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Li and Huang Journal of Inequalities and Applications (2016) 2016:326 DOI 10.1186/s13660-016-1264-0
R E S E A R C H Open Access
Stability and traveling fronts for a foodchain reaction-diffusion systems withnonlocal delaysChenglin Li1* and Guangchun Huang2
*Correspondence:[email protected] of Mathematics, HongheUniversity, Mengzi, Yunnan 661100,P.R. ChinaFull list of author information isavailable at the end of the article
AbstractThis paper is purported to investigate a food chain reaction-diffusion predator-preysystem with nonlocal delays in a bounded domain with no flux boundary condition.We investigate the global stability and find the sufficient conditions of global stabilityof the unique positive equilibrium for this system. The derived results show thatdelays often restrain stability. Using the method of linearizing this system, we see thatthe zero equilibrium is unstable. Moreover, by constructing upper-lower solutions, wefind that there exist traveling wavefronts which connect the zero equilibrium andpositive equilibrium when the wave speed is large enough and the prey intrinsicgrowth rate and the death rate of the predator are relatively big.
Keywords: stability; nonlocal delay; traveling waves
1 IntroductionThe work on dynamics of predator-prey systems is one of the dominant topics in math-ematical ecology. Among the relationships between the species living in the same envi-ronment, the predator-prey theory plays important role. The spatial content of the en-vironment has often been ignored in traditional predator-prey systems. These systemshave been formulated and investigated to reveal the time evolution of uniform populationdistributions in their habitats. However, the spatial distribution of the species is usually in-homogeneous, and ecologists and mathematical ecologists employ the reaction-diffusionpredator-prey systems to model the interaction and the tendency of movement betweenpredator and prey which imply that the species diffuse to areas of smaller population con-centration during the process of evolution, mainly due to resource limitation. In the pasttwo decades, reaction-diffusion predator-prey systems have been extensively discussed[–], but what these models reveal is that the future state of the models is determinedonly by the present, that is, it is independent of the past. For these systems, the predatorsmust take time to digest their food (preys) before further responses and activities takeplace. Therefore, the models of species without delays are approximate at best. A realisticmodel must incorporate the past history of the system, that is, it must include the delay.Recently, some work has studied a delayed diffusive predator-prey system [–].
The theory of traveling fronts for the reaction-diffusion equations is one of the fastestdeveloping areas of modern mathematics and has already attracted much attention due
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 2 of 15
to its significance in physics, chemistry, biology and epidemiology. The traveling waveproblem for reaction-diffusion systems has been studied by many authors [–] in thepast ten years. The food chain is a common phenomenon in population ecology. It is alsocentral to understanding of the community structure in ecology. We will explore a basicexample of a food chain, namely, a three species food chain in which a resource speciesis preyed upon by an intermediate predator which in turn is preyed upon by a dominantpredator, and this model includes nonlocal delays.
Motivated by the above work, we mainly take into account the following food chainreaction-diffusion predator-prey system with nonlocal delays:
∂u
∂t– d�u = u(r – au – bu),
∂u
∂t– d�u = u
(r – bu – au + a
∫�
∫ t
–∞K(x, y, t – s)u(s, y) ds dy
),
∂u
∂t– d�u = u
(–α – bu + a
∫�
∫ t
–∞K(x, y, t – s)u(s, y) ds dy
)
in (,∞) × �,
∂u
∂ν=
∂u
∂ν=
∂u
∂ν= on (,∞) × ∂�,
ui(θ , x) = φi(θ , x) ≥ (i = , , ) in [–∞, ] × �,
(.)
where � is bounded domain in RN (N ≥ is an integer) with a smooth boundary ∂�; u
represents the densities of the prey; u is for the density of the prey, and the same for thepredator; u represents the densities of predator; the positive constants d, d, and d arethe diffusion coefficients of the corresponding species; the positive constants r, r, and α
represent the prey intrinsic growth rates and the death rate of predator, respectively. a anda represent interaction rates, respectively; b, b, and b represent self-limitation rates,respectively; the initial functions ui(t, x) (i = , , ) are Hölder continuous on [–∞, ]×�.
The terms∫�
∫ t–∞ Ki(x, y, t – s)ui(s, y) ds dy (i = , ) represent a time delay because of
gestation, that is, predator contributes to the reproduction of predator biomass. In system(.), we suppose that the kernels Ki(x, y, t) depend on both the temporal and the spatialvariables. The delays in these formulations are nonlocal delays. These formulations revealthat the species drift to their present position (at time t) from all possible positions at allprevious times (see []). Here, we suppose this drift cannot be viewed as being sufficientlysmall so as to be purely a local phenomenon.
In this article, we assume that
Ki(x, y, t) = Gi(x, y, t)ki(t), x, y ∈ �, ki(t) ≥ ,∫�
Gi(x, y, t) dx =∫
�
Gi(x, y, t) dy = , t ≥ , (.)∫ ∞
ki(t) dt = , tki(t) ∈ L((,∞); R
), i = , ,
where Gi(x, y, t) are nonnegative functions which are continuous in (x, y) ∈ �×� for eacht ∈ [,∞) and measurable in t ∈ [,∞) for each pair (x, y) ∈ � × �.
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 3 of 15
In this article, by employing the method of eigenvalue and Lyapunov function, we in-vestigate the stability of the unique positive constant solutions and find the sufficient con-ditions of stability which indicate that nonlocal delays often make impact on the stabilityof positive constant solution, but they do not impact that of a trivial solution. This resultgeneralizes partially the one proved in []. By constructing upper-lower solutions, we es-tablish the existence of the traveling wavefronts when the wave speed is large enough. Thenovelty of this article is that the system (.) incorporates two nonlocal delay terms withwhich are difficult to deal.
This paper is organized into four sections. In Section , the stability of the positive con-stant solution and the instability of trivial solution of system (.) are studied. In Section ,the existence of traveling waves is established by constructing the upper and lower solu-tion. In the final section, we give a short comment and conclusion.
2 Stability of positive equilibriumIt is easy to check that (, , ) and (M, M, M) are a pair of coupled lower-upper solutionsof system (.), where
M = max
{r
b, supθ≤
∥∥φ(θ , ·)∥∥C(�,R)
},
M = max
{r + aM
b, supθ≤
∥∥φ(θ , ·)∥∥C(�,R)
}, (.)
M = max
{aM – α
b, supθ≤
∥∥φ(θ , ·)∥∥C(�,R)
}.
Hence, there exists a unique global solution (u(t, x), u(t, x), u(t, x)) satisfying ≤ u ≤M, ≤ u ≤ M, ≤ u ≤ M to system (.) (see []).
Note that (.) admits the following six equilibria: (, , ), (r/b, , ), (, r/b, ),( br–ar
aa+bb, ar+br
aa+bb, ), (, rb+αa
aa+bb, ra–αb
aa+bb), and (k∗
, k∗ , k∗
), where
k∗ =
(ra – αa)a + (rb – ra)b
aab + (aa + bb)b, k∗
=(rb + αa)b + rab
aab + (aa + bb)b,
k∗ =
(ra – αa)a + (ra – αb)b
aab + (aa + bb)b.
For the existence of a positive constant solution (k∗ , k∗
, k∗ ), it is necessary to assume that
ra > αa, rb > ra, ra > αb.
Let = μ < μ < · · · → +∞ denote the eigenvalues of –� in � under a homogeneousNeuman boundary condition and ϕ be the set of eigenfunctions corresponding to μ.
Notation .(i) Xij := {Cϕij : C ∈ R}, where ϕij are orthonormal basis of S(μi) for
j = , . . . , dim[S(μi)].(ii) X := {(u, v, w) ∈ C(�) × C(�) : ∂u
∂ν= ∂u
∂ν= ∂w
∂ν= on ∂�}, so that
X =∞⊕i=
dim[S(μi)]⊕j=
Xij.
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 4 of 15
Lemma . ([, ]) Let a and b be positive constants. Suppose that ξ ,η ∈ C[a, +∞),η ≥ , and ξ is bounded from below. If ξ ′(t) ≤ –bη(t) and η′(t) is bounded from above in[a, +∞), then lim tt→∞η(t) = .
The following theorem is the global stability result of the positive constant solution(k∗
, k∗ , k∗
) of (.). This result extends partially the one proved in [].
Theorem . Suppose that bb > aa, bb > aa. Then the positive constant solution(k∗
, k∗ , k∗
) is globally stable for the system (.).
Proof We use the approach developed by [] to find the proof. Let (u(t, x), u(t, x),u(t, x)) be positive solution of (.) and define the following Lyapunov function:
V(t) =∑
i=
βi
∫�
(ui – k∗
i – k∗i ln
ui
k∗i
)dx,
where β = , β and β are positive constant to be determined. By calculating the deriva-tive of V(t) along positive solutions of system (.), we obtain
dV(t)dt
=∑
i=
βi
∫�
∂ui
∂t
( –
k∗i
ui
)dx +
∫�
∂u
∂t
( –
k∗
u
)dx
= –∑
i=
βidik∗i
∫�
|∇ui|u
idx +
∫�
β(u – k∗
)(r – au – bu) dx
+∫
�
β(u – k∗
)(
r – bu – au + a
∫�
∫ t
–∞K(x, y, t – s)u(s, y) ds dy
)dx
+∫
�
(u – k∗
)(
–α – bu + a
∫�
∫ t
–∞K(x, y, t – s)u(s, y) ds dy
)dx. (.)
By using the inequality ab ≤ λa +
λb, we obtain
dV(t)dt
≤ –∑
i=
βidik∗i
∫�
|∇ui|u
idx – βb
∫�
(u – k∗
) dx
+ βa
∫�
[λ
(u – k∗
) +
λ
(u – k∗
)
]dx – βb
∫�
(u – k∗
) dx
+ βa
∫�
[λ
(u – k∗
) +
λ
(u – k∗
)
]dx – b
∫�
(u – k∗
) dx
+λaβ
∫�
∫�
∫ t
–∞K(x, y, t – s)
(u(s, y) – k∗
) ds dy dx
+
λaβ
∫�
∫�
∫ t
–∞K(x, y, t – s)
(u(t, y) – k∗
) ds dy dx
+λa
∫�
∫�
∫ t
–∞K(x, y, t – s)
(u(s, y) – k∗
) ds dy dx
+
λa
∫�
∫�
∫ t
–∞K(x, y, t – s)
(u(t, y) – k∗
) ds dy dx. (.)
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 5 of 15
Employing the property of Ki(x, y, t) (i = , ) as described in (.), we obtain
dV(t)dt
≤ –∑
i=
βidik∗i
∫�
|∇ui|u
idx – β(b – aλ/)
∫�
(u – k∗
) dx
– (βb – βa/λ – λβa/ – aβ/λ)∫
�
(u – k∗
) dx
– (b – βa/λ – a/λ)∫
�
(u – k∗
) dx
+λaβ
∫�
∫�
∫ ∞
K(x, y, r)
(u(t – r, y) – k∗
) dr dy dx
+λa
∫�
∫�
∫ ∞
K(x, y, r)
(u(t – r, y) – k∗
) dr dy dx. (.)
Define a new Lyapunov function
V (t) = V(t) +λaβ
∫�
∫�
∫ ∞
∫ t
t–rK(x, y, r)
(u(l, y) – k∗
) dl dr dy dx
+λa
∫�
∫�
∫ ∞
∫ t
t–rK(x, y, r)
(u(l, y) – k∗
) dl dr dy dx. (.)
Then, combining (.) and (.), we get
dV(t)dt
≤ –∑
i=
βidik∗i
∫�
|∇ui|u
idx – β(b – aλ/)
∫�
(u – k∗
) dx
– (βb – βa/λ – λβa/ – aβ/λ)∫
�
(u – k∗
) dx
– (b – βa/λ – a/λ)∫
�
(u – k∗
) dx
+λaβ
∫�
∫�
∫ ∞
K(x, y, r)
(u(t, y) – k∗
) dr dy dx
+λa
∫�
∫�
∫ ∞
K(x, y, r)
(u(t, y) – k∗
) dr dy dx. (.)
Since
∫�
∫�
∫ ∞
Ki(x, y, r)
(ui(t, y) – k∗
i) dr dy dx
=∫
�
(ui(t, y) – k∗
i) dy (i = , ),
it follows from (.) that
dV(t)dt
≤ –∑
i=
βidik∗i
∫�
|∇ui|u
idx
–(β(b – aλ/) – λaβ/
)∫�
(u – k∗
) dx
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 6 of 15
– (βb – βa/λ – λβa/ – aβ/λ – λa/)∫
�
(u – k∗
) dx
– (b – βa/λ – a/λ)∫
�
(u – k∗
) dx. (.)
Integrating (.) over [, T] (T > ), we obtain
∑i=
βidik∗i
∥∥∥∥ |∇ui|ui
∥∥∥∥
L(�T )
+ βb∥∥u – k∗
∥∥
L(�T )
+ βb∥∥u – k∗
∥∥
L(�T )
+ b∥∥u – k∗
∥∥
L(�T )
≤ V () +
(βaλ + λβa)∥∥u – k∗
∥∥
L(�T )
+
(βa/λ + λβa + aβ/λ
+ λa)∥∥u – k∗
∥∥
L(�T )
+
(βa/λ + a/λ)∥∥u – k∗
∥∥
L(�T )
. (.)
Taking
λ = λ =βb
aβ + aand λ = λ =
βa + a
b,
it is derived from (.) that
∑i=
βidik∗i
Mi
∥∥|∇ui|∥∥
L(�T )
+ βb∥∥u – k∗
∥∥
L(�T )
≤ V () +((aβ + aβ)/βb + (aβ + a)/b
)∥∥u – k∗∥∥
L(�T )
. (.)
Using the conditions bb > aa, bb > aa, one can choose β,β > such that
βb >
[(aβ + aβ)/βb + (aβ + a)/b
],
because the inequalities bbββ > aβ
+ aaββ + a
β and bbβ > a
β +
aaβ + a hold for certain β and β.
Therefore, we obtain
∥∥|∇ui|∥∥
L(�T )
≤ C,∥∥u – k∗
∥∥
L(�T )
≤ C, (.)
where C is constant independent of T . In a similar way, by taking
λ = λ =βa + aβ
βband λ = λ =
βb
aβ + a,
it is derived from (.) that
∑i=
βidik∗i
Mi
∥∥|∇ui|∥∥
L(�T )
+ βb∥∥u – k∗
∥∥
L(�T )
+ b∥∥u – k∗
∥∥
L(�T )
≤ V () +((βa + aβ)/βb
)∥∥u – k∗∥∥
L(�T )
+((βa + a)/βb
)∥∥u – k∗∥∥
L(�T )
. (.)
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 7 of 15
Using the conditions bb > aa, bb > aa, one can choose β,β > again such that
βb >(aβ + aβ)
βband b >
(aβ + a)
βb.
Therefore, we see that
∥∥u – k∗∥∥
L(�T )
≤ C and∥∥u – k∗
∥∥
L(�T )
≤ C, (.)
where C is constant independent of T .Choosing
λ = λ =βa + aβ
βb+ ε, λ = λ =
βb
aβ + a,
where ε is sufficiently small positive constant. Using the conditions of Theorem . and(.), one can easily verify that there exists a positive constant δ (δ > ) such that
dVdt
≤ –δ
∫�
[(u – k∗
) +
(u – k∗
) +
(u – k∗
)]dx,
dVdt
≤ , (u, u, u) = (k∗
, k∗ , k∗
).
(.)
Using integration by parts, the Hölder inequality, (.), and (.), one can easily check thatddt
∫�
[(u – k∗ ) + (u – k∗
) + (u – k∗ )] dx is bounded from above. Then, using Lemma .,
(.), and (.), we see that
∥∥u(t, ·) – k∗∥∥
L(�)
→ ,
∥∥u(t, ·) – k∗∥∥
L(�)
→ , (.)
∥∥u(t, ·) – k∗∥∥
L(�)
→ .
Obviously,
∥∥u(t, x)∥∥
L∞(�)
≤ C‖u‖ W
‖u‖L(�). (.)
It follows from (.), (.), (.), (.), and (.) that
∥∥u(t, ·) – k∗∥∥
L∞(�)
→ ,
∥∥u(t, ·) – k∗∥∥
L∞(�)
→ ,
∥∥u(t, ·) – k∗∥∥
L∞(�)
→ .
Namely, (u, u, u) converges uniformly to (k∗ , k∗
, k∗ ). Using the fact that V (u, u, u)
is decreasing for t, one can derive that (k∗ , k∗
, k∗ ) is globally stable. This completes the
proof. �
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 8 of 15
Theorem . The trivial equilibrium (, , ) is unstable for the system (.).
Proof The linearized problem of (.) at (, , ) can be expressed by
wt =(D� + Fw(, , )
)w,
where w = (u(t, x), u(t, x), u(t, x))T , and F = (u(r – au – bu), u(r – bu – au +a
∫�
∫ t–∞ K(x, y, t – s)u(s, y) ds dy), u(–α – bu + a
∫�
∫ t–∞ K(x, y, t – s)u(s, y) ds dy)). By
direct calculations, we obtain
D =
⎛⎜⎝
d d d
⎞⎟⎠ , Fw(, , ) =
⎛⎜⎝
r r α
⎞⎟⎠ .
Consider the following eigenvalue problem:
(D� + Fw(w∗)
)⎛⎜⎝
φ
ϕ
ψ
⎞⎟⎠ = λ̃
⎛⎜⎝
φ
ϕ
ψ
⎞⎟⎠ ,
where w∗ is constant solution of (.). Using the eigenfunction expansions (.) in [] forφ, ϕ and ψ , λ̃ is an eigenvalue of D�+ Fw(w∗) if and only if λ̃ is an eigenvalue of the matrix–μkD + Fw(, , ) for each k ≥ . Therefore, to study the local stability at (, , ), it isnecessary to investigate the characteristic equation
det(λ̃I + μkD – Fw(, , )
)= (λ̃ + μkd – r)(λ̃ + μkd – r)(λ̃ + μkd + α) = .
If i = , then μk = . Therefore, there exist two positive characteristic roots, which, in viewof Theorem . in [], yields the desired result. �
3 The existence of traveling wavesIn this section, we assume that � ⊂ R. Denote f(u, u, u) = u(r – au – bu),f(u, u, u) = u(r – bu – au + a
∫�
∫ t–∞ K(x, y, t – s)u(s, y) ds dy), f(u, u, u) =
u(–α – bu + a∫�
∫ t–∞ K(x, y, t – s)u(s, y) ds dy). Let (u(t, x), u(t, x), u(t, x)) = (φ(x +
ct),ϕ(x + ct),ψ(x + ct)) be a traveling wave solution of (.), where φ,ϕ,ψ ∈ C(R, R) andc > is a constant accounting for the wave speed, and denote the traveling wave coordinatex + ct still by t. Then the system (.) can be rewritten in the form
dφ′′(t) – cφ′(t) + fc(φt ,ϕt ,ψt) = ,
dϕ′′(t) – cϕ′(t) + fc(φt ,ϕt ,ψt) = , (.)
dψ′′(t) – cψ ′(t) + fc(φt ,ϕt ,ψt) = ,
where fci (i = , , ) are defined by
fci(φ,ϕ,ψ) = fi(φc,ϕc,ψ c), φc(s) = φ(cs), ϕc(s) = ϕ(cs),
ψ c(s) = ψ(cs), s ∈ (–∞, ], i = , , .
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 9 of 15
If (.) has a solution satisfying the following asymptotic boundary conditions:
limt→–∞φ(t) = φ–, lim
t→–∞ϕ(t) = ϕ–, limt→–∞ψ(t) = ψ–,
limt→+∞φ(t) = φ+, lim
t→+∞ϕ(t) = ϕ+, limt→+∞ψ(t) = ψ+,
then system (.) has a traveling wave solution (see [, ]). Without loss of generality, weassume that (φ–,ϕ–,ψ–) = (, , ) and (φ+,ϕ+,ψ+) = (k∗
, k∗ , k∗
).According to basic theory of the existence of traveling wave solutions (see [, ]), we
mainly need to check that the system (.) satisfies partial quasi-monotonicity conditions,that is, there exist three positive constants ρ,ρ,ρ > such that
f(φ,ϕ,ψ) – f(φ,ϕ,ψ) + ρ[φ() – φ()
] ≥ ,
f(φ,ϕ,ψ) – f(φ,ϕ,ψ) ≤ ,
f(φ,ϕ,ψ) – f(φ,ϕ,ψ) + ρ[ϕ() – ϕ()
] ≥ , (.)
f(φ,ϕ,ψ) – f(φ,ϕ,ψ) ≤ ,
f(φ,ϕ,ψ) – f(φ,ϕ,ψ) + ρ[ψ() – ψ()
] ≥ ,
with ≤ φ(s) ≤ φ(s) ≤ M, ≤ ϕ(s) ≤ ϕ(s) ≤ M, ≤ ψ(s) ≤ ψ(s) ≤ M, and we alsoneed to check that a pair of continuous functions (φ,ϕ,ψ) and (φ,ϕ,ψ) is a pair of upper-lower solution of system (.), that is,
dφ′′(t) – cφ′(t) + fc(φt ,ϕt ,ψ t) ≤ ,
dϕ′′(t) – cϕ′(t) + fc(ϕt ,ϕt ,ψ t) ≤ , (.)
dψ′′(t) – cψ ′(t) + fc(ψ t ,ϕt ,ψ t) ≤
and
dφ′′(t) – cφ′(t) + fc(φt ,ϕt ,ψ t) ≥ ,
dϕ′′(t) – cϕ′(t) + fc(ϕt ,ϕt ,ψ t) ≥ , (.)
dψ′′(t) – cψ ′(t) + fc(ψ t ,ϕt ,ψ t) ≥ ,
where (, , ) ≤ (φ,ϕ,ψ) ≤ (φ,ϕ,ψ) ≤ (M, M, M), t ∈ R.
Lemma . fc(φt ,ϕt ,ψt), fc(φt ,ϕt ,ψt), and fc(φt ,ϕt ,ψt) of system (.) satisfy (.).
Proof Let φ(s), φ(s), ϕ(s), ϕ(s), ψ(s), ψ(s) satisfy ≤ φ(s) ≤ φ(s) ≤ M, ≤ ϕ(s) ≤ϕ(s) ≤ M, ≤ ψ(s) ≤ ψ(s) ≤ M, s ∈ (–∞, ].
For any φi,ϕi,ψi ∈ ((–∞, ], R), i = , , we have
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt)
= φ()(r – aϕ() – bφ()
)– φ()
(r – aϕ() – bφ()
)
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 10 of 15
≥ r(φ() – φ()
)– aϕ()
(φ() – φ()
)– bM
(φ() – φ()
)≥ (–aM – bM)
(φ() – φ()
). (.)
Let ρ = aM + bM > , then it is easy to see that
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt) + ρ(φ() – φ()
) ≥ ,
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt)
= φ()(r – aϕ() – bφ()
)– φ()
(r – aϕ() – bφ()
)= –aφ()
(ϕ() – ϕ()
) ≤ .
(.)
For fc(φt ,ϕt ,ψt), we have
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt)
= ϕ()(r – bϕ() – aψ() + aφ()
)– ϕ()
(r – bϕ()
– aψ() + aφ())
≥ r(ϕ() – ϕ()
)– bM
(ϕ() – ϕ()
)– aM
(ϕ() – ϕ()
)+ aφ()
(ϕ() – ϕ()
)≥ (–aM – bM)
(ϕ() – ϕ()
). (.)
Let ρ = aM + bM > , then it is easy to see that
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt) + ρ(ϕ() – ϕ()
) ≥ ,
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt)
= ϕ()(r – bϕ() – aψ() + aφ()
)– ϕ()
(r – bϕ() (.)
– aψ() + aφ())
= –aϕ()(ψ() – ψ()
).
For fc(φt ,ϕt ,ψt), we have
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt)
= ψ()(–α – bψ() + aϕ()
)– ψ()
(–α – bψ() + aϕ()
)≥ –α
(ψ() – ψ()
)– bM
(ψ() – ψ()
)= (–α – bM)
(ψ() – ψ()
). (.)
Let ρ = bM + α > , then it is easy to see that
fc(φt ,ϕt ,ψt) – fc(φt ,ϕt ,ψt) + ρ(ψ() – ψ()
) ≥ .
This completes the proof of Lemma .. �
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 11 of 15
We assume that c > c∗ � max{dr, d(r + aM), d(α + aM)}, c∗ > . Using thisassumption, one can see that there exist ηi > (i = , , ) such that
dη – cη + r = ,
dη – cη + r + aM = , (.)
dη – cη + α + aM = .
Assume that r > max{ak∗ , ak∗
} and α > bbk∗
a, then one can choose positive constants
ε and ε such that
a(ε – k∗
)
> bk∗ , a
(ε – k∗
)
> bk∗ , α > b
(ε – k∗
),
rb > aa(ε – k∗
), r + b
(ε – k∗
)
> ak∗ ,
(.)
and
(k∗
+ ε)(
r – a(k∗
– ε)
– b(k∗
+ ε))
< ,(k∗
+ ε)(
r – b(k∗
+ ε)
– a(k∗
– ε)
+ aM)
< ,(k∗
+ ε)(
–α – b(k∗
+ ε)
+ aM)
< ,(k∗
– ε)[
r – a(k∗
+ ε)
– b(k∗
– ε)]
> ,(k∗
– ε)[
r – b(k∗
– ε)
– a(k∗
+ ε)]
> ,(k∗
– ε)(
–α – b(k∗
– ε))
> ,
(.)
for εi (i = , , ) being relatively big.For the above constants and suitable constants t̃i > (i = , , , , , ) satisfying t̃ <
min{t̃, t̃}, t̃ > max{t̃, t̃}, we define the continuous functions �(t) = (φ(t),ϕ(t),ψ(t)) and�(t) = (φ(t),ϕ(t),ψ(t)) as follows:
φ(t) =
{k∗
eηt , t ≤ t̃,k∗
+ εe–ηt , t > t̃,ϕ(t) =
{k∗
eηt , t ≤ t̃,k∗
+ εe–ηt , t > t̃,
ψ(t) =
{k∗
eηt , t ≤ t̃,k∗
+ εe–ηt , t > t̃,φ(t) =
{, t ≤ t̃,k∗
– εe–ηt , t > t̃,
ϕ(t) =
{, t ≤ t̃,k∗
– εe–ηt , t > t̃,ψ(t) =
{, t ≤ t̃,k∗
– εe–ηt , t > t̃,
where
t̃ =η
lnr
ak∗
, t̃ =η
lna(ε – k∗
)bk∗
,
t̃ = max
{η
lna(ε – k∗
)bk∗
,
η
lnr + b(ε – k∗
)ak∗
}.
(.)
Lemma . Assume that r > max{ak∗ , ak∗
} and α > bbk∗
a, then (φ,ϕ,ψ) is an upper
solution of system (.).
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 12 of 15
Proof If t ≤ t̃ < t̃, then φ(t) = k∗ eηt , and ϕ(t) = . Therefore, we have
dφ′′(t) – cφ′(t) + φ
(r – aϕ(t) – bφ(t)
)≤ (
dη – cη + r
)k∗
eηt = . (.)
If t > t̃ and t ≤ t̃, then φ(t) = k∗ eηt and ϕ(t) = k∗
– εe–ηt . Therefore, we obtain
dφ′′(t) – cφ′(t) + φ(t)
(r – aϕ(t) – bφ(t)
)= I(η), (.)
where I(η) = k∗ eηt[a(εe–ηt – k∗
) – bk∗ eηt], I() = k∗
eηt[a(ε – k∗ ) – bk∗
eηt] < whent > t̃. Therefore, there exists η∗
> such that dφ′′(t)–cφ′(t)+φ(t)(r –aϕ(t)–bφ(t)) ≤
for all η ∈ (,η∗ ).
If t > t̃ > t̃, then φ(t) = k∗ + εe–ηt and ϕ(t) = k∗
– εe–ηt . Therefore, we have
dφ′′(t) – cφ′(t) + φ(t)
(r – aϕ(t) – bφ(t)
)= I(η), (.)
where I(η) = (dεη +cεη)e–ηt +(k∗
+εe–ηt)(r –a(k∗ –εe–ηt)–b(k∗
+εe–ηt)). It followsfrom (.) that I() = (k∗
+ ε)(r – a(k∗ – ε) – b(k∗
+ ε)) < . Therefore, there existsη∗
> such that dφ′′(t) – cφ′(t) + φ(t)(r – aϕ(t) – bφ(t)) ≤ for all η ∈ (,η∗
).If t ≤ t̃ < t̃, then ϕ(t) = k∗
eηt , and ψ(t) = . It follows that
dϕ′′(t) – cϕ′(t) + fc
(φ(t),ϕ(t),ψ(t)
)≤ (
dη – cη + r
)k∗
eηt + k∗ eηt(–bk∗
eηt + aM)
≤ (dη
– cη + r + aM
)k∗
eηt = . (.)
If t > t̃ and t ≤ t̃, then ϕ(t) = k∗ eηt and ψ = k∗
– εe–ηt . By calculating, we have
dϕ′′(t) – cϕ′(t) + fc
(φ(t),ϕ(t),ψ(t)
) ≤ I(η), (.)
where I(η) = k∗ eηt(a(εe–ηt – k∗
) – bk∗ eηt). Thus, I() = k∗
eηt(a(ε – k∗ ) – bk∗
eηt) < when t > t̃. Hence, there exists η∗
> such that dϕ′′(t) – cϕ′(t) + fc(φ(t),ϕ(t),ψ(t)) ≤
for all η ∈ (,η∗).
If t > t̃, then ϕ(t) = k∗ + εe–ηt and ψ(t) = k∗
– εe–ηt . By calculating, we have
dϕ′′(t) – cϕ′(t) + fc
(φ(t),ϕ(t),ψ(t)
) ≤ I(η), (.)
where I(η) = (dεη + cεη)e–ηt + (k∗
+εe–ηt)(r + aM – b(k∗ +εe–ηt) – a(k∗
–εe–ηt)).It follows from (.) that I() = (k∗
+ ε)(r + aM – b(k∗ + ε) – a(k∗
– ε)) < . Hence,there exists η∗
> such that dϕ′′(t) – cϕ′(t) + fc(φ(t),ϕ(t),ψ(t)) ≤ for all η ∈ (,η∗
).If t ≤ t̃, then ψ(t) = k∗
eηt . By calculating, we have
dψ′′(t) – cψ ′(t) + fc
(φ(t),ϕ(t),ψ(t)
)≤ (
dη – cη
)k∗
eηt + k∗ eηt(–α – bk∗
eηt + aM)
≤ (dη
– cη – α + aM
)k∗
eηt = . (.)
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 13 of 15
If t > t̃, then ψ(t) = k∗ + εe–ηt . By calculating, we get
dψ′′(t) – cψ ′(t) + fc
(φ(t),ϕ(t),ψ(t)
) ≤ I(η), (.)
where I(η) = (dεη +cεη)e–ηt +(k∗
+εe–ηt)(–α –b(k∗ +εe–ηt)+aM). It follows from
(.) that I() = (k∗ + ε)(–α – b(k∗
+ ε) + aM) < . Hence, there exists η∗ > such
that dψ′′(t) – cψ ′(t) + fc(φ(t),ϕ(t),ψ(t)) ≤ for all η ∈ (,η∗
).Finally, for any η ∈ (, min{η∗
,η∗,η∗
,η∗,η∗
}), we see that (.) holds. This completes theproof. �
Lemma . Assume that r > max{ak∗ , ak∗
} and α > bbk∗
a, then (φ(t),ϕ(t),ψ(t)) is a
pair of lower solution of system (.).
Proof If t ≤ t̃, then φ(t) = . We have dφ′′(t) – cφ′(t) + φ(t)(r – aϕ(t) – bφ(t)) = .
If t̃ < t ≤ t̃, then φ(t) = k∗ – εe–ηt , ϕ = k∗
eηt . We have
dφ′′(t) – cφ′(t) + φ(t)
(r – aϕ(t) – bφ(t)
)= (–dη – c)εηe–ηt +
(k∗
– εe–ηt)(r – ak∗ eηt – b
(k∗
– εe–ηt))� I(η). (.)
Using (.), we see that I() = (k∗ – ε)(r – ak∗
eηt – b(k∗ – ε)) > when t̃ < t ≤ t̃.
Therefore, there exists η∗ such that dφ
′′(t) – cφ′(t) + φ(t)(r – aϕ(t) – bφ(t)) ≥ forη ∈ (,η∗
).If t > t̃, then φ(t) = k∗
– εe–ηt , ϕ = k∗ + εe–ηt . We have
dφ′′(t) – cφ′(t) + φ(t)
(r – aϕ(t) – bφ(t)
)≥ (–dη – c)εηe–ηt +
(k∗
– εe–ηt)(r – a(k∗
+ εe–ηt) – b(k∗
– εe–ηt))� I(η). (.)
Using (.), we see that I() = (k∗ – ε)(r – a(k∗
+ ε) – b(k∗ – ε)) > . Therefore, there
exists η∗ such that dφ
′′(t) – cφ′(t) + φ(t)(r – aϕ(t) – bφ(t)) ≥ for η ∈ (,η∗).
If t ≤ t̃, then ϕ = . Therefore we have dϕ′′(t) – cϕ′ + fc(φ(t),ϕ(t),ψ(t)) = .
If t̃ < t ≤ t̃, then ϕ(t) = k∗ – εe–ηt , ψ = k∗
eηt . We get
dϕ′′(t) – cϕ′ + fc
(φ(t),ϕ(t),ψ(t)
)= (–dη – c)εηe–ηt +
(k∗
– εe–ηt)(r – b(k∗
– εe–ηt) – ak∗ eηt)
� I(η). (.)
Using (.), we see that I() = (k∗ – ε)(r – b(k∗
– ε) – ak∗ eηt) > when t > t̃. Hence,
there exists η∗ such that dϕ
′′(t) – cϕ′(t) + fc(φ(t),ϕ(t),ψ(t)) ≥ for all η ∈ (,η∗).
If t > t̃, then ϕ(t) = k∗ – εe–ηt . We get
dϕ′′(t) – cϕ′ + fc
(φ(t),ϕ(t),ψ(t)
)≥ (–dη – c)εηe–ηt +
(k∗
– εe–ηt)(r – b(k∗
– εe–ηt) – a(k∗
+ εe–ηt))� I(η). (.)
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 14 of 15
Using (.), we see that I() = (k∗ – ε)(r – b(k∗
– ε) – a(k∗ + ε)) > . Hence, there
exists η∗ such that dϕ
′′(t) – cϕ′(t) + fc(φ(t),ϕ(t),ψ(t)) ≥ for all η ∈ (,η∗).
If t ≤ t̃, then ψ = . Therefore, we have dψ′′(t) – cψ ′ + fc(φ(t),ϕ(t),ψ(t)) = .
If t > t̃, then ψ(t) = k∗ – εe–ηt . Hence, we get
dψ′′(t) – cψ ′ + fc
(φ(t),ϕ(t),ψ(t)
)≥ (–dη – c)εηe–ηt +
(k∗
– εe–ηt)(–α – b(k∗
– εe–ηt))� I(η). (.)
Using (.), we see that I() = (k∗ –ε)(–α – b(k∗
–ε)) > . Hence, there exists η∗ such
that dψ′′(t) – cψ ′(t) + fc(φ(t),ϕ(t),ψ(t)) ≥ for all η ∈ (,η∗
).Finally, for any η ∈ (, min{η∗
,η∗,η∗
,η∗,η∗
}), we see that (.) holds. This completes theproof. �
By using Lemmas .-., we have the following conclusion.
Theorem . Assume that r > max{ak∗ , ak∗
} and α > bbk∗
a, then, for any c > c∗ > ,
system (.) always has a traveling wave solution with speed c connecting the trivial steadystate (, ) and the positive steady state (k∗
, k∗ , k∗
).
4 Results and discussionIn this article, we investigate a food chain reaction-diffusion predator-prey systems withnonlocal delay in a bounded domain with no flux boundary condition incorporating delayrespecting gestation of the predators. By using the methods of the Lyapunov function, weprove global stability of positive equilibrium of the system (.). If the nonlocal delay termsare replaced by the general terms without delay, one can easily verify that the positiveconstant solution of (.) is globally stable without any condition when a positive constantequilibrium exists.
The above result shows that if the intra-specific competitions of the predators and preysdominate their inter-specific interaction, then the unique positive equilibrium is globallystable if it exists, which implies that predators and preys are permanent from biologic view.If the nonlocal delay terms are replaced by a general term without delay, without any con-dition the positive equilibrium is globally stable when it exists. This implies that nonlocaldelays often impact the global stability of a positive constant solution. By using the methodof the upper-lower solutions, we also see that there exists a traveling wavefront connectingthe zero solution to the positive equilibrium of the system when the wave speed is largeenough and the prey intrinsic growth rate and the death rate of predator are relatively big.
In modern mathematics, the theory and methods of traveling waves solutions developquickly and they have attracted much attention due to their significance in the real word.We would like to extend this theory and these methods to the integral equation, and werefer to [–], but this question is still an open problem.
Competing interestsThe authors declare that they have no competing interests.
Authors’ contributionsCL carried out collecting data, participated in computation and drafted the manuscript. GH helped to draft themanuscript. All authors read and approved the final manuscript.
Li and Huang Journal of Inequalities and Applications (2016) 2016:326 Page 15 of 15
Author details1College of Mathematics, Honghe University, Mengzi, Yunnan 661100, P.R. China. 2College of Science, Honghe University,Mengzi, Yunnan 661199, P.R. China.
AcknowledgementsThis work is supported by the NSF of China (11461023) and the research funds of PhD for Honghe University (14bs19).
Received: 31 May 2016 Accepted: 29 November 2016
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