
t
=
Dφ
xx
+
aρ
−
bφ,
(1)where
χ,D,a
and
b
are given positive parameters. The cells move followingthe direction of the gradient of the concentration of chemoattractant with aresponse coeﬃcient
χ
; they also diﬀuse and
P
is a phenomenological, densitydependent function, which is usually given by a pressure law for isentropicgases, such as
P
(
ρ
)=
κρ
γ
, γ>
1
, κ>
0
,
(2)which is intended to prevent the overcrowding of cells. Besides, the evolutionof chemoattractant is still given by a linear diﬀusion equation with a sourceterm which depends on
ρ
. The chemoattractant is released by the cells,diﬀuses in the environment and it is degraded in ﬁnite time. The positiveparameters
D,a,b
are respectively its diﬀusion coeﬃcient, the productionrate, which is proportional to the cell density, and the degradation rate. If
δ
=1, we consider a parabolic-parabolic model and in the case where
δ
=0,we deal with a parabolic-elliptic model.Now, it is also expected that a hyperbolic model will enable us to observeintermediate organized structures, like aggregation patterns, at a ﬁner scale[37]. In [14, 24] the advantage of the hyperbolic approach over the parabolic
one was considered in the case of a semilinear model of chemotaxis basedon the Cattaneo law. In particular, the authors described qualitativelysome experiments of patterns formation. Here, we focus on a quasilinearhyperbolic model of chemotaxis introduced by Gamba et al. [17] to describethe early stages of the vasculogenesis process. This model writes as ahyperbolic-parabolic system for the following unknowns: the density of cells
ρ
(
x,t
), their momentum
ρu
(
x,t
) and the concentration
φ
(
x,t
) of achemoattractant:
ρ
t
+(
ρu
)
x
=0
,
(
ρu
)
t
+
ρu
2
+
P
(
ρ
)
x
=
−
αρu
+
χρφ
x
,φ
t
=
Dφ
xx
+
aρ
−
bφ.
(3)
COMPARISON OF HYPERBOLIC AND PARABOLIC CHEMOTAXIS MODELS 3
The positive constants
χ
and
α
measure respectively the strength of the cellsresponse to the concentration of the chemical substance and the strengthof the damping forces. The pressure
P
is still given by the pressure lawfor isentropic gases (2). This model of chemotaxis has been introduced todescribe the results of in vitro experiments performed by Serini et al. [41]using human endothelial cells which, randomly seeded on a matrigel, formedcomplex patterns with structures depending on the initial number of cells.Although analytical results about this model are still far from beingcomplete, for the Cauchy problem on the whole space and in all spacedimensions, so with no boundary conditions, it is possible to prove the globalexistence of smooth solutions if the initial datum is a small perturbation of asmall enough constant state, see [12, 13]. In the case of the one dimensionalboundary value problem, when the diﬀerential part is linearized, the globalexistence and the time asymptotic decay of the solutions were proved in[22], if the initial data are small perturbations of stable constant stationarystates. To complete the analytical study of the quasilinear model there aresome clear diﬃculties. The ﬁrst one lies in the appearance of regions of vacuum during the evolution of the time solution, since the hyperbolic partof the model degenerates as the eigenvalues coincide; as far as we know,the only related results are given in [26, 21], and they are about the localexistence of solutions for the Euler equations with damping and vacuum,but without chemotaxis.From a more biological point of view, the appearance of non constantsolutions with a succession of regions with high density of cells and regionsof vacuum, can be put in correspondence with the formation of patterns,such as a network of blood capillaries. In [34], present authors analyzedthe existence of some non-constant steady states to model (3) on a onedimensional bounded domain. In particular, for the pressure law (2) with
γ
=2, a complete description of the stationary solutions formed of one regionof positive density near the boundary and one region of vacuum was given.Numerical simulations also shown that such solutions are stable and can befound as asymptotic states of the system (3) even for strictly positive initialdata. In the following, we will call ”bump” a region with a nonnegativedensity surrounded by two regions of vacuum, as shown in blue in Figure1, and a ”lateral half bump” will be a bump cut in its middle and stuckto an extremity of the interval, as shown in red in the same Figure 1.Other stationary conﬁgurations with several bumps have been also observednumerically as asymptotic states of the model in [34]. These conﬁgurationsare described in details with a comparison of their energy values in [3].Remark that in the case of bounded domains with no-ﬂux boundaryconditions, stationary solutions for both systems (3) and (1) coincide and
it is worth exploring if the asymptotic states of the two systems are thesame or not. Actually, one may expect the Keller-Segel type model (1)with
δ
=0 (i.e.: the parabolic-elliptic case) to be the large time and largedamping limit of the hyperbolic system (3), and this is actually the casein [29] for the case without chemotaxis or in [11] for our case, both results
being proved only on unbounded domains. In this paper, our main goalis to make a careful comparison of the two models (3) and (1) with
δ
=1,
4 M. TWAROGOWSKA, R. NATALINI, AND M. RIBOT
by analyzing numerically their actual asymptotic behavior. In particular,we are able to exhibit some sets of initial data and some parameters suchthat the two systems converge asymptotically to two diﬀerent stationarysolutions, namely two solutions with a diﬀerent number of bumps. In thatcase, the diﬀusive Keller-Segel model (1) seems to be more inclined to mergebumps together, so that the asymptotic solution contains a smaller numberof bumps than the asymptotic solution for the hyperbolic system (3), oftenafter a long transient where nothing happens. Some similar phenomenawhich are referred as metastability of patterns, were observed in the case of a Keller-Segel type model with linear diﬀusion and a logistic chemosensitivefunction in [23, 40, 15], see Subsection 5.3 for more details.
In order to perform such a comparison, we need ﬁrst to ﬁnd an accuratescheme for the hyperbolic system to make a reliable comparison of the twomodels. The approximation of this system needs special care due to thepresence of vacuum states and emergence of non-constant steady states.More precisely, the discretization procedure has to generate non-negativesolutions with ﬁnite speed of propagation and should resolve properly thenon constant equilibria, characterized by a vanishing ﬂux. Such problemsare well-known when dealing with hyperbolic equations with sources, see forinstance [33, 18, 20, 19]. For that purpose, we consider the well-balanced
scheme proposed in [34] and based on the Upwinding Sources at Interfacesmethodology [5, 6, 39]. In [34], we used a hydrostatic reconstruction,
introduced by Audusse et al. [2] in the case of shallow-water equationsand by Bouchut, Ounaissa and Perthame [7] in the case of Euler equationswith large damping. To use this approach, we compute the reconstructedinterface variables by integrating the equation for stationary solutions witha constant velocity. According to the form of the equation we integrate,two diﬀerent reconstructions can be found; both lead to schemes that areconsistent with the hyperbolic problem, preserve the non-negativity of thedensity, and are exact on non-constant steady states. However, we show inthis paper that only one of these two schemes is asymptotically consistentwith a conservative scheme for the parabolic model in the large time andlarge damping limit, and therefore it will be the one used in our comparison.Another improvement of the scheme described here, with respect with [34],is the implicit treatment of the damping term, which solves more accuratelythe vacuum states and the ﬂux on the non constant equilibria.This paper is organized as follows: after a brief recall in Section 2 aboutthe structure of the stationary solutions with vacuum, found in [34] and [3],
we propose in Section 3 two diﬀerent numerical schemes for the quasilinearhyperbolic system (3) based on well-balanced techniques, with a particularcare for their asymptotic preserving property. Then, in Section 4, we showsome numerical evidences of the behavior of these schemes in order to choosea well adapted scheme. Finally, in Section 5, we present an accurate schemefor the parabolic system (1), based on the diﬀusive relaxation techniques of [1], and we perform a careful numerical comparison between the asymptoticsolutions for systems (3) and for system (1).
COMPARISON OF HYPERBOLIC AND PARABOLIC CHEMOTAXIS MODELS 5
2.
Stationary solutions with vacuum
In [34] and [3], we noticed that in the particular case
γ
=2, it is possibleto compute explicitly and classify the stationary solutions with vacuum of the two systems (3) and (1), which obviously coincide. Let us recall brieﬂy
these results to make the paper almost self-contained.Consider system (3) on a one dimensional bounded domain [0
,L
] withno-ﬂux boundary conditions, that is
ρ
x
(0
,
·
)=
ρ
x
(
L,
·
)=0
, ρu
(0
,
·
)=
ρu
(
L,
·
)=0
, φ
x
(0
,
·
)=
φ
x
(
L,
·
)=0
.
(4)Notice that, under these conditions, the stationary solutions of system (3)and system (1) coincide. Remark also that, when considering the evolutionproblem (3) or (1) with the previous boundary conditions (4), the mass of
the density is constant in time, namely
M
=
[0
,L
]
ρ
(
x,
0)
dx
=
[0
,L
]
ρ
(
x,t
)
dx,
for all
t
≥
0
.
(5)Therefore, the mass
M
will be considered in what follows as a parameterwhich characterizes stationary solutions.2.1.
Constant solution.
The ﬁrst type of solutions is given by the constantsolutions, that is to say, for all domain length
L>
0 and all mass
M >
0, thereis a solution deﬁned by (
ρ,u,φ
)=(
M L ,
0
,aM bL
), which is the only constantsolution in the space of stationary states. This kind of solution is displayedin green in Figure 1.2.2.
One lateral half bump.
Let us denote by
ω
= 1
D
aχ
2
κ
−
b
. Weassume that
ω>
0 and
L> π
√
ω
. Then there exists a unique, positive solution(up to symmetry) of mass
M
with only one region of positive density andone region of vacuum, given by the following expression :
ρ
(
x
)=
χ
2
κφ
(
x
)+
K,
for
x
∈
[0
,
¯
x
]
,
0
,
for
x
∈
(¯
x,L
]
,
(6a)and
φ
(
x
)=
2
κbK ωχD
cos(
√
ωx
)cos(
√
ω
¯
x
)
−
aK ωD,
for
x
∈
[0
,
¯
x
]
,
−
2
κK χ
cosh(
bD
(
x
−
L
))cosh(
bD
(¯
x
−
L
))
,
for
x
∈
(¯
x,L
]
.
(6b)The free boundary point ¯
x
is given by the only value ¯
x
∈
1
√
ω
(
π/
2
,π
) whichsolves the equation
bωD
tan(
√
ω
¯
x
)=tanh(
bD
(¯
x
−
L
))
,
(6c)

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