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1ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №1
УДК 517.956.4
https://doi.org/10.52754/16948645 2023 1 287
ON THE PERIODIC SOLUTION OF THE KELLER-SEGEL MODEL OF CHEMOTAXIS WITH A LOGISTIC SOURCE
Таkhirov Jozil Ostanovich, Dr Sc,professor, prof.takhirov@yahoo.com Boborakhimova Makhbuba Ikhtiyorovna, PhD student
kamina9314@mail.ru Institute of Mathematics, Acad.Sc., Tashkent, Uzbekistan Abstract. The paper investigates a system of three parabolic equations, which is a model of the spatiotemporal state of two competing populations of species, both of which are chemotactically attracted by the same signal substance. Individuals move according to random diffusion and chemotaxis, and both populations reproduce themselves and mutually compete with each other according to the classical Lotka-Volterra kinetics. The global existence and uniqueness of the classical solutions of this system is proved by the contraction mapping principle using a priori Lp estimates and Schauder-type estimates for parabolic equations.
Key words: Keller—Segel model, chemotaxis, a priori estimates, global solution.
O ПЕРИОДИЧЕСКОМ РЕШЕНИИ МОДЕЛИ ХЕМОТАКСИСА КЕЛЛЕРА-СЕГЕЛЯ С ЛОГИСТИЧЕСКОМ ИСТОЧНИКОМ
ТахировЖозил Останович, д.ф.-м.н.,профессор, prof.takhirov@yahoo.com Боборахимова Махбуба Ихтиёровна, базовый докторант
kamina9314@mail.ru Институт математики АН РУз, Ташкент, Узбекистан Аннотация. В статье исследуется система трех параболических уравнений, представляющая собой модель пространственно-временного состояния двух конкурирующих популяций видов, хемотаксически притягиваемых одним и тем же сигнальным веществом. Особи перемещаются в соответствии со случайной диффузией и хемотаксисом, и обе популяции воспроизводятся и взаимно конкурируют друг с другом согласно классической кинетике Лотка-Вольтерра. Глобальное существование и единственность классических решений этой системы доказывается принципом сжимающих отображений с использованием априорных оценок Lp и оценок типа Шаудера для параболических уравнений.
Ключевые слова: модель Келлера—Сегеля, хемотаксис, априорные оценки, глобальное решение .
1. Introduction ( Введение). It is known that in the mathematical modeling of the self-organization of living cells, the system of equations "Keller-Segel" is used [1],
ut =Au -V(wVv), (1)
vt = Av—v + u
The system describes the general behavior of a set of cells under the influence of chemotaxis. Under such conditions, the movement of each individual cell, although not entirely predictable, follows a preferred direction, namely to higher concentrations of a certain signaling chemical. If u (x, t) is the cell density, and v(x, t) is the chemical concentration, then the first
equation in (1) reflects the interaction of non-directional diffusion motion, on the one hand, and the "chemotactic motion" controlled by Vv, with others. The second equation expresses the assumption of the model that the signaling substance, in addition to diffusion and degradation, like most chemicals, is constantly produced by living cells. This association is known to be present in many other biologically significant situations associated with chemotaxis [2]. The striking feature of (1) is that, despite its simple mathematical structure, it turned out to be able to describe the phenomenon of spatial self-organization of cells.
Usually, consider two competing populations of biological species that are attracted to the same chemical stimulus. All individuals move according to the laws of random diffusion and chemotaxis, and both populations reproduce and mutually compete with each other according to the classical Lotka-Volterra kinetics .
In this note, we study a problem with periodic boundary conditions for a quasilinear system proposed by J. Tello and M. Winkler [3] and investigated in [4], which models the dynamics of populations of two competing species in the regionQ = {(x,t) ; 0 <x <L,t >0|,bothof which are chemotactically attracted by the same signal substance ut = {dxux -%uwx)x + nx (l - u - alv)u,
V = (d2vx "^x)x + Ml (! - a2u - v)v, (x, t) G Q>
wt = wxx -Aw + u + v , (2)
u(x, 0) = u0 (x), v(x, 0) = v0 (x), w(x, 0) = w0 (x), x e(0, L), U(0, t) = U(L,t), Ux (0, t) = Ux (L,t), t > 0,
where U(x, t) = (u, v, w), u (x, t)and v(x, t) are the population densities of two competing species in space-time (x, t) and w( x, t) are concentration of the attracting substance, di, Mi, ai , i = 1, 2 A are positive constants, / and £ are assumed to be non-negative constants. It is assumed that both species u and v direct their movements chemotactically along the concentration gradient of the chemical above the habitat. This is modeled by taking both % > 0 and £ > 0. Biologically, x and £ measure the strength of chemical attraction to species u andv,
respectively. Species kinetics is assumed to be of the classical Lotka-Volterra type, where Ml M2
measure the internal growth rate, u, v and ^ a2 interpret the strength of interspecific competition. In addition, the chemical is produced by both species at the same rate with no saturation effect and at the same time consumed by a particular enzyme in the environment at a rate of ^.
In the paper, first, some a priori Lp estimates and Schauder-type estimates are established. Next, we prove that model (1) has a unique classical global solution for any chemotactic
coefficients %, ^ > 0 .
2. A priori estimates and global existence
Let us now turn to establishing Lp -estimates (u, v, w) under the above conditions. Global existence (2) is a consequence of several lemmas.
The no negativity of (u, v, w) follows from maximum principles [5].
Lemma 1. If m1, M2 > 0 and (u, v, w) is the only non-negative solution of equation (2) in (0, Tmax),then there is a constant C>0 depending on ||u0, v0, w0||Ll and L such that
l|u(•, t)Ll) + l|v(•, t^L1(0,L) + |w(•, t^L1(0,L) ^ C, for an t - i0, Tmx). (3) Proof. We integrate the u -equation of (1) over (0, L) and have that d
d |u (x, t}dx = |Mi (1 _ u - a1vv)udx < Jm1 (1 _ u }udx
then it follows from the Gronwall's lemma that
L L
| u (x, t)dx < e U1' | u0 (x)dx + L,
0 0
similarly we can show
L L
J v(x, t)dx < e ~"2' J v0 (x)dx + L,
0 0
Integrating the w-equation over(0, L), we easily see that ||w(-, tis uniformly
bounded for all t e (0, »). This completes the proof of Lemma 1.
To obtain their L00 -bounds, we shall see that it is sufficient to obtain the boundedness of ||wx (', t)||. For this purpose, we first convert the w -equation into the following abstract form
'(-, t) = e (A-1JiW0 + fe (A-1}t ((1-A)w(,s) + u(,s) + v(,s))ds, (4)
w\> , -
0
2
where A = d-T . To estimate w (x, t) in (4), we apply the well-known smoothing properties of
dx
operator -A + 1 and estimates between the linear analytic semigroups generated by |e tA} . We have for all 1 < p < q < &, there exists a positive constant C dependent on ¡u1, ¡u2, and
IK IU(0,L) such that
\\w (•> t )IL M* C(! + i * ^ - s r 2 " 2 " (•' s ) + « (-. s ) + V (-, s )lds), (5)
0
7T
where t e (0, T), T e (0, œ], v = — is the first Neumann eigenvalue of -A .
L
Lemma 2. Let us assume the same conditions (u0, v0, w0 ) as in Lemma 1. For any q e (1, œ) ,there exists a positive constant C (q) such that
|wt)|U(0,L) * C(q). vt e(0, ). (6)
Proof. Let p = 1in (5), then we have that
i i ( i i
|w (■, C(1 + je - s)- 2 - 2 lp - iJ (||u (, s|l1(„,l) +
0
l|v(-. s^L(0,L) +llw(-. s)d) . (7)
On the other hand, there exists a constant C0 > 0 such that for any q e (1, ro)
t -1+ i
sup f e -v(t-ss (t - s) 2q ds < C
ie(0,®) I
then we conclude from (7) that
Ik (•' OIL(o^ cd+5P(|u C' *)IU)+llv (•' *)IU)+l k (•' s )ILl ))) . (8)
Finally, (6) is an immediate consequence of (1) and (8).
Lemma 3. Ifp e (2, ,then there is a constant C(p) > 0 such that
||u t t)\\LP +||v(s t)^ < c(p),V/ e(0, ) . (9)
Proof. We shall only show that ||u (•, t)|| < C (p), since ||v (•, t)|| < C (p), can be proved by the same arguments. For p > 2, we multiply the first equation of (1) by u p 1 and integrate it over (0, L) by parts, then it follows from simple calculations that 1 d L
p dt
L L L
Jup = Jup~lut = Jup 1 (djux - %uw
)x + j"Mi (1 " u - aiv)
u
< -
4dj (p - 1) P2
4dj (p - 1)
P2
u
u
+ Z\(up-j)xuwx - MJup+1 + Cj
2
Zju 2 (u P )
P 0
w -
x x
Mi L ,p+i
j up+1 +Cj, (10)
Where C1 is a positive constant that depends onp . It follows from Holder's and Young's inequality that
L p p
J u2 (u2)Xwxdx <
p P p
< u 2 2(p+1) L p (u 2)x ||wx|| L = ||u|| 2Lp+i L (u 2)x Wj L2p L
2d1 px
(u p ) x
r \\p + c2 ||u||lp+1 :
In light of (11), we obtain from (10) that
L d L p dt 0
Jupdx < C3 ||u||Lp+1 -uu1 +C1 .
(11)
(12)
LJ
Denotingyp (t) = | up (x, t}dx, one can apply Holder's on (12) to obtain that
p+1
yp (t) < -CAyJ (t) + C5, yp (0) = ||u0|\PLP .
We conclude that yp (t) < C(p)for all t g(0, «). Similarly, we can show that
J up (x, t)dx < C(p). This completes the proof of Lemma 3.
Theorem 4. Suppose at, Mi, i = 1, 2, and X are positive constants. Then for positive initial data (u0, v0, w0)gH1 (0, L)xH1 (0, L)xH1 (0, L)and any constantsx, % G R, problem (1) has a unique bounded positive solution {u(x, t), v(x, t), w(x, t)) defined on [0, L] x [0, such that (u(•,t), v(•,t), w(•,t))e C([0,H1 (0, L)xH1 (0, L)xH1 (0,L))
and (u, v) e C^a 2+a1+a ([0, L]x[0, »)) for some a e ( 0, 1 |.
References
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2. T. Hillen, A users' guide to PDE models for chemotaxis[Text]/ T. Hillen, K. Painter// J. Math. Biol. ,2009. V.58 . pp. 183-217.
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2
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3. J.Tello. Stabilization in a two-species chemotaxis system with a logistic source[Text]/ J.Tello, M.Winkler// Nonlinearity.2012. v.25 .pp. 1413-1425.
4. Qi Wang . Global existence and steady states of a two competing species Keller -Segel chemotaxis model[Text]/ Qi Wang, Lu Zhang, Jingyue Yang and Jia Hu// American Institute of Math. Sc..2015.V. 8, N. 4, pp.777-807.
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