Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Stability and travelling waves for a time-delayed population systemwith stage structureLiang Zhang a,∗, Bingtuan Li b, Jin Shang b

a Institute of Applied Mathematics, College of Science, Northwest A&F University, Yangling, Shaanxi 712100, PR Chinab Department of Mathematics, University of Louisville, Louisville, KY 40292, USA

a r t i c l e i n f o

Article history:Received 5 June 2011Accepted 11 November 2011

Keywords:CompetitionStage structureTime delayStabilityTravelling waves

a b s t r a c t

We study a time-delayed population system with stage structure for the interactionbetween two species, the adult members of which are in competition. For each of thetwo species the model incorporates a time delay which represents the time from birth tomaturity of that species. The global stability results are established for each equilibrium.The criteria for global convergence to each equilibrium are sharp and involve these delays.By using lower and upper travelling wave solutions, we show that the model has travellingwave solutions that connect the origin and the coexistence equilibriumwith speeds greaterthan the spreading speed of each species in the absence of its rival.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

This paper is concernedwith the followingdelayed reaction–diffusion Lotka–Volterra-type two-species competitionwithage structure

∂ui

∂t= dui

∂2ui

∂x2+ α1um − γ1ui − α1e−γ1τ1

∫R

14πduiτ1

e−(x−y)24dui τ1 × um(t − τ1, y)dy,

∂um

∂t= dum

∂2um

∂x2+ α1e−γ1τ1

∫R

14πduiτ1

e−(x−y)24dui τ1 × um(t − τ1, y)dy − η1u2

m − p1umvm,

∂vi

∂t= dvi

∂2vi

∂x2+ α2vm − γ2vi − α2e−γ2τ2

∫R

14πdviτ2

e−(x−y)24dvi τ2 × vm(t − τ2, y)dy,

∂vm

∂t= dvm

∂2vm

∂x2+ α2e−γ2τ2

∫R

14πdviτ2

e−(x−y)24dvi τ2 × vm(t − τ2, y)dy − η2v

2m − p2umvm.

(1.1)

Here um(t, x) and vm(t, x) (ui(t, x) and vi(t, x)) represent densities of adult (immature) members of two species u and vat time t and point x, respectively. dum > 0 and dui > 0 (dvm > 0 and dvi > 0) are the diffusion coefficients of the adultpopulation um (vm) and the immature population ui (vi). The delay τ1 (τ2) describes the time taken from birth to maturity ofthe population u (v). α1 (α2) is the birth rate of population u (v), and γ1 (γ2) is the death rate of the immature of population

∗ Correspondence to: Institute of AppliedMathematics, College of Science, NW A&F University, No. 22 Xinong Road, Yangling, Shaanxi 712100, PR China.Tel.: +86 29 87092226; fax: +86 29 87092226.

E-mail address: lzhang1979@yahoo.com.cn (L. Zhang).

1468-1218/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2011.11.007

1430 L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440

u (v). The death of the mature for each population is modelled by quadratic terms, as in the logistic equation. The terms

α1e−γ1τ1

∫R

14πduiτ1

e−(x−y)24dui τ1 × um(t − τ1, y)dy

and

α2e−γ2τ2

∫R

14πdviτ2

e−(x−y)24dvi τ2 × vm(t − τ2, y)dy

total up the individuals of the populations u and v born at time t−τ1 and t−τ2 in all parts of the domain that are still alive attime t and have just reachedmaturity and arrived at x, respectively. In themodel (1.1) it is assumed that competition effectsare of the classical Lotka–Volterra kind, and the effects of vm on um, and um on vm are measured by p1 and p2, respectively.

The growth and dispersal of adult numbers of the populations are described by the equations

∂um

∂t= dum

∂2um

∂x2+ α1e−γ1τ1

∫R

14πduiτ1

e−(x−y)24dui τ1 × um(t − τ1, y)dy − η1u2

m (1.2)

and

∂vm

∂t= dvm

∂2vm

∂x2+ α2e−γ2τ2

∫R

14πdviτ2

e−(x−y)24dvi τ2 × um(t − τ1, y)dy − η2u2

m. (1.3)

Eq. (1.2) or (1.3) is a generalization made in Gourley and Kuang [1] of the second equation of the system

dui

dt= αum(t)− γ ui(t)− αe−γ τum(t − τ),

dum

dt= αe−γ τum(t − τ)− βu2

m,

(1.4)

which was proposed by Aiello and Freedman [2] as a model of a single species, where ui denotes the number of sexuallyimmature members of the species and um the number of mature adult members.

Note that, in system (1.1), the second and the fourth equations are uncoupled from the first and the third, and thus it issufficient to consider these second and fourth equation on its own

∂u∂t

= d1∂2u∂x2

+ α1

∫RG1(y)u(t − τ1, x − y)dy − η1u2

− p1uv,

∂v

∂t= d2

∂2v

∂x2+ α2

∫RG2(y)v(t − τ2, x − y)dy − η2v

2− p2uv.

(1.5)

Here u(t, x) and v(t, x) represent densities of adult members of two species u and v at time t and point x, respectively.d1 > 0 (d2 > 0) is the diffusion coefficient of the adult population u (v). α1 (α2) is made up of two factors, the per capitabirth rate and the survival rate of immature for the population u (v) during the immature stage. The two probability kernelsG1 and G2 are given by

G1(y) =e−y2/4di(u)τ14πdi(u)τ1

, G2(y) =e−y2/4di(v)τ24πdi(v)τ2

.

For a more detailed description of the model (1.5), the reader is referred to [1–3]. For studies on delayed models related to(1.5), see [4–6].

Reaction–diffusion models can explain three types of spatial phenomena that are relevant in ecology: wave of invasionby exotic species; the formation of patters in homogeneous space; and the effects of the size, shape, and heterogeneity ofthe spatial environment on the persistence of species and the structure of communities. The idea that reaction–diffusionmodels can support travelling waves was introduced by Fisher (1937) to describe the spatial spread of advantageous gene.In the context of spatial ecology, reaction–diffusion equations are used to describe the growth, spread, and distributionsof species in space, and there are many noteworthy findings that have come out of this field (e.g., Canterll and Cosner [7],Murray [8,9], Okubo and Levin [10], Tilman and Kareiva (ed.) [11], Shigesada and Kawasaki [12]).

Since time delays occur frequently in nature,models in ecology can be often formulated as systems of delayed differentialequations. One important problem is to analyse the effect of time delays on the stability of a system. Time delays can affectstability of a system in differentways [13–17]. For some delay systems, stability switchmay occur as the time delay increase;and for other systems the stability of the system will not change as the time delay varies. Most ecological models involveinteractions of multiple species. There are a number of papers that have discussed the spreading speeds and travelling wavesolutions of multi-species models in terms of reaction–diffusion equations and integro–difference equations [6,18–23]. Forthe two-species diffusion Lotka–Volterra competition model, Tang and Fife [18] showed that there exist travelling wave

L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440 1431

solutions connecting the origin and the coexistence equilibriumwith speeds greater than or equal to the spreading speed ofeach competitor species in the absence of its rival. For the integro–difference model, Weinberger et al. [23], Lewis et al. [19],and Li et al. [22] showed that there is a spreading speed at which a species invades an environment pre-occupied by itscompetitor, and the spreading speed can be characterized as the slowest speed of a class of travelling wave solutions. Li andZhang [6] established the existence of travelling wave solutions for delayed cooperative recursions that are allowed to havemore than two equilibria. For other related references see e.g. [24–29].

The purpose of this paper is to study the stability ofmodel (1.5) aswell as the existence of travellingwaves connecting theorigin with the coexistence equilibrium. We use the Fluctuation Lemma, and convergence of solutions to obtain the criteriaof the global stability of the themono-culture equilibria and the coexistence equilibrium.We use lower and upper travellingwave solutions to show that there are travelling wave solutions with appropriate speeds. The approach different from thatin [18] has been proven successful in establishing the existence of travelling wave solutions for spatial models [30–33].

This paper is organized as follows. The stability results are given in Section 2. The existence of travelling wave solutionsis established in Section 3. Some concluding remarks are given in Section 4.

2. Equilibria and stability analysis

Let τ = max{τ1, τ2}. Consider

dudt

= α1

∫RGu(y)u(t − τ1, x − y)dy − η1u2

− p1uv,

dvdt

= α2

∫RGv(y)v(t − τ2, x − y)dy − η2v

2− p2uv

(2.1)

with initial conditions

u(t) ≥ 0 and u(t) = 0, v(t) ≥ 0 and v(t) = 0, v(t) ≥ 0 for − τ ≤ t ≤ 0. (2.2)

It is easy to see that the solutions of system (2.1), subject to (2.2), satisfy u(t), v(t) > 0 on (0,∞). Positivity implies thatthe system is persistent.

It is easy to show that system (1.5) has the trivial equilibrium E0 = (0, 0), the mono-culture equilibria Eu = (u∗, 0) andEv = (0, v∗)with

u∗=α1

η1, v∗

=α2

η2,

and the coexistence equilibrium E+ = (e1, e2)with

e1 =α2p1 − α1η2

p1p2 − η1η2, e2 =

α1p2 − α2η1

p1p2 − η1η2.

It is easily seen that E+ exists if and only if α2p1 < α1η2 and α1p2 < α2η1 or α2p1 > α1η2 and α1p2 > α2η1.By a analogous argument in [3] we can obtain the local stability results of E0, Eu, Ev and E+, which are summarized in

Table 1.The following theorem gives the criteria for global stability of the equilibria.

Theorem 2.1. Let Eu, Ev and E+ be defined as above, then the following statements are valid:

i. If α1p2 > α2η1, and α2p1 < α1η2, then the mono-culture equilibrium Eu is globally asymptotically stable;ii. If α1p2 < α2η1£ and α2p1 > α1η2£ then the mono-culture equilibrium Ev is globally asymptotically stable;iii. If α1p2 < α2η1£ and α2p1 < α1η2£ then the unique coexistence E+ is globally asymptotically stable.

Remark 2.1. Al-Omari and Gourley [3] obtained global stability results for a model that is slightly different from (1.5). Herewe give some results on global stability about (1.5).

The following two lemmas are elementary but useful in our discussion, which can be found in Gopalsamy [14] and Hirsh,Hanisch and Gabriel [34].

Lemma 2.1 (Barbălat Lemma). Let a be a finite number and f : [a,∞) → R be a differentiable function. If limt→∞ f (t) exists(finite) and f ′ is uniformly continuous on [a,∞), then limt→∞ f ′(t) = 0.

Lemma 2.2 (Fluctuation Lemma). Let a be a finite number, and f : [a,∞) → R be a differentiable function. If lim inft→∞ f (t) <lim supt→∞ f (t), then there exist sequences {tm} ↑ ∞ and {sm} ↑ ∞ such that limm→∞ f (tm) = lim supt→∞ f (t), f ′(tm) = 0,and limm→∞ f (sm) = lim inft→∞ f (t), f ′(sm) = 0.

1432 L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440

Table 1Summary of local stability of system (1.5).

Steady state Criteria for existence Criteria for stability

E0 Always exists UnstableEu Always exists α1p2 > α2η1Ev Always exists α2p1 > α1η2E+ α2p1 < α1η2 , α1p2 < α2η1 or α2p1 > α1η2 , α1p2 > α2η1 α1p2 < α2η1 and α2p1 < α1η2

Proof of Theorem 2.1. To show the global stability i, ii and iii hold, we shall first show that limt→∞ u(t) and limt→∞ v(t)exist.

Suppose that limt→∞ u(t) and limt→∞ v(t) do not exist. Then there exist two subsequences {tm} ↑ ∞ and {sm} ↑ ∞

such that limm→∞ u(tm) = lim supt→∞ u(t) := u, u′(tm) = 0, and limm→∞ u(sm) = lim inft→∞ u(t) := u, u′(sm) = 0. Andthere also exist two subsequences tn ↑ ∞ and sn ↑ ∞ such that limn→∞ v(tn) = lim supt→∞ v(t) := v, u′(tn) = 0, andlimn→∞ v(sn) = lim inft→∞ v(t) := v, u′(sn) = 0.

Consider the following system:

dudt

= α1u(t − τ1)− η1u2(t)− p1u(t)v(t),

dvdt

= α2v(t − τ2)− η2v2(t)− p2u(t)v(t).

(2.3)

From the first equation of (2.3) we obtain α1u(tm − τ1)− η1u2(tm)− p1u(tm)v(tm) = 0.Which implies

α1u − η1u2≤ p1uv. (2.4)

Again from the first equation of (2.3) we obtain

α1u(sm − τ1)− η1u2(sm)− p1u(sm)v(sm) = 0. (2.5)

We shall distinguish two cases to discuss (2.5).case i. u = 0. It is easily obtained from (2.5) that α1 − η1u ≤ p1v.Case ii. u = 0. Then there exists a subsequence {s∗m} ↑ ∞ of {sm} ↑ ∞ satisfying

0 < u(s∗m+1) ≤ u(s∗m) and u(s∗m) ≤ u(t) for all t ∈ [−τ1, s∗m].

The above inequality combined with (2.5) yields

0 = α1u(s∗m − τ1)− η1u2(s∗m)− p1u(s∗m)v(s∗

m)

≥ α1u(s∗m)− η1u2(s∗m)− c1u(s∗m)v(s∗

m).

Which implies that α1 − η1u ≤ p1v. We thus can claim that either u = 0 or u = 0 always holds

α1 − η1u ≤ p1v. (2.6)

If u = 0, then u = u. If u = 0, combining (2.4) with (2.6) we obtain

u ≥α1

η1

1 −

uu

+ u,

this implies that u ≥ u. Note that u ≥ u, we get immediately u = u, which is a contradiction. We thus have shownlimt→∞ u(t) exists. In a similar way, we also can show limt→∞ v(t) exists.

i. Since α1p2 > α2η1, Eu is local stable. By the above discussion, to show Eu is globally stable, it suffices to show u > 0.Assume that u = 0, then from (2.6) we obtain

v ≥α1

p1. (2.7)

From the second equation of (2.3) we get v′(t) ≤ α2v(t − τ2) − η2v2(t).Which together with the comparison principle

yields

v ≤α2

η2. (2.8)

Combining (2.7) with (2.8) we get α1η2 ≤ α2p1. This contradicts our assumption α1η2 > α2p1. Thus u > 0. Which impliesthat Eu is globally stable if α1p2 > α2η1 and α1η2 > α2p1.

The proof for ii is similar to i, we thus omit it.

L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440 1433

iii. If α1p2 < α2η1 and α2p1 < α2η1, then Eu and Ev are all unstable. To show the unique coexistence equilibrium E+ isglobally stable, it is sufficient to show u > 0 and v > 0. The left proofs are similar to i and ii, so we omit them. The proof forTheorem 2.1 is complete. �

3. Existence of travelling waves

A travelling wave solution of (1.5) is a special translation invariant solution of the form u(t, x) = φ(x + ct), v(x, t) =

ψ(x + ct), where (φ, ψ) ∈ C2(R,R2) are the profiles of the wave that propagates through the one-dimensional spatialdomain at a constant velocity c > 0.

Substituting u(x, t) = φ(x + ct), v(x, t) = ψ(x + ct) into (1.5) and denoting x + ct by ξ , we obtain the wave equationof (1.5)

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ) = 0,

d2ψ ′′(ξ)− cψ ′(ξ)+ α2

∫RG2(y)ψ(ξ − y − cτ1)dy − η2ψ

2(ξ)− p2φ(ξ)ψ(ξ) = 0,(3.1)

with asymptotic boundary conditions

limξ→−∞

(φ(ξ), ψ(ξ)) = E0 and limξ→∞

(φ(ξ), ψ(ξ)) = E+. (3.2)

To establish the existence of travelling solutions for (1.5), we first consider the linearisation of the wave Equation (3.1),which is given by

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy = 0,

d2ψ ′′(ξ)− cψ ′(ξ)+ α2

∫RG2(y)ψ(ξ − y − cτ2)dy = 0.

(3.3)

We substitute φ(ξ) = eλξ and ψ(ξ) = eλξ into (3.3) to obtain the characteristic equations of (3.3) as follows

∆i(λ, c) := diλ2 − cλ+ αie−cλτi

∫RGi(y)e−λydy, i = 1, 2. (3.4)

Then it is easy to verify the following properties:

i. ∆i(0, c) = αi

R Gi(y)e−λydy > 0;ii. limλ→∞∆i(λ, c) = ∞ for all c ≥ 0;iii. ∂2∆i(λ,c)

∂λ2= 2di > 0 and ∂∆i(λ,c)

∂c = −λ− λαiτi

R Gi(y)e−λydy < 0 for all λ > 0;iv. limc→∞∆i(λ, c) = −∞ for all λ > 0 and∆i(λ, 0) > 0.

Define

c∗

1 := infλ>0

d1λ2 + α1e−cλτ1

R G1(y)e−λydyλ

and

cast2 := infλ>0

d2λ2 + α2e−cλτ2

R G2(y)e−λydyλ

.

Clearly, c∗

i > 0, i = 1, 2 are well defined. By Liang and Zhao [35], c∗

i may be viewed as the spreading speeds of species u ifi = 1 and species v if i = 2 in the absence of its rival.

These properties lead to the following lemma which similar to Lemma 3.1 in [36] and Lemma 2.5 in [37].

Lemma 3.1. Let c∗

i > 0, i = 1, 2 be defined as above, then the following statements are valid.

i. If c ≥ c∗

i , then there exist four positive numbers Λi1, Λi2, i = 1, 2 (which are independent on c) with Λi1 ≤ Λi2 such that∆i(Λi1, c) = ∆i(Λi2, c) = 0.

ii. If c < c∗

i , then∆i(λ, c) > 0 for all λ > 0.iii. If c = c∗

i , then Λi1 = Λi2; and if c > c∗

i , then Λi1 < Λi2, ∆i(λ, c) < 0 for all λ ∈ (Λi1,Λi2), ∆i(λ, c) > 0 for allλ ∈ [0,∞) \ [Λi1,Λi2].

Define

c∗= max{c∗

1 , c∗

2 }.

We now are in the position to state and prove our main results on the existence of travelling wave solutions in (1.5).

1434 L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440

Theorem 3.1. Suppose that α1p2 < α2η1, and α2p1 < α1η2. Then for c > c∗, the system (1.5) has a travelling wave solution(u(t, x), v(t, x)) = (φ(x + ct), ψ(x + ct)) with limξ→∞(φ(ξ), ψ(ξ)) = (0, 0), limξ→−∞(φ(ξ), ψ(ξ)) = (e1, e2), andlimξ→∞ φ(ξ)eΛ11(ξ) = limξ→∞ ψ(ξ)eΛ21(ξ) = 1.

For a fixed c > c∗, we choose

η ∈

1,min

2,Λ12

Λ11,Λ22

Λ21

so thatΛi1 < ηΛi1 < Λi2, which implies that∆i(ηΛi1, c) < 0.

Consider gi(ξ) = eΛi1ξ − qieηΛi1ξ , for i = 1, 2, where each qi > 1 is a large number. An exercise in calculus shows thatgi(ξ) has the global maximum

mi =

1 −

1η

η−1

1qiη

> 0

at ξmi := − ln(qiη)/(η − 1)Λi1 due to the fact that g ′

i (ξ) > 0 on (−∞, ξmi ) and g ′

i (ξ) < 0 on (ξmi ,∞). For N > 1, define

ξi := maxξ : gi(ξ) =

mi

N

i = 1, 2.

Since gi(0) < 0, ξi < 0. For sufficiently small λ > 0, we thus can choose a suitable N > 1 and a large qi > 1 such thatϵi ∈ (0, ei) for i = 1, 2 and

ei − ϵie−λξi =mi

N, ξmi ≤

− ln eiϵi

λ< ξi,

where − ln eiϵi/λ is the solution of each equation ei − ϵie−λξ

= 0.Hypotheses α1p2 < α2η1 and α2p1 < α1η2 imply that there exist ϵ0 > 0, ϵ3 > 0 and ϵ4 > 0 such that

η1ϵ1 − p1ϵ4 > ϵ0, η2ϵ4 − p2ϵ1 > ϵ0,

η2ϵ2 − p2ϵ3 > ϵ0, η1ϵ3 − p1ϵ2 > ϵ0.(3.5)

We now construct two pairs of functions (φ(z), ψ(z)) and (φ(z), ψ(z)) as follows:

φ(ξ) =

eΛ11ξ , ξ ≤ ξ3,

e1 + ϵ3e−λξ , ξ ≥ ξ3,ψ(ξ) =

eΛ21ξ , ξ ≤ ξ4,

e2 + ϵ4e−λξ , ξ ≥ ξ4,(3.6)

and

φ(ξ) =

eΛ11ξ − q1eηΛ11ξ , ξ ≤ ξ1,

e1 − ϵ1e−λξ , ξ ≥ ξ1,ψ(ξ) =

eΛ21ξ − q2eηΛ21ξ , ξ ≤ ξ2,

e2 − ϵ2e−λξ , ξ ≥ ξ2,(3.7)

where each qi > 1 is sufficiently large and λ > 0 is sufficiently small.We summarise some useful properties of (φ(z), ψ(z)) and (φ(z), ψ(z)) in the following proposition. The proofs of these

properties are straightforward, and shall be omitted.

Proposition 3.1. Let (φ(z), ψ(z)) and (φ(z), ψ(z)) be constructed in (3.6) and (3.7). Then the following statements are valid.

i. min{ξ3, ξ4} ≥ max{ξ1, ξ2}, and there exist constants B1 > 0 and B2 > 0 such that 0 ≤ φ(ξ) < φ(ξ) ≤ B1 and0 ≤ ψ(ξ) < ψ(ξ) ≤ B2.

ii. φ(ξ) ≤ eΛ11ξ and φ(ξ) ≤ e1 + ϵ3e−λξ for ξ ∈ R, and ψ(z) ≤ eΛ21ξ and ψ(ξ) ≤ e2 + ϵ4e−λξ for ξ ∈ R.iii. φ(ξ) ≥ eΛ11ξ − q1eηΛ11ξ and φ(ξ) ≥ e1 − ϵ1e−λξ for ξ ∈ R, and ψ(ξ) ≥ eΛ21ξ − q2eηΛ21ξ and ψ(ξ) ≥ e2 − ϵ2e−λξ forξ ∈ R.

Definition 3.1. A pair of twice continuously differentiable functions (φ, ψ) and (φ, ψ) ∈ C(R,R2) are called an upper anda lower solution of (3.1), respectively, if (φ, ψ) and (φ, ψ) ∈ C(R,R2) satisfy

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ) ≤ 0 on R,

d2ψ ′′(ξ)− cψ ′(ξ)+ α2

∫RG2(y)ψ(ξ − y − cτ1)dy − η2ψ

2(ξ)− p2φ(ξ)ψ(ξ) ≤ 0 on R,(3.8)

and

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ) ≥ 0 on R,

d2ψ ′′(ξ)− cψ ′(ξ)+ α2

∫RG2(y)ψ(ξ − y − cτ1)dy − η2ψ

2(ξ)− p2φ(ξ)ψ(ξ) ≥ 0 on R.(3.9)

L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440 1435

Lemma 3.2. Suppose that α1p2 < α2η1 andα2p1 < α1η2. Then (φ(z), ψ(z)) defined in (3.6) is a pair of upper solutions of (3.1).

Proof. If ξ ≤ ξ3, then φ(ξ) = eΛ11ξ . Proposition 3.1 ii shows that φ(ξ) ≤ eΛ11ξ for ξ ∈ R. It follows that

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ)

≤ d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy

=

d1Λ2

11 − cΛ11 + α1e−cΛ11τ1

∫RG1(y)e−Λ11ydy

eΛ11ξ

= ∆1(Λ11, c)= 0.

If ξ ≥ ξ3, then φ(ξ) = e1+ϵ3e−λξ . Proposition 3.1 ii and iii show that φ(ξ) ≤ e1+ϵ3e−λξ for ξ ∈ R andψ(ξ) ≥ e2−ϵ2e−λξ

for ξ ∈ R. It follows from this that

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ)

≤ d1e1 + ϵ3e−λξ

′′− c

e1 + ϵ3e−λξ

′+ α1

∫RG1(y)

e1 + ϵ3e−λ(ξ−y−cτ1)

dy

− η1e1 + ϵ3e−λξ

2− p1

e1 + ϵ3e−λξ

e2 − ϵ2e−λξ

= d1ϵ3λ2e−λξ

+ cϵ3λe−λξ+ α1

∫RG1(y)(e1 + ϵ3e−λ(ξ−y−cτ1))dy

− η1e1 + ϵ3e−λξ

2− p1

e1 + ϵ3e−λξ

e2 − ϵ2e−λξ

=: H1(λ).

Noting that α1 − p1e2 − η1e1 = 0 we have by (3.5) that H1(0) = (e1 + ϵ3)(p1ϵ2 − η1ϵ3) < 0. The continuity of H1(λ)shows that there exists λ∗

1 > 0 such that H1(λ) < 0 for λ ∈ (0, λ∗

1).We now conclude that for λ ∈ (0, λ∗

1)

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ) ≤ 0 on R.

Similarly, we can also show that there exists λ∗

2 > 0 such that for λ ∈ (0, λ∗

2)

d2ψ ′′(ξ)− cψ ′(ξ)+ α2

∫RG2(y)ψ(ξ − y − cτ1)dy − η1ψ

2(ξ)− p2φ(ξ)ψ(ξ) ≤ 0 on R.

It follows that the statements of Lemma 3.2 hold for λ ∈ (0,min{λ∗

1, λ∗

2}). The proof of Lemma 2.2 is complete. �

Lemma 3.3. Suppose that α1p2 < α2η1 and α2p1 < α1η2 are satisfied. Then (φ(z), ψ(z)) defined in (3.7) is a pair of lowersolutions of (3.1).

Proof. If ξ ≤ ξ1, since ξ ≤ ξ1 implies that ξ ≤ ξ4, φ(ξ) = eΛ11ξ − q1eηΛ11ξ and ψ(ξ) = eΛ21ξ . By Proposition 3.1 iii wehave that φ(ξ) ≥ eΛ11ξ − q1eηΛ11ξ for ξ ∈ R. We thus obtain that

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ)

≥ d1Λ2

11eΛ11ξ − q1η2Λ2

11eηΛ11ξ

− c

Λ11eΛ11ξ − q1ηΛ11eηΛ11ξ

+α1

∫RG1(y)

eΛ11(ξ−y−cτ1) − q1eηΛ11(ξ−y−cτ1)

dy − η1

eΛ11ξ − q1eηΛ11ξ

2− p1

eΛ11ξ − q1eηΛ11ξ

eΛ21ξ

= ∆1(Λ11, c)eΛ11ξ − d1q1η2Λ211e

ηΛ11ξ + cq1ηΛ11eηΛ11ξ − α1q1

∫RG1(y)eηΛ11(ξ−y−cτ1)dy

− η1eΛ11ξ − q1eηΛ11ξ

2− p1

eΛ11ξ − q1eηΛ11ξ

eΛ21ξ

= eηΛ11ξ−q1∆1(ηΛ11, c)− η1e(2−η)Λ11ξ − p1e(Λ11+Λ21−2Λ11)ξ

≥ eηΛ11ξ

−q1∆1(ηΛ11, c)− η1e(2−η)Λ11ξ1 − p1e(Λ11+Λ21−2Λ11)ξ1

≥ eηΛ11ξ (−q1∆1(ηΛ11, c)− η1 − p1)≥ 0

for all q1 ≥ −(η1 + p1)/∆1(ηΛ11, c). Here we have used the simple fact that (a − b)2 ≤ a2 for a > b > 0.

1436 L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440

If ξ ≥ ξ1, then φ(ξ) = e1 − ϵ1e−λξ . By Proposition 3.1 ii and iii we have that ψ(ξ) ≤ e2 + ϵ4e−λξ for ξ ∈ R andφ(ξ) ≥ e1 − ϵ1e−λξ for ξ ∈ R. We thus have that

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ)

≥ d1e1 − ϵ1e−λξ

′′− c

e1 − ϵ1e−λξ

′+ α1

∫RG1(y)

e1 − ϵ1e−λ(ξ−y−cτ1)

dy

− η1e1 − ϵ1e−λξ

2− p1

e1 − ϵ1e−λξ

e2 + ϵ4e−λξ

= −d1ϵ1λ2e−λξ

− cϵ1λe−λξ+ α1

∫RG1(y)

e1 − ϵ1e−λ(ξ−y−cτ1)

dy

− η1e1 − ϵ1e−λξ

2− p1

e1 − ϵ1e−λξ

e2 + ϵ4e−λξ

=: H3(λ).

Since 0 < ϵ1 < e1 and (3.5), H3(0) = r1(e1 − ϵ1)(η1ϵ1 − p1ϵ4) > 0. It follows that there exists a constant λ∗

3 > 0 such thatfor λ ∈ (0, λ∗

3) and ξ ∈ R

d1φ′′(ξ)− cφ′(ξ)+ α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ) ≥ 0.

Similarly, we also can show that there exists a constant λ∗

4 > 0 such that for λ ∈ (0, λ∗

4) and ξ ∈ R

d2ψ ′′(ξ)− cψ ′(ξ)+ α2

∫RG2(y)ψ(ξ − y − cτ1)dy − η2ψ

2(ξ)− p2φ(ξ)ψ(ξ) ≥ 0.

It follows that the statements of Lemma 3.3 hold for λ ∈ (0,min{λ∗

3, λ∗

4}). The proof for Lemma 2.2 is complete. �

We shall use the usual Banach spaceB := C(R,R2) of bounded continuous functions endowedwith themaximumnorm‖(φ, ψ)‖ = supξ∈R(|φ(ξ)| + |ψ(ξ)|). For any c > c∗, let

Sc =

(φ, ψ) : (φ, ψ) ∈ B, φ(ξ) ≤ φ(ξ) ≤ φ(ξ), ψ(ξ) ≤ ψ(ξ) ≤ ψ(ξ)

.

Clearly, Sc is a bounded nonempty closed convex subset of B.Define the operator F = (F1, F2) : Sc → B by

F1(φ, ψ)(ξ) := α1

∫RG1(y)φ(ξ − y − cτ1)dy − η1φ

2(ξ)− p1φ(ξ)ψ(ξ)+ β1φ(ξ),

F2(φ, ψ)(ξ) := α2

∫RG2(y)ψ(ξ − y − cτ1)dy − η2ψ

2(ξ)− p2φ(ξ)ψ(ξ)+ β2ψ(ξ),

(3.10)

where each βi is a large positive number so that β1 > η1(B1 + B2)+ p1B2 and β2 > η2(B1 + B2)+ p2B1. Then system (3.1)now can be rewritten as

d1φ′′(ξ)− cφ′(ξ)− β1φ(ξ)+ F1(φ, ψ)(ξ) = 0,

d2ψ ′′(ξ)− cψ ′(ξ)− β2ψ(ξ)+ F2(φ, ψ)(ξ) = 0.(3.11)

Let

λ11 =c −

c2 + 4β1d12d1

, λ12 =c +

c2 + 4β1d12d1

,

λ21 =c −

c2 + 4β2d22d2

, λ22 =c +

c2 + 4β2d22d2

.

Clearly, λ11 < 0 < λ12, λ21 < 0 < λ22, d1λ21j − cλ1j − β1 = 0, j = 1, 2 and d2λ22j − cλ2j − β2 = 0, j = 1, 2.Define the operator Q = (Q1,Q2) : Sc → B by

Q1(φ, ψ)(ξ) =1

d1(λ12 − λ11)

∫ ξ

−∞

eλ11(ξ−s)F1(φ, ψ)(s)ds +

∫∞

ξ

eλ12(ξ−s)F1(φ, ψ)(s)ds,

Q2(φ, ψ)(ξ) =1

d2(λ22 − λ21)

∫ ξ

−∞

eλ21(ξ−s)F2(φ, ψ)(s)ds +

∫∞

ξ

eλ22(ξ−s)F2(φ, ψ)(s)ds.

(3.12)

L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440 1437

It easily verified that the operator Q is well defined for (φ, ψ) ∈ Sc and

d1Q1(φ, ψ)′′(ξ)− cQ1(φ, ψ)

′(ξ)− β1Q1(φ, ψ)(ξ)+ F1(φ, ψ)(ξ) = 0,d2Q2(φ, ψ)

′′(ξ)− cQ2(φ, ψ)′(ξ)− β2Q2(φ, ψ)(ξ)+ F2(φ, ψ)(ξ) = 0.

Thus the fixed of Q is the solution of (3.11), which is the travelling solution of (1.5).

Lemma 3.4. For (φi, ψi) ∈ Sc , Q1(φ, ψ) is nondecreasing in φ and nonincreasing in ψ , and Q2(φ, ψ) is nondecreasing in ψand nonincreasing in φ.

Proof. It suffices to show that for (φ, ψ) ∈ Sc , F1(φ, ψ) is nondecreasing in φ and nonincreasing in ψ , and F2(φ, ψ) isnondecreasing in ψ and nonincreasing in φ.

For (φ2, ψ2) ≥ (φ1, ψ1)where (φi, ψi) ∈ Sc , we have that

F1(φ2, ψ)(ξ)− F1(φ1, ψ)(ξ)

= α1

∫RG1(y)φ2(ξ − y − cτ1)dy − η1φ

22(ξ)− p1φ2(ξ)ψ(ξ)+ β1φ2(ξ)

−

α1

∫RG1(y)φ1(ξ − y − cτ1)dy − η1φ

21(ξ)− p1φ1(ξ)ψ(ξ)+ β1φ1(ξ)

= α1

∫RG1(y)(φ2 − φ1)(ξ − y − cτ1)dy + (φ2 − φ1)(ξ)[β1 − η1(φ1 + φ2)(ξ)− p1ψ(ξ)]

≥ 0

since β1 ≥ η1(B1+B2)+p1B2. Moreover, it is obvious that F1(φ, ψ) is nonincreasing inψ . The proof of the second statementof Lemma 3.4 is similar, and is omitted. The proof is complete. �

Lemma 3.5. (φ(ξ), ψ(ξ)) is an upper solution and (φ(ξ), ψ(ξ)) is a lower solution of the operator Q defined by (3.12) in thesense that

Q1(φ, ψ)(ξ) ≤ φ(ξ), Q2(φ, ψ)(ξ) ≤ ψ(ξ);

Q1(φ, ψ)(ξ) ≥ φ(ξ), Q2(φ, ψ)(ξ) ≥ ψ(ξ).

Proof. From Lemma 3.2 and the first equation of (3.11) we have that

F1(φ, ψ)(ξ) ≤ −d1φ′′(ξ)+ cφ′(ξ)+ β1φ(ξ),

which together with the first equation of (3.12) gives

Q1(φ, ψ)(ξ) =

∫ ξ

−∞

eλ11(ξ−s)+

∫ ξ

−∞

eλ12(ξ−s)F1

φ, ψ

(s)ds

≤

∫ ξ

−∞

eλ11(ξ−s)+

∫ ξ

−∞

eλ12(ξ−s)

−d1φ′′(s)+ cφ′(s)+ β1φ(s)(s)ds

= φ(ξ), ξ ∈ R.

An analogous argument shows Q2(φ, ψ)(ξ) ≤ ψ(ξ). Finally using Lemma 3.3, (3.11) and (3.12) one can also show thatQ1(φ, ψ)(ξ) ≥ φ(ξ) and Q2(φ, ψ)(ξ) ≥ ψ(ξ). The proof is complete. �

Lemma 3.6. The operator Q = (Q1,Q2) : Sc → Sc is continuous with respect to the norm ‖ · ‖ in Sc .

Proof. We first show that Q1 : Sc → Sc is continuous with respect to the norm ‖ · ‖. Let (φ1, ψ1), (φ2, ψ2) ∈ Sc with‖(φ1, ψ1)− (φ2, ψ2)‖ = supξ∈R(|φ1 − φ2| + |ψ1 − ψ2|) < δ. It is easily verified that

|F1(φ1, ψ1)(ξ)− F1(φ2, ψ2)(ξ)|

≤ α1

∫RG1(y)|φ1(ξ − y − cτ1)− φ2(ξ − y − cτ1)|dy + η1(φ1(ξ)+ φ2(ξ))|φ1(ξ)− φ2(ξ)|

+β1|φ1(ξ)− φ2(ξ)| + p1φ2(ξ)|ψ2(ξ)− ψ1(ξ)| + p1ψ1(ξ)|φ2(ξ)− φ1(ξ)|

≤ (α1 + 2η1B1 + β − 1 + p1B1 + p1B2)‖(φ1, ψ1)− (φ2, ψ2)‖.

1438 L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440

Define C1 := (α1 + 2η1B1 + β1 + p1B1 + p1B2). Noting that λ11 < 0 < λ12, we thus get by (3.12) that if ξ ≥ 0

|Q1(φ1, ψ1)(t)− Q1(φ2, ψ − 2)(t)|

≤1

d1(λ12 − λ11)

∫ ξ

−∞

eλ11(ξ−s)|F1(φ1, ψ1)(s)− F1(φ2, ψ2)(s)|ds

+

∫∞

ξ

eλ12(ξ−s)|F1(φ1, ψ1)(s)− F1(φ2, ψ2)(s)|ds

≤

C1

d1(λ12 − λ11)

∫ ξ

−∞

eλ11(ξ−s)ds +

∫∞

ξ

eλ12(ξ−s)ds

‖(φ1, ψ1)− (φ2, ψ2)‖

=C1

d1(λ12 − λ11)

∫ 0

−∞

eλ11(ξ−s)ds +

∫ ξ

0eλ11(ξ−s)ds +

∫∞

ξ

eλ12(ξ−s)ds

‖(φ1, ψ1)− (φ2, ψ2)‖

=C1

d1(λ12 − λ11)eλ11ξ

1λ12

e−λ12ξ −1λ11

e−λ11ξ

‖(φ1, ψ1)− (φ2, ψ2)‖

≤ −C1

λ11λ12‖(φ1, ψ1)− (φ2, ψ2)‖.

For any ε > 0, we set δ = −λ11λ12

C1ε, if ‖(φ1, ψ1) − (φ2, ψ2)‖ < δ£ then |Q1(φ1, ψ1)(t) − Q1(φ2, ψ − 2)(t)| < ε. Which

indicates that operator Q1 is continuous with respect to the maximum norm ‖ · ‖.Similarly, we also can show that the operator Q2 : Sc → Sc is continuous with respect to the norm ‖ · ‖. The proof for

Lemma 3.6 is complete. �

Proof of Theorem 3.1. By Lemmas 3.4 and 3.5 we obtain that for (φ, ψ) ∈ Sc

φ(ξ) ≥ Q1[φ, ψ](ξ) ≥ Q1[φ, ψ](ξ) ≥ Q1[φ,ψ](ξ) ≥ Q1[φ, ψ](ξ) ≥ Q1[φ, ψ](ξ) ≥ φ(ξ),

ψ(ξ) ≥ Q2[φ, ψ](ξ) ≥ Q2[φ, ψ](ξ) ≥ Q2[φ,ψ](ξ) ≥ Q2[φ,ψ](ξ) ≥ Q2[φ, ψ](ξ) ≥ ψ(ξ).

This shows that if (φ, ψ) ∈ Sc then Q (φ, ψ)(z) ∈ Sc . It follows that Sc is an invariant bounded nonempty closed convexsubset of B.

Let

Mi = sup(φ,ψ)∈Sc ,ξ∈R

|Fi(φ, ψ)(ξ)|.

For any given ε > 0 and (φ, ψ) ∈ Sc , keeping in mind that λ11 < 0 < λ12, by the definition of Q1 we obtain that

|Q1(φ, ψ)(ξ + ε)− Q1(φ, ψ)(ξ)| =1

d1(λ12 − λ11)

∫ ξ+ε

−∞

eλ11(ξ+ε−s)−

∫ ξ

−∞

eλ11(ξ−s)F1(φ, ψ)(s)ds

+

∫∞

ξ+ε

eλ12(ξ+ε−s)−

∫∞

ξ

eλ12(ξ+−s)F1(φ, ψ)(s)ds

≤

1d1(λ12 − λ11)

eλ11ε − 1 ∫ ξ

−∞

eλ11(ξ−s)|F1(φ, ψ)(s)|ds

+

∫ ξ+ε

ξ

eλ11(ξ+ε−s)|F1(φ, ψ)(s)|ds

+eλ12ε − 1

∫∞

ξ

eλ12(ξ−s)|F1(φ, ψ)(s)|ds +

∫ ξ+ε

ξ

eλ12(ξ−s)|F1(φ, ψ)(s)|ds

≤

M1

d1(λ12 − λ11)

[eλ11ξ

1 − eλ11ε

∫ ξ

−∞

e−λ11sds + eλ11(ξ+ϵ)∫ ξ+ε

ξ

e−λ11sds

+ eλ12ξeλ12ϵ − 1

∫∞

ξ+ε

e−λ12sds + eλ12ξ∫ ξ+ε

ξ

e−λ12sds]

=M1

d1(λ12 − λ11)

2

eλ11ε − 1

λ11

+e−λ12ε

eλ12ε − 1

+

1 − e−λ12ϵ

λ12

.

L. Zhang et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1429–1440 1439

Similarly, one also can show that

|Q2(φ, ψ)(ξ + ε)− Q2(φ, ψ)(ξ)| ≤M2

d2(λ22 − λ21)

2

eλ21ε − 1

λ21

+e−λ22ε

eλ22ε − 1

+

1 − e−λ22ϵ

λ22

.

It follows that {Q (φ, ψ)(ξ) : (φ, ψ) ∈ Sc} represents a family of equicontinuous functions. Then the Arzelà–Ascoli theoremimplies thatQ takes the bounded convex subset ofSc into a compact subset ofSc . An application of the Schauder–Tychonofffixed point (see [38]) shows that Q has a fixed point (φ, ψ) in Sc , which represents a travelling wave solution of (1.5).Since φ(ξ) ≤ φ(ξ) ≤ φ and ψ(ξ) ≤ ψ(ξ) ≤ ψ(ξ), the properties of (φ(ξ), ψ(ξ)) and (φ(ξ), ψ(ξ)) show thatlimξ→−∞(φ(ξ), ψ(ξ)) → (0, 0), limξ→∞(φ(ξ), ψ(ξ)) → (e1, e2), and limξ→−∞ φ(ξ)e−Λ11ξ = limξ→−∞ ψ(ξ)e−Λ21ξ = 1.The proof for Theorem 3.1 is complete. �

4. Concluding remarks

Dispersal is a ubiquitous process in nature. One of the principle goals of ecology is to understand invasion of species inspace. It has long been recognized that species spread is highly dynamic both in time and space. These changes related toseasonal, annual, and scandal cycles. Species expanded their geographical ranges as they crossed barriers, on land or in thesea, or as the climate changed. As species arrived at new places, they had to cope with a new physical environment, as wellas with other species that they encountered. They had to compete with them for resources. When competition occurs, howspeciesmove, distribute, and persist is an important biological andmathematical question. The existence of travellingwavesfor spatial systems provide a good answer to this question. It is shown that the interactions of unlike species can result inspatial heterogeneity and lead to the appearance of patterns out of a uniform state.

In the present paper we studied a delayed reaction–diffusion model that describes competition between adult membersof two species. We first studied the global stability of the system by using the Fluctuation Lemma. We then establishedthe existence of travelling wave solutions in the model. The travelling waves connect the the origin with the coexistenceequilibrium, and they can be used to describe the spread of two competing species that are introduced in spacesimultaneously. It was shown that the two species can spread to expand their spatial ranges at the same speed, whichis above the minimal travelling wave speed for each species when its rival is absent. We showed that for c > c∗ thereexists a travelling wave solution for the model that connects the origin with the coexistence equilibrium depending withc∗ described in terms of model parameters. We use lower and upper travelling wave solutions to construct a proper set ina function space, and show there is a travelling wave solution in the set. This approach is very technical, different from thatin [18], and has been proven successful in establishing the existence of travelling wave solutions for spatial models [30–33].

The existence of travelling waves connecting one unstable mono-culture equilibrium with a stable mono-cultureequilibrium or with the coexistence equilibrium was investigated in Al-Omari and Gourley [3] and Li and Zhang [6]. Thetravelling waves describe the situation that one species spreads into an environment pre-occupied by a resident species. Incontrast, the travelling waves discussed in this paper demonstrate simultaneous invasions of two competing species intoan unoccupied environment. The works in [3,6] and in the present paper provide important insight about spatial dynamicsof delayed reaction–diffusion systems.

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that improved ouroriginalmanuscript. The first author is partially supported by the NSF of China under Grant 11071042, Doctor Funding underGrant 201104054460, YZKC0917 andQN2009047 of Northwest A& FUniversity, and the project of science and technology ofthe Education department of Heilongjiang Province Project under Grant 12513096. The second author is partially supportedby the National Science Foundation under Grant DMS-616445.

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