Minimal wave speed for a class of non-cooperative reaction–diffusion systems of three equations

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.1 (1-47)

Available online at www.sciencedirect.com

ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Minimal wave speed for a class of non-cooperative reaction–diffusion systems of three equations Tianran Zhang School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China Received 11 April 2016; revised 30 November 2016

Abstract In this paper, we study the traveling wave solutions and minimal wave speed for a class of non-cooperative reaction–diffusion systems consisting of three equations. Based on the eigenvalues, a pair of upper–lower solutions connecting only the invasion-free equilibrium are constructed and the Schauder’s fixed-point theorem is applied to show the existence of traveling semi-fronts for an auxiliary system. Then the existence of traveling semi-fronts of original system is obtained by limit arguments. The traveling semi-fronts are proved to connect another equilibrium if natural birth and death rates are not considered and to be persistent if these rates are incorporated. Then non-existence of bounded traveling semi-fronts is obtained by two-sided Laplace transform. Then the above results are applied to some disease-transmission models and a predator–prey model. © 2016 Elsevier Inc. All rights reserved.

Keywords: Three equations; Minimal wave speed; Persistence theory; LaSalle’s invariance principle; Applications

1. Introduction In this paper, we will study the following reaction–diffusion system:

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jde.2016.12.017 0022-0396/© 2016 Elsevier Inc. All rights reserved.

JID:YJDEQ AID:8654 /FLA

2

[m1+; v1.241; Prn:30/12/2016; 12:35] P.2 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

⎧ ∂ ⎪ ⎪ u1 = d1 u1 + f (u1 ) − g1 (u), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ u = d2 u2 + g2 (u) − δ2 u2 , ⎪ ∂t 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ u3 = d3 u3 + g3 (u) − δ3 u3 , ∂t

(1.1)

where x ∈ Rn , ui = ui (x, t), i = 1, 2, 3, u = (u1 , u2 , u3 ),  =

n  ∂2 , ∂xj2 j =1

and all the parameters are positive. The unknowns ui , i = 1, 2, 3 denote the densities of population, virus, etc. We first present some basic biological assumptions on functions f (u1 ) and gi (u), i = 1, 2, 3. Some notations are first needed to give these assumptions. Define R3+ := {(u1 , u2 , u3 ) : u1 > 0, u2 > 0, u3 > 0}, gi,j (u) :=

∂gi (u) ∂ 2 gi (u) , gi,j k (u) := , i, j, k = 1, 2, 3. ∂uj ∂uj ∂uk

Then the first three assumptions are as follows. (A1) f (·) ∈ C 1 ([0, ∞)), gi (·) ∈ C 2 (cl(R3+ )), i = 1, 2, 3, where cl(R3+ ) denotes the closure of R3+ . (A2) (I) Either f (u1 ) ≡ 0 or f (u1 ) satisfies f (K) = 0, f  (K) < 0, f (u1 )(u1 − K) < 0 for u1 ∈ (0, K) ∪ (K, +∞), and f  (0) > 0 if f (0) = 0. (II) gi (u) > 0, gi (u1 , 0, 0) = 0 and g1 (0, u2 , u3 ) = 0 for u ∈ R3+ , i = 1, 2, 3. (A3) (I) gi,j (u) ≥ 0, g2,3 (u1 , 0, 0) > 0, g3,2 (u1 , 0, 0) > 0, i, j = 1, 2, 3 for u ∈ R3+ . (II) gi,j 1 (u) ≥ 0, i, j = 2, 3, g2,21 (u) + g2,31 (u) > 0, u ∈ R3+ . The Hessian matrices H[gi (u1 , ·)](u), i = 1, 2, 3 are negative semi-definite for u ∈ R3+ , where  H[gi (u1 , ·)](u) :=

gi,22 (u) gi,32 (u)

 gi,23 (u) . gi,33 (u)

Remark 1.1. From (A1) it follows that gi,j k (u), i, j, k = 2, 3 are bounded in a small neighborhood U of the equilibrium E0 (K, 0, 0). In this paper we always suppose (A1)–(A3) are satisfied. It follows from (A2) and (A3) that the interaction between u2 and u3 is cooperative and that both of u2 and u3 have negative or non-positive effects on u1 . System (1.1) is consequently non-cooperative or non-monotonic. If f (u1 ) = d(K − u1 ) or f (u1 ) ≡ 0, system (1.1) can serve as an SIR model with two progression stages [23, model (16)], an SEIR model [25] or a virus

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.3 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

3

model [41], and an influenza model with treatment [47]. If f (u1 ) = ru1 (K − u1 ), system (1.1) can model the predator–prey interaction with predators having stage structure [24]. Traveling wave solutions for reaction–diffusion systems are used to study the invasion speed of predators into preys or the spreading speed of infectious diseases. To continue this paper, we first present some basic concepts and corresponding backgrounds on traveling wave solutions. A solution u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)) of (1.1) is called a traveling wave solution if it has the form u(x, t) = (U1 (s), U2 (s), U3 (s)) =: U (s), s = x · ν + ct,

(1.2)

where c > 0 is the wave speed and the unit vector ν denotes the traveling direction. A positive solution (1.2) is called a traveling semi-front if it satisfies U (−∞) = E0 (K, 0, 0),

(1.3)

where E0 is the population state before the invasion or the spread of diseases, and thus is called a invasion-free equilibrium. Here K is the initial state of u1 before invasion. However, the final state U (+∞) of traveling wave solutions is needed to make sure that the population invasion or the spread of diseases is successful. The traveling semi-fronts were therefore proved to connect another equilibrium [7,47] or a periodic solution [8,20] for some diffusive models. The concept of persistence of traveling semi-fronts is introduced by Zhang et al. [48] since it is difficult to prove that the traveling semi-fronts connect another equilibrium or a periodic solution for the models with natural birth and death rates and with complex interaction functions [22,47]. The traveling semi-front (1.2) is called to be persistent if 0 < lim inf Ui (s), lim sup Ui (s) < +∞, i = 1, 2, 3. s→+∞

(1.4)

s→+∞

A positive constant c∗ is called the minimal wave speed if (1.1) has a traveling semi-front with wave speed c if and only if c ≥ c∗ . The minimal wave speed c∗ is called strong if the final state U (+∞) of traveling semi-fronts is given and weak if the traveling semi-fronts with wave speed c ≥ c∗ are persistent [48]. For many cooperative population models, it has been proved that the minimal wave speed coincides with the spreading speed, which is the asymptotic speed at which the population spreads in the direction the traveling waves spread to [5,26,30,43]. However, this has not been proved for non-cooperative models, such as the predator–prey models and SI disease-transmission models, due to the difficulty in theoretical analysis of non-cooperative systems [6,22]. Of course, it is also difficult to show the stability of traveling wave with minimal wave speed for non-cooperative models. In Hilker and Lewis [12], the (weak) minimal wave speed of traveling waves was used to approximate the invasion speed of the species for a predator–prey model in river environments. Actually, Hilker and Lewis did not even show the existence of the traveling semi-fronts. We adopt this idea in this paper and also use the (strong or weak) minimal wave speed to approximate the invasion speed. It thus follows from the definition that the weak minimal wave speed can tell the information same as the strong minimal wave speed does except the final detailed information. Next we review the frequently used methods for the existence of traveling wave solutions for non-cooperative systems. Liang and Zhao [30] and Wu and Zou [44] set up the general theory on the existence of traveling wave solutions for monotonic (or cooperative) systems by monotonic theories. This method can not be applied to (1.1) since system (1.1) is non-monotonic. Earliest

JID:YJDEQ AID:8654 /FLA

4

[m1+; v1.241; Prn:30/12/2016; 12:35] P.4 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

method, i.e. shooting method, for the existence of traveling wave solutions for the Lotka–Volterra predator–prey system (non-monotonic) is proposed by Dunbar [6,7] and is applied by many researchers, e.g. [13–15,17,28,31]. Recently, Huang [21] proposed a geometric method for a class of non-monotonic systems consisting of two equations with general interaction functions and developed it in [22], which abandons the restriction condition on the diffusion coefficients. The Schauder’s fixed-point theorem is also widely used for the existence of traveling wave solutions, e.g. [18,19,27,33,42]. However, for non-monotonic models it is not easy to construct a pair of appropriate upper–lower solutions connecting two equilibria for the application of the Schauder’s fixed-point theorem. The above methods for traveling wave solutions and the minimal wave speed are important in studying the population invasion or the spread of infectious diseases. In real world the interactions among different populations or different subclasses of humans with infectious diseases are complicated. This means that the functional responses or the disease incidence rates should be complex and, therefore, they were improved by subsequent researchers [1]. In 2016, Zhang et al. [48] proposed another method to deal with a class of non-cooperative systems with complex functional responses or disease incidence rates. In [48] the persistence theory for dynamical systems is first introduced into the studies of traveling wave solutions and is applied to show the persistence of traveling semi-fronts, which implies the existence of weak minimal wave speed. This enables us to obtain the population invasion speed or the spreading speed of infectious diseases even if the detailed final state U (+∞) (i.e. equilibria, periodic solutions etc.) is unknown and allow us to avoid the difficulties in studying U (+∞). At the same time, this also allows us to incorporate more complicated nonlinear interaction functions in noncooperative reaction–diffusion models. The non-cooperative models in above literatures mostly consist of two equations. According to what I know, there are few literatures about the existence of minimal wave speed for non-cooperative systems consisting of three equations, except some simple epidemic diseasetransmission models [45,47]. Actually, as what is pointed out in [22], there was no any extension of Dunbar’s work to general predator–prey models consisting of two equations for almost twenty years after the publication of Dunbar’s papers and before the publication of [22,48], except a few papers [9–11,34] in which singular perturbation method, Conley index theory and bifurcation approach were applied. There are, however, some shortcomings for singular perturbation method, Conley index theory and bifurcation approach in modeling the real world though they are powerful in the studies of traveling wave solutions (refer to the arguments in [22] for specific reasons). It is, consequently, important to develop some methods for traveling wave solutions and minimal wave speed for the non-cooperative system (1.1), which is constituted by three equations and has general interaction functions. We use the methods in [6,18,19,47,48] and list the ideas of the proofs for (1.1). An auxiliary system is introduced for (1.1) since it is difficult to construct a pair of upper–lower solutions for (1.1). Based on the analysis of the eigenvalues, a pair of upper–lower solutions connecting only the invasion-free equilibrium E0 for this auxiliary system are constructed and are applied, together with the Schauder’s fixed-point theorem, to prove the existence of traveling semi-fronts of the auxiliary system. Limit arguments then imply the existence of traveling semifronts of the original system (1.1). These traveling semi-fronts of (1.1) are proved to connect another equilibrium by a method similar to that in [47] if f (u1 ) ≡ 0, and to be persistent by applying the persistent theory for dynamical systems in [39] if f (u1 ) ≡ 0. We use two-sided Laplace transform, which was used to study traveling wave solutions by Carr and Chmaj [4], to show the non-existence of traveling semi-fronts of (1.1). Then the weak minimal wave speed is obtained. Finally, theses theoretical results are applied to a class of (endemic or epidemic)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.5 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

5

disease-transmission models and a predator–prey model with stage structure for predators. To show that the minimal wave speed is strong for the endemic disease-transmission models, a Lyapunov function is constructed and the LaSalle’s invariance principle is used to show that the persistent traveling semi-fronts connect an endemic equilibrium. Though the basic ideas in this paper come from [6,18,19,47,48], some crucial improvements are necessary because the increase of the equation number results in further complexity of solution behaviors of (1.1). Four improvements are listed as follows. (I) Though c∗ is called the minimal wave speed, the existence of positive traveling wave solution with wave speed c∗ has rarely been studied for non-cooperative models except, as I know, [22,29] in which the models consist of two equations. However, the method in [22] is geometric and it seems difficult to apply this method to model (1.1). A Lyapunov function is needed in [29] to get the traveling wave solution with wave speed c∗ . It is also difficult to construct a Lyapunov function for model (1.1) with the general interaction functions. It is especially difficult to study the disease-transmission models without recruitment (that is, f (u1 ) ≡ 0) and there are few literatures for this case. We will show the existence of traveling wave solutions with minimal wave speed by two different methods according to f (u1 ) ≡ 0 or f (u1 ) ≡ 0. (II) The minimal wave speeds in [47,48] can be expressed by parameters by solving corresponding characteristic equations since the model in [48] consists of two equations and there is a restriction condition d2 = d3 on the model in [47]. It is very technical to handle the characteristic equation for (1.1) since it is a quartic polynomial without the restriction d2 = d3 and there is no good tools to analyze a general quartic polynomial at present. (III) The upper–lower solutions in [47] must be extended since they can only be used to handle the case of d2 = d3 . Furthermore, the upper–lower solutions for (1.1) in this paper are constructed by using the smallest positive eigenvalue or the secondly smallest positive eigenvalue according to the classification of the parameters. This is very different from the construction in the case of the systems consisting of two equations since only the smallest positive eigenvalue is used in those cases, and this shows the complexity of (1.1). (IV) The proof for the persistence of traveling wave solutions in [48] should be generalized since the boundedness of the ratios of traveling wave solutions’ derivatives to the corresponding traveling wave solutions (Lemma 2.10 in [48]) is needed for the application of the persistence theory. However, it is difficult to show the existence of such ratios for (1.1). We, hence, abandon the corresponding Lemma 2.10 in [48] and improve the proof of the persistence of traveling wave solutions for (1.1). This improvement can let us abandon the corresponding last inequality of the assumption (C3) in [48]. This paper is organized as follows. In the next section we give the existence of traveling semifronts with wave speed c > c∗ for an auxiliary system by constructing a pair of upper–lower solutions together with the application of the Schauder’s fixed-point theorem. In Section 3 the traveling semi-fronts of (1.1) are proved to exist by limit arguments and to connect another equilibrium if f (u1 ) ≡ 0. In Section 4 the traveling semi-fronts of (1.1) are proved to exist by similar arguments and shown to be persistent if f (u1 ) ≡ 0. Section 5 is devoted to the non-existence of bounded traveling semi-fronts by using two-sided Laplace transform. In Section 6 the results obtained in Sections 3, 4 and 5 are applied to some disease-transmission models and a predator– prey model with stage structure for predators. The LaSalle’s invariance principle is used to show that the persistent traveling semi-fronts connect an endemic equilibrium for endemic diseasetransmission models. 2. Existence of traveling semi-fronts for an auxiliary system Define

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.6 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

6

 A(u1 ) :=

g2,2 (u) − δ2 g3,2 (u)

g2,3 (u) g3,3 (u) − δ3

 , A(u1 ) := det(A(u1 )).

(2.1)

u=(u1 ,0,0)

For simplicity we denote Gi (u) := gi (u) − δi ui , G0ij := Gi,j (E0 ), i = 2, 3.

(2.2)

Note that we also have G0 A(K) = 22 G032

G023 . G033

To give the main results the following classifications are needed: (C1) A(K) < 0; (C2) A(K) > 0, G022 > 0 and G033 > 0; (C3) A(K) > 0, G022 < 0 and G033 < 0. Note that A(K) > 0 implies G022 G033 > 0 since G023 > 0, G032 > 0 by (A3). It also can be shown that the Perron–Frobenius principle eigenvalue of A(K) is positive if (C1) or (C2) holds and is negative if (C3) holds. Substituting the traveling profile U (s) defined by (1.2) into system (1.1) yields the following equations: ⎧  cU1 = d1 U1 + f (U1 ) − g1 (U ), ⎪ ⎪ ⎨ cU2 = d2 U2 + g2 (U ) − δ2 U2 , ⎪ ⎪ ⎩  cU3 = d3 U3 + g3 (U ) − δ3 U3 ,

(2.3)

where  denotes the derivative with respect to s. 2.1. Linearization of (2.3) at E0 Linearizing (2.3) at E0 (K, 0, 0) and considering the last two equations of the linearized system, we have ⎧ ⎨ cφ2 = d2 φ2 + G022 φ2 + G023 φ3 , (2.4) ⎩ cφ  = d φ  + G0 φ + G0 φ . 3 3 3 32 2 33 3 Substituting (φ2 (s), φ3 (s)) = eλs (κ2 , κ3 ) into (2.4) we get ⎧ ⎨ cλκ2 = d2 λ2 κ2 + G022 κ2 + G023 κ3 , ⎩ cλκ = d λ2 κ + G0 κ + G0 κ . 3 3 3 32 2 33 3 This system can be rewritten as

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.7 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

Aλ K = 0,

7

(2.5)

where

Aλ :=

P2 (λ) G032

   G023 κ2 , , K := κ3 P3 (λ)

P2 (λ) := d2 λ2 − cλ + G022 , P3 (λ) := d3 λ2 − cλ + G033 . Note that we have A(K) = det A0 . Therefore, the characteristic equation is H (λ) := det Aλ = P (λ) − γ = 0,

(2.6)

where P (λ) := P2 (λ)P3 (λ) and γ := G023 G032 > 0 by (A3). Clearly, we have A(K) = H (0). For convenience we set ± ± λ± M = max{λ2 , λ3 }, c ± c2 − 4d2 G022 ± , λ2 = 2d2

± ± λ± m = min{λ2 , λ3 }, c ± c2 − 4d3 G033 ± λ3 = . 2d3

± ± ± Obviously, λ± i , i = 2, 3 are the roots of Pi (λ), and both of λm and λM make sense only if λ2 and ± λ3 are real. We use the basic idea in [16] to prove the following lemma. However, we have to do more since our characteristic equation (2.6) is more general (e.g. both G022 and G033 may be positive, which is most complex).

Lemma 2.1. (a) Assume (C1) or (C2) holds. Then there exists a positive constant c∗ such that for any c > c∗ we obtain that H (λ1 + ω) > 0, P2 (λ1 ) < 0, P3 (λ1 ) < 0

(2.7)

for ω > 0 small enough, where λ1 is the smallest positive root of H (λ) if (C1) holds and the secondly smallest positive root if (C2) holds. Furthermore, there exist positive constants κ2 and κ3 such that (2.5) holds for λ = λ1 . (b) Assume (C1) or (C2) holds. Then for any c < c∗ there exists no positive constant λ∗ such that H (λ∗ ) = 0, P2 (λ∗ ) < 0, P3 (λ∗ ) < 0.

(2.8)

(c) Assume (C3) holds. Then for any c > 0, there exists no positive constant λ∗ satisfying (2.8) and H (λ) < 0 for λ ∈ (λ∗ − , λ∗ ), where the positive constant  is small enough. (d) If A(K) = 0, (2.6) has no roots with zero real parts for any c > 0 or c < 0.

JID:YJDEQ AID:8654 /FLA

8

[m1+; v1.241; Prn:30/12/2016; 12:35] P.8 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

− − − + + + + Fig. 1. The distribution of the roots of H (λ), P (λ), P2 (λ) and P3 (λ), where λ− m = λ3 , λM = λ2 , λm = λ2 , λM = λ3 .

(e) Suppose c ≥ c∗ . If (C1) holds, (2.6) has only one root with negative real part and this root, denoted by λ− , is real such that Pi (λ− ) > 0, i = 2, 3. If (C2) holds, (2.6) has no roots with negative real parts. Proof. Set

 cM := max 2 d2 G022 , 2 d3 G033 . We firstly consider the case (C2) and suppose it is satisfied. To finish the proof in this case three steps are given when the speed c increases. Step 1. Let c ≤ cM . It is clear that there does not exist λ∗ > 0 such that P2 (λ∗ ) < 0, P3 (λ∗ ) < 0

(2.9)

since we have that P2 (λ) ≥ 0 for any real number λ or P3 (λ) ≥ 0 for any real number λ. Therefore, there does not exist λ∗ satisfying (2.8). ± Step 2. Let c > cM . Under this assumption both λ± 2 and λ3 are real. It is easy to verify that − + L(c) := λm − λM is an increasing function with respect to c. If L(cM ) < 0, then there exists − − + a positive constant cL (> cM ) such that λ+ m < λM for c ∈ (cM , cL ) and λm > λM for c > cL . ∗ If c ∈ (cM , cL ], there still does not exist λ satisfying (2.9) and, therefore, (2.8). + Step 3. Let c > cL if L(cM ) < 0 or c > cM if L(cM ) ≥ 0. It is obvious that λ− M < λm in this − step. Furthermore, we have λm > 0 due to the condition (C2). By the expression of P (λ) we obtain that P (λ) decreases with respect to λ ∈ (−∞, λ− m ) and increases with respect to − , λ− ) ∪ (λ+ , λ+ ) and P (λ) > 0 in (λ− , λ+ ) (see , +∞), and that P (λ) < 0 in (λ λ ∈ (λ+ m M m M M M m − + + Fig. 1). Consequently, P (λ) has exact one minimum in (λ− m , λM ) and (λm , λM ), respec+ tively, and exact one maximum in (λ− M , λm ) since P (λ) is a quartic polynomial. Obviously,

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.9 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

9

+ we must have λ1 ∈ (λ− M , λm ) if λ1 satisfying (2.7) exists. Hence, we need to study the prop+ ). Simple calculations yield , λ erty of P (λ) in (λ− M m

dP (λ) = −λ [P2 (λ) + P3 (λ)] . dc + Thus dPdc(λ) > 0 for λ ∈ (λ− M , λm ) and P (λ) is an increasing function with respect to c for − any fixed λ ∈ (λM , λ+ m ). Furthermore, we have

  1 lim P2 √ = −∞, c→+∞ c

  1 lim P3 √ = −∞, c→+∞ c

 lim P

c→+∞

1 √ c

 = +∞,

√ + which implies 1/ c ∈ (λ− M , λm ) for c large enough. Then the monotonicity of P (λ) with − + respect to c with λ ∈ (λM , λm ) implies that there exists a positive constant c∗ (> cM ) such − + ∗ + that (2.6) has two positive roots in (λ− M , λm ) if c > c and has no positive roots in (λM , λm ) ∗ if cM < c < c . Since H (0) = A(K) > 0, (2.6) has exact one positive root in (−∞, λ− m) − + and (λ+ M , +∞), respectively. Set λ1 to be the smallest positive root of (2.6) in (λM , λm ) if c > c∗ . Then λ1 satisfies (2.7) and is the secondly smallest positive root of (2.6). If c < c∗ with the assumption in Step 3, there does not exist λ∗ satisfying (2.8). Then based on the analysis in Steps 1–3 we know that there does not exist λ∗ satisfying (2.8) for any c < c∗ . Combination of Steps 1–3 completes the proof for (a) and (b) with the assumption (C2) except the existence of positive constants κ2 and κ3 . Since H (λ1 ) = 0, P2 (λ1 ) < 0, P3 (λ1 ) < 0, then κ2 = G023 , κ3 = −P2 (λ1 ) can satisfy (2.5). The proof for (a) and (b) with the assumption (C1) is similar to that for (a) and (b) with the assumption (C2) and is omitted. Actually, this case is simpler than the case (C2). From Step 3 it is easy to see that (e) holds. − ± + Now assume that (C3) holds. It is clear that both λ± M and λm are real and satisfy λM < 0 < λm . Since H (0) > 0, it can be concluded that (2.6) has exact one real root in each of the following intervals: − + + (−∞, λ− m ), (λM , 0), (0, λm ), (λM , +∞). + Set the (positive) roots in (0, λ+ m ) and (λM , +∞) to be λ21 and λ22 , respectively. Then it is easy to see that

H (λ) > 0 for λ ∈ (0, λ21 ), P2 (λ22 ) > 0, P3 (λ22 ) > 0, which implies that λ21 and λ22 can not satisfy (2.8). Thus the proof for (c) is completed. In this paragraph we prove (d) and suppose A(K) = 0. λ = 0 is not the root of (2.6) since A(K) = 0. Consider a quartic polynomial λ4 + a3 λ3 + a2 λ2 + a1 λ + a0 = 0,

(2.10)

where ai , i = 0, 1, 2, 3 are real. If λ = βi (β > 0) is the root of (2.10), substituting λ = βi into (2.10) and separating real and imaginary parts yields β 4 − a2 β 2 + a0 = 0, a1 = a3 β 2 .

(2.11)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.10 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

10

In the case of (2.6) we have a1 = −

c(G022 + G033 ) c(d2 + d3 ) , a3 = − , d2 d3 d2 d3

which, together with the second equality of (2.11), implies that G022 + G033 > 0. Eliminating β in (2.11) yields a12 + a0 a32 − a1 a2 a3 = 0. In the case of (2.6), it is easy to show that a12 + a0 a32 − a1 a2 a3 =−

c2  d23 d33

 (d2 + d3 )(G022 + G033 )c2 + (d2 G33 − d3 G22 )2 + γ (d2 + d3 )2 < 0,

a contradiction. The conclusion (d) is proved. 2 2.2. Positively invariant cone To prove the existence of positive solutions of (2.3) satisfying (1.3), we introduce an auxiliary system: ⎧  cU1 = d1 U1 + f (U1 ) − g1 (U ), ⎪ ⎪ ⎪ ⎨ cU2 = d2 U2 + g2 (U ) − δ2 U2 − χU22 , (2.12) ⎪ ⎪ ⎪ ⎩  cU3 = d3 U3 + g3 (U ) − δ3 U3 − χU32 , where χ is a small positive constant. For simplicity we denote G1 (u) := f (u1 ) − g1 (u), Gi (u) := gi (u) − δi ui − χu2i , i = 2, 3.

(2.13)

In the remainder of Section 2 we assume that (C1) or (C2) holds and that c > c∗ is satisfied. We will use Schauder’s fixed-point theorem on an invariant cone to show the existence of traveling wave solutions. To give such invariant cone we need to construct a pair of upper–lower solutions. Set U 1 (s) := K,

U 1 (s) := max{K − σ1 eαs , 0},

U 2 (s) := min{κ2 eλ1 s , κ2 K ∗ }, U 2 (s) := max{κ2 eλ1 s (1 − σ2 es ), 0}, U 3 (s) := min{κ3 eλ1 s , κ3 K ∗ }, U 3 (s) := max{κ3 eλ1 s (1 − σ3 es ), 0}. The constants κ2 , κ3 and λ1 have been determined in Lemma 2.1. The positive constants K ∗ , α, , σi , i = 1, 2, 3 will be determined later. Note that U 2 (s) and U 3 (s) were improved based on the upper–lower solutions in [47]. Such improvement can allow us to deal with the case of d2 = d3 . We first show that this pair of upper– lower solutions satisfy some inequalities in Lemmas 2.2, 2.3 and 2.5.

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.11 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

11

Lemma 2.2. There exists a positive constant K ∗ (> 1, large enough) such that the functions U 2 (s) and U 3 (s) satisfy inequalities 







cU 2 ≥ d2 U 2 + G2 (K, U 2 , U 3 ), cU 3 ≥ d3 U 3 + G3 (K, U 2 , U 3 ) for any s = s := (ln K ∗ )/λ1 . Proof. If s < s, from Lemma 2.1 (a), (A2) (II) and the negative semi-definite property in (A3) we have U 2 (s) = κ2 eλ1 s , U 3 (s) = κ3 eλ1 s , and 







d2 U 2 − cU 2 + G2 (K, U 2 , U 3 ) = d2 U 2 − cU 2 + G2,2 (E0 )U 2 + G2,3 (E0 )U 3 1 1 2 2 + G2,22 (E0∗ )U 2 + G2,23 (E0∗ )U 2 U 3 + G2,33 (E0∗ )U 3 2 2 = [κ2 P2 (λ1 ) + κ3 G2,3 (E0 )]eλ1 s 1 1 2 2 + G2,22 (E0∗ )U 2 + G2,23 (E0∗ )U 2 U 3 + G2,33 (E0∗ )U 3 2 2 1 1 2 2 2 = g2,22 (E0∗ )U 2 + g2,23 (E0∗ )U 2 U 3 + g2,33 (E0∗ )U 3 − χU 2 2 2 ≤ 0, where E0∗ is between E0 and (K, U 2 , U 3 ). Similar to (2.14), it can be concluded that 



d3 U 3 − cU 3 + G3 (K, U 2 , U 3 ) ≤ 0. If s > s, then 



d2 U 2 − cU 2 + G2 (K, U 2 , U 3 ) = G2,2 (E0 )U 2 + G2,3 (E0 )U 3 1 1 2 2 + G2,22 (E0∗ )U 2 + G2,23 (E0∗ )U 2 U 3 + G2,33 (E0∗ )U 3 2 2 = [κ2 G2,2 (E0 ) + κ3 G2,3 (E0 )]K ∗ − χκ22 K ∗ 2 1 1 2 2 + g2,22 (E0∗ )U 2 + g2,23 (E0∗ )U 2 U 3 + g2,33 (E0∗ )U 3 2 2 ≤ 0, where K∗ >

κ2 G2,2 (E0 ) + κ3 G2,3 (E0 ) . χκ22

(2.14)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.12 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

12

It can be similarly shown that 



d3 U 3 − cU 3 + G3 (K, U 2 , U 3 ) ≤ 0 for s > s.

2

Lemma 2.3. For 0<α<

  1 g1,2 (E0 )κ2 + g1,3 (E0 )κ3 c , λ1 , σ1 > max K, min , 2 d1 (c − d1 α)α

the function U 1 (s) satisfies inequality cU 1 ≤ d1 U 1 + f (U 1 ) − g1 (U 1 , U 2 , U 3 ) for any s = s 1 := (1/α) ln(K/σ1 ). Proof. If s > s 1 , then U 1 (s) = 0 and the conclusion obviously holds from (A2). Assume s < s 1 . From the choice of σ1 it follows that s<

1 K < 0 < s¯ . ln α σ1

Since f (U 1 (s)) ≥ 0 by (A2) and g1 (U 1 , U 2 , U 3 ) ≤ g1 (K, U 2 , U 3 ) by (A3), we have d1 U 1 − cU 1 + f (U 1 ) − g1 (U 1 , U 2 , U 3 ) ≥ cσ1 αeαs − d1 σ1 α 2 eαs − g1 (K, U 2 , U 3 ) = (c − d1 α)σ1 αeαs − g1,2 (E0 )U 2 − g1,3 (E0 )U 3 1 1 2 2 − g1,22 (E0∗ )U 2 − g1,23 (E0∗ )U 2 U 3 − g1,33 (E0∗ )U 3 2 2   ≥ (c − d1 α)σ1 αeαs − g1,2 (E0 )κ2 + g1,3 (E0 )κ3 eλ1 s     = (c − d1 α)σ1 α − g1,2 (E0 )κ2 + g1,3 (E0 )κ3 e(λ1 −α)s eαs    ≥ (c − d1 α)σ1 α − g1,2 (E0 )κ2 + g1,3 (E0 )κ3 eαs ≥ 0, where we used the fact e(λ1 −α)s < 1.

2

The following lemma is prepared for the proof of Lemma 2.5. Lemma 2.4. Let a12 , a21 be two positive constants and a11 (), a22 () two negative functions with  ∈ [0, 1) such that lim aii () = aii (0) < 0, i = 1, 2. Denote →0

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.13 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

 A() :=

a11 () a21

a12 a22 ()

13



such that det A(0) = 0, det A() > 0 for all  ∈ (0, 1). Then for any  ∈ (0, 1) there exist positive constants xi (), yi , i = 1, 2 such that 

a11 ()x1 () + a12 x2 () < 0,

(2.15)

a21 x1 () + a22 ()x2 () < 0, 

a11 (0)y1 + a12 y2 = 0,

(2.16)

a21 y1 + a22 (0)y2 = 0.

Set xi (), yi , i = 1, 2 to be any positive constants satisfying (2.15) and (2.16) for  ∈ (0, 1). Then we have x1 ()y2 = 1. →0 x2 ()y1 lim

(2.17)

Proof. It is obvious for the existence of y1 and y2 satisfying (2.16). The existence of xi (), i = 1, 2 satisfying (2.15) has been proved by Lemma 3.2 in [16]. Now let xi (), yi , i = 1, 2 be any positive constants satisfying (2.15) and (2.16) for  ∈ (0, 1). Then it is easy to see that −

a12 a12 x1 () a22 () y1 a22 (0) , =− , < <− =− a11 () x2 () a21 y2 a11 (0) a21

which implies (2.17).

2

Lemma 2.5. There exist positive constants  (small enough), σ2 and σ3 (large enough) such that functions U 2 (s) and U 3 (s) satisfy inequalities 1 cU 2 ≤ d2 U 2 + G2 (U 1 , U 2 , U 3 ), for s = s 2 := − ln σ2 ,  1 cU 3 ≤ d3 U 3 + G3 (U 1 , U 2 , U 3 ), for s = s 3 := − ln σ3 . 

(2.18) (2.19)

Proof. Without loss of generality we suppose s 3 < s 2 , which implies σ2 < σ3 . It is clear that (2.18) holds for s > s 2 and that (2.19) holds for s > s 3 . Since lim U 1 (s) = K,

s→−∞

lim U i (s) = 0,

s→−∞

lim

→0+ ,σi →+∞

s i = −∞, i = 2, 3,

we can assume (U 1 (s), U 2 (s), U 3 (s)) ∈ U for all s ≤ s 2 by setting  small enough and σ2 , σ3 large enough, where U is defined in Remark 1.1. We also assume s 2 < s 1 . In the remainder of this proof we assume s ≤ s 2 , which implies

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.14 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

14

U 1 (s) = K − σ1 eαs , U 2 (s) = κ2 eλ1 s (1 − σ2 es ), U 3 (s) ≥ κ3 eλ1 s (1 − σ3 es ) =: U 3 , where U 3 (s) = U 3 if and only if s ≤ s 3 . By Taylor’s theorem and assumption (A3) we have G2 (U 1 , U 2 , U 3 ) ≥ G2 (U 1 , U 2 , U 3 ) = G2,2 (U 1 , 0, 0)U 2 + G2,3 (U 1 , 0, 0)U 3 1 1 + G2,22 (P1 )U 22 + G2,23 (P1 )U 2 U 3 + G2,33 (P1 )U 3 2 2 2     = G2,2 (E0 ) + G2,21 (P2 )(U 1 − K) U 2 + G2,3 (E0 ) + G2,31 (P3 )(U 1 − K) U 3 1 1 + G2,22 (P1 )U 22 + G2,23 (P1 )U 2 U 3 + G2,33 (P1 )U 3 2 2 2 = G2,2 (E0 )U 2 + G2,3 (E0 )U 3 + (U 1 − K)[G2,21 (P2 )U 2 + G2,31 (P3 )U 3 ]

(2.20)

1 1 + G2,22 (P1 )U 22 + G2,23 (P1 )U 2 U 3 + G2,33 (P1 )U 3 2 , 2 2 where P1 := (U 1 , ξ1 U 2 , ξ1 U 3 ), P2 := (ξ2 U 1 , 0, 0), P3 := (ξ3 U 1 , 0, 0), 0 < ξi < 1, i = 1, 2, 3. Obviously, Pi ∈ U , i = 1, 2, 3. Then we obtain   e−λ1 s d2 U 2 − cU 2 + G2 (U 1 , U 2 , U 3 )   ≥ e−λ1 s d2 U 2 − cU 2 + G2 (U 1 , U 2 , U 3 ) = κ2 [d2 λ21 − cλ1 + G2,2 (E0 )] + κ3 G2,3 (E0 )   − κ2 σ2 [d2 (λ1 + )2 − c(λ1 + ) + G2,2 (E0 )] + κ3 σ3 G2,3 (E0 ) es − σ1 R1 (s)eαs − R2 (s)eλ1 s , where R1 (s) := G2,21 (P2 )κ2 (1 − σ2 es ) + G2,31 (P3 )κ3 (1 − σ3 es ), 1 R2 (s) := G2,22 (P1 )κ22 (1 − σ2 es )2 2 1 + G2,23 (P1 )κ2 κ3 (1 − σ2 es )(1 − σ3 es ) + G2,33 (P1 )κ32 (1 − σ3 es )2 . 2 Noting G2,3 (E0 ) = G023 and G2,2 (E0 ) = G022 , Lemma 2.1 (a) implies

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.15 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

15

κ2 [d2 λ21 − cλ1 + G2,2 (E0 )] + κ3 G2,3 (E0 ) = 0. Then we have   e−λ1 s d2 U 2 − cU 2 + G2 (U 1 , U 2 , U 3 )   ≥ − κ2 σ2 [d2 (λ1 + )2 − c(λ1 + ) + G2,2 (E0 )] + κ3 σ3 G2,3 (E0 ) es

(2.21)

− σ1 R1 (s)eαs − R2 (s)eλ1 s   = − κ2 σ2 P2 (λ1 + ) + κ3 σ3 G2,3 (E0 ) es − σ1 R1 (s)eαs − R2 (s)eλ1 s , where P2 (λ) is defined in (2.5). For s < s 2 , it is easy to show that 0 ≤ 1 − σ2 es ≤ 1, 1 −

σ3 ≤ 1 − σ3 es ≤ 1, σ2

which, together with the Remark 1.1, implies that there exists a positive constant M1 depending on the ratio σ3 /σ2 such that |R1 (s)| ≤ M1 , |R2 (s)| ≤ M1

(2.22)

  e−λ1 s d3 U 3 − cU 3 + G3 (U 1 , U 2 , U 3 )   = − κ3 σ3 P3 (λ1 + ) + κ2 σ2 G3,2 (E0 ) es − σ1 R3 (s)eαs − R4 (s)eλ1 s

(2.23)

for all s < s 2 . Similar to (2.21) we have

for s < s 3 such that |R3 (s)| ≤ M2 , |R4 (s)| ≤ M2 , where M2 does not depend on the ratio σ3 /σ2 . Consider the following inequalities 

P2 (λ1 + )ζ2 + G2,3 (E0 )ζ3 < 0, G3,2 (E0 )ζ2 + P3 (λ1 + )ζ3 < 0.

(2.24)

By (2.7) and Lemma 2.4, there exist positive constants ζ2 and ζ3 satisfying (2.24) such that lim

ζ3 κ2

→0 ζ2 κ3

= 1.

Set σ2 :=

ζ 0 ζ2 ζ0 ζ3 , σ3 := , κ2 κ3

where ζ0 is a large positive constant. Then for small  we have that

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.16 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

16

−1 < 1 −

σ3 ≤ 1 − σ3 es ≤ 1, σ2

and that M1 , defined in (2.22), can be selected such that M1 does not depend on the ratio σ3 /σ2 = ζ3 κ2 /(ζ2 κ3 ). By (2.21) it follows that   e−λ1 s d2 U 2 − cU 2 + G2 (U 1 , U 2 , U 3 )   ≥ − κ2 σ2 P2 (λ1 + ) + κ3 σ3 G2,3 (E0 ) es − σ1 R1 (s)eαs − R2 (s)eλ1 s = [ζ0 ι2 − σ1 R1 (s)e(α−)s − R2 (s)e(λ1 −)s ]es ≥ (ζ0 ι2 − σ1 M1 − M2 )es > 0, where ι2 = −ζ2 P2 (λ1 + ) − ζ3 G2,3 (E0 ) > 0, ζ0 >

σ1 M1 + M2 ,  < min{α, λ1 } ι2

from (2.24). We also used the fact s < s 2 < 0. Thus inequality (2.18) is proved. Inequality (2.19) can be similarly proved. 2 Remark 2.1. Obviously, we can select positive constants K ∗ , α, , σi , i = 1, 2, 3 such that Lemmas 2.2, 2.3 and 2.5 hold. Therefore, we always suppose the conditions in Lemmas 2.2, 2.3 and 2.5 hold. Furthermore, if c (c > c∗ ) is fixed and χ ∈ (0, 1], then s 1 , defined in Lemma 2.3, and both of s 2 and s 3 , defined in Lemma 2.5, can be chosen such that they do not depend on the choice of χ . We will find traveling wave solutions in the following profile set:

= {(U1 (·), U2 (·), U3 (·)) ∈ C(R, R3 ) : U i (s) ≤ Ui (s) ≤ U i (s), i = 1, 2, 3, s ∈ R}, which is constructed by the pair of upper–lower solutions. Furthermore, (2.12) can be rewritten as the following form ⎧ −d U  + cU1 + β1 U1 = H1 [U (·)](s), ⎪ ⎪ 1 1 ⎨ −d2 U2 + cU2 + β2 U2 = H2 [U (·)](s), ⎪ ⎪ ⎩ −d3 U3 + cU3 + β3 U3 = H3 [U (·)](s), where U (s) = (U1 (s), U2 (s), U3 (s)), Hi [U (·)](s) = βi Ui (s) + Gi (U (s)), i = 1, 2, 3. Positive constants βi , i = 1, 2, 3 will be determined later. Let i1 < 0 < i2 , i = 1, 2, 3 be the roots of

(2.25)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.17 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

17

di 2 − c − βi = 0, and define F = (F1 , F2 , F3 ) : → C(R, R3 ) by ⎤ ⎡ s  +∞ 1 ⎣ Fj [U (·)](s) = ej 1 (s−t) Hj [U (·)](t)dt + ej 2 (s−t) Hj [U (·)](t)dt ⎦ , dj  j −∞

s

where j := j 2 − j 1 > 0. By (A1), we can set βi > sup {−Gi,i (u)},

(2.26)

u∈ 0

where

0 := {(u1 , u2 , u3 ) : 0 < u1 ≤ K, 0 < u2 ≤ κ2 K ∗ , 0 < u3 ≤ κ3 K ∗ }. Then we show that is invariant under the effect of F . Lemma 2.6. F ( ) ⊂ . Proof. Assume U (·) ∈ , i.e., U i (s) ≤ Ui (s) ≤ U i (s) for any s ∈ R, i = 1, 2, 3. Then it suffices to prove U i (s) ≤ F [Ui (·)](s) ≤ U i (s) for any s ∈ R, i = 1, 2, 3. By (2.26) and (A2)–(A3) it follows that Hi [U (·)] increases with respect to Ui (·) and that Hi [U (·)](s) ≥ 0, i = 1, 2, 3 for s ∈ R. If s ≥ s 2 (s 2 is defined in (2.18)), then we have U 2 (s) = 0 and F2 [U (·)](s) ≥ 0 = U 2 (s) due to H2 [U (·)](s) ≥ 0. Now suppose s < s 2 . By (2.26), (A2)–(A3) and Lemma 2.5 we have −d2 U 2 + cU 2 + β2 U 2 ≤ β2 U 2 + G2 (U 1 , U 2 , U 3 ) ≤ β2 U2 + G2 (U1 , U2 , U3 ) = H [U (·)](s), implying F2 [U (·)](s) ⎤ ⎡ s  +∞ 1 ⎣ e21 (s−t) H2 [U (·)](t)dt + e22 (s−t) H2 [U (·)](t)dt ⎦ = d2  2 −∞

≥

1 d2  2

s −∞

s

e21 (s−t) [−d2 U 2 (t) + cU 2 (t) + β2 U 2 (t)]dt

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.18 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

18

1 + d2  2 1 + d2  2 = U 2 (s) +

s 2

e22 (s−t) [−d2 U 2 (t) + cU 2 (t) + β2 U 2 (t)]dt

s

+∞ e22 (s−t) [−d2 U 2 (t) + cU 2 (t) + β2 U 2 (t)]dt s2

1 22 (s−s )  2 [U (s + 0) − U  (s − 0)] e 2 2 2 2 2

≥ U 2 (s), where the final inequality is due to U 2 (s 2 + 0) = 0 and U 2 (s 2 − 0) < 0. In conclusion, F2 [U (·)](s) ≥ U 2 (s) for any s ∈ R. Other cases can be similarly proved. 2 2.3. Existence of traveling semi-fronts of (2.12) In this subsection we first prove that the conditions for Schauder’s fixed-point theorem are satisfied by the following two lemmas. Then the existence of traveling semi-fronts of (2.12) is proved in Lemma 2.9 by using Schauder’s fixed-point theorem. Set μ to be a small positive constant, which will be determined later. For (s) = (φ1 (s), φ2 (s), φ3 (s)) we define

||(·)|| := max sup |φ1 (s)|e s∈R

−μ|s|

, sup |φ2 (s)|e s∈R

−μ|s|

, sup |φ3 (s)|e

−μ|s|



s∈R

and Bμ (R, R3 ) := {(·) ∈ C(R, R3 ) : ||(·)|| < +∞}. Obviously, is closed and convex in Bμ (R, R3 ). Lemma 2.7. For μ small enough, map F = (F1 , F2 , F3 ) : → is continuous with respect to the norm || · ||. Proof. Assume φ(·), ψ(·) ∈ and φ(·) = ψ(·), where φ(·) = (φ1 (·), φ2 (·), φ3 (·)), ψ(·) = (ψ1 (·), ψ2 (·), ψ3 (·)). Then we have

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.19 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

|H2 [φ(·)](s) − H2 [ψ(·)](s)| = |β2 [φ2 (s) − ψ2 (s)] + G2 (φ(s)) − G2 (ψ(s))| 3  = β2 [φ2 (s) − ψ2 (s)] + G2,i (φ(s) + θi (ψ(s) − φ(s)))[φi (s) − ψi (s)] , i=1

where 0 < θi < 1. By (A3) there exists a positive constant M3 such that β2 +

3 

|G2,i (φ(s) + θi (ψ(s) − φ(s)))| ≤ M3 ,

i=1

which implies |H2 [φ(·)](s) − H2 [ψ(·)](s)|e−μ|s| ≤ M3 ||φ(·) − ψ(·)||. Furthermore, we have |F2 [φ(·)](s) − F2 [ψ(·)](s)|e−μ|s| ⎡ s  −μ|s| e ⎣ ≤ e21 (s−t)+μ|t| |H2 [φ(·)](t) − H2 [ψ(·)](t)e−μ|t| |dt d2  2 −∞ ⎤ +∞ + e22 (s−t)+μ|t| |H2 [φ(·)](t) − H2 [ψ(·)](t)|e−μ|t| dt ⎦ s

⎤ ⎡ s  +∞ M3 e−μ|s| ⎣ e21 (s−t)+μ|t| dt + e22 (s−t)+μ|t| dt ⎦ ||φ(·) − ψ(·)||. ≤ d2  2 −∞

s

Now assume μ < min{−21 , 22 }. If s < 0, we have |F2 [φ(·)](s) − F2 [ψ(·)](s)|e−μ|s| ⎡ s 0 M3 eμs ⎣ 21 s −(21 +μ)t 22 s ≤ e dt + e e−(22 +μ)t dt e d2  2 s −∞ ⎤ +∞ + e22 s e(μ−22 )t dt ⎦ ||φ(·) − ψ(·)|| 0

 1 − e(22 +μ)s e(22 +μ)s + + ||φ(·) − ψ(·)|| −21 − μ 22 + μ 22 − μ   1 1 1 M3 + + ||φ(·) − ψ(·)||. ≤ d2 2 −21 − μ 22 + μ 22 − μ

M3 = d2  2

If s ≥ 0, we have



1

19

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.20 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

20

|F2 [φ(·)](s) − F2 [ψ(·)](s)|e−μ|s| ⎡ 0 s −μs M3 e ⎣e21 s ≤ e−(21 +μ)t dt + e21 s e(μ−21 )t dt d2  2 −∞

+∞

0

⎤

e(μ−22 )t dt ⎦ ||φ(·) − ψ(·)||

+ e22 s s

 1 1 − e(21 −μ)s e(21 −μ)s + + ||φ(·) − ψ(·)|| −21 − μ μ − 21 22 − μ   1 1 1 M3 + + ||φ(·) − ψ(·)||. ≤ d2 2 −21 − μ μ − 21 22 − μ

M3 = d2  2



In conclusion, we have shown that ||F2 [φ(·)](s) − F2 [ψ(·)](s)|| ≤ M4 ||φ(·) − ψ(·)||, where M4 :=

 M3 1 1 1 1 1 1 max + + + , + . d2  2 −21 − μ 22 + μ 22 − μ −21 − μ μ − 21 22 − μ

Thus F2 : → C(R, R) is continuous with respect to the norm || · ||. Similarly, it can be proved that Fi : → C(R, R), i = 1, 3 are also continuous with respect to the norm || · ||. 2 Lemma 2.8. Map F = (F1 , F2 , F3 ) : → is compact with respect to the norm || · ||. Proof. Since is a closed subset of Bμ (R, R3 ) and F ( ) ⊂ (Lemma 2.6), it suffices to prove that F : → Bμ (R, R3 ) is compact with respect to the norm || · ||. Assume (·) = (U1 (·), U2 (·), U3 (·)) ∈ . Then there exists a positive constant M5 such that |H2 [(·)](s)| = |β2 U2 (s) + G2 (U (s))| ≤ M5 for all s ∈ R. Consequently, it follows that d F2 [(·)](s) ds s +∞ 1 21 (s−t) 22 (s−t) e H [(·)](t)dt +  e H [(·)](t)dt =  21 2 22 2 d2  2 s −∞ ⎤ ⎡ s +∞ M5 ⎣ 21 (s−t) e dt + 22 e22 (s−t) dt ⎦ |21 | ≤ d2  2 −∞

=

2M5 , d2  2

s

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.21 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

21

which implies d F2 [(·)](·) < 2M5 . d  ds 2 2 d d Therefore, ds F2 [(·)](·) is bounded. Similarly, ds Fi [(·)](·) , i = 1, 3 are also bounded, which shows that F ( ) is uniformly bounded and equicontinuous with respect to the norm || · || in any compact interval. Moreover, for fixed positive integer n, we define ⎧ ⎪ ⎨ F [(·)](s), F n [(·)](s) := F [(·)](−n), ⎪ ⎩ F [(·)](n),

s ∈ [−n, n], s ∈ (−∞, −n], s ∈ [n, +∞).

Obviously, F n : → Bμ (R, R3 ) is compact with respect to the norm || · ||. Since ⎡ |F2 [(·)](s)| ≤

M5 ⎣ d2  2

s

⎤ +∞ e21 (s−t) dt + e22 (s−t) dt ⎦ =

−∞

s

M5 , d2 |21 |22

we have ||F2n [(·)](·) − F2 [(·)](·)|| = sup |F2n [(·)](s) − F2 [(·)](s)|e−μ|s| s∈R

=

sup s∈(−∞,−n]∪[n,+∞)

≤

|F2n [(·)](s) − F2 [(·)](s)|e−μ|s|

2M5 e−μn → 0, as n → +∞. d2 |21 |22

Similarly, we can prove that ||Fin [(·)](·) − Fi [(·)](·)|| → 0, i = 1, 3 when n → +∞. Thus, ||F n [(·)](·) − F [(·)](·)|| → 0 when n → +∞. By Proposition 2.12 in [46], F : → Bμ (R, R3 ) is compact. 2 Lemma 2.9. If (C1) or (C2) holds and c∗ is defined by Lemma 2.1, system (2.12) has a bounded positive solution U (s) for c > c∗ satisfying (1.3) and U1 (s) < K. Proof. Combination of Schauder’s fixed-point theorem, Lemmas 2.6, 2.7 and 2.8 implies that (2.12) has a non-negative bounded solution U (·) = (U1 (·), U2 (·), U3 (·)) ∈ satisfying (1.3) and U1 (s) ≤ K. Next we show that U (s) = (U1 (s), U2 (s), U3 (s)) is a positive solution of (2.12). Setting Vi (s) = Ui (s), i = 1, 2, 3, together with equations (2.12), yields

JID:YJDEQ AID:8654 /FLA

22

[m1+; v1.241; Prn:30/12/2016; 12:35] P.22 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

⎧ U1 ⎪ ⎪ ⎪ ⎪ d1 V1 ⎪ ⎪ ⎪ ⎨ U 2  ⎪ V d 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ U3 ⎪ ⎩ d3 V3

= = = = = =

V1 , cV1 − f (U1 ) + g1 (U ), V2 , cV2 − g2 (U ) + δ2 U2 + χU22 , V3 , cV3 − g3 (U ) + δ3 U3 + χU32 .

(2.27)

Since U1 (s) ≥ U 1 (s) for any s ∈ R, we find that there is a constant s0 such that U1 (s) > 0 for any s < s0 . Then maximum principle shows that U1 (s) > 0 for any s ∈ R. Similarly, it can be shown U2 (s) > 0, U3 (s) > 0, U1 (s) < K for any s ∈ R. 2 3. Existence of traveling semi-fronts of (1.1) if f (u1 ) ≡ 0 3.1. Traveling semi-fronts when c > c∗ In this section we show that system (1.1) has a traveling semi-front with wave speed c > c∗ connecting two equilibria if f (u1 ) ≡ 0. For this aim we give the following assumption. (A4) f (u1 ) ≡ 0, g2 (u) ≤ α1 g1 (u). g3 (u) ≤ α2 g1 (u) or g3 (u) = h(u2 ) ≤ α2 u2 . Theorem 3.1. Assume (A4) holds. If (C1) or (C2) is satisfied, then for c > c∗ system (1.1) has a traveling semi-front U (x + ct) such that U1 (s) is decreasing and U (+∞) = (K0 , 0, 0),

(3.1)

where 0 ≤ K0 < K, and K0 depends on wave speed c. Proof. Let U (s) := (U1 (s), U2 (s), U3 (s)) be the positive solution of (2.12) in Lemma 2.9. From the proof of Lemma 2.9, we know that U (·) ∈ is the fixed point of operator F : → . Then applying L’Hospital principal to operator F yields Ui (−∞) = 0, i = 1, 2, 3, which, together with equations (2.12), implies Ui (−∞) = 0, i = 1, 2, 3. The first equation of (2.12) can be rewritten as c  g1 (U (s)) U1 (s) − U1 (s) = − . d1 d1 Multiplying this equation by e−c/d1 s , we have −[e−c/d1 s U1 (s)] = −

1 g1 (U (s))e−c/d1 s . d1

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.23 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

23

Since U1 (s) is bounded, integrating this equation from s to +∞ yields 1 c/d1 s e d1

U1 (s) = −

+∞ g1 (U (η))e−c/d1 η dη < 0,

(3.2)

s

which implies that U1 (s) is decreasing in R and that there exists a constant 0 ≤ K1 < K such that U1 (+∞) = K1 . Integrating the first equation of (2.12) from −∞ to s gives s

g1 (U (η))dη = d1 U1 (s) − c[U1 (s) − K],

(3.3)

−∞

which, together with the boundedness of U1 (s) and U1 (s) in R, implies ∞ g1 (U (η))dη < +∞. −∞

By integrating the second equation of (2.12) from −∞ to s, we have cU2 (s) = d2 U2 (s) +

s

s g2 (U (η))dη − δ2

−∞

s U2 (η)dη − χ

−∞

U22 (η)dη.

(3.4)

−∞

From (A4) we know +∞ +∞ g2 (U (η))dη ≤ α1 g1 (U (η))dη < +∞, −∞

−∞

which, together with the boundedness of U2 (s) and U2 (s) in R, implies +∞ U2 (η)dη < +∞, −∞

lim U2 (s) = 0.

s→+∞

Similarly, we have lim U3 (s) = 0. Because of U2 (s) = F2 [U (·)](s), L’Hospital principal ims→+∞

plies lim U2 (s) = 0. Consequently, using (3.3) and (3.4) we obtain s→+∞

+∞ [δ2 U2 (η) + χU22 (η)]dη ≤ α1 c(K − K1 ), −∞

which implies

(3.5)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.24 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

24

+∞ δ2 U2 (η)dη ≤ α1 c(K − K1 ). −∞

From this inequality and by similar arguments we have +∞ [δ3 U3 (η) + χU32 (η)]dη ≤ α3 c(K − K1 ),

(3.6)

−∞

where α3 = α2 or α3 = α1 α2 /δ2 according to g3 (u) ≤ α2 g1 (u) or g3 (u) ≤ α2 u2 . Next, we prove U2 (s) ≤ α1 K, U3 (s) ≤ α3 K for any s ∈ R. Set 1 G(s) = c

s [δ2 U2 (η) + χU22 (η)]dη + −∞

1 c

+∞ ec(s−η)/d2 [δ2 U2 (η) + χU22 (η)]dη. s

It is easy to verify cG  (s) − d2 G  (s) = δ2 U2 (s) + χU22 (s) and G(−∞) = 0. From (3.5), it can be concluded that 1 G(+∞) = c

∞ [δ2 U2 (η) + χU22 (η)]dη ≤ α1 (K − K1 ) < α1 K. −∞

Define W(s) = U2 (s) + G(s). Simple calculations show cW  (s) − d2 W  (s) = g2 (U (s)). Since [−d2 e−cs/d2 W  (s)] = e−cs/d2 [cW  (s) − d2 W  (s)], multiplying (3.7) by d2 e−cs/d2 and then integrating from s to +∞, we have 1 W (s) = d2 

+∞ e(c/d2 )(s−η) g2 (U (η))dη ≥ 0 s

for any s ∈ R. Consequently, W(s) is non-decreasing in R. Since

(3.7)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.25 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

25

W(+∞) = G(+∞) ≤ α1 K, we have 0 ≤ U2 (s) ≤ α1 K for any s ∈ R. Similarly, it can be proved that 0 ≤ U3 (s) ≤ α3 K for any s ∈ R. In conclusion, we have proved that (2.12) has a positive solution U (s) = (U1 (s), U2 (s), U3 (s)) satisfying 0 < U1 (s) ≤ K, 0 < U2 (s) ≤ α1 K, 0 < U3 (s) ≤ α3 K.

(3.8)

Set χ = χn := 1/n. By above arguments, there is a positive solution (n)

(n)

(n)

n (·) := (U1 (·), U2 (·), U3 (·)) ∈ of system (2.12) with χ = χn satisfying (1.3) and U1 (+∞) = K1n . From (3.8), it follows that n (s) is uniformly bounded with respect to n, which implies that n (s) is uniformly bounded since n (·) is the fixed point of operator F . In addition, n (s) and  n (s) are also uniformly bounded since n (s) is the solution of (2.12) with χ = χn . Therefore, {n (s)}, {n (s)}, {n (s)} are equicontinuous and uniformly bounded in R. Then Arzelà–Ascoli’s theorem implies that there exists a subsequence {χnk } such that nk (s) → (s), nk (s) →   (s), nk (s) →   (s) uniformly in any bounded closed interval when k → +∞, and pointwise on R, where (s) = (ψ1 (s), ψ2 (s), ψ3 (s)). Since nk (s) is the solution of (2.12) and χnk → 0, we get ⎧   ⎪ ⎨ cψ1 (s) = d1 ψ1 (s) + f (ψ1 ) − g1 (),  cψ2 (s) = d2 ψ2 (s) + g2 () − δ2 ψ2 (s), ⎪ ⎩ cψ  (s) = d ψ  (s) + g () − δ ψ (s). 3 3 3 3 3 3 Consequently, (s) is a non-negative solution of (2.3) satisfying (1.3). From (3.5) it follows  +∞ that −∞ ψ2 (η)dη ≤ α1 cK/δ2 , which implies ψ2 (+∞) = 0. It can be similarly shown that (n)

ψ3 (+∞) = 0. Since U1 (s) is monotonic with respect to s, it follows that ψ1 (+∞) exists. Thus (s) satisfies (3.1). By Remark 2.1, there exist a constant s0 such that ψi (s) > 0, i = 1, 2, 3 for any s < s0 . Then similar to the proof of Lemma 2.9, it can be shown that ψi (s) > 0, i = 1, 2, 3 for any s ∈ R. 2 3.2. Traveling semi-front with wave speed c∗ In this subsection we show the existence of traveling semi-front with minimal wave speed c∗ .

Theorem 3.2. Assume (A4) holds. If (C1) or (C2) is satisfied, then system (1.1) has a traveling semi-front U (x + c∗ t) such that U1 (s) is decreasing and satisfies (3.1), where K0 depends on wave speed c∗ . Proof. We will prove this theorem by four steps. Step 1 Since K satisfies conditions (C1) or (C2), it follows from assumption (A1) that there exists a positive constant 0 , which does not depend on the choice of c, such that (C1) or (C2) still

JID:YJDEQ AID:8654 /FLA

26

[m1+; v1.241; Prn:30/12/2016; 12:35] P.26 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

holds if K is replaced by ρ ∈ [K, K − 0 ]. Let U (s) = (U1 (s), U2 (s), U3 (s)) be a positive solution of (2.3) obtained in Theorem 3.1 and suppose that K0 = K0 (c), determined in Theorem 3.1, satisfies K0 (c) ∈ [K, K − 0 ]. It is easy to verify that W (x − ct) = U (−x + ct) is a positive traveling wave solution of (1.1) with wave speed −c < 0 and satisfies W (−∞) = (K0 (c), 0, 0). However, this contradicts Theorem 5.1 (note that the proof of Theorem 5.1 only relies on Lemma 2.1). Therefore, we have shown that K0 (c) < K − 0 for any c > c∗ . Step 2 Let n (s) = (U1n (s), U2n (s), U3n (s)) be the positive solution of (2.3) obtained in Theorem 3.1 with c = cn , where cn > c∗ and cn → c∗ . Since system (2.3) is autonomous, U1n (s) is decreasing, and K0 (cn ) < K − 0 for any n, we can suppose by a possible translation that U1n (0) = K − , U1n (s) > K −  for some positive constant  (< 0 ) and for any s < 0 and n. Integrating the two sides of the first equality of (2.3) from −∞ to 0 yields

 cn  = −d1 U1n (0) +

0 g1 (n (ξ ))dξ.

(3.9)

−∞

However, we have from Taylor’s theorem that g1 (n (s)) = g1,2 (U1n , 0, 0)U2n + g1,3 (U1n , 0, 0)U3n 1 1 2 2 + g1,22 (P ∗ )U2n + g1,33 (P ∗ )U3n + g1,23 (P ∗ )U2n U3n 2 2   1 ∗ ∗ = g1,2 (U1n , 0, 0) + g1,22 (P )U2n + g1,23 (P )U3n U2n 2   1 + g1,3 (U1n , 0, 0) + g1,33 (P ∗ )U3n U3n , 2 where P ∗ = (U1n , θ U2n , θ U3n ), 0 < θ < 1. Now we suppose that sup U2n (ξ ) → 0, sup U3n (ξ ) → 0 as n → ∞ ξ ≤0

ξ ≤0

(3.10)

(we will get a contradiction later). Then the first equality of (2.3) implies  U1n (s) → K − , U1n (s) → 0, s ≤ 0

since U1n (s) is bounded on R. It follows from (3.10) and assumption (A1) that there exists a positive constant M such that |gk,ij (P ∗ )| ≤ 2M, k = 1, 2, 3, i, j = 2, 3 for any n and s ≤ 0. It thus follows that

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.27 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

27

    g 1,2 () − M U¯ 2n (0) − 2M U¯ 3n (0) U2n + g 1,3 () − M U¯ 3n U3n (3.11)

≤ g1 (n (s))     ≤ g¯ 1,2 () + M U¯ 2n (0) + 2M U¯ 3n (0) U2n + g¯ 1,3 () + M U¯ 3n (0) U3n , where g¯ i,j () =

max

K−≤u1 ≤K

gi,j (u1 , 0, 0), g i,j () =

min

K−≤u1 ≤K

gi,j (u1 , 0, 0), U¯ j n (s) = sup Uj n (ξ ) ξ ≤s

for i = 1, 2, 3, j = 2, 3. The equality (3.9) furthermore gives that

cn  ≤

 − d1 U1n (0) + [g¯ 1,2 () + M U¯ 2n (0) + 2M U¯ 3n (0)]

0 U2n (ξ )dξ

−∞

+ [g¯ 1,3 () + M U¯ 3n (0)]

0

(3.12)

U3n (ξ )dξ.

−∞

We have by setting n → ∞ and from (3.10) that c∗  ≤ g¯ 1,2 ()κ2 (0) + g¯ 1,3 ()κ3 (0), where s κi (s) := lim

n→∞ −∞

Uin (ξ )dξ, i = 2, 3.

We therefore have κ2 (0) > 0 or κ3 (0) > 0.

(3.13)

By expressions similar to (3.11), it follows that ! g 2,2 () − δ2 κ2 (0) + g 2,3 ()κ3 (0) ≤ 0, ! g 3,2 ()κ2 (0) + g 3,3 () − δ3 κ3 (0) ≤ 0. (3.14) can be rewritten as B κ(0) ≤ 0, where

B =

g 2,2 () − δ2

g 2,3 ()

g 3,2 ()

g 3,3 () − δ3



 , κ(0) =

 κ2 (0) . κ3 (0)

(3.14)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.28 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

28

Obviously, lim B = A(K), where A(K) was defined by (2.1). It follows from assumption (A3) →0+

that A(K) is irreducible and from both of (C2) and (C3) that the principle eigenvalue ρ(K) of A(K) is positive. Consequently, B is irreducible and the principle eigenvalue ρ of B is positive by setting  small enough. (3.13) and (3.14) furthermore imply that κ2 (0) > 0 and κ3 (0) > 0. Then Theorem 2.6 of [35] gives that g 2,2 () − δ2 + g 2,3 ()

κ3 (0) ≥ ρ > 0 κ2 (0)

or g 3,2 ()

κ2 (0) + g 3,3 () − δ3 ≥ ρ > 0, κ3 (0)

contradicting (3.14). We have proved that (3.10) does not hold. Step 3 Since {n (·)} is uniformly bounded by (3.8), elliptic estimate shows that n (·) → ∗ (·) 2 (R) norm by passing to a subsequence, where ∗ (s) := (U ∗ (s), U ∗ (s), U ∗ (s)) is a nonin Cloc 1 2 3 negative solution of (2.3) with c = c∗ and satisfies U1∗ (s) ≥ K −  for s ≤ 0. U1∗ (s) is decreasing with respect to s ∈ R since U1n (s) is decreasing. It follows from the results in Step 2 that sup U2n (ξ )  0 or sup U3n (ξ )  0 ξ ≤0

ξ ≤0

as n → ∞. Since n (s) satisfies (1.3), we can assume by a rightward translation that K −  ≤ U1n (s) ≤ K, U2n (s) ≤ , U3n (s) ≤ 

(3.15)

hold for all s ≤ 0, and that either U2n (0) =  or U3n (0) =  is satisfied. Note that it is unnecessary that U1n (0) = K − . Without loss of generality we suppose U2n (0) =  for all n by passing to a subsequence. This implies that U2∗ (0) = , K −  ≤ U1∗ (s) ≤ K, U2∗ (s) ≤ , U3∗ (s) ≤  for all s ≤ 0. Then maximum principle gives that U2∗ (s) > 0 for all s ∈ R. If U3∗ (s) ≡ 0, the third equality of (2.3) yields that g3 (U1∗ (s), U2∗ (s), 0) ≡ 0, contradicting assumption (A3). Thus there exists s0 ∈ R such that U3∗ (s0 ) > 0 and maximum principle furthermore shows that U3∗ (s) > 0 for all s ∈ R. Step 4 In this step we will prove ∗ (−∞) = E0 (K, 0, 0). U1∗ (−∞) exists since U1∗ (s) is   decreasing with respect to s. The boundedness of U1∗ (s) and U1∗ (s) on R then yields   U1∗ (−∞) = U1∗ (−∞) = 0 and the first equality of (2.3) gives lim g1 (∗ (s)) = 0. This, tos→−∞

gether with assumption (A4), shows that lim g2 (∗ (s)) = 0. Then assumption (A3) implies s→−∞

that U3∗ (−∞) = 0. It can be similarly shown that U2∗ (−∞) = 0. Similar to the derivation of (3.12), we have

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.29 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

29

 cn (K − U1n (s)) ≤ − d1 U1n (s) + [g¯ 1,2 () + M U¯ 2n (s) + 2M U¯ 3n (s)]

s U2n (ξ )dξ

−∞

+ [g¯ 1,3 () + M U¯ 3n (s)]

s U3n (ξ )dξ

−∞

≤

 − d1 U1n (s) + [g¯ 1,2 () + 3M]

s U2n (ξ )dξ

−∞

s + [g¯ 1,3 () + M]

U3n (ξ )dξ

−∞

for any s < 0, where the fact (3.15) was used. Setting n → ∞ yields that 

c∗ (K − U1∗ (s)) ≤ −d1 U1∗ (s) + [g¯ 1,2 () + 3M]κ2 (s) + [g¯ 1,3 () + M]κ3 (s). Setting s → −∞ gives c∗ (K − U1∗ (−∞)) ≤ [g¯ 1,2 () + 3M]κ2 (−∞) + [g¯ 1,3 () + M]κ3 (−∞). If U1∗ (−∞) = K, then U1∗ (−∞) < K and we have κ2 (−∞) > 0 or κ3 (−∞) > 0. Completely similar to the arguments in Step 2, a contradiction can be obtained. It consequently follows that U1∗ (−∞) = K. Similar to the proof of Theorem 3.1, we can show ∗ (+∞) = (K0 , 0, 0). 2 4. Persistence of traveling semi-fronts of (1.1) if f (u1 ) ≡ 0 In this section we prove that (1.1) has a persistent traveling semi-front with wave speed c ≥ c∗ if f (u1 ) ≡ 0. The following assumption is needed. (A5)

(I) Let f (u1 ) ≡ 0. f (0) = 0 implies g1,i (0, 0, 0) = 0, g2,j (0, 0, 0) = 0, g3,3 (0, 0, 0) = 0, gi,1 (K, 0, 0) = 0, i = 1, 2, 3, j = 2, 3. (II) g2 (u) is bounded for all 0 < u1 ≤ K, u2 > 0, u3 > 0. g3 (u) is bounded for all 0 < u1 ≤ K, u3 > 0 and fixed u2 > 0. (III) g2 (0, u2 , u3 ) = 0, g3 (0, 0, u3 ) = 0.

We will apply the uniform persistence Theorem 4.5 in [39] and restate it as a lemma as follows. Lemma 4.1. Let X be locally compact, and let X2 be compact in X and X1 be forward invariant under the continuous semiflow  on X. Assume that 2 ,

JID:YJDEQ AID:8654 /FLA

30

[m1+; v1.241; Prn:30/12/2016; 12:35] P.30 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

2 =

"

ω(y), Y2 = {x ∈ X2 : t (x) ∈ X2 ∀t > 0},

y∈Y2

# has an acyclic isolated covering M = m k=1 Mk . If each part Mk of M is a weak repeller for X1 , then X2 is a uniform strong repeller for X1 . To use Lemma 4.1 we denote U1 = {(U1 , V1 , 0, 0, 0, 0) : 0 ≤ U1 ≤ K, |V1 | ≤ B}, U2 = {(0, 0, U2 , V2 , U3 , V3 ) : 0 ≤ Ui ≤ K 0 , |Vi | ≤ B, i = 2, 3}, X1 = {(U1 , V1 , U2 , V2 , U3 , V3 ) : (U1 , U2 , U3 ) is the traveling semi-front of (2.3) and Ui (s) = Vi (s), i = 1, 2, 3}, X2 = U1 ∪ U2 , where K 0 := K ∗ (κ2 + κ3 ) (K ∗ is determined in Remark 2.1) and B is a positive constant that will be determined later. Obviously, system (2.3) is equivalent to the following system ⎧ U1 ⎪ ⎪ ⎪ ⎪ d1 V1 ⎪ ⎪ ⎪ ⎨ U 2 ⎪ d2 V2 ⎪ ⎪ ⎪ ⎪ U3 ⎪ ⎪ ⎩ d3 V3

= = = = = =

V1 , cV1 − f (U1 ) + g1 (U ), V2 , cV2 − g2 (U ) + δ2 U2 , V3 , cV3 − g3 (U ) + δ3 U3 .

(4.1)

Next we study the local behaviors of (4.1) at boundary equilibria. ¯ denote the stable manifold of system Lemma 4.2. Suppose (A5) holds and c ≥ c∗ . Let W s (E) ¯ (4.1) at the equilibrium E. Then we have W s (E¯ K ) ∩ X1 = φ, where E¯ K := (K, 0, 0, 0, 0, 0) and φ denotes the empty set. Furthermore, if f (0) = 0, it follows that W s (E¯ 0 ) ∩ X1 = φ, where E¯ 0 := (0, 0, 0, 0, 0, 0). Proof. We first consider the case W s (E¯ 0 ) and assume f (0) = 0. From (A5) (I) the Jacobian matrix of (4.1) at E¯ 0 has the form

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.31 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

⎡

⎤ 0 ⎥ 0 ⎦, J33

0 J22 ∗

J11 ⎢ ⎣ ∗ ∗

31

(4.2)

where

J11 =

0

1



− f d(0) 1



 , J22 =

c d1

0

1

δ2 d2

c d2



 , J33 =

0

1

δ3 d3

c d3

 ,

and ∗ denotes the elements that are not clearly expressed. Thus (A2) (I) implies that the eigenvalues of J11 have positive real parts and, therefore, the (generalized) eigenvectors of (4.2) corresponding to the eigenvalues with negative real parts have the form (0, 0, ∗, ∗, ∗, ∗)T where T denotes the transposition. Since (4.2) is hyperbolic and 1 := {(0, 0, u2 , v2 , u3 , v3 ) : ui , vi ∈ R, i = 2, 3} is invariant for (4.1), we have W s (E¯ 0 ) ⊂ 1 , which implies W s (E¯ 0 ) ∩ X1 = φ. Now consider W s (E¯ K ). From (A2) (II) the Jacobian matrix of (4.1) at E¯ K has the form  J=

 ∗ , J55

J44 0

(4.3)

where ⎡

J44 =

0 

− f d(K) 1

1 c d1



0

⎢ G0 ⎢ − 22 ⎢ , J55 = ⎢ d2 ⎢ 0 ⎣ G0 − d32 3

1 c d2

0 −

0 0

G023 d2

0 −

G033 d3

0

⎤

⎥ 0 ⎥ ⎥ ⎥. 1 ⎥ ⎦

c d3

Simple calculations show that (2.6) is the characteristic equation of J55 . In addition, it is easy to see that J44 has one positive root and one negative root. Let λ− 44 denote the negative root. T Obviously, the eigenvector of matrix J corresponding to λ− has the form (1, λ− 44 44 , 0, 0, 0, 0) . If (C2) holds it follows from Lemma 2.1 (d), (e) and the invariance of the set 2 := {(u1 , v1 , 0, 0, 0, 0) : u1 , v1 ∈ R} for (4.1) that W s (E¯ K ) ⊂ 2 and W s (E¯ K ) ∩ X1 = φ. In the rest of this proof we suppose (C1) holds. From Lemma 2.1 (e) it follows that J55 has a negative eigenvalue λ− 55 . It is easy to show from the first and second lines of J55 that the − 0 0 T eigenvector of J55 corresponding to λ− 55 has the form (G23 , ∗, −P2 (λ55 ), ∗) , where G23 > 0, − − − P2 (λ55 ) > 0 from (A3) and Lemma 2.1 (e). If λ44 = λ55 , the stable subspace of the linearized system of (4.1) at E¯ K is spanned by − T 0 T (1, λ− 44 , 0, 0, 0, 0) and (∗, ∗, G23 , ∗, −P2 (λ55 ), ∗) .

JID:YJDEQ AID:8654 /FLA

32

[m1+; v1.241; Prn:30/12/2016; 12:35] P.32 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

Then we have W s (E¯ K ) ∩ X1 = φ by the invariance of 2 and the tangency of stable manifold in − − stable manifold theorem [36]. Now suppose λ− 44 = λ55 . Since λ55 is a simple eigenvalue of J55 and a multiple eigenvalue of J with multiplicity 2, the stable subspace of the linearized system of (4.1) at E¯ K is still spanned by − T 0 T (1, λ− 44 , 0, 0, 0, 0) and (∗, ∗, G23 , ∗, −P2 (λ55 ), ∗) . − However, the notation ∗ in this case is different from those in the case of λ− 44 = λ55 . Then we s similarly obtain W (E¯ K ) ∩ X1 = φ. 2

Now we give the main theorem in this section. Theorem 4.1. Assume (A5) holds. If (C1) or (C2) holds and c∗ is defined by Lemma 2.1, system (1.1) has a persistent traveling semi-front U (s) = (U1 (s), U2 (s), U3 (s)), s = x + ct for c ≥ c∗ . Proof. We first consider the case c > c∗ . Let U˜ (s) = (U˜ 1 (s), U˜ 2 (s), U˜ 3 (s)) be the traveling semi-front in Lemma 2.9. We next show that U˜ i (s), U˜ i (s), U˜ i (s), U˜ i (s), i = 1, 2, 3 are uniformly bounded with respect to parameter χ ∈ (0, 1] for fixed c > c∗ . From (A5) there exists a positive constant B1 such that g2 (U˜ (s)) < B1 for any s ∈ R, where B1 does not depend on χ . Then we will prove U˜ 2 (s) ≤ B1 /δ2 for any s ∈ R. On the contrary, we suppose that there exists s1∗ such that U˜ 2 (s1∗ ) > B1 /δ2 . If U˜ 2 (s1∗ ) is a maximum, it follows that U˜ 2 (s1∗ ) = 0, U˜ 2 (s1∗ ) ≤ 0, which, together with the second equality of (2.12), imply 0 = d2 U˜ 2 (s1∗ ) + g2 (U˜ (s1∗ )) − δ2 U˜ 2 (s1∗ ) − χ U˜ 22 (s1∗ ) < B1 − δ2 U˜ 2 (s1∗ ) < 0, a contradiction. Now suppose that U˜ 2 (s1∗ ) is not a maximum and, without loss of generality, that U˜ 2 (s) is increasing in s ∈ [s1∗ , +∞). By the boundedness of U˜ 2 (s), U˜ 2 (s), U˜ 2 (s) and U˜ 2 (s), it follows from Lemma 2.3 in [44] that the limit U˜ 2 (+∞) exists (> B1 /δ2 ) and U˜ 2 (+∞) = U˜  (+∞) = 0, which, together with the second equality of (2.12), imply 2

0 = g2 (U˜ (+∞)) − δ2 U˜ 2 (+∞) − χ U˜ 22 (+∞) ≤ B1 − δ2 U˜ 2 (+∞) < 0, a contradiction. Thus we proved that U˜ 2 (s) is uniformly bounded by B1 /δ2 . Similarly, U˜ 3 (s) is also uniformly bounded. At the same time, U˜ 1 (s) < K implies that U˜ 1 (s) is uniformly bounded. Since U˜ (·) is the fixed point of the operator F , similar to the proof of Lemma 2.8, it can be shown that U˜  (s) is uniformly bounded with respect to χ ∈ (0, 1]. Furthermore, (2.12) implies that U˜  (s) and U˜  (s) are uniformly bounded. Now we let B denote one of the uniform upper bounds of |U˜ i (s)|, |U˜ i (s)|, i = 1, 2, 3. Set χ = 1/n. Then similar to the proof of Theorem 3.1, we can show that (2.3) has a positive solution U (s) := (U1 (s), U2 (s), U3 (s)) satisfying (1.3), which is the limit of U˜ (s) when χ → 0. Suppose

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.33 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

33

(U1 (si ), V1 (si ), U2 (si ), V2 (si ), U3 (si ), V3 (si )) → (U1∗ , V1∗ , U2∗ , V2∗ , U3∗ , V3∗ ), si → +∞ as i → ∞, U1∗ > 0, where Vj (s) = Uj (s), j = 1, 2, 3. We will prove that U2∗ = 0 implies U3∗ = 0 = V2∗ = V3∗ and now assume U2∗ = 0. It is easy to show V2∗ = 0 since V2∗ = 0, together with Taylor’s theorem, implies that there exists s2∗ such that U2 (s2∗ ) < 0, a contradiction. We may assume that lim U2 (si ) i→∞

exists by choosing a subsequence of {si } and denoting this subsequence still by {si }. Again by Taylor’s theorem, we conclude that U2 (+∞) ≥ 0. Then the second equality of (2.3) implies & ' g2 U1∗ , 0, U3∗ ≤ 0. From (A2) and (A3) it follows that U3∗ = 0. Then similar to the proof of V2∗ = 0 it can be shown that V3∗ = 0. In summary, U2∗ = 0 implies U3∗ = 0 = V2∗ = V3∗ . We can similarly prove that U3∗ = 0 implies U2∗ = 0 = V2∗ = V3∗ and that U1∗ = 0 implies V1∗ = 0. Consequently, it is sufficient to prove that X2 repels X1 if we want to show that U (s) is persistent. We only consider the case f (0) = 0 and omit the case f (0) > 0 since the proof for the latter case is simpler. We need to study the dynamics of (4.1) in X2 . However, it is sufficient to study the dynamics of (4.1) in U1 and U2 , respectively, since both of the sets {(U1 , V1 , 0, 0, 0, 0) : U1 ∈ R, V1 ∈ R} and {(0, 0, U2 , V2 , U3 , V3 ) : Ui ∈ R, Vi ∈ R, i = 2, 3} are invariant for (4.1). Consequently, by (A2) (II) and (A5) (III) we need to study the subsystems

u1 = v1 , d1 v1 = cv1 − f (u1 ),

(4.4)

and ⎧ ⎪ ⎪ ⎪ ⎨

u2 d2 v2 ⎪ u3 ⎪ ⎪ ⎩ d3 v3

= = = =

v2 , cv2 + δ2 u2 , v3 , cv3 − g3 (0, u2 , u3 ) + δ3 u3 .

(4.5)

Since (4.4) is equivalent to the classical Fisher equation it is easy to show (also referring to the proof of Lemma 2.11 in [48]) that 10 := {ω(u1 , v1 ) : t (u1 , v1 ) ∈ U10 , ∀t ≥ 0} = (0, 0) ∪ (K, 0), where U10 := {(u1 , v1 ) : 0 ≤ u1 ≤ K, |v1 | ≤ B}. Since the first and second equations of (4.5) are linear and both of u3 and v3 do not appear in them, it follows that the flow generated by (4.5) satisfies 20 := {ω(u2 , v2 , u3 , v3 ) : t (u2 , v2 , u3 , v3 ) ∈ U20 , ∀t ≥ 0} ⊂ 30 ,

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.34 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

34

where 30 := {(0, 0, u3 , v3 ) : 0 ≤ u3 ≤ K 0 , |v3 | ≤ B}, U20 := {(u2 , v2 , u3 , v3 ) : 0 ≤ ui ≤ K 0 , |vi | ≤ B, i = 2, 3}. From (A5) (III) we have that the limit system of (4.5) restricted in 30 has the form

u3 = v3 , d3 v3 = cv3 + δ3 u3 .

Then by 20 ⊂ 30 and Theorem 1.6 in [38] it follows that 20 = (0, 0, 0, 0). From the arguments about 10 and 20 we obtain that the limit set 2 defined in Lemma 4.1 for (4.1) satisfies 2 = E¯ 0 ∪ E¯ K . Since E¯ 0 ∪ E¯ K is isolated and acyclic, Lemmas 4.1 and 4.2 complete the proof in the case of c > c∗ . Now we consider the case of c = c∗ . Let n (·) := (U1n (·), U2n (·), U3n (·)) denote the traveling semi-front of system (2.3) with c = cn := c∗ + 1/n. Suppose sup(K − U1n (ξ )) → 0, sup Uin (ξ ) → 0 as n → ∞, i = 2, 3.

ξ ∈R

ξ ∈R

However, since Lemma 2.1 (d) implies that (K, 0, 0, 0, 0, 0) is a hyperbolic equilibrium of (4.1), Hartman–Grobman theorem in [36] shows that n (+∞) = E0 (K, 0, 0) for large n, contradicting the fact that n (s) is persistent. Thus, we can assume by passing to a subsequence that for certain positive constant  small enough, one of sup(K − U1n (ξ )) > , sup U2n (ξ ) >  and sup U3n (ξ ) > 

ξ ∈R

ξ ∈R

ξ ∈R

holds. Since n (−∞) = E0 (K, 0, 0), we can suppose by a translation that K −  ≤ U1n (s) ≤ K, U2n (s) ≤ , U3n (s) ≤ , ∀s ≤ 0 hold and one of U1n (0) = K − , U2n (0) = , U3n (0) =  holds. Set n (·) → ∗ (·) := (U1∗ (·), U2∗ (·), U3∗ (·)). Then we obtain that

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.35 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

35

K −  ≤ U1∗ (s) ≤ K, U2∗ (s) ≤ , Un∗ (s) ≤ , ∀s ≤ 0 and that one of U1∗ (0) = K − , U2∗ (0) = , U3∗ (0) =  holds. By setting  small enough, it follows from Hartman–Grobman theorem that ∗ (−∞) = E0 (K, 0, 0) and maximum principle gives that ∗ (s) is a positive solution of (2.3). Similar to the proof in the case of c > c∗ , we have that ∗ (s) is persistent. 2 5. Non-existence of traveling semi-fronts of (1.1) In this section we prove the non-existence of traveling semi-fronts. For a non-negative function V (s) we define the two-sided Laplace transform [4] by +∞ L [V (·)](λ) := e−λs V (s)ds −∞

for λ ≥ 0. It is obvious that L [V (·)](λ) = L − [V (·)](λ) + L + [V (·)](λ) where L − [V (·)](λ) :=

0

e−λs V (s)ds, L + [V (·)](λ) :=

−∞

+∞ e−λs V (s)ds. 0

Obviously, if V (s) is bounded in [0, +∞), L + [V (·)](λ) is convergent for any λ > 0 and the convergence of L [V (·)](λ) is equivalent to that of L − [V (·)](λ). It is also evident that for a positive constant λ0 and the function V (s) = eλ0 s , L − [V (·)](λ) is convergent if and only if λ < λ0 . In addition, since L − [V (·)](λ) is increasing with respect to λ, L [V (·)](λ) is defined in [0, λ∗ ) such that λ∗ < +∞ satisfying lim L [V (·)](λ) = +∞ or λ∗ = +∞. λ→λ∗−

In the following we will exclude bounded traveling semi-fronts by two-sided Laplace transform. To apply Laplace transform the prior estimate of exponential decay is needed. Lemma 5.1. Let c = 0. Assume that (C1) or (C2) is satisfied with c < c∗ or that (C3) holds. And assume (A4) holds if f (u1 ) ≡ 0 or (A5) holds if f (u1 ) ≡ 0. If U (s) = (U1 (s), U2 (s), U3 (s)) is a non-negative solution of (2.3) satisfying (1.3), then there exists a positive constant ω such that sup{|K − U1 (s)|e−ωs } < +∞, sup{Ui (s)e−ωs } < +∞, i = 2, 3. s∈R

(5.1)

s∈R

Proof. Since U (s) = (U1 (s), U2 (s), U3 (s)) satisfies (1.3) and (4.1), it is easy to see from the second equation of (4.1) that lim V1 (s) = 0. Similarly, we have lim Vi (s) = 0, i = 2, 3, s→−∞

which implies

s→−∞

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.36 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

36

(U1 (s), V1 (s), U2 (s), V2 (s), U3 (s), V3 (s)) → E¯ 0 (K, 0, 0, 0, 0, 0). If f (u1 ) ≡ 0, (5.1) can be similarly proved as Lemma 3.1 and the proof of Theorem 3.1(I) in [47]. Now suppose that f (u1 ) ≡ 0. It is easy to show that the characteristic equation of the linearization of (4.1) at the equilibrium E¯ 0 is [d1 λ2 − cλ + f  (K)]H (λ) = 0, where H (λ) is defined by (2.6). Since f  (K) < 0, d1 λ2 − cλ + f  (K) = 0 has no roots with zero real parts. Moreover, H (λ) also has no roots with zero real parts by Lemma 2.1 (d), which implies that E¯ 0 is hyperbolic. Then Stable Manifold theorem in [36] implies that there exists a positive constant ω such that (5.1) holds. 2 Theorem 5.1. Assume (A4) holds if f (u1 ) ≡ 0 or (A5) holds if f (u1 ) ≡ 0. If (C1) or (C2) is satisfied and c < c∗ , c = 0, then system (1.1) has no bounded traveling semi-fronts with speed c. If (C3) holds, then for any c > 0, system (1.1) has no bounded traveling semi-fronts with speed c. Proof. By the expression (2.20) of G2 (U 1 , U 2 , U 3 ), it can be similarly shown that there exists a positive constants N such that |gi,2 (E0 )u2 + gi,3 (E0 )u3 − gi (u)| ≤ N[|K − u1 |(u2 + u3 ) + u22 + u23 ], i = 2, 3

(5.2)

for any |u1 − K|  1, 0 < u2  1, 0 < u3  1. If u1 < K, we also have g2,2 (E0 )u2 + g2,3 (E0 )u3 − g2 (u) > 0, g3,2 (E0 )u2 + g3,3 (E0 )u3 − g3 (u) ≥ 0

(5.3)

from (A3) (II). We will finish the proof by a contradiction and thus suppose system (1.1) has a bounded traveling semi-front U (s) = (U1 (s), U2 (s), U3 (s)) with s = x + ct, which satisfies (2.3) and (4.1). By Lemma 5.1, U (s) also satisfies (5.1). We will complete the proof by showing the existence of positive constant λ∗ satisfying (2.8). Denote Ji (λ) := L [Ui (·)](λ) for λ ∈ [0, λ∗i ), i = 2, 3. By Lemma 5.1, it is clear that λ∗i ≥ ω, i = 2, 3. Then the second and third equations of (2.3) can be rewritten as

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.37 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••



d2 U2 (s) − cU2 (s) + G022 U2 (s) = Q2 (s) − G023 U3 (s), d3 U3 (s) − cU3 (s) + G033 U3 (s) = Q3 (s) − G032 U2 (s),

37

(5.4)

where Qi (s) := G0i2 U2 (s) + G0i3 U3 (s) − Gi (U (s)) = gi,2 (E0 )U2 (s) + gi,3 (E0 )U3 (s) − gi (U (s)). Here, Gi (U (s)) is defined by (2.2). Taking the two-sided Laplace transform of (5.4) yields 

P2 (λ)J2 (λ) = Q2 (λ) − G023 J3 (λ), P3 (λ)J3 (λ) = Q3 (λ) − G032 J2 (λ),

(5.5)

where Qi (λ) = L [Qi (·)](λ). In addition, multiplying the first equality by the second one of (5.5) yields H (λ)J2 (λ)J3 (λ) = Q2 (λ)Q3 (λ) − Q2 (λ)G032 J2 (λ) − Q3 (λ)G023 J3 (λ).

(5.6)

Next, we prove λ∗i < +∞, i = 2, 3. By the third equation of (2.3), the second equality of (5.5) can be rewritten as (λ) := (d3 λ2 − cλ − δ3 )J3 (λ) + L [g3 (U (·))](λ) = 0. Since J3 (λ) > 0 and L [g3 (U (·))](λ) > 0 for λ ∈ [0, λ∗3 ), then λ∗3 = +∞ implies (+∞) = +∞, a contradiction. Therefore, λ∗3 < +∞ holds. Similarly, we have λ∗2 < +∞. Moreover, we show λ∗2 = λ∗3 . On the contrary, we suppose λ∗2 < λ∗3 , which implies lim J2 (λ) = +∞ and λ→λ∗− 2 ∗ lim J3 (λ) = J3 (λ2 ) < +∞. This contradicts the second equality of (5.5) since G032 > 0 from λ→λ∗− 2 (A3) and |Q3 (λ∗2 )| < +∞ from (5.2) and Lemma 5.1. Thus λ∗2 ≥ λ∗3 . It can be similarly proved that λ∗2 ≤ λ∗3 and, therefore, λ∗ := λ∗2 = λ∗3 . We now consider P2 (λ∗ ) and P3 (λ∗ ). If P3 (λ∗ ) ≥ 0, we have from (5.2) that

P3 (λ∗ )J3 (λ∗ ) + G032 J2 (λ∗ ) = +∞ > |Q3 (λ∗ )|, which contradicts the second equality of (5.5). Therefore, we have P3 (λ∗ ) < 0 and, similarly, P2 (λ∗ ) < 0. From (5.2) it follows that |Q2 (λ∗ )| < +∞ and |Q3 (λ∗ )| < +∞, which, together with (5.6), imply H (λ∗ ) = lim

λ→λ∗−

Q2 (λ)Q3 (λ) − Q2 (λ)G032 J2 (λ) − Q3 (λ)G023 J3 (λ) = 0, J2 (λ)J3 (λ)

contradicting Lemma 2.1 (b) if (C1) or (C2) is satisfied. If U1 (s) < K, c > 0 and (C3) are satisfied, we furthermore have from (5.3) that H (λ) < 0 for λ ∈ (λ∗ − , λ∗ ), where positive constant  is small enough, contradicting Lemma 2.1 (c).

JID:YJDEQ AID:8654 /FLA

38

[m1+; v1.241; Prn:30/12/2016; 12:35] P.38 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

To complete the proof we only need to show that U1 (s) < K for all s ∈ R. On the contrary, we suppose that there exists an s0 such that U1 (s0 ) ≥ K. If there exists a local maximum value U1 (s1 ) of U1 (s) such that U1 (s1 ) ≥ K, it follows that U1 (s1 ) = 0, U1 (s1 ) ≤ 0 and, consequently, 0 = U1 (s1 ) + f (U1 (s1 )) − g1 (U (s1 )), contradicting assumption (A2). Otherwise, there exists s2 such that U1 (s2 ) > K and that U1 (s) is increasing on (s2 , +∞). It follows from the boundedness of U1 (s) that there exists s3 (> s2 ) such that U1 (s3 ) ≥ 0, U1 (s3 ) ≤ 0, again reaching a contradiction to the first equality of (2.3). In conclusion, we have U1 (s) < K for all s ∈ R. 2 Remark 5.1. From Theorems 3.1, 3.2 and 5.1 it follows that c∗ is the strong minimal wave speed of (1.1) if (A1)–(A4) hold. From Theorems 4.1 and 5.1 it follows that c∗ is the weak minimal wave speed of (1.1) if (A1)–(A3) and (A5) hold. 6. Applications In this section above results will be applied to a class of disease-transmission models and a predator–prey model with stage-structure. The disease-transmission models are classified into two types: epidemic and endemic. 6.1. Endemic disease-transmission models Infectious diseases can be classified into two types: epidemic and endemic diseases [2]. Epidemic diseases are prevalent in a population only at particular times or under particular circumstances, and endemic diseases are habitually prevalent. Therefore, natural birth and death rates are considered in endemic disease models and omitted in epidemic disease models. However, some diseases can be treated as both epidemic and endemic types according to the research aims [1,2,37]. 6.1.1. An SIR model with two progression stages A non-diffusive endemic SIR model with two progression stages is studied by [23, model (16)]. We incorporate diffusions into that model and obtain the following general model: ⎧ ∂ ⎪ ⎪ u1 = d1 u1 + d(K − u1 ) − ψ1 (u1 )[ψ2 (u2 ) + ψ3 (u3 )], ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ u2 = d2 u2 + ψ1 (u1 )[ψ2 (u2 ) + ψ3 (u3 )] − δ2 u2 , ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ u3 = d3 u3 + hu2 − δ3 u3 , ∂t

(6.1)

where all the parameters are positive. Here, u1 denotes the density of susceptible individuals, u2 and u3 the densities of infected individuals in the first and second progression stages, respectively. ψ2 (u2 ) and ψ3 (u3 ) stand for the rates of infection from u2 and u3 , respectively. Some assumptions are needed for model (6.1).

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.39 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

39

(H1) ψi (·) ∈ C 2 ([0, +∞)), ψi (0) = 0, ψ1 (ξ ) > 0, ψ3 (0) > 0, ψi (ξ ) ≥ 0, ψj (ξ ) ≤ 0, i = 1, 2, 3, j = 2, 3 for any ξ > 0. ψj (ξ ), j = 2, 3 are bounded on ξ ∈ [0, +∞). Remark 6.1. The boundedness of ψj (ξ ), j = 2, 3 means that the disease incidence rates are saturated and these saturated rates can better describe the transmission of diseases than the bilinear disease incidence rate [3,32]. It is clear that (H1) implies (A1)–(A3) and (A5). By the method in [40] we can define the basic reproduction number R01 :=

ψ1 (K)ψ2 (0) hψ1 (K)ψ3 (0) + . δ2 δ2 δ3

By the results in [37] it follows that system (6.1) has a positive equilibrium E1 (u∗1 , u∗2 , u∗3 ) if and only if R01 > 1. Therefore, we suppose that R01 > 1 holds for (6.1) to study the spread of the disease. It is evident that E0 (K, 0, 0) is the disease-free equilibrium of (6.1). (H2) [ψi (ui ) − ψi (u∗i )]



 ψi (ui ) ψi (u∗i ) − ≤ 0, ui > 0, i = 2, 3. ui u∗i

Assumption (H2) is needed in the inequalities (5.2)–(5.3) (or (5.15)–(5.16)) of [37]. Theorem 6.1. Suppose (H1), (H2) and R01 > 1 are satisfied. Then there exists a positive constant c∗ such that system (6.1) has a positive traveling wave solution with wave speed c > 0 connecting E0 and E1 if and only if c ≥ c∗ . Proof. It is easy to show that R01 > 1 implies that (C1) holds for (6.1). Then it follows from Theorem 4.1 that there exists a positive constant c∗ such that (6.1) has a persistent traveling semi-front U (s) = (U1 (s), U2 (s), U3 (s)), s = x + ct satisfying U (−∞) = E0 if c ≥ c∗ . Obviously, U (s) = (U1 (s), U2 (s), U3 (s)) also satisfies ⎧ U1 (s) ⎪ ⎪ ⎪ ⎪ d1 V1 (s) ⎪ ⎪ ⎪ ⎨ U  (s) 2 ⎪ d2 V2 (s) ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ U3 (s) ⎩ d3 V3 (s)

= = = = = =

V1 (s), cV1 (s) − F1 (U (s)), V2 (s), cV2 (s) − F2 (U (s)), V3 (s), cV3 (s) − F3 (U (s)),

where F1 (u) := d(K − u1 ) − ψ1 (u1 )[ψ2 (u2 ) + ψ3 (u3 )], F2 (u) := ψ1 (u1 )[ψ2 (u2 ) + ψ3 (u3 )] − δ2 u2 , F3 (u) := hu2 − δ3 u3 .

(6.2)

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.40 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

40

Next we need to show U (+∞) = E1 and this aim will be achieved by LaSalle’s invariance principle. Define L(s) := L1 (s) + L2 (s) +

ψ1 (u∗1 )ψ3 (u∗3 ) L3 (s), hu∗2

ψ1 (u∗1 )d1 V1 (s) L1 (s) := cU1 (s) − d1 V1 (s) + −c ψ1 (U1 (s))

U1 (s) u∗1

ψ1 (u∗1 ) dξ, ψ1 (ξ )

u∗2 d2 V2 (s)

− cu∗2 ln U2 (s), U2 (s) u∗ d3 V3 (s) L3 (s) := cU3 (s) − d3 V3 (s) + 3 − cu∗3 ln U3 (s). U3 (s)

L2 (s) := cU2 (s) − d2 V2 (s) +

L(s) is obtained by improving the Lyapunov function (5.6) of [37]. It follows from Theorem 4.1 and its proof that L(s) is bounded. Then simple calculations yield that the derivative of L1 (s) satisfies ψ1 (U1 (s)) − ψ1 (u∗1 ) d1 ψ1 (u∗1 )ψ1 (U1 (s))[V1 (s)]2 dL1 (s) = [cV1 (s) − d1 V1 (s)] − ds ψ1 (U1 (s)) [ψ1 (U1 (s))]2 = J11 − J12 , where J11 := [cV1 (s) − d1 V1 (s)] J12 :=

ψ1 (U1 (s)) − ψ1 (u∗1 ) ψ1 (U1 (s)) − ψ1 (u∗1 ) = F1 (U (s)) , ψ1 (U1 (s)) ψ1 (U1 (s))

d1 ψ1 (u∗1 )ψ1 (U1 (s))[V1 (s)]2 ≥ 0. [ψ1 (U1 (s))]2

Similarly, we have Ui (s) − u∗i di u∗i [Vi (s)]2 dLi (s) ≥ 0, i = 2, 3. = Ji1 − Ji2 , Ji1 := Fi (U (s)) , Ji2 := ds Ui (s) [Ui (s)]2 In conclusion, we obtain L (s) = J1 − J2 , where J1 := J11 + J21 +

ψ1 (u∗1 )ψ3 (u∗3 ) ψ1 (u∗1 )ψ3 (u∗3 ) J31 , J2 := J12 + J22 + J32 ≥ 0. ∗ hu2 hu∗2

Furthermore, we have 

 ψ1 (u∗1 ) − U1 ) 1 − ψ1 (U1 )   ψ1 (u∗1 ) ψ2 (U2 ) U2 ψ1 (U1 )ψ2 (U2 )u∗2 ∗ ∗ + ψ1 (u1 )ψ2 (u2 ) 2 − + − ∗ − ψ1 (U1 ) ψ2 (u∗2 ) u2 ψ1 (u∗1 )ψ2 (u∗2 )U2

J1 = d(u∗1

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.41 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

41

  ψ1 (u∗1 ) ψ3 (U3 ) U2 ψ1 (U1 )ψ3 (U3 )u∗2 + ψ1 (u∗1 )ψ3 (u∗3 ) 2 − − + − ψ1 (U1 ) ψ3 (u∗3 ) u∗2 ψ1 (u∗1 )ψ3 (u∗3 )U2   U2 U3 U2 u∗ + ψ1 (u∗1 )ψ3 (u∗3 ) ∗ − ∗ − ∗ 3 + 1 . u2 u3 u2 U3 Therefore, J1 has the expression similar to (5.7) of [37]. It follows from (H1) that   ψ1 (u∗1 ) (u∗1 − U1 ) 1 − ≤ 0. ψ1 (U1 ) Then by the proof of Theorem 5.1 of [37] we have that J1 ≤ 0 and that J1 = 0 if and only if U (s) = E1 , which implies that L (s) ≤ 0 and that L (s) = 0 if and only if (U1 (s), V1 (s), U2 (s), V2 (s), U3 (s), V3 (s)) = (u∗1 , 0, u∗2 , 0, u∗3 , 0). Then LaSalle’s invariance principle implies (U1 (+∞), V1 (+∞), U2 (+∞), V2 (+∞), U3 (+∞), V3 (+∞)) = (u∗1 , 0, u∗2 , 0, u∗3 , 0), and U (+∞) = E1 . To complete the proof we need to exclude the existence of traveling semi-fronts for 0 < c < c∗ . This can be directly obtained by using Theorem 5.1. 2 6.1.2. An endemic SEIR model Consider the following model: ⎧ ∂ ⎪ ⎪ u1 = d1 u1 + d(K − u1 ) − φ(u1 , u3 ), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ u2 = d2 u2 + φ(u1 , u3 ) − δ2 u2 , ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ u3 = d3 u3 + hu2 − δ3 u3 , ∂t

(6.3)

where all the parameters are positive. Let u1 , u2 and u3 denote the densities of susceptible, exposed and infectious individuals, respectively. Then system (6.3) stands for a diffusive SEIR model [25]. System (6.3) can also serve as a diffusive HBV model [41] with general nonlinear incidence. The following assumption is needed. (H3) φ(·) ∈ C 2 ([0, +∞) × [0, +∞)). φ(u1 , u3 ) is bounded for (u1 , u3 ) ∈ (0, K] × (0, +∞). φ(0, u3 ) = φ(u1 , 0) = 0, φu 3 (u1 , 0) > 0, φu 1 (u1 , u3 ) ≥ 0, φu 3 (u1 , u3 ) ≥ 0, for any u1 > 0, u3 > 0.

∂ 2 φ(u1 , u3 ) ∂ 2 φ(u1 , u3 ) ≤ 0, ≥0 ∂u1 ∂u3 ∂u23

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.42 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

42

Remark 6.2. It is evident that (H3) holds if we set φ(u1 , u3 ) = ψ1 (u1 )ψ3 (u3 ) and suppose that both of ψ1 (u1 ) and ψ3 (u3 ) satisfy (H1). Define the basic reproduction number by R02 :=

h ∂φ(K, 0) . δ2 δ3 ∂u3

From the results in [25] it follows that (6.3) has a positive equilibrium E1 (u∗1 , u∗2 , u∗3 ) if and only if R02 > 1. Therefore, we suppose that R02 > 1 holds for (6.3). It is evident that E0 (K, 0, 0) is the disease-free equilibrium of (6.3). (H4) [φ(u1 , u3 ) − φ(u1 , u∗3 )]



 φ(u1 , u3 ) φ(u1 , u∗3 ) − ≤ 0. u3 u∗3

Assumption (H4) is needed in [25]. Theorem 6.2. Suppose (H3), (H4) and R02 > 1 are satisfied. There exists a positive constant c∗ such that (6.3) has a positive traveling wave solution with wave speed c > 0 connecting E0 and E1 if and only if c ≥ c∗ . Proof. This proof is similar to that of Theorem 6.1. Thus we only give the Lyapunov function and its derivative. Define L(s) := L1 (s) + L2 (s) +

δ2 L3 (s), h

φ(u∗1 , u∗3 )d1 V1 (s) −c L1 (s) := cU1 (s) − d1 V1 (s) + φ(U1 (s), u∗3 )

U1 (s) u∗1

φ(u∗1 , u∗3 ) dξ, φ(ξ, u∗3 )

u∗2 d2 V2 (s)

− cu∗2 ln U2 (s), U2 (s) u∗ d3 V3 (s) − cu∗3 ln U3 (s). L3 (s) := cU3 (s) − d3 V3 (s) + 3 U3 (s)

L2 (s) := cU2 (s) − d2 V2 (s) +

L(s) is obtained by improving the Lyapunov function (10) of [25]. Then simple calculations yield L (s) = J1 − J2 , where 

 φ(u∗1 , u∗3 ) 1− φ(U1 (s), u∗3 ) U3 (s) − u∗3 U2 (s) − u∗2 + δ2 [cV3 (s) − d3 V3 (s)] , + [cV2 (s) − d2 V2 (s)] U2 (s) hU3 (s) d1 φ(u∗1 , u∗3 )φU 1 (U1 (s), u∗3 )[V1 (s)]2 d2 u∗2 [V2 (s)]2 δ2 d3 u∗3 [V3 (s)]2 J2 = + + . [φ(U1 (s), u∗3 )]2 [U2 (s)]2 h[U3 (s)]2

J1 = [cV1 (s) − d1 V1 (s)]

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.43 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

43

Then by the proof of Theorem 3.3 of [25] we have that J1 ≤ 0 and that J1 = 0 if and only if U (s) = E1 , which implies that L (s) ≤ 0 and that L (s) = 0 if and only if (U1 (s), V1 (s), U2 (s), V2 (s), U3 (s), V3 (s)) = (u∗1 , 0, u∗2 , 0, u∗3 , 0). Then LaSalle’s invariance principle implies (U1 (+∞), V1 (+∞), U2 (+∞), V2 (+∞), U3 (+∞), V3 (+∞)) = (u∗1 , 0, u∗2 , 0, u∗3 , 0), and U (+∞) = E1 .

2

Remark 6.3. From Subsection 6.1 it can be found that the persistent traveling wave solution can be proved to connect the coexistence equilibrium of (1.1) if a Lyapunov function can be constructed to show that the coexistence equilibrium is globally asymptotically stable for the non-diffusive version of (1.1). Therefore, the key to prove that the persistent traveling wave solution connects the coexistence equilibrium of (1.1) lies in the construction of the Lyapunov function for the non-diffusive version of (1.1). However, it is not easy to construct a Lyapunov function for the non-diffusive version of (1.1) with general reaction functions. 6.2. Epidemic disease-transmission models In this subsection we study two types of epidemic disease-transmission models. 6.2.1. An epidemic model with general disease incidence The following model is an epidemic model with general disease incidence: ⎧ ∂ ⎪ ⎪ u1 = d1 u1 − g(u), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ u = d2 u2 + g(u) − δ2 u2 , ⎪ ∂t 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ u3 = d3 u3 + hu2 − δ3 u3 , ∂t

(6.4)

where all the parameters are positive. We firs give some biological assumptions. (H5) g(·) ∈ C 2 (cl(R3+ )). The Hessian matrix H[g(u1 , ·)](u) is negative semi-definite. g(u1 , 0, 0) = 0, g(0, u2 , u3 ) = 0, g( u) > 0, gu i (u) ≥ 0, gu 3 (u1 , 0, 0) > 0, i = 1, 2, 3, ∂ 2 g(u) ∂ 2 g(u) ≥ 0, ≥0 ∂u1 ∂u2 ∂u1 ∂u3 for any u ∈ R3+ .

JID:YJDEQ AID:8654 /FLA

44

[m1+; v1.241; Prn:30/12/2016; 12:35] P.44 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

Obviously, the epidemic SEIR model in [45] and both of (6.1) and (6.3) with d = 0 are special cases of (6.4). However, we will improve the results in [45] by abandoning the restriction condition d2 = d3 and by considering the traveling wave solution with minimal wave speed. Define the basic reproduction number R03 :=

gu 2 (E0 ) δ2

+

hgu 3 (E0 ) δ2 δ3

,

where E0 = (K, 0, 0). Then we have the following theorem. Theorem 6.3. Assume that (H5) holds and that K is a positive constant. If R03 > 1, there exists a positive constants c∗ and system (6.4) has a positive traveling wave solution U (x + ct), c > 0 satisfying U (−∞) = (K, 0, 0), U (+∞) = (K+ , 0, 0) if and only if c ≥ c∗ , where 0 ≤ K+ < K and K+ depends on wave speed c. Proof. It is easy to verify that (H5) implies (A1)–(A3), that (A4) holds for (6.4), that R03 > 1 if and only if A(K) < 0 where A(K) is defined by (2.1), that R03 < 1 implies (C3), and that G033 = −δ3 < 0. With above conclusions, Theorems 3.1, 3.2 and 5.1 complete the proof. 2 6.2.2. An epidemic influenza model with treatment In this part we study the following epidemic influenza model with treatment: ⎧ ∂ 2S ∂S ⎪ ⎪ ⎪ = ds 2 − β(Iu + δIh )S, ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎨ ∂Iu ∂ 2 Iu + (1 − μ)β(Iu + δIh )S − ku Iu , = d u ⎪ ∂t ∂x 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ∂Ih = dh ∂ Ih + μβ(Iu + δIh )S − kh Ih , ∂t ∂x 2

(6.5)

where 0 < μ < 1, and all the constants are positive. Here, the total density of humans is divided into four subclasses: susceptible S, infected and untreated Iu , treated Ih , and recovered R (subclasses S, Iu , Ih have no relation with R). This model was studied by [47] with the restriction condition du = dh . In this paper this restriction condition will be abandoned. We can tell more information about traveling wave solutions than the results in [47] except abandoning the restriction condition. For example, we consider the traveling wave solution with wave speed c = c∗ , which was not studied in [47]. Define R04 :=

(1 − μ)βS 0 μδβS 0 + . ku kh

Note that R04 > 1 if and only if S 0 > S ∗ , where

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.45 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

S ∗ :=

1 (1−μ)β ku

+

μδβ kh

45

.

Theorem 6.4. Assume S 0 is a positive constant. If S 0 > S ∗ , then there exists a positive constant c∗ such that system (6.5) has a positive traveling wave solution U (s) = (S(ξ ), Iu (ξ ), Ih (ξ )), ξ = x + ct, c > 0 satisfying U (−∞) = (S 0 , 0, 0), U (+∞) = (S0 , 0, 0), if and only if c ≥ c∗ , where S0 is a non-negative constant depending on c. If S 0 < S ∗ , (6.5) has no bounded traveling semi-fronts with wave speed c for any c > 0. Proof. The proof for this theorem is similar to that for Theorem 6.3.

2

6.3. A predator–prey model with stage structure Khajanchi [24] studied a Beddington–DeAngelis type, stage-structure predator–prey model without diffusions. The corresponding diffusive model is as follows: ⎧ ∂ β1 u1 u3 ⎪ , u1 = d1 u1 + ru1 (K − u1 ) − ⎪ ⎪ ⎪ ∂t 1 + h1 u1 + h3 u3 ⎪ ⎪ ⎪ ⎨ β2 u1 u3 ∂ − α1 u2 − δ2 u2 , u2 = d2 u2 + ⎪ ∂t 1 + h 1 u1 + h3 u3 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎩ u = d u + α u − δ u , 3 3 3 2 2 3 3 ∂t

(6.6)

where all the parameters are positive. u1 denotes the density of the prey population, u2 and u3 the densities of juvenile and adult predators, respectively. If δ3 (α1 + δ2 )(Kh1 + 1) > α2 β2 K, the predators die out according to [24]. Therefore, we suppose δ3 (α1 + δ2 )(Kh1 + 1) < α2 β2 K.

(6.7)

It is easy to show that (6.7) holds if and only if (C1) holds and that model (6.6) satisfies (A1)–(A3) and (A5). From Theorems 4.1 and 5.1 we have the following theorem. Theorem 6.5. There exists a positive constant c∗ such that system (6.6) has a persistent traveling semi-front U (s), s = x + ct, c > 0 satisfying U (−∞) = (K, 0, 0) if and only if c ≥ c∗ . The global stability of coexistence equilibrium under certain conditions for non-diffusive version of (6.6) is obtained by using the theory on monotone dynamical systems since it is difficult

JID:YJDEQ AID:8654 /FLA

46

[m1+; v1.241; Prn:30/12/2016; 12:35] P.46 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

to construct a Lyapunov function for the non-diffusive version of (6.6) [24]. However, Theorem 6.5 can still tell us the invasion speed of predators into preys though we can not know the final state after invasion. From [24] we know that preys and predators may coexist at constant levels or periodically fluctuating state. Acknowledgments T. Zhang was supported by the National Science Fund of China (11571284) and the Fundamental Research Funds for the Central Universities (XDJK2015C141). T. Zhang also thanks the Department of Mathematics of The Ohio State University for the hospitality when he visited there. References [1] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, second edition, Springer, New York, 2012. [2] N.F. Britton, Essential Mathematical Biology, Springer, London, 2003. [3] V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978) 43–61. [4] J. Carr, A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004) 2433–2439. [5] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations 33 (1979) 58–73. [6] S.R. Dunbar, Travelling wave solutions of diffusive Lotka–Volterra equations, J. Math. Biol. 17 (1983) 11–32. [7] S.R. Dunbar, Traveling wave solutions of diffusive Lotka–Volterra equations: a heteroclinic connection in R4 , Trans. Amer. Math. Soc. 286 (1984) 557–594. [8] S.R. Dunbar, Traveling waves in diffusive predator–prey equations: periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math. 46 (1986) 1057–1078. [9] T. Faria, W.Z. Huang, J.H. Wu, Travelling waves for delayed reaction–diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006) 229–261. [10] R. Gardner, C.K.R.T. Jone, Stability of travelling wave solutions of diffusive predator–prey systems, Trans. Amer. Math. Soc. 327 (1991) 465–524. [11] R.A. Gardner, Existence of travelling wave solutions of predator–prey systems via the connection index, SIAM J. Appl. Math. 44 (1984) 56–79. [12] F.M. Hilker, M.A. Lewis, Predator–prey systems in streams and rivers, Theor. Ecol. 3 (2010) 175–193. [13] Y. Hosono, B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World 1 (1994) 277–290. [14] Y. Hosono, B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci. 5 (1995) 935–966. [15] C.H. Hsu, C.R. Yang, T.H. Yang, T.S. Yang, Existence of traveling wave solutions for diffusive predator–prey type systems, J. Differential Equations 252 (2012) 3040–3075. [16] C.H. Hsu, T.S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity 26 (2013) 121–139. [17] J.H. Huang, G. Lu, S.G. Ruan, Existence of traveling wave solutions in a diffusive predator–prey model, J. Math. Biol. 46 (2003) 132–152. [18] J.H. Huang, X.F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity, Acta Math. Appl. Sin. 22 (2006) 243–256. [19] J.H. Huang, X.F. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Syst. 9 (2003) 925–936. [20] W.Z. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction–diffusion equations with time delay and nonlocal response, J. Differential Equations 244 (2008) 1230–1254. [21] W.Z. Huang, Traveling wave solutions for a class of predator–prey systems, J. Dynam. Differential Equations 24 (2012) 633–644.

JID:YJDEQ AID:8654 /FLA

[m1+; v1.241; Prn:30/12/2016; 12:35] P.47 (1-47)

T. Zhang / J. Differential Equations ••• (••••) •••–•••

47

[22] W.Z. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction– diffusion systems, J. Differential Equations 260 (2016) 2190–2224. [23] J.M. Hyman, J. Li, E.A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci. 155 (1999) 77–109. [24] S. Khajanchi, Dynamic behavior of a Beddington–DeAngelis type stage structured predator–prey model, Appl. Math. Comput. 244 (2014) 344–360. [25] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol. 69 (2007) 1871–1886. [26] B.T. Li, H.F. Weinberger, M.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci. 196 (2005) 82–98. [27] W.T. Li, G. Lin, S.G. Ruan, Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems, Nonlinearity 19 (2006) 1253–1273. [28] W.T. Li, S.L. Wu, Traveling waves in a diffusive predator–prey model with Holling type-III functional response, Chaos Solitons Fractals 37 (2008) 476–486. [29] Y. Li, W.T. Li, G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal. 14 (2015) 1001–1022. [30] X. Liang, X.Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007) 1–40. [31] X.B. Lin, P.X. Weng, C.F. Wu, Traveling wave solutions for a predator–prey system with sigmoidal response function, J. Dynam. Differential Equations 23 (2011) 903–921. [32] W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987) 359–380. [33] S.W. Ma, Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem, J. Differential Equations 171 (2001) 294–314. [34] K. Mischaikow, J.F. Reineck, Travelling waves in predator–prey systems, SIAM J. Math. Anal. 24 (1993) 1179–1214. [35] D. Noutsos, On Perron–Frobenius property of matrices having some negative entries, Linear Algebra Appl. 412 (2006) 132–153. [36] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001. [37] Z.S. Shuai, P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci. 234 (2011) 118–126. [38] H.R. Thieme, Convergence results and a Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 30 (1992) 755–763. [39] H.R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal. 24 (1993) 407–435. [40] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. [41] K.F. Wang, W.D. Wang, Propagation of HBV with spatial dependence, Math. Biosci. 210 (2007) 78–95. [42] Z.C. Wang, J.H. Wu, Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010) 237–261. [43] H.F. Weinberger, Long time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982) 353–396. [44] J.H. Wu, X.F. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651–687. [45] Z.T. Xu, Traveling waves for a diffusive SEIR epidemic model, Commun. Pure Appl. Anal. 15 (2016) 871–892. [46] E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer, New York, 1986. [47] T.R. Zhang, W.D. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl. 419 (2014) 469–495. [48] T.R. Zhang, W.D. Wang, K.F. Wang, Minimal wave speed for a class of non-cooperative diffusion–reaction system, J. Differential Equations 260 (2016) 2763–2791.