Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 936952, 14 pages
http://dx.doi.org/10.1155/2013/936952
Research Article
Stability in a Simple Food Chain System with Michaelis-Menten
Functional Response and Nonlocal Delays
Wenzhen Gan,1 Canrong Tian,2 Qunying Zhang,3 and Zhigui Lin3
1
School of Mathematics and Physics, Jiangsu Teachers University of Technology, Changzhou 213001, China
Basic Department, Yancheng Institute of Technology, Yangcheng 224003, China
3
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2
Correspondence should be addressed to Zhigui Lin; [email protected]
Received 26 April 2013; Accepted 8 July 2013
Academic Editor: Rodrigo Lopez Pouso
Copyright © 2013 Wenzhen Gan et al. 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.
This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann
boundary condition. By taking food ingestion and species’ moving into account, the model is further coupled with MichaelisMenten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady
state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific
competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional
response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative
steady states of the system. Numerical simulations are carried out to illustrate the main results.
1. Introduction
The overall behavior of ecological systems continues to be
of great interest to both applied mathematicians and ecologists. Two species predator-prey models have been extensively investigated in the literature. But recently more and
more attention has been focused on systems with three or
more trophic levels. For example, the predator-prey system
for three species with Michaelis-Menten type functional
response was studied by many authors [1–4]. However, the
systems in [1–4] are either with discrete delay or without
delay or without diffusion. In view of individuals taking
time to move, spatial dispersal was dealt with by introducing
diffusion term to corresponding delayed ODE model in
previous literatures, namely, adding a Laplacian term to the
ODE model. In recent years, it has been recognized that there
are modelling difficulties with this approach. The difficulty is
that diffusion and time delay are independent of each other,
since individuals have not been at the same point at previous
times. Britton [5] made a first comprehensive attempt to
address this difficulty by introducing a nonlocal delay; that
is, the delay term involves a weighted-temporal average over
the whole of the infinite domain and the whole of the previous
times.
There are many results for reaction-diffusion equations
with nonlocal delays [5–18]. The existence and stability of
traveling wave fronts were studied in reaction-diffusion equations with nonlocal delay [5–9]. The stability of impulsive
cellular neural networks with time varying was discussed
in [10] by means of new Poincare integral inequality. The
asymptotic behavior of solutions of the reaction-diffusion
equations with nonlocal delay was investigated in [11, 12] by
using an iterative technique and in [13–15] by the Lyapunov
functional. The stability and Hopf bifurcation were discussed
in [16] for a diffusive logistic population model with nonlocal
delay effect.
Motivated by the work above, we are concerned with
the following food chain model with Michaelis-Menten type
functional response:
𝜕𝑢1
𝑎 𝑢
− 𝑑1 Δ𝑢1 = 𝑢1 (𝑎1 − 𝑎11 𝑢1 − 12 2 ) ,
𝜕𝑡
𝑚1 + 𝑢1
2
Abstract and Applied Analysis
𝜕𝑢2
− 𝑑2 Δ𝑢2
𝜕𝑡
= 𝑢2 (−𝑎2 + 𝑎21 ∫ ∫
𝑡
Ω −∞
only contribute to the production of predator biomass. The
boundary condition in (2) implies that there is no migration
across the boundary of Ω.
The main purpose of this paper is to study the global
asymptotic behavior of the solution of system (1)–(3). The
preliminary results are presented in Section 2. Section 3 contains sufficient conditions for the global asymptotic behaviors
of the equilibria of system (1)–(3) by means of the Lyapunov
functional. Numerical simulations are carried out to show the
feasibility of the conditions in Theorems 8–10 in Section 4.
Finally, a brief discussion is given to conclude this work.
𝑢1 (𝑠, 𝑦)
𝑚1 + 𝑢1 (𝑠, 𝑦)
× 𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠) 𝑑𝑠 𝑑𝑦
−𝑎22 𝑢2 −
𝑎23 𝑢3
),
𝑚2 + 𝑢2
𝜕𝑢3
− 𝑑3 Δ𝑢3
𝜕𝑡
= 𝑢3 (−𝑎3 + 𝑎32 ∫ ∫
𝑡
Ω −∞
𝑢2 (𝑠, 𝑦)
𝑚2 + 𝑢2 (𝑠, 𝑦)
2. Preliminary Results
In this section, we present several preliminary results that will
be employed in the sequel.
× 𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠) 𝑑𝑠 𝑑𝑦
−𝑎33 𝑢3 ) ,
(1)
for 𝑡 > 0, 𝑥 ∈ Ω with homogeneous Neumann boundary
conditions
𝜕𝑢1 𝜕𝑢2 𝜕𝑢3
=
=
= 0,
𝜕]
𝜕]
𝜕]
𝑡 > 0, 𝑥 ∈ 𝜕Ω,
(2)
Lemma 1 (see [3]). Let 𝑎 and 𝑏 be positive constants. Assume
that 𝜙, 𝜑 ∈ 𝐶1 (𝑎, +∞), 𝜑(𝑡) ≥ 0, and 𝜙 is bounded from
below. If 𝜙󸀠 (𝑡) ≤ −𝑏𝜑(𝑡) and 𝜑󸀠 (𝑡) ≤ 𝐾 in [𝑎, +∞) for some
positive constant 𝐾, then lim𝑡 → +∞ 𝜑(𝑡) = 0.
The following lemma is the Positivity Lemma in [20].
Lemma 2. Let 𝑢𝑖 ∈ 𝐶([0, 𝑇] × Ω) ⋂ 𝐶1,2 ((0, 𝑇] × Ω) (𝑖 =
1, 2, 3) and satisfy
and initial conditions
𝑢𝑖 (𝑡, 𝑥) = 𝜙𝑖 (𝑡, 𝑥) ≥ 0
(𝑖 = 1, 2) , (𝑡, 𝑥) ∈ (−∞, 0] × Ω,
𝑢3 (0, 𝑥) = 𝜙3 (𝑥) ≥ 0,
𝑥 ∈ Ω,
3
𝑢𝑖𝑡 − 𝑑𝑖 Δ𝑢𝑖 ≥ ∑𝑏𝑖𝑗 (𝑡, 𝑥) 𝑢𝑗 (𝑡, 𝑥)
𝑗=1
3
(3)
where 𝜙𝑖 is bounded, Hölder continuous function and satisfies 𝜕𝜙𝑖 /𝜕] = 0 (𝑖 = 1, 2, 3) on (−∞, 0] × 𝜕Ω. Here, Ω is a
bounded domain in R𝑛 with smooth boundary 𝜕Ω and 𝜕/𝜕]
is the outward normal derivative on 𝜕Ω. 𝑢𝑖 (𝑡, 𝑥) represents
the density of the 𝑖th species (prey, predator, and top predator
resp.) at time 𝑡 and location 𝑥 and thus only nonnegative
𝑢𝑖 (𝑡, 𝑥) is of interest. The parameter 𝑎1 is the intrinsic growth
rate of the prey, and 𝑎2 and 𝑎3 are the death rates of the
predator and top-predator. 𝑎𝑖𝑖 is the intraspecific competitive
rate of the 𝑖th species. 𝑎12 is the maximum predation rate. 𝑎23
and 𝑎32 are the efficiencies of food utilization of the predator
and top predator, respectively. We assume the predator and
top predator show the Michaelis-Menten (or Holling type
II) functional response with 𝑢1 /(𝑚1 + 𝑢1 ) and 𝑢2 /(𝑚2 + 𝑢2 ),
respectively, where 𝑚1 and 𝑚2 are half-saturation constants.
For a through biological background of similar models, see
[18, 19]. As our most knowledge, the tritrophic food chain
model has been found to have many interesting biological
properties, such as the coexistence and the Hopf bifurcation.
However, the effect of nonlocal time delays on the coexistence
has not been reported. Our paper mainly concerns this
perspective.
𝑡
Additionally, ∫Ω ∫−∞ (𝑢𝑖 (𝑠, 𝑦)/(𝑚𝑖 + 𝑢𝑖 (𝑠, 𝑦)))𝐾𝑖 (𝑥, 𝑦, 𝑡 −
𝑠)𝑑𝑠 𝑑𝑦 (𝑖 = 1, 2) represents the nonlocal delay due to
the ingestion of predator; that is, mature adult predator can
+ ∑ 𝑐𝑖𝑗 𝑢𝑗 (𝑡 − 𝜏𝑗 , 𝑥)
in (0, 𝑇] × Ω,
𝑗=1
𝜕𝑢𝑖 (𝑡, 𝑥)
≥0
𝜕]
𝑢𝑖 (𝑡, 𝑥) ≥ 0
(4)
on (0, 𝑇] × 𝜕Ω,
in [−𝜏𝑖 , 0] × Ω,
where 𝑏𝑖𝑗 , 𝑐𝑖𝑗 ∈ 𝐶([0, 𝑇] × Ω), 𝑢𝑖𝑡 = (𝑢𝑖 )𝑡 . If 𝑏𝑖𝑗 ≥ 0 for 𝑗 ≠ 𝑖
and 𝑐𝑖𝑗 ≥ 0 for all 𝑖, 𝑗 = 1, 2, 3. Then 𝑢𝑖 (𝑡, 𝑥) ≥ 0 on [0, 𝑇] × Ω.
Moreover, if the initial function is nontrivial, then 𝑢𝑖 > 0 in
(0, 𝑇] × Ω.
Lemma 3 (see [20]). Let ̂c and ̃c be a pair of constant vector
satisfying ̃c ≥ ̂c and let the reaction functions satisfy local
Lipschitz condition with Λ = ⟨̂c, ̃c⟩. Then system (1)–(3) admits
a unique global solution u(𝑡, 𝑥) such that
̂c ≤ u (𝑡, 𝑥) ≤ ̃c,
∀𝑡 > 0, 𝑥 ∈ Ω,
(5)
whenever ̂c ≤ 𝜙(𝑡, 𝑥) ≤ ̃c, (𝑡, 𝑥) ∈ (−∞, 0] × Ω.
The following result was obtained by the method of upper
and lower solutions and the associated iterations in [21].
Abstract and Applied Analysis
3
Lemma 4 (see [21]). Let 𝑢(𝑡, 𝑥) ∈ 𝐶([0, ∞) × Ω) ∩
𝐶2,1 ((0, ∞)×Ω) be the nontrivial positive solution of the system
𝑢𝑡 − 𝑑Δ𝑢 = 𝐵𝑢 (𝑡 − 𝜏, 𝑥) ± 𝐴 1 𝑢 (𝑡, 𝑥)
− 𝐴 2 𝑢2 (𝑡, 𝑥) ,
𝜕𝑢
= 0,
𝜕]
𝐴 1 = 𝑎1 − 𝑎11 𝑢1 −
𝑎12 𝑢2
,
𝑚1 + 𝑢1
𝐴 2 = −𝑎2 + 𝑎21 ∫ ∫
𝑎 𝑢
− 𝑎22 𝑢2 − 23 3 ,
𝑚2 + 𝑢3
𝐴 3 = −𝑎3 + 𝑎32 ∫ ∫
(2) if 𝐵 ± 𝐴 1 ≤ 0, then 𝑢 → 0 uniformly on Ω as 𝑡 → ∞.
Throughout this paper, we assume that
∫
+∞
0
𝑘𝑖 (𝑡) 𝑑𝑡 = 1,
𝑡 ≥ 0;
(7)
𝑡𝑘𝑖 (𝑡) ∈ 𝐿1 ((0, +∞) ; 𝑅) ,
where 𝐺𝑖 (𝑥, 𝑦, 𝑡) is a nonnegative function, which is continuous in (𝑥, 𝑦) ∈ Ω × Ω for each 𝑡 ∈ [0, +∞) and measurable in
𝑡 ∈ [0, +∞) for each pair (𝑥, 𝑦) ∈ Ω × Ω.
Now we prove the following propositions which will be
used in the sequel.
Proposition 5. For any nonnegative initial function, the
corresponding solution of system (1)–(3) is nonnegative.
Proof. Suppose that (𝑢1 , 𝑢2 , 𝑢3 ) is a solution and satisfies 𝑢𝑖 ∈
𝐶([0, 𝑇) × Ω) ∩ 𝐶1,2 ((0, 𝑇) × Ω) with 𝑇 ≤ +∞. Choose 0 <
𝜏 < 𝑇. Then
𝑢1𝑡 − 𝑑1 Δ𝑢1 = 𝐴 1 𝑢1 ,
0 < 𝑡 < 𝜏, 𝑥 ∈ Ω,
𝑢2𝑡 − 𝑑2 Δ𝑢2 = 𝐴 2 𝑢2 ,
0 < 𝑡 < 𝜏, 𝑥 ∈ Ω,
𝑢3𝑡 − 𝑑3 Δ𝑢3 = 𝐴 3 𝑢3 ,
0 < 𝑡 < 𝜏, 𝑥 ∈ Ω,
𝜕𝑢𝑖
= 0,
𝜕]
0 < 𝑡 < 𝜏, 𝑥 ∈ 𝜕Ω,
𝑢𝑖 (𝑡, 𝑥) ≥ 0
(𝑖 = 1, 2, 3) , 𝑡 ≤ 0, 𝑥 ∈ Ω,
𝑢2 (𝑠, 𝑦)
𝑚2 + 𝑢2 (𝑠, 𝑦)
It follows from Lemma 2 that 𝑢𝑖 ≥ 0, (𝑡, 𝑥) ∈ [0, 𝜏] × Ω. Due
to the arbitrariness of 𝜏, we have 𝑢𝑖 ≥ 0 for (𝑡, 𝑥) ∈ [0, 𝑇) ×
Ω.
Proposition 6. System (1)–(3) with initial functions 𝜙𝑖 admits
a unique global solution (𝑢1 , 𝑢2 , 𝑢3 ) satisfying 0 ≤ 𝑢𝑖 (𝑡, 𝑥) ≤
𝑀𝑖 for 𝑖 = 1, 2, 3, where 𝑀𝑖 is defined as
𝑀1 = max {
𝑎1 󵄩󵄩
󵄩
, 󵄩𝜙 (𝑡, 𝑥)󵄩󵄩󵄩𝐿∞ ((−∞,0]×Ω) } ,
𝑎11 󵄩 1
𝑀2 = max {
(𝑎21 − 𝑎2 ) 𝑀1 − 𝑚1 𝑎2 󵄩󵄩
󵄩
, 󵄩󵄩𝜙2 (𝑡, 𝑥)󵄩󵄩󵄩𝐿∞ ((−∞,0]×Ω) } ,
𝑎22 (𝑚1 + 𝑀1 )
𝑀3 = max {
(𝑎32 − 𝑎3 ) 𝑀2 − 𝑚2 𝑎3 󵄩󵄩
󵄩
, 󵄩󵄩𝜙3 (𝑥)󵄩󵄩󵄩𝐿∞ (Ω) } .
𝑎22 (𝑚2 + 𝑀2 )
𝑥, 𝑦 ∈ Ω, 𝑘𝑖 (𝑡) ≥ 0;
Ω
(9)
× 𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠) 𝑑𝑠 𝑑𝑦 − 𝑎33 𝑢3 .
(1) if 𝐵 ± 𝐴 1 > 0, then 𝑢 → (𝐵 ± 𝐴 1 )/𝐴 2 uniformly on Ω
as 𝑡 → ∞;
Ω
𝑡
Ω −∞
where 𝐴 1 ≥ 0, 𝐵, 𝐴 2 , 𝜏 > 0. The following results hold:
∫ 𝐺𝑖 (𝑥, 𝑦, 𝑡) 𝑑𝑥 = ∫ 𝐺𝑖 (𝑥, 𝑦, 𝑡) 𝑑𝑦 = 1,
𝑢1 (𝑠, 𝑦)
𝑚1 + 𝑢1 (𝑠, 𝑦)
× 𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠) 𝑑𝑠 𝑑𝑦
(6)
(𝑡, 𝑥) ∈ (−𝜏, 0) × Ω,
𝐾𝑖 (𝑥, 𝑦, 𝑡) = 𝐺𝑖 (𝑥, 𝑦, 𝑡) 𝑘𝑖 (𝑡) ,
𝑡
Ω −∞
(𝑡, 𝑥) ∈ (0, ∞) × Ω,
(𝑡, 𝑥) ∈ (0, ∞) × 𝜕Ω,
𝑢 (𝑡, 𝑥) = 𝜙 (𝑡, 𝑥) ≥ 0,
where
(8)
(10)
Proof. It follows from standard PDE theory that there exists
a 𝑇 > 0 such that problem (1)–(3) admits a unique solution
in [0, 𝑇) × Ω. From Lemma 2 we know that 𝑢𝑖 (𝑡, 𝑥) ≥ 0 for
(𝑡, 𝑥) ∈ [0, 𝑇) × Ω. Considering the first equation in (1), we
have
𝜕𝑢1
− 𝑑1 Δ𝑢1 ≤ 𝑢1 (𝑎1 − 𝑎11 𝑢1 ) .
𝜕𝑡
(11)
Using the maximum principle gives that 𝑢1 ≤ 𝑀1 . In a similar
way, we have 𝑢𝑖 ≤ 𝑀𝑖 for 𝑖 = 2, 3. It is easy to see that (0, 0, 0)
and (𝑀1 , 𝑀2 , 𝑀3 ) are a pair of coupled upper and lower
solutions to system (1)–(3) from the direct computation. In
virtue of Lemma 3, system (1)–(3) admits a unique global
solution (𝑢1 , 𝑢2 , 𝑢3 ) satisfying 0 ≤ 𝑢𝑖 (𝑡, 𝑥) ≤ 𝑀𝑖 for 𝑖 = 1, 2, 3.
In addition, if 𝜙1 (𝑡, 𝑥), 𝜙2 (𝑡, 𝑥), 𝜙3 (𝑥), (𝑡, 𝑥) ∈ (−∞, 0] × Ω is
nontrivial, it follows from the strong maximum principle that
𝑢𝑖 (𝑡, 𝑥) > 0 (𝑖 = 1, 2, 3) for all 𝑡 > 0, 𝑥 ∈ Ω.
3. Global Stability
In this section, we study the asymptotic behavior of the
equilibrium of system (1)–(3). In the beginning, we show the
existence and uniqueness of positive steady state.
4
Abstract and Applied Analysis
by continuously increasing the value of 𝑢3 , 𝐿 3 goes up, and
meanwhile the intersection point between 𝐿 1 and 𝐿 2 also
goes up. However, 𝐿 3 goes up faster than 𝐿 2 , while 𝐿 1 keeps
still. In other words, there exists a critical value 𝑢3 such
that the intersection point lies in 𝐿 3 , which implies that
there is a unique positive solution 𝐸∗ (𝑢1∗ , 𝑢2∗ , 𝑢3∗ ) to (15), or
equivalently (1)–(3).
0.25
L1
0.2
2
0.15
Lemma 7. Assume that 𝐻2 and 𝐺2 hold, and then the positive
steady state 𝐸∗ of system (1)–(3) is locally asymptotically stable.
L2
0.1
Proof. We get the local stability of 𝐸∗ by performing a
linearization and analyzing the corresponding characteristic
equation. Similarly as in [22], let 0 < 𝜇1 < 𝜇2 < 𝜇3 <
. . . be the eigenvalues of −Δ on Ω with the homogeneous
Neumann boundary condition. Let 𝐸(𝜇𝑖 ) be the eigenspace
corresponding to 𝜇𝑖 in 𝐶1 (Ω), for 𝑖 = 1, 2, 3, . . . . Let
0.05
L3
0
0
0.2
0.4
1
0.6
0.8
Figure 1: Graphs of the equations in (12). 𝐿 3 is the boundary
condition (𝑎3 = 𝑎32 V2 /(𝑚2 + V2 )) and 𝐿 2 corresponds to −𝑎2 +
𝑎21 (V1 /(𝑚1 + V1 )) − 𝑎22 V2 = 0, while 𝐿 1 corresponds to 𝑎1 − 𝑎11 V1 −
𝑎12 V2 /(𝑚1 + V1 ) = 0. Intersection point between 𝐿 1 and 𝐿 2 gives the
positive solution (V1∗ , V2∗ ) to (12). Parameter values are listed in the
example in Section 4.
3
X = {u = (𝑢1 , 𝑢2 , 𝑢3 ) ∈ [𝐶1 (Ω)] 𝜕𝜂 u = 0, 𝑥 ∈ 𝜕Ω} , (16)
{𝜑𝑖𝑗 , 𝑗 = 1, . . . , dim 𝐸(𝜇𝑖 )} be an orthonormal basis of 𝐸(𝜇𝑖 ),
and X𝑖𝑗 = {𝑐 ⋅ 𝜑𝑖𝑗 | 𝑐 ∈ 𝑅3 }. Then
dim 𝐸(𝜇𝑖 )
∞
X𝑖 = ⨁ X𝑖𝑗 ,
𝑗=1
Let us consider the following equations:
𝑎1 − 𝑎11 V1 −
− 𝑎2 +
𝑎12 V2
= 0,
𝑚1 + V1
𝑎21 V1
− 𝑎22 V2 = 0.
𝑚1 + V1
(12)
A direct computation shows that the above equations have
only one positive solution (V1∗ , V2∗ ) if and only if
𝐻1 : 𝑎1 (𝑎21 − 𝑎2 ) > 𝑚1 𝑎2 𝑎11 ,
𝐻2 : 𝑎3 𝑚2 < (𝑎32 −
𝑎12 𝑢2
= 0,
𝑚1 + 𝑢1
− 𝑎2 +
𝑎 𝑢
𝑎21 𝑢1
− 𝑎22 𝑢2 − 23 3 = 0,
𝑚1 + 𝑢1
𝑚2 + 𝑢2
− 𝑎3 +
𝑎32 𝑢2
− 𝑎33 𝑢3 = 0.
𝑚2 + 𝑢2
Let 𝐷 = diag(𝐷1 , 𝐷2 , 𝐷3 ) and k = (V1 , V2 , V3 ), V𝑖 =
𝑢𝑖 − 𝑢𝑖∗ , (𝑖 = 1, 2, 3). Then the linearization of (1)
is k𝑡 = 𝐿k = 𝐷Δk + Fk (E∗ )k, Fk (E∗ )k = {𝑐𝑖𝑗 },
where 𝑐𝑖𝑖 = 𝑏𝑖𝑖 V𝑖 , 𝑖 = 1, 2, 3, 𝑐12 = 𝑏12 V2 , 𝑐23 =
𝑡
𝑏23 V3 , 𝑐21 = 𝑏21 ∫Ω ∫−∞ V1 (𝑠, 𝑦)𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)𝑑𝑠 𝑑𝑦, and 𝑐32 =
𝑡
𝑏32 ∫Ω ∫−∞ V2 (𝑠, 𝑦)𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)𝑑𝑠 𝑑𝑦. The coefficients 𝑏𝑖𝑗 are
defined as follows:
𝑏11 = −𝑎11 𝑢1∗ +
𝑏12 = −
(14)
Suppose that 𝐻1 and 𝐻2 hold. We take 𝑢3 as a parameter and
consider the following system:
𝑎1 − 𝑎11 𝑢1 −
(17)
𝑖=1
(13)
is satisfied. Taking 𝐻1 into account, we consider the equation
𝑎3 = 𝑎32 V2∗ /(𝑚2 + V2∗ ), which corresponds to 𝐿 3 in Figure 1.
It is clear that if 𝑎3 < 𝑎32 V2∗ /(𝑚2 + V2∗ ), the intersection
point (V1∗ , V2∗ ) always lies in 𝐿 3 . Let
𝑎3 ) V2∗ .
X =⨁ X𝑖 .
𝑏21 =
𝑎21 𝑚1 𝑢2∗
When the parameter 𝑢3 is sufficiently small, the first two
equations in (15) can be approximated by (12). Moreover,
2
(𝑚1 + 𝑢1∗ )
𝑏31 = 0,
2
(𝑚1 + 𝑢1∗ )
𝑎12 𝑢1∗
,
𝑚1 + 𝑢1∗
𝑏22 = −𝑎22 𝑢2∗ +
(15)
𝑎12 𝑢1∗ 𝑢2∗
,
𝑏13 = 0,
,
(18)
𝑎23 𝑢2∗ 𝑢3∗
,
2
(𝑚2 + 𝑢2∗ )
𝑏32 =
𝑎32 𝑚2 𝑢3∗
𝑏23 = −
2
(𝑚2 + 𝑢2∗ )
,
𝑎23 𝑢2∗
,
𝑚2 + 𝑢2∗
𝑏33 = −𝑎33 𝑢3∗ .
Since X𝑖 is invariant under the operator 𝐿 for each 𝑖 ≥ 1, then
the operator 𝐿 on X𝑖 is k𝑡 = 𝐿k = 𝐷𝜇𝑖 k + Fk (E∗ )k. Let V𝑖 =
𝑐𝑖 𝑒𝜆𝑡 (𝑖 = 1, 2, 3) and we can get the characteristic equation
Abstract and Applied Analysis
5
𝜑𝑖 (𝜆) = 𝜆3 + 𝐴 𝑖 𝜆2 + 𝐵𝑖 𝜆 + 𝐶𝑖 = 0, where 𝐴 𝑖 , 𝐵𝑖 , and 𝐶𝑖 are
defined as follows:
𝐴 𝑖 = (𝑑1 + 𝑑2 + 𝑑3 ) 𝜇𝑖 − 𝑏11 − 𝑏22 − 𝑏33 ,
Proof. It is easy to see that the equations in (1) can be rewritten
as
𝜕𝑢1
− 𝑑1 Δ𝑢1 = 𝑢1 [ − 𝑎11 (𝑢1 − 𝑢1∗ )
𝜕𝑡
𝐵𝑖 = (𝑑1 𝑑2 + 𝑑2 𝑑3 + 𝑑3 𝑑1 ) 𝜇𝑖2
−
+ [𝑑3 (−𝑏11 − 𝑏22 ) + 𝑑1 (−𝑏33 − 𝑏22 )
+𝑑2 (−𝑏11 − 𝑏33 )] 𝜇𝑖
𝑎12 (𝑢2 − 𝑢2∗ )
𝑎12 𝑢2∗ (𝑢1 − 𝑢1∗ )
+
],
𝑚1 + 𝑢1
(𝑚1 + 𝑢1 ) (𝑚1 + 𝑢1∗ )
𝑡
𝜕𝑢2
− 𝑑2 Δ𝑢2 = 𝑢2 [ ∫ ∫ 𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
𝜕𝑡
Ω −∞
+ 𝑏11 𝑏22 + 𝑏11 𝑏33 + 𝑏22 𝑏33
∞
∞
− 𝑏12 𝑏21 ∫ 𝑘1 (𝑠) 𝑒−𝜆𝑠 𝑑𝑠 − 𝑏23 𝑏32 ∫ 𝑘2 (𝑠) 𝑒−𝜆𝑠 𝑑𝑠,
0
×
0
𝐶𝑖 = 𝑑1 𝑑2 𝑑3 𝜇𝑖3 + [−𝑏33 𝑑1 𝑑2 − 𝑏11 𝑑2 𝑑3 − 𝑏22 𝑑1 𝑑3 ] 𝜇𝑖2
− 𝑎22 (𝑢2 − 𝑢2∗ ) −
(19)
+ [𝑏11 𝑏22 𝑑3 + 𝑏11 𝑏33 𝑑2 + 𝑏22 𝑏33 𝑑1 + 𝑏22 𝑏33 𝑑1
+
∞
− 𝑑3 𝑏12 𝑏21 ∫ 𝑘1 (𝑠) 𝑒−𝜆𝑠 𝑑𝑠
0
𝑎21 𝑚1 (𝑢1 (𝑠, 𝑦) − 𝑢1∗ )
𝑑𝑠 𝑑𝑦
(𝑚1 + 𝑢1 (𝑠, 𝑦)) (𝑚1 + 𝑢1∗ )
𝑎23 (𝑢3 − 𝑢3∗ )
𝑚2 + 𝑢2
𝑎23 𝑢3∗ (𝑢2 − 𝑢2∗ )
],
(𝑚2 + 𝑢2 ) (𝑚2 + 𝑢2∗ )
𝑡
𝜕𝑢3
− 𝑑3 Δ𝑢3 = 𝑢3 [∫ ∫ 𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
𝜕𝑡
Ω −∞
∞
−𝑑1 𝑏23 𝑏32 ∫ 𝑘2 (𝑠) 𝑒−𝜆𝑠 𝑑𝑠] 𝜇𝑖
0
×
∞
+ [−𝑏11 𝑏22 𝑏33 + 𝑏12 𝑏21 𝑏33 ∫ 𝑘1 (𝑠) 𝑒−𝜆𝑠 𝑑𝑠
0
𝑎32 𝑚2 (𝑢2 (𝑠, 𝑦) − 𝑢2∗ )
𝑑𝑠 𝑑𝑦
(𝑚2 + 𝑢2 (𝑠, 𝑦)) (𝑚2 + 𝑢2∗ )
−𝑎33 (𝑢3 − 𝑢3∗ ) ] .
∞
+𝑏11 𝑏23 𝑏32 ∫ 𝑘2 (𝑠) 𝑒−𝜆𝑠 𝑑𝑠] .
0
(22)
It is easy to see that 𝑏12 , 𝑏23 , 𝑏33 < 0 and 𝑏21 , 𝑏32 > 0. It
follows from assumption 𝐺1 and 𝐺2 that 𝑎11 < 0, 𝑎22 < 0.
So 𝐴 𝑖 > 0, 𝐵𝑖 > 0, 𝐶𝑖 > 0 and 𝐴 𝑖 𝐵𝑖 − 𝐶𝑖 > 0 for 𝑖 ≥ 1
from the direct calculation. According to the Routh-Hurwitz
criterion, the three roots 𝜆 𝑖,1 , 𝜆 𝑖,2 , 𝜆 𝑖,3 of 𝜑𝑖 (𝜆) = 0 all have
negative real parts.
By continuity of the roots with respect to 𝜇𝑖 and RouthHurwitz criterion, we can conclude that there exists a positive
constant 𝜀 such that
Re {𝜆 𝑖,1 } , Re {𝜆 𝑖,2 } , Re {𝜆 𝑖,3 } ≤ −𝜀,
𝑖 ≥ 1.
Define
𝑉 (𝑡) = 𝛼 ∫ (𝑢1 − 𝑢1∗ − 𝑢1∗ ln
𝑢1
) 𝑑𝑥
𝑢1∗
+ ∫ (𝑢2 − 𝑢2∗ − 𝑢2∗ ln
𝑢2
) 𝑑𝑥
𝑢2∗
Ω
Ω
+ 𝛽 ∫ (𝑢2 − 𝑢3∗ − 𝑢3∗ ln
Ω
(20)
Consequently, the spectrum of 𝐿, consisting only of eigenvalues, lies in {Re 𝜆 ≤ −𝜀}. It is easy to see that E∗ is
locally asymptotically stable and follows from Theorem 5.1.1
of [23].
𝑢3
) 𝑑𝑥,
𝑢3∗
where 𝛼 and 𝛽 are positive constants to be determined.
Calculating the derivatives 𝑉(𝑡) along the positive solution
to the system (1)–(3) yields
𝑉󸀠 (𝑡) = Φ1 (𝑡) + Φ2 (𝑡) ,
Theorem 8. Assume that
𝑎 (𝑎 − 𝑎2 )
𝑎 𝑢∗
𝐻3 : 1 21
> 𝑎11 > 12 2 2 ,
𝑚1 𝑎2
𝑚1
where
(21)
𝐻2 and 𝐺2 hold, and the positive steady state 𝐸∗ of system (1)–
(3) with nontrivial initial function is globally asymptotically
stable.
(23)
Φ1 (𝑡) = − ∫ (
Ω
∗
𝛼𝑑1 𝑢1∗ 󵄨󵄨
󵄨2 𝑑2 𝑢 󵄨
󵄨2
󵄨󵄨∇𝑢1 󵄨󵄨󵄨 + 2 2 󵄨󵄨󵄨∇𝑢2 󵄨󵄨󵄨
2
𝑢1
𝑢2
+
𝛽𝑑3 𝑢3∗ 󵄨󵄨
󵄨2
󵄨∇𝑢 󵄨󵄨 ) 𝑑𝑥,
𝑢32 󵄨 3 󵄨
(24)
6
Abstract and Applied Analysis
Φ2 (𝑡) = − ∫ 𝛼 [𝑎11 −
Ω
− ∫ [𝑎22 −
Ω
𝑎12 𝑢2∗
2
] (𝑢1 − 𝑢1∗ ) 𝑑𝑥
(𝑚1 + 𝑢1 ) (𝑚1 + 𝑢1∗ )
𝑎23 𝑢3∗
2
] (𝑢2 − 𝑢2∗ ) 𝑑𝑥
∗
(𝑚2 + 𝑢2 ) (𝑚2 + 𝑢2 )
2
− ∫ 𝑎33 𝛽(𝑢3 − 𝑢3∗ ) 𝑑𝑥
𝑎23 (𝑢3 − 𝑢3∗ ) (𝑢2 − 𝑢2∗ )
𝑑𝑥
𝑚2 + 𝑢2
−∫
𝛼𝑎12 (𝑢2 − 𝑢2∗ ) (𝑢1 − 𝑢1∗ )
𝑑𝑥
𝑚1 + 𝑢1
Ω
Ω
𝑡
+∬ ∫
Ω −∞
×
Ω −∞
𝑎32 𝛽𝑚2
(𝑚2 + 𝑢2 (𝑠, 𝑦)) (𝑚2 + 𝑢2∗ )
1
2
(𝑢 (𝑠, 𝑦) − 𝑢2∗ )
4𝜖4 2
2
(26)
According to the property of the Kernel functions 𝐾𝑖 (𝑥, 𝑦, 𝑡),
(𝑖 = 1, 2), we know that
𝑎21 𝑚1 (𝑢1 (𝑠, 𝑦) − 𝑢1∗ ) (𝑢2 (𝑡, 𝑥) − 𝑢2∗ )
𝑑𝑠 𝑑𝑦 𝑑𝑥
(𝑚1 + 𝑢1 (𝑠, 𝑦)) (𝑚1 + 𝑢1∗ )
+ 𝛽∬ ∫
×
Φ2 (𝑡) ≤ − ∫ [𝛼𝑎11 −
Ω
𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
− ∫ [𝑎22 −
𝑎 𝑚 (𝑢 (𝑠, 𝑦) − 𝑢2∗ ) (𝑢3 (𝑡, 𝑥) − 𝑢3∗ )
× 32 2 2
𝑑𝑠 𝑑𝑦 𝑑𝑥.
(𝑚2 + 𝑢2 (𝑠, 𝑦)) (𝑚2 + 𝑢2∗ )
(25)
Ω
−
Ω
+
+
2
2
] (𝑢2 − 𝑢2∗ ) 𝑑𝑥
𝛼𝑎12
1
2
2
[𝜖 (𝑢 − 𝑢1∗ ) +
(𝑢 − 𝑢2∗ ) ] 𝑑𝑥
𝑚1 + 𝑢1 1 1
4𝜖1 2
+∫
𝑎23
1
2
2
[
(𝑢 − 𝑢2∗ ) + 𝜖2 (𝑢3 − 𝑢3∗ ) ] 𝑑𝑥
𝑚2 + 𝑢2 4𝜖2 2
Ω
𝑡
+∬ ∫
Ω −∞
×
𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
Define a new Lyapunov functional as follows:
𝐸 (𝑡) = 𝑉 (𝑡)
+
𝑎21 𝑚1
(𝑚1 + 𝑢1 (𝑠, 𝑦)) (𝑚1 + 𝑢1∗ )
× [𝜖3 (𝑢1 (𝑠, 𝑦) −
𝑡
𝑎32 𝛽
∬ ∫ 𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
4𝑚2 𝜖4 Ω −∞
2
+∫
Ω
𝑡
𝑎21 𝜖3
∬ ∫ 𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
𝑚1
Ω −∞
× (𝑢2 (𝑠, 𝑦) − 𝑢2∗ ) 𝑑𝑠 𝑑𝑦 𝑑𝑥.
(27)
Ω
(𝑚2 + 𝑢2 ) (𝑚2 + 𝑢2∗ )
𝑎23 𝜖2 𝑎32 𝛽𝜖4
2
−
] (𝑢3 − 𝑢3∗ ) 𝑑𝑥
𝑚2
𝑚2
2
− ∫ 𝑎33 𝛽(𝑢3 − 𝑢3∗ ) 𝑑𝑥
Ω
𝑎21
2
] (𝑢2 − 𝑢2∗ ) 𝑑𝑥
4𝑚1 𝜖3
× (𝑢1 (𝑠, 𝑦) − 𝑢1∗ ) 𝑑𝑠 𝑑𝑦 𝑑𝑥
𝑎12 𝑢2∗
2
] (𝑢1 − 𝑢1∗ ) 𝑑𝑥
Φ2 (𝑡) ≤ − ∫ 𝛼 [𝑎11 −
(𝑚1 + 𝑢1 ) (𝑚1 + 𝑢1∗ )
Ω
− ∫ [𝑎22 −
𝛼𝑎12 𝑢2∗ 𝛼𝑎12 𝜖1
2
−
] (𝑢1 − 𝑢1∗ ) 𝑑𝑥
𝑚1
𝑚12
𝑎23 𝑢3∗
𝑎
𝛼𝑎12
−
− 23
4𝑚1 𝜖1 4𝑚2 𝜖2
𝑚22
− ∫ [𝑎33 𝛽 −
Applying the inequality 𝑎𝑏 ≤ 𝜖𝑎2 + (1/4𝜖)𝑏2 , we derive from
(25) that
𝑎23 𝑢3∗
𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
+𝜖4 (𝑢3 (𝑡, 𝑥) − 𝑢3∗ ) ] 𝑑𝑠 𝑑𝑦 𝑑𝑥.
𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
𝑡
𝑡
Ω −∞
×[
Ω
−∫
+∬ ∫
2
𝑢1∗ )
1
2
+
(𝑢 (𝑡, 𝑥) − 𝑢2∗ ) ] 𝑑𝑠 𝑑𝑦 𝑑𝑥
4𝜖3 2
+∞ 𝑡
𝑎21 𝜖3
∬ ∫ ∫ 𝐾1 (𝑥, 𝑦, 𝑟)
𝑚1
𝑡−𝑟
Ω 0
2
× (𝑢1 (𝑠, 𝑦) − 𝑢1∗ ) 𝑑𝑠 𝑑𝑟 𝑑𝑦 𝑑𝑥
+
+∞ 𝑡
𝑎32 𝛽
∬ ∫ ∫ 𝐾2 (𝑥, 𝑦, 𝑟)
4𝑚2 𝜖4 Ω 0
𝑡−𝑟
2
× (𝑢2 (𝑠, 𝑦) − 𝑢2∗ ) 𝑑𝑠 𝑑𝑟 𝑑𝑦 𝑑𝑥.
(28)
Abstract and Applied Analysis
7
It is derived from (27) and (28) that
If we choose
∗
𝛼𝑑 𝑢∗ 󵄨
󵄨2 𝑑 𝑢 󵄨
󵄨2
𝐸 (𝑡) ≤ − ∫ ( 12 1 󵄨󵄨󵄨∇𝑢1 󵄨󵄨󵄨 + 2 2 2 󵄨󵄨󵄨∇𝑢2 󵄨󵄨󵄨
𝑢1
𝑢2
Ω
𝛼=
󸀠
+
𝛽𝑑3 𝑢3∗ 󵄨󵄨
󵄨2
󵄨∇𝑢 󵄨󵄨 ) 𝑑𝑥
𝑢32 󵄨 3 󵄨
−
3
(29)
𝐸󸀠 (𝑡) ≤ − ∫ (
(30)
𝐺1 : 𝑎11 𝑚12 > 𝑎12 𝑢2∗ ,
2𝑎12 𝑎21 𝑚22
2𝑎23 𝑎32
+
,
𝑎33
𝑎11 𝑚12 − 𝑎12 𝑢2∗
(35)
(𝑖 = 1, 2, 3) .
(36)
Recomputing 𝐸󸀠 (𝑡) gives
Assume that
(31)
Ω
∗
𝛼𝑑1 𝑢1∗ 󵄨󵄨
󵄨󵄨2 𝑑2 𝑢2 󵄨󵄨
󵄨2
∇𝑢
+
󵄨
󵄨
󵄨󵄨∇𝑢2 󵄨󵄨󵄨
1
󵄨
󵄨
2
2
𝑢1
𝑢2
+
𝛽𝑑3 𝑢3∗ 󵄨󵄨
󵄨2
󵄨∇𝑢 󵄨󵄨 ) 𝑑𝑥
𝑢32 󵄨 3 󵄨
(37)
󵄨2 󵄨
󵄨2 󵄨
󵄨2
󵄨
≤ −𝑐 ∫ (󵄨󵄨󵄨∇𝑢1 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇𝑢2 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇𝑢3 󵄨󵄨󵄨 ) 𝑑𝑥 = −𝑔 (𝑡) ,
Ω
where 𝑐 = min{𝛼𝑑1 𝑢1∗ /𝑀12 , 𝑑2 𝑢2∗ /𝑀22 , 𝛽𝑑3 𝑢3∗ /𝑀32 }. Using (30)
and (1), we obtain that the derivative of 𝑔(𝑡) is bounded in
[1, +∞). From Lemma 1, we conclude that 𝑔(𝑡) → 0 as 𝑡 →
∞. Therefore
󵄨2 󵄨
󵄨2 󵄨
󵄨2
󵄨
lim ∫ (󵄨󵄨󵄨∇𝑢1 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇𝑢2 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇𝑢3 󵄨󵄨󵄨 ) 𝑑𝑥 = 0.
𝑡→∞ Ω
(38)
Applying the Poincaré inequality
and denote
∫ 𝜆|𝑢 − 𝑢|2 𝑑𝑥 ≤ ∫ |∇𝑢|2 𝑑𝑥,
𝛼𝑎12 𝑢2∗ 𝛼𝑎12 𝜖1 𝑎21 𝜖3
−
−
,
𝑚1
𝑚1
𝑚12
Ω
𝑎 𝑢∗
𝑎 𝛽
𝑎
𝛼𝑎12
𝑎
𝑙2 = 𝑎22 − 23 2 3 −
− 23 − 21 − 32 ,
4𝑚1 𝜖1 4𝑚2 𝜖2 4𝑚1 𝜖3 4𝑚2 𝜖4
𝑚2
𝑙3 = 𝑎33 𝛽 −
(34)
From Proposition 6 we can see that the solution of system
(1) and (3) is bounded, and so are the derivatives of (𝑢𝑖 − 𝑢𝑖∗ )
(𝑖 = 1, 2, 3) by the equations in (1). Applying Lemma 1, we
obtain that
lim ∫ (𝑢𝑖 − 𝑢𝑖∗ ) 𝑑𝑥 = 0,
Since (𝑢1 , 𝑢2 , 𝑢3 ) is the unique positive solution of system (1).
Using Proposition 5, there exists a constant 𝐶 which does not
depend on 𝑥 ∈ Ω or 𝑡 ≥ 0 such that ‖𝑢𝑖 (⋅, 𝑡)‖∞ ≤ 𝐶 (𝑖 = 1, 2, 3)
for 𝑡 ≥ 0. By Theorem 𝐴 2 in [24], we have
∀𝑡 ≥ 1.
Ω
𝑡→∞ Ω
𝛽𝑎 𝜖
𝑎 𝜖
2
− [𝑎33 𝛽 − 23 2 − 32 4 ] ∫ (𝑢3 − 𝑢3∗ ) 𝑑𝑥.
𝑚2
𝑚2
Ω
𝑙1 = 𝛼𝑎11 −
𝑎11 𝑚12 − 𝑎12 𝑢2∗
,
4𝑚1 𝑎12
2
𝑖=1
𝑎32 𝛽
2
] ∫ (𝑢 − 𝑢2∗ ) 𝑑𝑥
4𝑚2 𝜖4 Ω 2
𝐺2 : 𝑎22 𝑚22 > 𝑎23 𝑢3∗ +
𝜖1 = 𝜖3 =
𝐸󸀠 (𝑡) ≤ −∑𝑙𝑖 ∫ (𝑢𝑖 − 𝑢𝑖∗ ) 𝑑𝑥.
𝑎23 𝑢3∗
𝑎
𝛼𝑎12
𝑎
−
− 23 − 21
2
4𝑚1 𝜖1 4𝑚2 𝜖2 4𝑚1 𝜖3
𝑚2
󵄩
󵄩󵄩
󵄩󵄩𝑢𝑖 (⋅, 𝑡)󵄩󵄩󵄩𝐶2,𝛼 (Ω) ≤ 𝐶,
𝑎23
,
𝑎32
then 𝑙𝑖 > 0 (𝑖 = 1, 2, 3). Therefore, we have
𝑎21 𝜖3
2
] ∫ (𝑢1 − 𝑢1∗ ) 𝑑𝑥
𝑚1
Ω
− [𝑎22 −
𝛽=
𝑚𝑎
𝜖2 = 𝜖4 = 2 33 ,
4𝑎32
𝛼𝑎 𝑢∗ 𝛼𝑎 𝜖
− [𝛼𝑎11 − 122 2 − 12 1
𝑚1
𝑚1
−
𝑎21
,
𝑎12
𝑎23 𝜖2 𝛽𝑎32 𝜖4
−
.
𝑚2
𝑚2
(32)
Ω
𝛽𝑑 𝑢∗ 󵄨
󵄨2
+ 32 3 󵄨󵄨󵄨∇𝑢3 󵄨󵄨󵄨 ) 𝑑𝑥
𝑢3
− ∑𝑙𝑖 ∫ (𝑢𝑖 −
𝑖=1
Ω
leads to
2
lim ∫ (𝑢𝑖 − 𝑢𝑖 ) 𝑑𝑥 = 0,
(𝑖 = 1, 2, 3) ,
𝑡→∞ Ω
(40)
where 𝑢𝑖 = (1/|Ω|) ∫Ω 𝑢𝑖 𝑑𝑥 and 𝜆 is the smallest positive
eigenvalue of −Δ with the homogeneous Neumann condition.
Therefore,
2
2
Ω
∗
𝛼𝑑1 𝑢1∗ 󵄨󵄨
󵄨󵄨2 𝑑2 𝑢2 󵄨󵄨
󵄨2
∇𝑢
+
󵄨
󵄨
󵄨∇𝑢 󵄨󵄨
𝑢12 󵄨 1 󵄨
𝑢22 󵄨 2 󵄨
3
(39)
|Ω| (𝑢1 (𝑡) − 𝑢1∗ ) = ∫ (𝑢1 (𝑡) − 𝑢1∗ ) 𝑑𝑥
Then (29) is transformed into
𝐸󸀠 (𝑡) ≤ − ∫ (
Ω
2
𝑢𝑖∗ ) 𝑑𝑥.
2
= ∫ (𝑢1 (𝑡) − 𝑢1 (𝑡, 𝑥) + 𝑢1 (𝑡, 𝑥) − 𝑢1∗ ) 𝑑𝑥
Ω
2
(33)
≤ 2 ∫ (𝑢1 (𝑡) − 𝑢1 (𝑡, 𝑥)) 𝑑𝑥
Ω
2
+ 2 ∫ (𝑢1 (𝑡, 𝑥) − 𝑢1∗ ) 𝑑𝑥.
Ω
(41)
8
Abstract and Applied Analysis
So we have 𝑢1 (𝑡) → 𝑢1∗ as 𝑡 → ∞. Similarly, 𝑢2 (𝑡) → 𝑢2∗
and 𝑢3 (𝑡) → 𝑢3∗ as 𝑡 → ∞. According to (30), there exists
a subsequence 𝑡𝑚 , and non-negative functions 𝑤𝑖 ∈ 𝐶2 (Ω),
such that
󵄩
󵄩
lim 󵄩󵄩𝑢 (⋅, 𝑡𝑚 ) − 𝑤𝑖 (⋅)󵄩󵄩󵄩𝐶2 (Ω) = 0 (𝑖 = 1, 2, 3) .
(42)
𝑚 → ∞󵄩 𝑖
Applying (40) and noting that 𝑢𝑖 (𝑡) → 𝑢𝑖∗ , we then have 𝑤𝑖 =
𝑢𝑖∗ , (𝑖 = 1, 2, 3). That is,
󵄩
󵄩
lim 󵄩󵄩𝑢 (⋅, 𝑡𝑚 ) − 𝑢𝑖∗ 󵄩󵄩󵄩𝐶2 (Ω) = 0 (𝑖 = 1, 2, 3) .
(43)
𝑚 → ∞󵄩 𝑖
Furthermore, the local stability of E∗ combining with (43)
gives the following global stability.
Similar to the argument of Theorem 8, we have 𝑢1 (𝑥, 𝑡) → V1∗
and 𝑢2 (𝑥, 𝑡) → V2∗ as 𝑡 → ∞ uniformly on Ω provided that
the following additional condition holds:
𝐺3 : 𝑎11 𝑚12 > 𝑎12 V2∗ ,
𝐺4 : 𝑎22 𝑚22 >
𝑎1 (𝑎21 − 𝑎2 )
𝑎 V∗
> 𝑎11 > 12 2 2 ,
𝑚1 𝑎2
𝑚1
(44)
𝐻4 and 𝐺4 hold, and then the semi-trivial steady state 𝐸2
of system (1)–(3) with non-trivial initial functions is globally
asymptotically stable.
Proof. It is obvious that system (1)–(3) always has two nonnegative equilibria (𝑢1 , 𝑢2 , 𝑢3 ) as follows: 𝐸0 = (0, 0, 0) and
𝐸1 = (𝑎1 /𝑎11 , 0, 0). If 𝐻1 and 𝐻4 are satisfied, system (1) has
the other semitrivial solution denoted by 𝐸2 (V1∗ , V2∗ , 0), where
𝐻4 : 𝑎3 𝑚2 > (𝑎32 − 𝑎3 ) V2∗ .
(45)
We consider the stability of 𝐸2 under condition 𝐻1 and
𝐻4 . Equation (1) can be rewritten as
𝑎 (𝑢 − V2∗ )
𝜕𝑢1
− 𝑑1 Δ𝑢1 = 𝑢1 [−𝑎11 (𝑢1 − V1∗ ) − 12 2
𝜕𝑡
𝑚1 + 𝑢1
+
𝜃=
𝑡
Ω −∞
𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
×
𝑎21 𝑚1 (𝑢1 (𝑠, 𝑦) − V1∗ )
𝑑𝑠 𝑑𝑦
(𝑚1 + 𝑢1 (𝑠, 𝑦)) (𝑚1 + V1∗ )
𝑎 𝑢
−𝑎22 (𝑢2 − V2∗ ) − 23 3 ] ,
𝑚2 + 𝑢2
𝜕𝑢3
− 𝑑3 Δ𝑢3
𝜕𝑡
= 𝑢3 [∫ ∫
󵄨
󵄨
𝛿 = 󵄨󵄨󵄨𝑎32 − 𝑎3 󵄨󵄨󵄨 .
(48)
∀𝑡 > 𝑡1 , 𝑥 ∈ Ω.
(49)
Then there exists 𝑡1 > 0 such that
V2∗ − 𝜃 < 𝑢2 (𝑡, 𝑥) < V2∗ + 𝜃,
Consider the following two systems:
𝑤3𝑡 − 𝑑3 Δ𝑤3 = 𝑤3 [−𝑎3 +
𝑎32 (V2∗ + 𝜃)
− 𝑎33 𝑤3 ] ,
𝑚2 + V2∗ + 𝜃
(𝑡, 𝑥) ∈ [𝑡1 , +∞) × Ω,
𝜕𝑤3
= 0,
𝜕]
𝑤3 = 𝑢3 ,
(𝑡, 𝑥) ∈ [𝑡1 , +∞) × 𝜕Ω,
(𝑡, 𝑥) ∈ (−∞, 𝑡1 ] × Ω,
(50)
(𝑡, 𝑥) ∈ [𝑡1 , +∞) × Ω,
𝜕𝑢2
− 𝑑2 Δ𝑢2
𝜕𝑡
= 𝑢2 [∫ ∫
𝑎3 − (𝑎32 − 𝑎3 ) V2∗
,
2𝛿
𝑎 (V∗ − 𝜃)
𝑊3𝑡 − 𝑑3 Δ𝑊3 = 𝑊3 [−𝑎3 + 32 2 ∗
− 𝑎33 V3 ] ,
𝑚2 + V2 − 𝜃
𝑎12 V2∗ (𝑢1 − V1∗ )
],
(𝑚1 + 𝑢1 ) (𝑚1 + V1∗ )
(47)
Next, we consider the asymptotic behavior of 𝑢3 (𝑡, 𝑥). Let
Theorem 9. Assume that
𝐻5 :
2𝑎12 𝑎21 𝑚22
𝑎23 𝑎32
+
.
𝑎33
𝑎11 𝑚12 − 𝑎12 V2∗
𝜕𝑊3
= 0,
𝜕]
(𝑡, 𝑥) ∈ [𝑡1 , +∞) × 𝜕Ω,
𝑊3 = 𝑢3 ,
(𝑡, 𝑥) ∈ (−∞, 𝑡1 ] × Ω.
Combining comparison principle with (50), we obtain that
(46)
𝑊3 (𝑡, 𝑥) ≤ 𝑢3 (𝑡, 𝑥) ≤ 𝑤3 (𝑡, 𝑥) ,
∀𝑡 > 𝑡1 , 𝑥 ∈ Ω.
(51)
By Lemma 4, we obtain
0 = lim 𝑊3 (𝑡, 𝑥) ≤ inf 𝑢3 (𝑡, 𝑥) ≤ sup 𝑢3 (𝑡, 𝑥)
𝑡→∞
𝑡
Ω −∞
𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
×
𝑎32 𝑚2 (𝑢2 (𝑠, 𝑦) − V2∗ )
𝑑𝑠 𝑑𝑦
(𝑚2 + 𝑢2 (𝑠, 𝑦)) (𝑚2 + V2∗ )
−𝑎33 𝑢3 ] .
≤ lim 𝑤3 (𝑡, 𝑥) = 0,
𝑡→∞
∀𝑥 ∈ Ω,
(52)
which implies that lim𝑡 → ∞ 𝑢3 (𝑡, 𝑥) = 0 uniformly on Ω.
Theorem 10. Suppose that 𝐺5 and 𝐺6 hold, and then the semitrivial steady state 𝐸1 of system (1)–(3) with non-trivial initial
functions is globally asymptotically stable.
Abstract and Applied Analysis
9
Proof. We study the stability of the semi-trivial solution 𝐸1 =
(̃
𝑢1 , 0, 0). Similarly, the equations in (1) can be written as
Define
𝐸 (𝑡) = 𝑉 (𝑡)
𝑎 𝑢
𝜕𝑢1
− 𝑑1 Δ𝑢1 = 𝑢1 [−𝑎11 (𝑢1 − 𝑢̃1 ) − 12 2 ] ,
𝜕𝑡
𝑚1 + 𝑢1
+
+∞ 𝑡
𝑎21 𝜖3
∬ ∫ ∫ 𝐾1 (𝑥, 𝑦, 𝑟)
𝑚1
𝑡−𝑟
Ω 0
2
𝜕𝑢2
− 𝑑2 Δ𝑢2
𝜕𝑡
× (𝑢1 (𝑠, 𝑦) − 𝑢1∗ ) 𝑑𝑠 𝑑𝑟 𝑑𝑦 𝑑𝑥
= 𝑢2 [−𝑎2 + ∫ ∫
𝑡
+
𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
Ω −∞
𝑎21 𝑚1 (𝑢1 (𝑠, 𝑦) − 𝑢̃1 )
×
𝑑𝑠 𝑑𝑦
(𝑚1 + 𝑢1 (𝑠, 𝑦)) (𝑚1 + 𝑢̃1 )
𝑎 𝑢
𝑎 𝑢̃
+ 21 1 − 𝑎22 𝑢2 − 23 3 ] ,
𝑚1 + 𝑢̃1
𝑚2 + 𝑢2
2
× (𝑢2 (𝑠, 𝑦) − 𝑢2∗ ) 𝑑𝑠 𝑑𝑟 𝑑𝑦 𝑑𝑥.
(56)
It is easy to see that
󵄨󵄨
󵄨2
󵄨󵄨∇𝑢1 󵄨󵄨󵄨
𝐸 (𝑡) ≤ −𝛼𝑑1 𝑢̃1 ∫
𝑑𝑥
𝑢12
Ω
󸀠
𝜕𝑢3
− 𝑑3 Δ𝑢3
𝜕𝑡
= 𝑢3 [−𝑎3 + ∫ ∫
+∞ 𝑡
𝑎32 𝛽
∬ ∫ ∫ 𝐾2 (𝑥, 𝑦, 𝑟)
4𝜖4 𝑚2 Ω 0
𝑡−𝑟
− ∫ [𝛼 (𝑎11 −
𝑡
Ω
𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
Ω −∞
×
𝑎
𝛼𝑎12
𝑎
− 23 − 21
4𝑚1 𝜖1 4𝑚2 𝜖2 4𝑚1 𝜖3
+ ∫ [𝑎22 −
Ω
𝑎32 𝑢2 (𝑠, 𝑦)
𝑑𝑠 𝑑𝑦 − 𝑎33 𝑢3 ] .
𝑚2 + 𝑢2 (𝑠, 𝑦)
(53)
𝑎
− 32 ] 𝑢22 𝑑𝑥
4𝑚2 𝜖4
+ ∫ [𝛽𝑎33 −
Define
Ω
𝑢
𝑉 (𝑡) = 𝛼 ∫ [𝑢1 − 𝑢̃1 − 𝑢̃1 log 1 ] 𝑑𝑥 + ∫ 𝑢2 𝑑𝑥
𝑢̃1
Ω
Ω
− ∫ [(𝑎2 −
Ω
(54)
Ω
𝑉󸀠 (𝑡) ≤ − ∫ [𝛼 (𝑎11 −
Ω
𝑎12 𝜖1
2
)] (𝑢1 − 𝑢̃1 ) 𝑑𝑥
𝑚1
Ω
𝑎21 𝑢̃1
) 𝑢 + 𝑎3 𝑢3 ] 𝑑𝑥
𝑚1 + 𝑢̃1 2
𝐺6 : 𝑎22 𝑚22 >
𝑙1 = 𝛼𝑎11 −
𝑎
𝛼𝑎12
𝑎
− 23 − 21 ] 𝑢2 𝑑𝑥
4𝑚1 𝜖1 4𝜖2 𝑚2 4𝑚1 𝜖3 2
𝑙2 = 𝑎22 −
(55)
𝛼𝑎12 𝜖1 𝑎21 𝜖3
−
,
𝑚1
𝑚1
(59)
𝑎23 𝜖2 𝛽𝑎32 𝜖4
−
.
𝑚2
𝑚2
𝑎21
,
𝑎12
𝛽=
𝑎23
,
𝑎32
𝜖2 = 𝜖4 =
2
× 𝑢22 (𝑠, 𝑦) 𝑑𝑠 𝑑𝑦 𝑑𝑥.
(58)
Choose
× (𝑢1 (𝑠, 𝑦) − 𝑢̃1 ) 𝑑𝑠 𝑑𝑦 𝑑𝑥
𝑡
𝑎32 𝛽
∬ ∫ 𝐾2 (𝑥, 𝑦, 𝑡 − 𝑠)
4𝜖4 𝑚2 Ω −∞
𝑎23 𝑎32 2𝑎12 𝑎21 𝑚22
+
.
𝑎33
𝑎11 𝑚12
𝑎 𝛽
𝑎
𝛼𝑎12
𝑎
− 23 − 21 − 32 ,
4𝑚1 𝜖1 4𝑚2 𝜖2 4𝑚1 𝜖3 4𝑚2 𝜖4
𝑙3 = 𝑎33 𝛽 −
𝛼=
𝑡
𝑎 𝜖
+ 21 3 ∫ ∫ 𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠)
𝑚1 Ω −∞
+
𝑎21 𝑢̃1
) 𝑢 − 𝑎3 𝛽𝑢3 ] 𝑑𝑥.
𝑚1 + 𝑢̃1 2
Let
𝑎 𝜖
+ ∫ [𝛽𝑎33 − 23 2 ] 𝑢32 𝑑𝑥
𝑚2
Ω
− ∫ [(𝑎2 −
𝑎23 𝜖2 𝑎32 𝛽𝜖4 2
−
] 𝑢3 𝑑𝑥
𝑚2
𝑚2
𝐺5 : 𝑎1 (𝑎21 − 𝑎2 ) < 𝑚1 𝑎2 𝑎11 ,
Calculating the derivative of 𝑉(𝑡) along 𝐸1 , we get from (54)
that
+ ∫ [𝑎22 −
(57)
Assume that
+ 𝛽 ∫ 𝑢3 𝑑𝑥.
Ω
𝑎 𝜖
𝑎12 𝜖1
2
) − 21 3 ] (𝑢1 − 𝑢̃1 ) 𝑑𝑥
𝑚1
𝑚1
𝜖1 = 𝜖3 =
𝑎11 𝑚1
,
4𝑎12
𝑚2 𝑎33
.
4𝑎32
(60)
Then we get 𝑙𝑖 > 0 (𝑖 = 1, 2, 3). Therefore we have
lim 𝑢
𝑡→∞ 1
uniformly on Ω.
(𝑡, 𝑥) = 𝑢̃1 ,
(61)
Abstract and Applied Analysis
4
4
3
3
2
2
u2
u1
10
1
1
0
4
0
4
3
2
x
1
0
4
2
0
8
6
3
2
1
x
t
(a)
4
2
0 0
6
8
t
(b)
4
u3
3
2
1
0
4
3
2
x
1
0
0
4
2
8
6
t
(c)
Figure 2: Global stability of the positive equilibrium 𝐸∗ with an initial condition (𝜙1 , 𝜙2 , 𝜙3 ) = (2.7 + 0.5 sin 𝑥, 2.7 + 0.5 cos 𝑥, 2.7 + 0.5 sin 𝑥).
𝑇
Next, we consider the asymptotic behavior of 𝑢2 (𝑡, 𝑥) and
𝑢3 (𝑡, 𝑥). For any 𝑇 > 0, integrating (57) over [0, 𝑇] yields
󵄩󵄩󵄨󵄨 ∇𝑢 󵄨󵄨󵄩󵄩2
󵄩󵄨
󵄨󵄩
󵄩󵄨
󵄨󵄩2
𝐸 (𝑇) + 𝛼𝑑1 󵄩󵄩󵄩󵄨󵄨󵄨 1 󵄨󵄨󵄨󵄩󵄩󵄩 + 𝑙1 󵄩󵄩󵄩󵄨󵄨󵄨𝑢1 − 𝑢̃1 󵄨󵄨󵄨󵄩󵄩󵄩2
󵄩󵄩󵄨󵄨 𝑢1 󵄨󵄨󵄩󵄩2
𝑇
− 𝑎22 ∫ ∫ 𝑢23 𝑑𝑥 𝑑𝑡 − 𝑎23 ∫ ∫
0
0
Ω
𝑇
+ 𝑎21 ∫ ∬ ∫
0
𝑢1 (𝑠, 𝑦) 𝑢22
𝑡
(𝑡, 𝑥)
𝑚1 + 𝑢1 (𝑠, 𝑦)
Ω −∞
(62)
Ω
𝑢22 𝑢3
𝑑𝑥 𝑑𝑡
𝑚2 + 𝑢3
× 𝐾1 (𝑥, 𝑦, 𝑡 − 𝑠) 𝑑𝑠 𝑑𝑦 𝑑𝑥 𝑑𝑡.
󵄩󵄨 󵄨󵄩2
󵄩󵄨 󵄨󵄩2
+ 𝑙2 󵄩󵄩󵄩󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨󵄩󵄩󵄩2 + 𝑙3 󵄩󵄩󵄩󵄨󵄨󵄨𝑢3 󵄨󵄨󵄨󵄩󵄩󵄩2 ≤ 𝐸 (0) ,
(63)
𝑇
where ‖|𝑢𝑖 |‖22 = ∫0 ∫Ω 𝑢𝑖2 𝑑𝑥 𝑑𝑡. It implies that ‖|𝑢𝑖 |‖2 ≤ 𝐶𝑖 (𝑖 =
2, 3) for the constant 𝐶𝑖 which is independent of 𝑇. Now we
consider the boundedness of ‖|∇𝑢2 |‖2 and ‖|∇𝑢3 |‖2 . From the
Green’s identity, we obtain
Note that
𝑇
∫ ∫ 𝑢2 (𝑡, 𝑥)
0
Ω
𝑇
𝑇
󵄨2
󵄨
𝑑2 ∫ ∫ 󵄨󵄨󵄨∇𝑢2 (𝑡, 𝑥)󵄨󵄨󵄨 𝑑𝑥 𝑑𝑡
0 Ω
=∫ ∫
0
𝑇
= −𝑑2 ∫ ∫ 𝑢2 (𝑡, 𝑥) Δ𝑢2 (𝑡, 𝑥) 𝑑𝑥 𝑑𝑡
0
Ω
𝑇
= − ∫ ∫ 𝑢2
0
=
Ω
𝑇
𝜕𝑢2
𝑑𝑥 𝑑𝑡 − 𝑎2 ∫ ∫ 𝑢22 𝑑𝑥 𝑑𝑡
𝜕𝑡
0 Ω
Ω
𝜕𝑢2 (𝑡, 𝑥)
𝑑𝑥 𝑑𝑡
𝜕𝑡
2
1 𝜕𝑢2
𝑑𝑡 𝑑𝑥
2 𝜕𝑡
1
1
∫ 𝑢2 (𝑇, 𝑥) 𝑑𝑥 − ∫ 𝑢22 (0, 𝑥) 𝑑𝑥 ≤ 𝑀22 |Ω| ,
2 Ω 2
2 Ω
𝑇
𝑇
∫ ∫ 𝑢22 𝑢3 𝑑𝑥 𝑑𝑡 ≤ 𝑀3 ∫ ∫ 𝑢22 𝑑𝑥 𝑑𝑡,
0
Ω
0
Ω
11
4
4
3
3
2
2
u2
u1
Abstract and Applied Analysis
1
1
0
4
0
4
3
2
x
1
0
4
2
0
8
6
3
2
x
t
1
(a)
0
0
2
4
t
6
8
(b)
4
u3
3
2
1
0
4
3
2
x
1
0
0
2
4
6
8
t
(c)
Figure 3: Global stability of the positive equilibrium 𝐸2 with an initial condition (𝜙1 , 𝜙2 , 𝜙3 ) = (2.7 + 0.5 sin 𝑥, 2.7 + 0.5 cos 𝑥, 2.7 + 0.5 sin 𝑥).
𝑇
𝑇
∫ ∫ 𝑢23 𝑑𝑥 𝑑𝑡 ≤ 𝑀2 ∫ ∫ 𝑢22 𝑑𝑥 𝑑𝑡,
0
0
Ω
𝑇
𝑡
∫ ∬ ∫
0
Ω −∞
Ω
𝑢1 (𝑠, 𝑦) 𝑢22 (𝑡, 𝑥)
𝐾 (𝑥, 𝑦, 𝑡 − 𝑠) 𝑑𝑠 𝑑𝑦 𝑑𝑥 𝑑𝑡
𝑚1 + 𝑢1 (𝑠, 𝑦) 1
𝑇
≤ ∫ ∫ 𝑢22 (𝑡, 𝑥) 𝑑𝑥 𝑑𝑡.
0
In the end, we show that the trivial solution 𝐸0 is an
unstable equilibrium. Similarly to the local stability to 𝐸∗ , we
can get the characteristic equation of 𝐸0 as
(𝜆 + 𝜇𝑖 𝐷1 − 𝑎1 ) (𝜆 + 𝜇𝑖 𝐷2 + 𝑎2 ) (𝜆 + 𝜇𝑖 𝐷3 + 𝑎3 ) = 0. (67)
Ω
(64)
Thus, we get ‖|∇𝑢2 |‖2 ≤ 𝐶4 . In a similar way, we have
‖|∇𝑢3 |‖2 ≤ 𝐶5 . Here 𝐶4 and 𝐶5 are independent of 𝑇.
It is easy to see that 𝑢2 (𝑡, 𝑥), 𝑢3 (𝑡, 𝑥) ∈ 𝐿2 ((0, ∞);
1,2
𝑊 (Ω)). These imply that
󵄩
󵄩
lim 󵄩󵄩𝑢 (⋅, 𝑡)󵄩󵄩󵄩𝑊1,2 (Ω) = 0,
𝑡 → ∞󵄩 𝑖
𝑖 = 2, 3.
󵄩
󵄩󵄩
(⋅, 𝑡)󵄩󵄩󵄩𝐶(Ω) = 0,
𝑖 = 2, 3.
Theorem 11. The trivial equilibrium 𝐸0 is an unstable equilibrium of system (1)–(3).
(65)
From the Sobolev compact embedding theorem, we know
lim 󵄩𝑢
𝑡 → ∞󵄩 𝑖
If 𝑖 = 1, then 𝜇1 = 0. It is easy to see that this equation admits
a positive solution 𝜆 = 𝑎1 . According to Theorem 5.1 in [23],
we have the following result.
(66)
4. Numerical Illustrations
In this section, we perform numerical simulations to illustrate
the theoretical results given in Section 3.
12
Abstract and Applied Analysis
4
4
3
u1
u2
3
2
1
1
0
4
2
0
4
3
2
x
1
0
0
6
4
t
2
3
8
2
1
x
(a)
0
0
2
4
6
8
t
(b)
4
u3
3
2
1
0
4
3
2
x
1
0
0
6
4
2
8
t
(c)
Figure 4: Global stability of the positive equilibrium 𝐸1 with an initial condition (𝜙1 , 𝜙2 , 𝜙3 ) = (2.7 + 0.5 sin 𝑥, 2.7 + 0.5 cos 𝑥, 2.7 + 0.5 sin 𝑥).
In the following, we always take Ω = [0, 𝜋], 𝐾𝑖 (𝑥, 𝑦, 𝑡) =
∗
𝐺𝑖 (𝑥, 𝑦, 𝑡)𝑘𝑖 (𝑡), where 𝑘𝑖 (𝑡) = (1/𝜏∗ )𝑒−𝑡/𝜏 (𝑖 = 1, 2) and
2
1 2∞
𝐺1 (𝑥, 𝑦, 𝑡) = + ∑ 𝑒−𝑑2 𝑛 𝑡 cos 𝑛𝑥 cos 𝑛𝑦,
𝜋 𝜋 𝑛=1
𝐺2 (𝑥, 𝑦, 𝑡) =
1 2 ∞ −𝑑3 𝑛2 𝑡
cos 𝑛𝑥 cos 𝑛𝑦.
+ ∑𝑒
𝜋 𝜋 𝑛=1
𝑢1
𝜕V1
1
− V1 ) ,
− 𝑑2 ΔV1 = ∗ (
𝜕𝑡
𝜏
𝑚1 + 𝑢1
𝑢2
𝜕V2
1
− V2 ) ,
− 𝑑3 ΔV2 = ∗ (
𝜕𝑡
𝜏
𝑚2 + 𝑢2
(69)
(68)
where
𝜋
𝑡
0
−∞
V𝑖 = ∫ ∫
𝐺𝑖 (𝑥, 𝑦, 𝑡 − 𝑠)
However, it is difficult for us to simulate our results directly
because of the nonlocal term. Similar to [25], the equations
in (1) can be rewritten as follows:
𝜕𝑢1
𝑎 𝑢
− 𝑑1 Δ𝑢1 = 𝑢1 (𝑎1 − 𝑎11 𝑢1 − 12 2 ) ,
𝜕𝑡
𝑚1 + 𝑢1
𝑎 𝑢
𝜕𝑢2
− 𝑑2 Δ𝑢2 = 𝑢2 (−𝑎2 + 𝑎21 V1 − 𝑎22 𝑢2 − 23 3 ) ,
𝜕𝑡
𝑚2 + 𝑢3
𝜕𝑢3
− 𝑑3 Δ𝑢3 = 𝑢3 (−𝑎3 + 𝑎32 V2 − 𝑎33 𝑢3 ) ,
𝜕𝑡
1 −(𝑡−𝑠)/𝜏∗ 𝑢𝑖 (𝑠, 𝑦)
𝑒
𝑑𝑦 𝑑𝑠,
𝜏∗
𝑚𝑖 + 𝑢𝑖 (𝑠, 𝑦)
𝑖 = 1, 2.
(70)
Each component is considered with homogeneous Neumann
boundary conditions, and the initial condition of V𝑖 is
𝜋
0
0
−∞
V𝑖 (0, 𝑥) = ∫ ∫
𝐺𝑖 (𝑥, 𝑦, −𝑠)
1 𝑠/𝜏∗ 𝑢𝑖 (𝑠, 𝑦)
𝑒
𝑑𝑦 𝑑𝑠,
𝜏∗
𝑚𝑖 + 𝑢𝑖 (𝑠, 𝑦)
𝑖 = 1, 2.
(71)
Abstract and Applied Analysis
In the following examples, we fix some coefficients and
assume that 𝑑1 = 𝑑2 = 𝑑3 = 1, 𝑎1 = 3, 𝑎12 = 1, 𝑚1 = 𝑚2 =
1, 𝑎2 = 1, 𝑎22 = 12, 𝑎23 = 1, 𝑎33 = 12, and 𝜏∗ = 1. The
asymptotic behaviors of system (1)–(3) are shown by choosing
different coefficients 𝑎11 , 𝑎21 , 𝑎3 , and 𝑎32 .
Example 12. Let 𝑎11 = 53/9, 𝑎21 = 81/13, 𝑎3 = 1/13 and
𝑎32 = 14. Then it is easy to see that the system admits a unique
positive equilibrium 𝐸∗ (1/2, 1/12, 1/12). By Theorem 8, we
see that the positive solution (𝑢1 (𝑡, 𝑥), 𝑢2 (𝑡, 𝑥), 𝑢3 (𝑡, 𝑥)) of
system (1)–(3) converges to 𝐸∗ as 𝑡 → ∞. See Figure 2.
Example 13. Let 𝑎11 = 6, 𝑎21 = 12, 𝑎3 = 1/4 and
𝑎32 = 1. Clearly, 𝐻2 does not hold. Hence, the positive
steady state is not feasible. System (1) admits two semitrivial steady state 𝐸2 (0.4732, 0.2372, 0) and 𝐸1 (1/2, 0, 0).
According to Theorem 9, we know that the positive solution
(𝑢1 (𝑡, 𝑥), 𝑢2 (𝑡, 𝑥), 𝑢3 (𝑡, 𝑥)) of system (1)–(3) converges to 𝐸2
as 𝑡 → ∞. See Figure 3.
Example 14. Let 𝑎11 = 6, 𝑎21 = 2, 𝑎3 = 1/4 and 𝑎32 = 1.
Clearly, 𝐻1 does not hold. Hence, 𝐸∗ and 𝐸2 are not feasible.
System (1) has a unique semi-trivial steady state 𝐸1 (1/2, 0, 0).
According to Theorem 10 we know that the positive solution
(𝑢1 (𝑡, 𝑥), 𝑢2 (𝑡, 𝑥), 𝑢3 (𝑡, 𝑥)) of system (1)–(3) converges to 𝐸1
as 𝑡 → ∞. See Figure 4.
5. Discussion
In this paper, we incorporate nonlocal delay into a threespecies food chain model with Michaelis-Menten functional
response to represent a delay due to the gestation of the predator. The conditions, under which the spatial homogeneous
equilibria are asymptotically stable, are given by using the
Lyapunov functional.
We now summarize the ecological meanings of our
theoretical results. Firstly, the positive equilibrium 𝐸∗ of
system (1)–(3) exists under the high birth rate of the prey
(𝑎1 ) and low death rates (𝑎2 and 𝑎3 ) of predator and top
predator. 𝐸∗ is globally stable if the intraspecific competition
𝑎11 is neither too big nor too small and the maximum harvest
rates 𝑎12 , 𝑎23 are small enough. Secondly, the semi-trivial
equilibrium 𝐸2 of system (1)–(3) exists if the birth rate of the
prey (𝑎1 ) is high, death rate of predator 𝑎2 is low, and the death
rate (𝑎3 ) exceeds the conversion rate from predator to top
predator (𝑎32 ). 𝐸2 is globally stable if the maximum harvest
rates 𝑎12 , 𝑎23 are small and the intra-specific competition 𝑎11
is neither too big nor too small. Thirdly, system (1)–(3) has
only one semi-trivial equilibrium 𝐸1 when the death rate (𝑎2 )
exceeds the conversion rate from prey to predator (𝑎21 ). 𝐸1
is globally stable if intra-specific competitions (𝑎11 , 𝑎22 , and
𝑎33 ) are strong. Finally, 𝐸0 is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations go
to extinction. Furthermore, our main results imply that the
nonlocal delay is harmless for stabilities of all non-negative
steady states of system (1)–(3).
There are still many interesting and challenging problems
with respect to system (1)–(3), for example, the permanence
13
and stability of periodic solution or almost periodic solution.
These problems are clearly worthy for further investigations.
Acknowledgments
This work is partially supported by PRC Grants NSFC
11102076 and 11071209 and NSF of the Higher Education Institutions of Jiangsu Province (12KJD110002 and 12KJD110008).
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