Boundary Value Problems
Volume 2010, Article ID 285961,24pages doi:10.1155/2010/285961
Research Article
Global Existence and Convergence of Solutions to
a Cross-Diffusion Cubic Predator-Prey System with
Stage Structure for the Prey
Huaihuo Cao
1and Shengmao Fu
21Department of Mathematics and Computer Science, Chizhou College, Chizhou 247000, China
2College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Huaihuo Cao,caguhh@yahoo.com.cn Received 3 December 2009; Accepted 30 March 2010
Academic Editor: Wenming Zou
Copyrightq2010 H. Cao and S. Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.
1. Introduction and Mathematical Model
The predator-prey model as, which follows, the ordinary differential equation system
du dt
b1b2u−b3u2
u−b4uv,
dv
dt −cv
αu−βvv
1.1
is said to be the general Lotka-Volterra predator-prey model in 1–3, and to be cubic predator-prey system in 4, where u, v are the population densities of prey and predator species at time t, respectively. b3, b4, c, α, β are positive constants, b1 is nonnegative as the
mortality rate of predator population, and the survival of predator species is dependent on the survival state of prey species, and b2u−b3u2, βv are the respective density restriction
terms of prey and predator species.b4uis the predation rate of the predator, andαuis the
conversion rate of the predator. In4, three questions about system1.1are discussed: the stability of nonnegative equilibrium points, and the existence, as well as numbers of limit cycle.
Referring to5, we establish cubic predator-prey system with stage structure for the prey as follows:
dx1
dt η1x2−r1x1−η2x1b2x
2
1−b3x13−b4x1x3,
dx2
dt η2x1−r2x2,
dx3
dt −cx3
αx1−βx3
x3,
1.2
where x1 and x2 are the population densities of the immature and mature prey species,
respectively, andx3 denotes the density of the predator species. The predators live only on
the immature prey species, as well as the survival of the predator species is dependent on the survival state of the immature prey species.η1, η2, r1, r2, b3, b4, c, α, βare positive constants,
and the sign ofb2is undetermined.η1 andr1are the birth rate and the mortality rate of the
immature prey species, respectively.r2 and care the net mortality rate of the mature prey
population and the predator population, andη2is the conversion rate of the immature prey
to the mature prey species.b2x1−b3x21andβx3are the respective density restriction terms of
the immature prey species and predator species.b4x1is the predation rate of the predator to
the immature prey population, andαx1is the conversion rate of the predator.
Using the scaling
u1
α r2
x1, u2
α η2
x2, u3
β r2
x3, dτr2dt, 1.3
and redenotingτbyt, system1.2reduces to
du1
dt a0u2−a1u1a2u
2
1−a3u31−ku1u3,
du2
dt u1−u2,
du3
dt −bu1−u3u3,
1.4
wherea0η1η2/r22, a1 r1η2/r2, a2b2/α, a3 b3/r2, kb4/β,andbc/r2are positive
constants, anda2b2/αis undetermined to the sign.
we derive the following PDE system of reaction-diffusion type:
u1t−d1Δu1a0u2−a1u1a2u21−a3u31−ku1u3, x∈Ω, t >0,
u2t−d2Δu2u1−u2, x∈Ω, t >0,
u3t−d3Δu3 −bu1−u3u3, x∈Ω, t >0,
∂ηui 0, i1,2,3, x∈∂Ω, t >0,
uix,0 ui0x≥0, i1,2,3, x∈Ω,
1.5
where∂η ∂/∂η,η is the unit outward normal vector of the boundary∂Ωwhich we will assume to be smooth. The homogeneous Neumann boundary condition indicates that the above system is self-contained with zero population flux across the boundary. The positive constantsd1,d2, andd3are said to be the diffusion coefficients, and the initial valuesui0x
i1,2,3are nonnegative smooth functions.
Note that, in recent years, there has been considerable interest to investigate the global behavior of a system of interacting populations by taking into account the effect of self as well as cross-diffusion. According to the ideas in6–13, especially to8,9, the cross-diffusion term will be only included in the third equation, that is, the following cross-diffusion system:
ut Δ
d1uα11u2
a0v−a1ua2u2−a3u3−kuw, x∈Ω, t >0,
vt Δ
d2vα22v2
u−v, x∈Ω, t >0,
wt Δ
d3wα31uwα32vwα33w2
−bu−ww, x∈Ω, t >0,
uηx, t vηx, t wηx, t 0, x∈∂Ω, t >0,
ux,0 u0x≥0, vx,0 v0x≥0, wx,0 w0x≥0, x∈Ω.
1.6
In the above, di, αii i 1,2,3, α31, and α32 are positive constants. d1, d2 and d3 are the
diffusion rates of the three species, respectively. αii i 1,2,3 are referred to as self-diffusion pressures.α31andα32are cross-diffusion pressures. The term self-diffusion implies
the movement of individuals from a higher to a lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. Generally, the value of the cross-diffusion coefficient may be positive, negative, or zero. The term positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species, and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species
9.
The paper will be organized as follows. In Section 2, we analyze the asymptotical stability of equilibrium points for the ODE system1.4via linearization and the Lyapunov method. In Section 3, we prove the global existence of solutions and the stability of the equilibrium points to the diffusion system1.5. InSection 4, we investigate the existence of nonnegative classical global solutions by assuminga0,a1,a2,a3,k,bto be positive constants
only for the simplicity of calculation, and the global asymptotic stability of unique positive equilibrium point to the cross-diffusion system1.6.
2. Equilibrium Solution of the ODE System
In this section we discuss the stability of unique positive equilibrium point for system1.4. The following theorem shows that the solution of system1.4is bounded.
Theorem 2.1. Let u1t, u2t, u3t be the solution of system 1.4with initial values ui0 > 0i 1,2,3, and let0, T be the maximal existence interval of the solution. Then0 < uit ≤
Mi i1,2,3, t∈0, T, where
M1max
u10 a0a1u20, C0
1 a0
a1 ,
M2max
u10
a0a1
,C0 a1
,
M3max{u30, M1−b}.
2.1
The aboveC0is a positive constant depending only ona0, a1, a2, a3, and furtherT ∞.
Proof. It is easy to see that1.4has a unique positive local solutionu1t, u2t, u3t. Let
T ∈0,∞be the maximal existence time of the solution, and combinu1andu2linearly, that
is,u1 a0a1u2, it follows from1.4that
d
dtu1 a0a1u2≤ −a1u2a0u1a2u
2
1−a3u31. 2.2
Using Young inequality, we can check that there exists a positive constantC0depending only
ona0, a1, a2,anda3such that
a0u1a2u21−a3u31≤C0− a1
a0a1
u1. 2.3
It follows that
d
dtu1 a0a1u2≤C0− a1
a0a1
which implies that there existM1andM2referring to2.1such that 0 < u1 ≤M1, 0 < u2 ≤
M2, andt∈0, T.
Finally, we note that du3/dt −b u1 − u3u3 ≤ −b M1 − u3u3. Let M3
max{u30, M1−b}, then
du3−M3
dt u3−M3u3≤0 u30−M3≤0. 2.5
From the comparison inequality for the ODE, we haveu3−M3≤0,t∈0, T.
Thus the solutions for system1.4are bounded. Further, from the extension theorem of solutions, we haveT ∞.
By the simple calculation, the sufficient conditions for system1.4having a unique positive equilibrium point as follows:
ia2−k 2
a3a1−a0−kb > 2a3b;iimax{a1−a0,a2−2a3bb} ≤ bk ≤ a2 −
a3bb, where the left equal sign holds if and only ifa1−a0 < a2 −2a3bb;iiia1 −a0 ≤
ba2 −a3b ≤ kb;iv a1 −a0 < kb ≤ a2 −2a3bb;v kb < a1 −a0 ≤ a2 −a3bband
a2−k ≥ max{2a3b,2
a3a1−a0−kb}, where the second equal sign holds if and only if
a3b2> a1−a0−kb;vi2
a3a1−a0−kb< a2−k≤2a3bandkb < a1−a0<a2−a3bb.
If one of the above conditions holds, then system 1.4 has the unique positive equilibrium pointu1, u2, u3, where
u1u2
a2−k
a2−k24a3kba0−a1
2a3 , u3u1−b.
2.6
Theorem 2.2. System1.4has the unique positive equilibrium point u1, u2, u3when one of the
above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. Ifa1−a0<a22−2ka2/4a3 kbholds, then
u1, u2, u3is locally asymptotically stable.
Theorem 2.2is easy to be obtained by using linearization; therefore, we omit its proof.
The objective of this section is to prove the following result.
Theorem 2.3. System1.4has the unique positive equilibrium point u1, u2, u3when one of the
above conditions (i), (ii), (iii), (iv), (v), and (vi) holds. Ifa1−a0<min{a22−2ka2/4a3 kb, kb−
a2k/a3}holds, thenu1, u2, u3is globally asymptotically stable.
Proof. We make use of the general Lyapunov function
Vut
3
i1
αi
ui−ui−uiln
ui
ui
, 2.7
dV dt
3
i1
αi
ui−ui
ui
dui
dt
α1
u1−u1
u1
×
a0
u1
u1u2−u2−u2u1−u1−u1u1−u1−a2a3u1a3u1−ku1u3−u3
α2
u2−u2
u2
1
u2
u2u1−u1−u1u2−u2 α3u3u1−u1−u3−u3
u3−u3
u3
−α1−a2a3u1a3u1u1−u12−α3u3−u32
α3−kα1u1−u1u3−u3 a0α1u1−u1
×
− u2
u1u1
u1−u1 1
u1
u2−u2
α2u2−u2u1−u1u2−u2−u2u1
u2u2
.
2.8
Letα2a0α1andα3kα1. Then
dV
dt −α1−a2a3u1a3u1u1−u1
2−α
3u3−u32
−a0α1
1
u1
u2
u1
u1−u1−
u1
u2
u2−u2 2
.
2.9
We observe that
a1−a0< kb−
a2k
a3
2.10
is a sufficient condition of−a2a3u1a3u1 >0. So, when condition2.10holds, we have
dV
dt ≤0. 2.11
Set D {u ∈ IntR3
: dV/dt 0} {u1, u2, u3}. According to the Lyapunov-LaSalle
invariance principle 14, u1, u2, u3 is global asymptotic stability if inequality 2.10and
all conditions ofTheorem 2.2are satisfied.Theorem 2.3is, thus, proved.
3. Stability of the PDE System without Cross-Diffusion
Denote thatFu f1, f2, f3T, whereu u1, u2, u3, f1a0u2−a1u1a2u21−a3u31−
ku1u3, f2 u1−u2,and f3 −bu1 −u3u3. It is easy to see thatf1, f2, f3 ∈ C1R 3
with
R3 {u1, u2, u3 | ui ≥ 0, i 1,2,3}. The standard PDE theory15shows that1.5has the unique solution u1, u2, u3 ∈ CΩ×0, T∩C2,1Ω×0, T
3
, where T ≤ ∞is the maximal existence time. The following theorem shows that the solution of1.5is uniformly bounded, and thusT ∞.
Theorem 3.1. Let u1, u2, u3 ∈ CΩ×0, T∩C2,1Ω×0, T 3
be the solution of system
1.5 with initial values ui0x ≥ 0 i 1,2,3, and let T be the maximal existence time.
Then 0 ≤ u1x, t, u2x, t ≤ M1, 0 ≤ u3x, t ≤ M2, and t ∈ 0, T, where M1 is a
positive constant depending only on Ω and all coefficients of 1.5 and ui0L∞Ωi 1,2,
M2 max{u30L∞Ω, M1−b}. Furthermore,T ∞and uix, t > 0on Ωfor any t > 0if
ui0≥/≡0 i1,2,3.
Proof. Letu1, u2, u3be the solution of1.5with initial valuesui0x≥ 0i 1,2,3. From
the maximum principle for parabolic equations16, it is not hard to verify thatuix, t ≥0 for x, t ∈ Ω×0, T i 1,2,3, whereT is the maximal existence time of the solution
u1, u2, u3. Furthermore, we know by the strong maximum principle thatuix, t > 0 onΩ for allt >0 ifui0 ≥/≡0i1,2,3. Next we prove that the solutionu1, u2, u3is bounded
onΩ×0, T.
Integrating the first two equations of1.5overΩand adding the results linearly, we have that, by Young inequality,
d dt
Ωu1 a0a1u2dx≤
Ωa0u1−a1u2dx
Ω
a2u21−a3u31
dx
≤ −a1
Ωu2dx
Ω
a0
a2 2
a3
u1−a2u21
dx
≤C− a1 a0a1
Ωu1 a0a1u2dx
3.1
for some positive constant C depending only on the coefficients of 1.5. Therefore,
u1,2tL1Ω is bounded in 0,∞. Using 17, Exercise 5 of Section 3.5, we obtain that
u1,2tL∞Ω is also bounded in 0,∞. Now note that supΩ×0,∞u1,2x, t ≤ M1. The maximum principle givesu3x, t≤max{u30L∞Ω, M1−b}:M2. The proof ofTheorem 3.1
is completed.
In order to prove the global stability of unique positive equilibrium solution for system
1.5, we first recall the following lemma which can be found in7,17.
Lemma 3.2. Leta and bbe positive constants. Assume that φ, ϕ ∈ C1a,∞,ϕt > 0,and
φ is bounded from below. Ifφt ≤ −bϕt, andϕt ≤ K in a,∞ for some constantK, then
Let 0 μ1 < μ2 < μ3 < · · · be the eigenvalues of the operator−Δ onΩ with the
homogeneous Neumann boundary condition, and letEμibe the eigenspace corresponding toμiinC1Ω. Denote thatX{u∈C1Ω
3
∂ηu0, x∈∂Ω},{φij;j1,2, . . . ,dimEμi}is an orthonormal basis ofEμi,andXij{c·φij|c∈R3}. Then
X∞
i1
Xi, Xi
dimEμi
j1
Xij. 3.2
Next we present the clear proof of the the global stability by two steps:
Step 1Local Stability. LetDdiagd1, d2, d3andLDFuuDΔ {aij}, where
Fuu ⎛ ⎜ ⎜ ⎝
−2a3u21a2u1−a0 a0 −ku1
1 −1 0
u3 0 −u3 ⎞ ⎟ ⎟
⎠
⎛ ⎜ ⎜ ⎝
a11 a12 a13
a21 a22 a23
a31 a32 a33 ⎞ ⎟ ⎟
⎠. 3.3
The linearization of1.5atuis
utLu. 3.4
For eachi≥1,Xiis invariant under the operatorL, andλis an eigenvalue ofLonXiif and only if it is an eigenvalue of the matrix−μiDFuu.
The characteristic polynomial of−μiDFuuis given by
ϕiλ λ3Aiλ2BiλCi, 3.5
where
Aiμid1d2d3−a11−a331,
Biμ2id1d2d1d3d2d3 μid11−a33−d2a11a33 d31−a11
a11a33−a13a31−a33−a11a0,
Ciμ3id1d2d3μ2id1d3−a11d2d3−a33d1d2
−μid1a33−d2a11a33−a13a31 d3a11a0 a33a11a0−a13a31.
3.6
Thus
whereP0, P1, P2,andP3are given by
P0 a11a33a13a31a33−a11a33 a11a0−a331a0−a11a0,
P1d1a11a33−a13a31−a11a0−d2a11a0 a33 d3a11a33−a33−a13a31 −a11a33−1d11−a33−d2a11a33 d31−a11,
P2 d1d2d3d11−a33−d2a11a33 d31−a11 −a11d1d2d3 d2d1d3−a33d3d1d2,
P3 d1d2d1d2d1d3d2d3 d23d1d2.
3.8
According to the Routh-Hurwitz criterion18, for eachi≥ 1, the three rootsλi,1, λi,2, λi,3 of
ϕiλ 0 all have negative real parts if and only if Ai > 0,Ci > 0, and Hi > 0. Noting thata33 < 0 anda13a31 < 0, the three roots have negative real parts ifa11a0 < 0. A direct
calculation shows thata11a0is negative if
a1−a0 <
a2 2−2a2k
4a3
kb. 3.9
Now we can conclude that there exists a positive constantδsuch that
Re{λi,1},Re{λi,2},Re{λi,3} ≤ −δ, i≥1. 3.10
In fact, letλμiξ, then
ϕiλ μ3iξ3Aiμ2iξ2BiμiξCi ϕiξ. 3.11
Sinceμi → ∞asi → ∞, it follows that
lim i→ ∞
ϕiξ
μ3
i
ξ3 d1d2d3ξ2 d1d2d2d3d1d3ξd1d2d3ϕξ. 3.12
It is easy to see thatd1, d2, d3 are the three roots ofϕξ 0. Thus, there exists a positive
constantδsuch that
Re{−d1},Re{−d2},Re{−d3} ≤ −δ. 3.13
By continuity, we see that there existsi0 such that the three rootsξi,1, ξi,2, ξi,3 ofϕiξ 0 satisfy
Re{ξi,1},Re{ξi,2},Re{ξi,3} ≤ −
δ
So
Re{λi,1},Re{λi,2},Re{λi,3} ≤ −
μiδ 2 ≤ −
δ
2, i≥i0. 3.15
Let
−δmax
1≤i≤i0
{Re{λi,1},Re{λi,2},Re{λi,3}}, 3.16
thenδ > 0, and3.10holds forδmin{δ, δ/ 2}.
Consequently, the spectrum ofL, consisting only of eigenvalues, lies in{Reλ≤ −δ}if
3.9holds, and the local stability ofufollows19, Theorem 5.1.1.
Step 2Global Stability. In the following,Cdenotes a generic positive constant which does not depend onx ∈Ωandt ≥0. Letube the unique positive solution. Then it follows from
Theorem 3.1thatu·, tis bounded uniformly onΩ, that is,ui·, t∞≤Cfor allt≥0. By20, TheoremA2,
ui·, tC2,αΩ≤C, ∀t≥1. 3.17
Define the Lyapunov function
Et
Ω
u1−u1−u1lnu1
u1
dxa0
Ω
u2−u2−u2lnu2
u2 dx k Ω
u3−u3−u3lnu3
u3
dx.
3.18
ThenEt≥0 for allt >0. Using1.5and integrating by parts, we have
Et
Ω
u1−u1
u1 u1ta0
u2−u2
u2 u2tk
u3−u3
u3 u3t dx − Ω
d1u1
u2 1
|∇u1|2a0
d2u2
u2 2
|∇u2|2kd3u3 u2
3 |∇u3|2
dx Ω
u1−u1
u1
f1u a0
u2−u2
u2
f2u k
u3−u3
u3
f3u dx ≤ Ω
−−a2a3u1a3u1u1−u12−ku3−u32
a0u1−u1
− u2
u1u1
u1−u1
1
u1
u2−u2
a0u2−u2
u1−u1u2−u2−u2u1
u2u2
Ω
−−a2a3u1a3u1u1−u12−ku3−u32
dx
−a0
u1 Ω u2 u1
u1−u1−
u1
u2
u2−u2 2
dx
≤ −
Ω−a2a3u1a3u1u1−u1
2dx−k
Ωu3−u3
2dx.
3.19
Takingl1−a2a3u1andl3k, we have that
Et≤ −l1
Ωu1−u1
2dx−l 3
Ωu3−u3
2dx, 3.20
wherel1>0 holds for
a1−a0< a3b−a2k
a3
. 3.21
FromTheorem 3.1the solutionuiof1.5is bounded, and so are the derivatives ofu1−u12
andu3−u32by equations in1.5. ApplyingLemma 3.2, we obtain
lim t→ ∞
Ωu1−u1
2dx0, lim
t→ ∞
Ωu3−u3
2dx0. 3.22
Asu2
i ≤C, it follows that
Et≤ −
Ω
d1u1
u2 1
|∇u1|2a0
d2u2
u2 2
|∇u2|2kd3u3 u2
3 |∇u3|2
dx
≤ −C
Ω
|∇u1|2|∇u2|2|∇u3|2dx
−ϕt.
3.23
Using inequality3.17and system1.5, the derivative ofϕtis bounded in1,∞. From
Lemma 3.2, we conclude thatϕt → 0 ast → ∞. Therefore
lim t→ ∞
Ω
|∇u1|2|∇u2|2|∇u3|2dx0. 3.24
Using the Poincar´e inequality yields
lim t→ ∞
Ωu1−u1
2dx lim
t→ ∞
Ωu2−u2
2dx lim
t→ ∞
Ωu3−u3
whereuit 1/|Ω| Ωuidx, i1,2,3.Thus, it follows from3.22and3.25that
|Ω|u1t−u12
Ωu1−u1
2dx≤2
Ωu1−u1
2dx2
Ωu1−u1
2dx−→0 3.26
ast → ∞. So we haveu1t → u1 ast → ∞. Similarly,u3t → u3 ast → ∞. Therefore,
there exists a sequence{tm}withtm → ∞such thatu1tm → 0. Asu2tmis bounded, there exists a subsequence of{tm}, still denoted by the same notation, and nonnegative constantu2
such that
u2tm−→u2. 3.27
Atttm, from the first equation of1.5, we have
|Ω|u1tm
Ωu1tdx !! !! tm
Ω
dΔu1f1u
dx!!!!
tm
Ω
a0
u1
u1u2−u2−u2u1−u1dx !! !! tm
Ω−u1u1−u1−a2a3u1u1dx !! !! tm
Ω−ku1u3−u3dx !! !! tm
−→0.
3.28
In view of3.22and3.27, it follows from3.28thatu2u2, thus
lim
m→ ∞u2tm u2. 3.29
According to 3.17, there exists a subsequence of {tm}, denoted still by {tm}, and nonnegative functionswi∈C2Ω, such that
lim
m→ ∞ui·, tm−wi·C2Ω0, i1,2,3. 3.30
In view of3.29and noting that in factu1t → u1andu3t → u3, we know thatwi≡ui, i 1,2,3. Therefore,
lim
m→ ∞ui·, tm−uiC2Ω0, i1,2,3. 3.31
The global asymptotic stability ofufollows from3.31and the local stability ofu.
Theorem 3.3. System1.5has the unique positive equilibrium point u1, u2, u3when one of the
conditions (i), (ii), (iii), (iv), (v), and (vi) inSection 2holds. If 3.9and3.21hold, thenu1, u2, u3
4. Global Existence of Classical Solutions and Convergence
In this section, we discuss the existence of nonnegative classical global solutions and the global asymptotic stability of unique positive equilibrium point of system1.6.
Some notations throughout this section are as follows: QT Ω × 0, T, u ∈
WpkΩ means that Dαu ∈ LpΩ for any |α| ≤ k with α α1, α2, . . . , αn, uWk pΩ
Ω"|α|≤k|Dαu|pdx
1/p
, u ∈ Wp2,1QT means that u, uxi, uxixji, j 1,2, . . . , n and ut
are in LpQ
T, uW2,1
p QT QT|u|
p|Du|p|D2u|p|u
t|pdx dt
1/p
, and uV2QT
sup0≤t≤Tu·, tL2Ω∇uL2Q
TwithV2QT L
∞0, T, L2Ω∩W1,0 2 QT. To obtainC2α,1α/2Q
Tnormal estimates of the solution for1.6, we present a series of lemmas in the following.
Lemma 4.1. Letu, v, wbe the solution of 1.6. Then there exists a positive constantM0(≥1) such
that
0≤u, v≤M0, 0≤w, ∀t≥0. 4.1
Proof. By applying the comparison principle20to system1.6, we haveu≥0, v≥0,and
w≥0 inQT. To prove thatu, v≤M0in the following, we consider the auxiliary problem
ut−d12α11uΔu2α11
n
i1
uxiuxi f1, x∈Ω, t >0,
vt−d22α22vΔv2α22
n
i1
vxivxi f2, x∈Ω, t >0,
∂ηux, t ∂ηvx, t 0, x∈∂Ω, t >0,
ux,0 u0x≥/≡0, vx,0 v0x≥/≡0, x∈Ω.
4.2
Notice that the functionsf1andf2are sufficiently smooth inR2, and are quasimonotone in
R2
. Let0,0andM, Nbe a pair of upper-lower solutions for4.2, whereMandNare positive constants. Direct calculation with inequalities
a0N−a1Ma2M2−a3M3≤0,
M−N≤0, u0≤M, v0≤N
4.3
yieldsM max{a2
a2
24a3|a0−a1|/2a3,u0L∞Ω}and N max{M,v0L∞Ω}. It follows that there existsM0 Kmax{M, N} ≥ 1 for anyt ≥ 0, whereK is a big enough
Lemma 4.2. LetX d1α11uu, andu∈L∞QTfor the solution to following equation:
ut Δd1α11uu f1, x, t∈Ω×0, T,
∂ηu0, x, t∈∂Ω×0, T,
ux,0 u0x≥0, x∈Ω,
4.4
whered1,α11are positive constants and0≤w∈L2QT. Then there exists a positive constantCT,
depending onu0W1
2Ωandu0L∞Ω, such that
XW2,1
2 QT≤CT. 4.5
Furthermore,
∇X∈V2QT, ∇u∈L2n2/nQT. 4.6
Proof. It is easy to check, fromX d1α11uu, that
Xt d12α11uΔXC1−C2w, 4.7
whereC1 d12α11ua0v−a1ua2u2 −a3u3andC2 kd12α11uu.C1and C2 are
bounded inQTfrom4.1. Multiplying4.7by−ΔX, and integrating by parts overQt, yields
1 2
Ω|∇Xx, t|
2dx−1
2
Ω|∇Xx,0|
2dxd 1
Qt
|ΔX|2dx dt
≤
QT
|C1C2wΔX|dx dt.
4.8
Using H ¨older inequality and Young inequality to estimate the right side of4.8, we have
C1C2wL2Q
TΔXL2QT≤m1
1wL2Q
T
ΔXL2Q
T
≤ m211M3 2
2d1
d1
2 ΔX
2
L2QT
4.9
with somem1>0. Substituting4.9into4.8yields
sup
0≤t≤T
Ω|∇Xx, t|
2dxd 1
Qt
where m2 depends on u0W1
2Ω and u0L∞Ω. Since X ∈ L
2Q
T, the elliptic regularity estimate10, Lemma 2.3yields
QT |Xxixj|
2dx dt≤m
3, i, j 1, . . . , n. 4.11
From 4.7, we have Xt ∈ L2QT. Hence, XW22,1QT ≤ CT. Moreover, the Sobolev
embedding theorem shows that4.6holds.
Lemma 4.3Lemma 4.3 can be presented by combining Lemmas 2.3 and 2.4 in11. Letp >
1,p24p/nq1, and letwsatisfy
sup
0≤t≤T
wL2p/p1Ω∇wL2QT<∞, 4.12
and there exist positive constantsβ ∈0,1andCT such that Ω|w·, t|βdx ≤CT ∀t≤ T. Then
there exists a positive constantMindependent ofw but possibly depending onn,Ω,p,β,andCT
such that
wLqQ T≤M
⎧ ⎨ ⎩1
sup
0≤t≤T
wtL2p/p1Ω
4p/np1p
∇w2L/2pQ
T ⎫ ⎬
⎭. 4.13
Finally, one proposes some standard embedding results which are important to obtain the C2α,1α/2Q
Tnormal estimates of the solution for1.6.
Lemma 4.4. LetΩ⊂RNbe a fixed bounded domain and∂Ω⊂C2. Then for allu∈Wq2,1QTwith
q≥1, one has
1∇uLpQ
T≤CuWq2,1QT, for all1≤p≤n2q/n2−q, q < n2,
2∇uLpQ
T≤CuWq2,1QT, for all1≤p≤ ∞, qn2,
3∇uCα,α/2QT ≤CuW2,1
q QT for all1−n2/q≤α≤1, q > n2,
whereCis a positive constant dependent onq, n, Ω,andT.
The main result about the global existence of nonnegative classical solution for the cross-diffusion system1.6is given as follows.
Theorem 4.5. Assume thatu0 ≥0, v0 ≥ 0,andw0 ≥0satisfy homogeneous Neumann boundary
conditions and belong toC2λΩfor someλ ∈ 0,1. Then system1.6has a unique nonnegative
Proof.
Step 1. L1-, L2-Estimates andLq-Estimates of w. Firstly, integrating the third equation of1.6 overΩ, we have
d dt
Ωw dx≤ 1 2
Ωu
2dx−1
2
Ωw
2dx
≤ |Ω|
2 M
2 0−
1 2|Ω|
Ωw dx 2
.
4.14
Thus
Ωw dx≤max
M0|Ω|,
Ωw0dx :M1, ∀t≥0. 4.15
Furthermore
wL1QT≤
T
0
M1dt:M2. 4.16
Integrating4.14in0, Tand moving terms yield
wL2QT≤
M20|Ω|T2w0L1Ω
1/2
:M3. 4.17
Secondly, multiplying the third equation of1.6byqwq−1 q >1and integrating over
Ω, we have
d dt
Ωw
qdx≤ −4
q−1d3
q
Ω !!
!∇wq/2!!!2dx−8q
q−1α33
q12
Ω !!
!∇wq1/2!!!2dx
−q−1α31
Ω∇u· ∇w
qdx−q−1α
32
Ω∇v· ∇w qdx
q
Ωw
q−bu−wdx.
Integrating the above expression in0, t t≤Tyields
Ωw
qx, tdx4
q−1d3
q
Qt !!
!∇wq/2!!!2dx dt8q
q−1α33
q12
Qt !!
!∇wq1/2!!!2dx dt
≤
Ωw q
0xdx−
q−1α31
Qt
∇u· ∇wqdx dt−q−1α32
Qt
∇v· ∇wqdx dt
q
Qt
wq−bu−wdx dt.
4.19
Since∇u∈L2n2/nQ
TfromLemma 4.2, and using H ¨older inequality and Young inequality,
we have
−q−1α31
Qt
∇u· ∇wqdx dt−q−1α32
Qt
∇v· ∇wqdx dt
≤ 2q
q−1α31
q1
!! !! !
Qt
wq−1/2∇wq1/2· ∇u dx dt!!!!!
2q
q−1α32
q1
!! !! !
Qt
wq−1/2∇wq1/2· ∇v dx dt!!!!!
≤ 2q
q−1
q1
α31∇uL2n2/nQ
Tα32∇vL2n2/nQT
× wq−1/2
Ln2QT∇wq1/2L2QT
≤C3)))∇
wq1/2)))
L2QTw q−1/2
Ln2QT
≤C3ε)))∇wq1/2))) 2
L2Q
T
C3
4εw
q−1/22
Ln2QT.
4.20
From4.1andαpβ≤p/p1αp1 1/p1βp1 α, β >0, it holds that
q
Qt
wq−bu−wdx dt≤q−bM0
Qt
wqdx dt−q
Qt
wq1dx
≤q−bM0|QT|1/q1wqLq1Q
t−qw q1
Lq1Q
t
≤−bM02|QT|C4.
Taking w wq1/2 and selecting a proper εsuch that C
3ε ≤ 4qq−1α33/q12, then
applying4.20and4.21to4.19yields
Ωw
2q/q1x, tdx4
q−1d3
q
Qt !!
!∇wq/q1!!!2dx dt4q
q−1α33
q12
Qt
|∇w|2dx dt
≤ w0qLqΩ
C3
4εw
2q−1/q1
Lq−1n2/q1QTC4 ≤C5
1wL2qq−−11/qn2/q11Q T
.
4.22
Denote thatE≡sup0<t<T Ωw2q/q1x, tdx Q
T|∇w|
2dx dt. Then it follows from4.22that
E≤C6
1wL2qq−−11/qn2/q11Q T
. 4.23
It is easy to see that 2q/q1<2<qandq−1n2/q1<q24q/nq1for any
q < nn4/n2−4; hence
E≤C7
1w2LqqQ−1/q1 T
. 4.24
Takeβ2/q1∈0,1. Then it follows fromL1Ω-estimates ofw,namely4.15, that
wLβΩ
Ω|w·, t|
βdx1/βw1/β L1Ω≤M
1/β
1 , ∀t≤T. 4.25
It follows fromLemma 4.3and4.24that
E≤C7 ⎡ ⎣1
MMsup
0<t<T
w·, t4Lq/nq2q/q1Ω1q∇w
2/q
L2Q
T
2q−1/q1⎤ ⎦
≤C8 ⎡ ⎣1
sup
0<t<T
w·, t2Lq/q2q/q11Ω
4q−1/nq1q
∇w2L2Q
T
2q−1/q1q ⎤ ⎦
≤C8
1E2n/nq·2q−1/q1−.
4.26
Since2n/nq·2q−1/q1 ∈ 0,1,Eis bounded by contrary proof. It follows thatwLqQ
T is bounded, that is, w ∈ L
q1q/ 2Q
T. It is easy to check that q1q/ 2 ∈
1,2n1/n−2for allq < nn4/n2−4still denoteq1q/ 2 byq, then
w∈LqQT, ∀q∈
1,2n1 n−2
Finally, we observe thatq2 satisfiesn2−4q < n24nwithn2,3,4,5. So takeq2 for
4.17and4.19. Then there exists a positive constantM4such that
wV2QT≤M4. 4.28
Step 2. L∞-Estimates ofw. We rewrite the third equation of1.6as a linear parabolic equation
∂w ∂t
n
i,j1
∂ ∂xi
aijx, t∂w
∂xj
n
i1
∂ ∂xi
aix, tw−b−uww, 4.29
whereaijx, t d3 α31uα32v2α33wδij,aix, t α31∂u/∂xi α32∂v/∂xi,δij are Kronecker symbols.
To apply the maximum principle15, Theorem 7.1, page 181to4.15to obtainw ∈ L∞QT, we need to verify that the following conditions hold:1wV2QTis bounded;2 "n
i,j1aijx, tξiξj ≥ ν "n
i1ξ2i; 3 "n
i1a2ix, t, b−uwLq,rQ
T ≤ μ1, where ν and μ1 are
positive constants, andqandrsatisfy
1
r n
2q 1−χ, 0< χ <1, q∈
n
21−χ,∞
r∈
1 1−χ,∞
, n≥2.
4.30
Next we verify conditions1–3in turn. From4.28, condition1is true forn≤5. One can chooseνd3such that condition2holds. To verify condition3, the first equation
of1.6is written in the divergence form
ut∇ ·d12α11u∇u a0v−a1ua2u2−a3u3−kuw, 4.31
whered12α11uis bounded inQTbyLemma 4.1, anda0v−a1ua2u2−a3u3−kuw∈LqQT forq∈ n2/2,2n1/n−2from4.27. Application of the H ¨older continuity result
15, Theorem 10.1, page 204to4.19yields
u∈Cβ,β/2QT, β∈0,1. 4.32
Returning to4.7, sinceC1C2w ∈LqQTfor anyq∈n2/2,2n1/n−2by4.1 and4.27, andd1α11u ∈ Cβ,β/2QTby4.32, then by applying the parabolic regularity theorem15, Theorem 9.1, pages 341-342to4.7we have
X∈Wq2,1QT, ∀q∈
n2 2 ,
2n1
n−2
. 4.33
Hence∇X ∈Ln2q/n2−qQ
TfromLemma 4.4, which shows that∇u∈Ln2q/n2−qQT.
Similarly,∇v ∈ Ln2q/n2−qQ
|∇u|2,|∇v|2 ∈ Ln2q/2n2−qQ
T, which imply that "ni1a2ix, t ∈ Ln2q/2n2−qQT. In addition, b−uw obviously belongs to LqQ
T. It follows that one can select q r
n2p/2n2−p. Now the above three conditions are satisfied, andw∈L∞QTfrom15, Theorem 7.1, page 181. RecallingLemma 4.1, thus there exists a positive constantM5for any
T >0 such that
uL∞QT, vL∞QT, wL∞QT≤M5. 4.34
Step 3. The Proof of the Classical Solution u, v, wof 1.6 in QTfor Any T > 0. Because
d1uα11u2αt d1 2α11uΔd1u α11u2 d1 2α11uf1, we have from 4.34 that
X d1uα11u2 ∈ Wq2,1QT for any q > 1. So ∇X ∈ Cβ ∗,β∗/2
QT for all β∗ ∈ 0,1. It follows from15, Lemma 3.3, page 80that X ∈ C1β∗,1β∗/2Q
T. And direct calculation
X d1α11uuyieldsu −d1
d2
14α11X/2α11. So we have
u∈C1β∗,1β∗/2QT, ∀β∗∈0,1. 4.35
The third equation of1.6can be written as
wt∇ ·d3α31uα32v2α33w∇w α31∇uα32∇vw −bu−ww. 4.36
Summarizing the above conclusions that are proved, we know that−bu−ww∈L∞QT andu, v, w,∇u,∇vare all bounded inQT. It follows from15, Theorem 10.1,page 204that there existsσ1∈0,1such that
w∈Cσ1,σ1/2Q T
. 4.37
The proof ofLemma 4.2is similar. Then we have∇v ∈ L2Q
T, that is,v ∈ V2QT. Applying the13, Theorem 10.1,page 204to the second equation1.6, there existsσ2∈0,1
such that
v∈Cσ2,σ2/2
QT. 4.38
Furthermore, applying Schauder estimate15, page 320-321yieldsv ∈C2σ∗,1σ∗/2Q
Tfor
σ∗min{σ2, λ}. Selectingαmin{σ1, σ∗}and using Sobolev embedding theorem, we have
C2σ∗,1σ∗/2Q
T→Cα,α/2QT. Still applying Schauder estimate, we have
v∈C2σ,1σ/2Q
T
, σmin{α, λ}. 4.39
Letw d3α31uα32vα33ww. Thenwsatisfies
wherefx, t d3α31uα32v2α33w−bu−wwα31utα32vtw. By4.35–4.38, we haved3α31uα32v2α33w, fx, t ∈Cα,α/2QT. So applying Schauder estimate to4.40 yieldsw∈C2σ,1σ/2QT. Sincew −d3α31uα32v
d3α31uα32v24α33w/2α33,
we have
w∈C2σ,1σ/2Q
T
, σmin{α, λ}. 4.41
The first equation of1.6can be written as
ut d12α11uΔugx, t, 4.42
wheregx, t 2α11|∇u|2a
0v−a1ua2u2−a3u3−kuw. By4.35,4.39, and4.41, we
haved12α11u, gx, t∈Cα,α/2QT. So applying Schauder estimate to4.42yields
u∈C2σ,1σ/2QT, σmin{α, λ}. 4.43
In particular, if λ < α, then σ λ; in other words, Theorem 4.5 is proved. For the caseα < λ, from Sobolev embedding theorem, we have C2σ,1σ/2Q
T → Cλ,λ/2QT. Repeating the above bootstrap and Shauder estimate arguments, this completes the proof of
Theorem 4.5. About space dimensionn1, see21.
Theorem 4.6. System 1.6 has the unique positive equilibrium point u, v, w when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. Let the space dimension be n ≤ 5, and let the initial values u0, v0, w0 be nonnegative smooth functions and satisfy the homogenous
Neumann boundary conditions. If the following condition 4.44holds, then the solutionu, v, w of 1.6converges tou, v, winL2Ω:
4αud1d2d3> βw
αuα232d12α11M0 α31M02d22α22M0
, 4.44
whereα2/a201/2a3u,andβ ka3uα.
Proof. Define the Lyapunov function
Ht α
Ω
u−u−ulnu
u
dx1
2
Ωv−v
2dxβ
Ω
w−w−wlnw
w
dx, 4.45
1.6,Htis well posed for allt ≥0 fromTheorem 4.5. According to system1.6, the time derivative ofHtsatisfies
dHt dt −
Ω
αu
u2d12α11u|∇u| 2 d
22α22v|∇v|2
βw
w2d3α31uα32v2α33w|∇w|
2βα31w
w ∇u∇wβ α32w
w ∇v∇w
dx − Ω .
αa0 u −a2
a3uu
u−u2 v−v2βw−w2
−1a0α
u
u−uv−v kα−βu−uw−w/dx
≤ −
Ω
αu
u2d12α11u|∇u| 2 d
22α22v|∇v|2
βw
w2d3α31uα32v2α33w|∇w|
2βα31w
w ∇u∇wβ α32w
w ∇v∇w
dx − Ω 0 α a0 u − a2
Ma0/a21/a2a20/4a2a3u
u
u−u2
−a0α
u u−uv−v−
1
uv−v
2 1 dx − Ω
αa3u
2 u−u
2−u−uv−v v−v2 dx − Ω
αa3u
2 u−u
2kα−βu−uw−w βw−w2dx.
4.46
It is easy to check that the final three integrands on the right side of the above expression are positive definite because of the electing ofα, β, and the sufficient and necessary conditions of the first integrand being positive definite are the following:
4αβu wd12α11ud22α22vd3α31uα32v2α33w
> αuβα32w2d12α11u
βα31wu 2
d22α22v.
4.47
Noticing that4.44is the sufficient conditions of4.47, so there exists a positive constant
δ >0 such that
dHt dt ≤ −δ
Ω
u−u2 v−v2 w−w2dx,
dHt
dt <0 u, v, w/ u, v, w.
Similar to the tedious calculations of dHt/dt, using integration by parts, H ¨older inequality, and 4.34, one can verify that d/dt Ωu−u2 v−v2 w−w2dx is bounded from above. Thus we have from4.48andLemma 3.2inSection 3that
|u·, t−u|2−→0, |v·, t−v|2−→0, |w·, t−w|2 −→0 t−→ ∞. 4.49
In addition,Htis decreasing for t ≥ 0, so we can conclude that the solution u, v, wis globally asymptotically stable. The proof ofTheorem 4.6is completed.
Acknowledgments
The work of this author was partially supported by the Natural Science Foundation of Anhui Province Education Department KJ2009B101 and the NSF of Chizhou CollegeXK0833
caguhh@yahoo.com.cn. The work of this author was partially supported by the China National Natural Science Foundation10871160, the NSF of Gansu Province096RJZA118, and NWNU-KJCXGC-03-47, 61 Foundationsfusm@nwnu.edu.cn.
References
1 E. Kuno, “Mathematical models for predator-prey interaction,”Advances in Ecological Research, vol. 16, pp. 252–265, 1987.
2 J. B. Zheng, Z. X. Yu, and J. T. Sun, “Existence and uniqueness of limit cycle for prey-predator systems with sparse effect,”Journal of Biomathematics, vol. 16, no. 2, pp. 156–161, 2001.
3 C. Shen and B. Q. Shen, “A necessary and sufficient condition of the existence and uniqueness of the limit cycle for a class of prey-predator systems with sparse effect,”Journal of Biomathematics, vol. 18, no. 2, pp. 207–210, 2003.
4 X. Huang, Y. Wang, and L. Zhu, “One and three limit cycles in a cubic predator-prey system,”
Mathematical Methods in the Applied Sciences, vol. 30, no. 5, pp. 501–511, 2007.
5 X. Zhang, L. Chen, and A. U. Neumann, “The stage-structured predator-prey model and optimal harvesting policy,”Mathematical Biosciences, vol. 168, no. 2, pp. 201–210, 2000.
6 N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,”Journal of Theoretical Biology, vol. 79, no. 1, pp. 83–99, 1979.
7 P. Y. H. Pang and M. Wang, “Strategy and stationary pattern in a three-species predator-prey model,”
Journal of Differential Equations, vol. 200, no. 2, pp. 245–273, 2004.
8 K. Kuto, “Stability of steady-state solutions to a prey-predator system with cross-diffusion,”Journal of Differential Equations, vol. 197, no. 2, pp. 293–314, 2004.
9 B. Dubey, B. Das, and J. Hussain, “A predator-prey interaction model with self and cross-diffusion,”
Ecological Modelling, vol. 141, no. 1–3, pp. 67–76, 2001.
10 Y. Lou, W.-M. Ni, and Y. Wu, “On the global existence of a cross-diffusion system,”Discrete and Continuous Dynamical Systems, vol. 4, no. 2, pp. 193–203, 1998.
11 Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,”Discrete and Continuous Dynamical Systems, vol. 10, no. 3, pp. 719–730, 2004.
12 R. Zhang, L. Guo, and S. M. Fu, “Global behavior for a diffusive predator-prey model with stage-structure and nonlinear density restriction-II: the case in R1,”Boundary Value Problems, vol. 2009,
Article ID 654539, 19 pages, 2009.
13 R. Zhang, L. Guo, and S. M. Fu, “Global behavior for a diffusive predator-prey model with stage-structure and nonlinear density restriction-I: the case inRn,”Boundary Value Problems, vol. 2009,
Article ID 378763, 26 pages, 2009.
15 O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva,Linear and Quasilinear Partial Differential Equations of Parabolic Type, vol. 23 ofTranslations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1968.
16 M. H. Protter and H. F. Weinberger,Maximum Principles in Differential Equations, Springer, New York, NY, USA, 2nd edition, 1984.
17 M. X. Wang,Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, China, 1993. 18 R. M. May,Stability and Complexity in Model Ecosystems, Princeton Univesity Press, Princeton, NJ, USA,
1974.
19 D. Henry,Geometric Theory of Semilinear Parabolic Equations, vol. 840 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1993.
20 M. H. Protter and H. F. Weinberger,Maximum Principles in Differential Equations, Springer, New York, NY, USA, 2nd edition, 1984.