- Email: [email protected]
Dynamical behavior for a cross-diffusion model of forests
216
Communications
in Nonlinear
Science
& Numerical
Simulation
Vo1.2,
No.4
(Dec.
1997)
References [l] [2] [3] [4] [5] [6] [7] [8]
F’risch, U., Hasslacher, B. and Pomeau, Y., Lattice gas automata for the Navier-Stokes equation. Phys. Lett., 1986, 56:1505-1508. Bernardin, D., Sero-Guillaume, O., and Sun, C.H., Multispecies 2D lattice gas with energy levels: Diffusive properties. Physica D, 1991, 47:169-188. Bernardin, D., Sero-Guillaume, 0. and Sun, C.H., Thermal conduction in 2D-lattice gases. in Discrete models of fluid dynamics, World Scientific, 1990, 72-84. Chen, H., Chen, S. and Matthaeus, W., Recovery of the Navier-Stokes equation using a lattice-gas Boltzmann method. Phys. Rev. A, 1992, 45(8):5339-5342. Qian, Y.H., d’Hum%res, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation. Europhys. Lett., 1992, 17:479-484. McNamara G. and Alder, B., Analysis of the Lattice Boltzmann treatment of hydrodynamics. Physica A, 1993, 194:218-228. Chen, Y., Ohashi, H. and Akiyama, M., Heat transfer in lattice BGK modeled fluid. J. Stat. Phys., 1995, 81(1/2):71-85. McNamara, G., Garcia, A. and Alder, B., Stabilization of thermal lattice Boltzmann models. J. Stat. Phys., 1995, 81(1/2):395-408.
Dynamical Forest?
Behavior
for a Cross-Diffusion
Model
of
Yuanqu LIN2 (School of Mathematical Sciences, Peking University, Beijing 100871, China) E-mail: [email protected] In this paper, the existence of a bounded global attractor for a Cross-Diffusion model of forest with homogeneous Dirichlet boundary condition is proved under some condition on the parameters. Key Words: Age-structured forest model, cross-diffusion, equivalent norm, Lyapunov function, a-contraction, gradient semigroup Abstract:
Introduction A simple mathematical account of seed production
model of mono-species forest with two age classes which takes and dispersal is first presented in [l], ut vt wt 1
=spw-y(v)u-fu = fu-hv =CYV-/3w+Dww,,
(1)
where 21and v are tree densities of “young” and “old” age classes, w is density of air-borne seeds, (Y, p, S and D are seed production, deposition, establishment and diffusion rates respectively; f and h are coefficients of aging and mortality of “old” tree respectively, while lThe paper 2Supported
was received on Mar. 28 rd, 1997 by the Science Foundation of Peking
University
LIN: Dynamical
No.4
Behavior for a Cross-Diffusion
Model of.. .
217
y(v) is a mortality rate of “young” trees. Considering the qualitative nature of the model, it is possible to choose T(V) = a (V - b)2 + c with positive a, b, c . By means of an asymptotic procedure and by a linear change of variables (time and space scales), we can transform system (1) into the following lower dimensional reaction cross-diffusion model with dimensionless form (see [l], 222-223 ): ‘Ilt 1 vt
=pv-(v-1)221-ss’1L+v,,
(2)
=u-hv
where p = (Y6 u = ~(t, z), and v = v(t, z) are the real scalar functions for z E (0, L) and t>o In this paper we consider system (2) with initial data 40, z) = 210(z), In addition,
v(O, x) = uo(x) (z E (0, L))
v satisfies the homogeneous Dirichlet boundary
(3)
condition
v(t, 0) = v(t, L) = 0 (t 2 0) Equivalently,
(2) can be written
(4)
as the following hyperbolic
equation
vtt + [(v - 1)2 + s + h]vt + [h (v - 1)2 + h s - p] v - v,, = 0
(5)
1. Preliminaries Let A = -$ and its domain is D(A) = H2(0,L) initial data (3) can be written in the vector form
nHi(O,L).
System
jl = cy + f(y) (t > 0) Y(O) = Yo
(2) with the
(6)
where
c=
1
-“;I
p!;;],
y=
[ ;]
where si = s + 1 and I is the identity in Hi(O,L) It has been found that the eigenvalues of C
/I? = -;
E E,
> 0,
[ -yy)]
(7)
1 = i (h - SI)
(8)
or L2(0,L).
(SI + h) f dm,
Suppose i (s+l-h)2+p-Xi
f(y)=
where
then Z$ are real, and the corresponding eigenvectors
are cl:
*w,
l
= [ ‘:‘I
gsin(yz)
where
d=dm We introduce a new inner (., .)* on E, i.e., for V yi = Ui e 1 + vi e2 E E (YI,Y~)*
=
(Vv1,Vv2)
+ Cd2 -Al>
(v1,v2)
+
(w
-
Zv1,212
where Z and d defined in (8) and (9), respectively. The norm I].(] * = dm to the usual norm ]]e(]E.
-
(9) (i = 1,2) let
Zv2)
is equivalent
Communications
218
2. Main
in Nonlinear
Science & Numerical
Simulation
Vo1.2, No.4 (Dec. 1997)
results
Our main results are as follows: Theorem (i)
1 Assume d” - X1 > 0, then
For V ya E E, there exists a unique vector-valued function Y(-) = Y(., YO) E (X0, +m); E)
such that
~(0, YO) = YO
and y(t) satisfies the integral equation y(t) = ecu@‘yo +
J0
(10)
t eCup lt--‘)fu, (y(T)) dr
where fwp(y) = f(y) + wpy. In this case, y(t) is called a mild solution of (6). (ii) Ifyo
th ere exists y(a) E C([O, +oo); D(C)) n C1 (R+; E) which satisfies (6)
E D(C),
and y(0) = yc. (iii) Ifd’---1
>0
and O
then
IlleCtl[* 5 embt
(V t > 0)
vlhere o = 1~: I. Theorem 2 For system (2), there is a bounded attractive from bounded sets are bounded. Theorem 3 Suppose p < 26 toticallp stable.
region in E and the orbits starting
- 1, then the zero solution of (2) is exponentially
asymp-
For any t 2 0, we introduce a map S(t) : yc ++ y(t,ya), where y(t, yo) is the mild solution (or solution) of (6) with y(O,yc) = ys, then S(t) : E + E (or D(C) --+ D(C)) is a continuous map. Therefore, S(t) I t > 0 is a continuous semidynamical system on E or D(C). Let
GEWA), 4 = W,
E=N+El then E is the set of equilibrium
-@‘+[h(+1)2+hs-p]+o points of (2) in E.
Theorem4 Assume da--Ax1 >O, Ol+$. Then (2), (4) defines a gradient semigroup S(t) on E, the equilibrium set & is bounded, and there is a connected global attractor A in E. We are interested in non-negative nonconstant
E, =
v E D(A)
steady state solutions of (5), i.e., v E &P,
w(0) = w(L) = 0, w(x)
$
$$ + f(w) = 0 (0 < x < L)
0 ‘I
(11)
I
where f(w)=-hw[v’-2w+s+
(12)
No.4
LIN: Dynamical
T(a)
Behavior for a Cross-Diffusion
d5
= F(a)
219
Model of.. .
(O
F(t)
and Lo=&(;),
L1=
7i-
(13)
l/w=3 Theorem
5 Suppose that LO, L1 and E, are defined as in (13) and (ll),
respectively.
Case 1. p < (s + 1)h. In this case, E, = 0. Case 2. p = (s + 1)h. In this case, (i) if L > LO, th ere are precisely two solutions 211,212E E,; (ii) if L = LO, t here is precisely one solution z E E,; (iii) ifL
< LO, E, = 0.
Case 3. p > (s + 1)h. In this case, (i) i;fLc < L < LI, th ere are precisely two solutions VI, 02 E E,; (ii ) if L = LO or L > L1 , there is precisely one solution V E E,; (iii) if L < LO, I,, =
0.
Every v E EP satisfies v,(x)>0
(O
v,(x)
< 0 (f
L
?Jz(f L) = 0
Acknowledgement The author would like to thank Profs. Qian Min and Ye Qi-xiao for their encouragement.
References [l]
Kuznetsov, Yu.A., Antonovsky, M.Ya.,Biktashev, V.N. and Aponina, E.A., A cross diffusion model of forest boundary dynamics, J. Math. Biol., 1994, 32:219-232. [2] Wu Ya-ping, The Existence of Travelling Waves for a Cross-Diffusion Model of Forest Boundary Dynamics, Beiijing Mathematics, 1995, l(l):278287. [3] Hale, J.K., Asymptotic Behavior of Dissipative Systems, Amer. Math. Sot., Providence, RI., 1988. [4] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., Vol. 44, Springer-Verlag, 1983.
Communications
in Nonlinear
Science
& Numerical
Simulation
Vo1.2,
No.4
(Dec.
1997)
References [l] [2] [3] [4] [5] [6] [7] [8]
F’risch, U., Hasslacher, B. and Pomeau, Y., Lattice gas automata for the Navier-Stokes equation. Phys. Lett., 1986, 56:1505-1508. Bernardin, D., Sero-Guillaume, O., and Sun, C.H., Multispecies 2D lattice gas with energy levels: Diffusive properties. Physica D, 1991, 47:169-188. Bernardin, D., Sero-Guillaume, 0. and Sun, C.H., Thermal conduction in 2D-lattice gases. in Discrete models of fluid dynamics, World Scientific, 1990, 72-84. Chen, H., Chen, S. and Matthaeus, W., Recovery of the Navier-Stokes equation using a lattice-gas Boltzmann method. Phys. Rev. A, 1992, 45(8):5339-5342. Qian, Y.H., d’Hum%res, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation. Europhys. Lett., 1992, 17:479-484. McNamara G. and Alder, B., Analysis of the Lattice Boltzmann treatment of hydrodynamics. Physica A, 1993, 194:218-228. Chen, Y., Ohashi, H. and Akiyama, M., Heat transfer in lattice BGK modeled fluid. J. Stat. Phys., 1995, 81(1/2):71-85. McNamara, G., Garcia, A. and Alder, B., Stabilization of thermal lattice Boltzmann models. J. Stat. Phys., 1995, 81(1/2):395-408.
Dynamical Forest?
Behavior
for a Cross-Diffusion
Model
of
Yuanqu LIN2 (School of Mathematical Sciences, Peking University, Beijing 100871, China) E-mail: [email protected] In this paper, the existence of a bounded global attractor for a Cross-Diffusion model of forest with homogeneous Dirichlet boundary condition is proved under some condition on the parameters. Key Words: Age-structured forest model, cross-diffusion, equivalent norm, Lyapunov function, a-contraction, gradient semigroup Abstract:
Introduction A simple mathematical account of seed production
model of mono-species forest with two age classes which takes and dispersal is first presented in [l], ut vt wt 1
=spw-y(v)u-fu = fu-hv =CYV-/3w+Dww,,
(1)
where 21and v are tree densities of “young” and “old” age classes, w is density of air-borne seeds, (Y, p, S and D are seed production, deposition, establishment and diffusion rates respectively; f and h are coefficients of aging and mortality of “old” tree respectively, while lThe paper 2Supported
was received on Mar. 28 rd, 1997 by the Science Foundation of Peking
University
LIN: Dynamical
No.4
Behavior for a Cross-Diffusion
Model of.. .
217
y(v) is a mortality rate of “young” trees. Considering the qualitative nature of the model, it is possible to choose T(V) = a (V - b)2 + c with positive a, b, c . By means of an asymptotic procedure and by a linear change of variables (time and space scales), we can transform system (1) into the following lower dimensional reaction cross-diffusion model with dimensionless form (see [l], 222-223 ): ‘Ilt 1 vt
=pv-(v-1)221-ss’1L+v,,
(2)
=u-hv
where p = (Y6 u = ~(t, z), and v = v(t, z) are the real scalar functions for z E (0, L) and t>o In this paper we consider system (2) with initial data 40, z) = 210(z), In addition,
v(O, x) = uo(x) (z E (0, L))
v satisfies the homogeneous Dirichlet boundary
(3)
condition
v(t, 0) = v(t, L) = 0 (t 2 0) Equivalently,
(2) can be written
(4)
as the following hyperbolic
equation
vtt + [(v - 1)2 + s + h]vt + [h (v - 1)2 + h s - p] v - v,, = 0
(5)
1. Preliminaries Let A = -$ and its domain is D(A) = H2(0,L) initial data (3) can be written in the vector form
nHi(O,L).
System
jl = cy + f(y) (t > 0) Y(O) = Yo
(2) with the
(6)
where
c=
1
-“;I
p!;;],
y=
[ ;]
where si = s + 1 and I is the identity in Hi(O,L) It has been found that the eigenvalues of C
/I? = -;
E E,
> 0,
[ -yy)]
(7)
1 = i (h - SI)
(8)
or L2(0,L).
(SI + h) f dm,
Suppose i (s+l-h)2+p-Xi
f(y)=
where
then Z$ are real, and the corresponding eigenvectors
are cl:
*w,
l
= [ ‘:‘I
gsin(yz)
where
d=dm We introduce a new inner (., .)* on E, i.e., for V yi = Ui e 1 + vi e2 E E (YI,Y~)*
=
(Vv1,Vv2)
+ Cd2 -Al>
(v1,v2)
+
(w
-
Zv1,212
where Z and d defined in (8) and (9), respectively. The norm I].(] * = dm to the usual norm ]]e(]E.
-
(9) (i = 1,2) let
Zv2)
is equivalent
Communications
218
2. Main
in Nonlinear
Science & Numerical
Simulation
Vo1.2, No.4 (Dec. 1997)
results
Our main results are as follows: Theorem (i)
1 Assume d” - X1 > 0, then
For V ya E E, there exists a unique vector-valued function Y(-) = Y(., YO) E (X0, +m); E)
such that
~(0, YO) = YO
and y(t) satisfies the integral equation y(t) = ecu@‘yo +
J0
(10)
t eCup lt--‘)fu, (y(T)) dr
where fwp(y) = f(y) + wpy. In this case, y(t) is called a mild solution of (6). (ii) Ifyo
th ere exists y(a) E C([O, +oo); D(C)) n C1 (R+; E) which satisfies (6)
E D(C),
and y(0) = yc. (iii) Ifd’---1
>0
and O
then
IlleCtl[* 5 embt
(V t > 0)
vlhere o = 1~: I. Theorem 2 For system (2), there is a bounded attractive from bounded sets are bounded. Theorem 3 Suppose p < 26 toticallp stable.
region in E and the orbits starting
- 1, then the zero solution of (2) is exponentially
asymp-
For any t 2 0, we introduce a map S(t) : yc ++ y(t,ya), where y(t, yo) is the mild solution (or solution) of (6) with y(O,yc) = ys, then S(t) : E + E (or D(C) --+ D(C)) is a continuous map. Therefore, S(t) I t > 0 is a continuous semidynamical system on E or D(C). Let
GEWA), 4 = W,
E=N+El then E is the set of equilibrium
-@‘+[h(+1)2+hs-p]+o points of (2) in E.
Theorem4 Assume da--Ax1 >O, O
E, =
v E D(A)
steady state solutions of (5), i.e., v E &P,
w(0) = w(L) = 0, w(x)
$
$$ + f(w) = 0 (0 < x < L)
0 ‘I
(11)
I
where f(w)=-hw[v’-2w+s+
(12)
No.4
LIN: Dynamical
T(a)
Behavior for a Cross-Diffusion
d5
= F(a)
219
Model of.. .
(O
F(t)
and Lo=&(;),
L1=
7i-
(13)
l/w=3 Theorem
5 Suppose that LO, L1 and E, are defined as in (13) and (ll),
respectively.
Case 1. p < (s + 1)h. In this case, E, = 0. Case 2. p = (s + 1)h. In this case, (i) if L > LO, th ere are precisely two solutions 211,212E E,; (ii) if L = LO, t here is precisely one solution z E E,; (iii) ifL
< LO, E, = 0.
Case 3. p > (s + 1)h. In this case, (i) i;fLc < L < LI, th ere are precisely two solutions VI, 02 E E,; (ii ) if L = LO or L > L1 , there is precisely one solution V E E,; (iii) if L < LO, I,, =
0.
Every v E EP satisfies v,(x)>0
(O
v,(x)
< 0 (f
L
?Jz(f L) = 0
Acknowledgement The author would like to thank Profs. Qian Min and Ye Qi-xiao for their encouragement.
References [l]
Kuznetsov, Yu.A., Antonovsky, M.Ya.,Biktashev, V.N. and Aponina, E.A., A cross diffusion model of forest boundary dynamics, J. Math. Biol., 1994, 32:219-232. [2] Wu Ya-ping, The Existence of Travelling Waves for a Cross-Diffusion Model of Forest Boundary Dynamics, Beiijing Mathematics, 1995, l(l):278287. [3] Hale, J.K., Asymptotic Behavior of Dissipative Systems, Amer. Math. Sot., Providence, RI., 1988. [4] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., Vol. 44, Springer-Verlag, 1983.