Analysis of a Model Arising from Invasion by Precursor and Differentiated Cells

Abstract

We study the wave solutions for a degenerated reaction diffusion system arising from the invasion of cells. We show that there exists a family of waves for the wave speed larger than or equal a certain number and below which there are no monotonic wave solutions. We also investigate the monotonicity, uniqueness, and asymptotics of the waves.

1. Introduction

In [1], the following coupled partial differential equation system was proposed to study the invasion by precursor and differentiated cells: 𝑢_𝑡=𝐷𝑢_𝑥𝑥+𝛼𝑢1−𝑢+𝜈𝑣𝑘₁𝑣−𝛽𝑢1−𝑘₂,𝑣_𝑡𝑣=𝛽𝑢1−𝑘₂, (1) where 𝑢(𝑥,𝑡) denotes the population densities of the precursor cells. The constant 𝐷>0 is the diffusion rate of the cell 𝑢, which has proliferation rate 𝛼>0, and 𝑘₁>0 is the carrying capacity of 𝑢. The parameter 𝜈 measures the relative contribution that the differentiated cell with population density 𝑣(𝑥,𝑡) makes to the carrying capacity 𝑘₁. The cell population density 𝑣 is limited by its carrying capacity 𝑘₂ and has a maximum differentiation rate 𝛽>0. The model assumes that the differentiated cells do not have mobility.

By letting (see [1]) 𝑢𝑢=𝑘₁,̂𝑣𝑣=𝑘₁,𝑡̂=𝛼𝑡,𝑥=𝛼𝐷𝑥 (2) and dropping the hat notation for convenience, system (1) is changed into 𝑢_𝑡=𝑢_𝑥𝑥()(),𝑣+𝑢1−𝑢−𝜈𝑣−𝜆𝑢1−𝐾𝑣_𝑡=𝜆𝑢(1−𝐾𝑣), (3) where 𝜆=𝛽/𝛼 and 𝐾=𝑘₁/𝑘₂.

System (1) or (3) belongs to reaction diffusion systems of degenerate type, and such systems have attracted much attention in research fields such as epidemics and wound healing [2–4] as well as combustion and calcium wave problems [5–8]. However, system (3) differs from the above systems in the appearance of degenerate reaction terms. In fact, 𝑢=0 coupling with any 𝑣=constant consists of a constant solution of (3). This resembles the combustion wave equation considered in [9]; however, our method in proving the existence of the fronts of (3) differs from theirs.

If the parameters satisfy 0≤𝜈<𝐾, (4) then system (3) admits an additional equilibrium: 𝐵∶(1−𝜈/𝐾,1/𝐾) representing the state that the spatial domain is successfully invaded. We also separate the equilibrium 𝐴∶(0,0) from the rest of the line of equilibria, 𝑢=0. The unstable equilibrium (0,0) represents the state before the invasion.

We are interested in the existence of the wave solutions connecting 𝐴 with 𝐵 as time and space evolve from −∞ to +∞. Setting 𝜉=𝑥+𝑐𝑡, 𝑥∈ℝ, 𝑡∈ℝ⁺, a traveling wave solution to (3) solves 𝑢_𝜉𝜉−𝑐𝑢_𝜉+𝑢(1−𝑢−𝜈𝑣)−𝜆𝑢(1−𝐾𝑣)=0,−𝑐𝑣_𝜉+𝜆𝑢(1−𝐾𝑣)=0, (5) with boundary conditions: ()=(),()=𝜈𝑢,𝑣)(−∞0,0𝑢,𝑣)(+∞1−𝐾,1𝐾. (6) For the notational convenience, we further set 𝑢𝑢=,1−𝜈/𝐾 (7) and drop the bar on 𝑢 to have 𝑢_𝜉𝜉−𝑐𝑢_𝜉𝜈+𝑢1−𝜆−1−𝐾𝑢+(𝜆𝐾−𝜈)𝑣=0,−𝑐𝑣_𝜉𝜈+𝜆1−𝐾𝑢()()=(),()=11−𝐾𝑣=0,𝑢,𝑣)(−∞0,0𝑢,𝑣)(+∞1,𝐾. (8)(9)

Numerical investigations [1] strongly suggest that system (10) and (9) admit traveling wave solutions for 𝜈=0 and 𝜈=1. When the differentiated cell density does not affect the proliferation of the precursor cells, we have 𝜈=0; when the total cell population contributes to the proliferation carrying capacity, we have 𝜈=1. Numerically, however, when 𝜈=1, (8) may have nonmonotone traveling wave solutions and requires a different treatment. Hence, in this paper we only study the wave solutions for 𝜈=0. The system (8) in this case can be further reduced to 𝑢_𝜉𝜉−𝑐𝑢_𝜉+𝑢(1−𝜆−𝑢+𝜆𝐾𝑣)=0,−𝑐𝑣_𝜉+𝜆𝑢(1−𝐾𝑣)=0. (10)

The computations in [1] show that the wave may exist for 𝑐≥2√1−𝜆, but a rigorous existence proof is still lacking. We will confirm this observation by a mathematical analysis of the model. The system is of cooperative type, and we can use the monotone iteration scheme developed in [10] for the existence proof. Such method reduces the existence of the wave solutions to that of the ordered upper and lower solution pairs for (10) and (9). The upper and lower solutions in this paper come directly from two KPP type equations, which are constructed so that they have the same decay rate at −∞. Such information is also relevant to the monotonicity and uniqueness of the wave solutions. Indeed, since we have a good understanding of the decay properties of the solutions at infinities, we then can study the properties of the solutions on finite domain, in which the powerful sliding domain method (see [11]) can be used to have the desired results. We remark that the methods we used in the proofs of the monotonicity and the uniqueness have subtle difference from the ones used in [12].

For a comprehensive study and interesting applications of the traveling wave solutions arising in various degenerate or nondegenerate parabolic equations and systems, please see [13].

2. The Main Result

In this section, we will use monotone iteration method to set up the upper and lower solutions for system (10) and (9).

Definition 1. A 𝐶²(ℝ)×𝐶¹(ℝ) function (𝑢(𝜉),𝑣(𝜉))^𝑇, 𝜉∈ℝ is an upper solution of (10) and (9) if it satisfies 𝑢_𝜉𝜉−𝑐𝑢_𝜉+𝑢(1−𝜆−𝑢+𝜆𝐾𝑣)≤0,−𝑐𝑣_𝜉+𝜆𝑢(1−𝐾𝑣)≤0 (11) and the boundary conditions 𝑢𝑣()≥00,𝑢𝑣()≥11−∞+∞𝐾. (12)
We can similarly define the lower solution (𝑢,𝑣)(𝜉), 𝜉∈ℝ by reversing the inequalities (11) and (12).

The following known result [14] is needed in the construction of the upper and lower solutions.

Consider the following form of the KPP equation: 𝜔^′′−𝑐𝜔^′(𝜔)𝜔()()+𝑓=0,−∞=0,𝜔+∞=𝑏, (13) where 𝑓∈C²([0,𝑏]) and 𝑓>0 on the open interval (0,𝑏) with 𝑓(0)=𝑓(𝑏)=0, 𝑓^′(0)=𝑎>0, and 𝑓^′(𝑏)=−𝑏₁<0.

Lemma 2. Corresponding to every 𝑐≥2√𝑎, system (13) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution 𝜔(𝜉) for 𝜉∈ℝ. The traveling wave solution 𝜔 has the following asymptotic behaviors.
For the wave solution with noncritical speed 𝑐>2√𝑎, one has 𝜔(𝜉)=𝑎_𝜔𝑒^{√((𝑐−𝑐2−4𝑎)/2)𝜉}+𝑜𝑒^{√((𝑐−𝑐2−4𝑎)/2)𝜉}𝜔(𝜉)𝑎𝑠𝜉⟶−∞,=𝑏−𝑏_𝜔𝑒^{((𝑐−𝑐2+4𝑏1)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝑏1)/2)𝜉}𝑎𝑠𝜉⟶+∞, (14) where 𝑎_𝜔 and 𝑏_𝜔 are positive constants.
For the wave with critical speed 𝑐=2√𝑎, one has 𝜔(𝜉)=𝑑_𝑐𝜉𝑒^√𝑎𝜉+𝑜𝜉𝑒^√𝑎𝜉𝜔(𝜉)𝑎𝑠𝜉⟶−∞,=𝑏−𝑏_𝑐𝑒^{(√𝑎−𝑎+𝑏1)𝜉}+𝑜𝑒^{(√𝑎−𝑎+𝑏1)𝜉}𝑎𝑠𝜉⟶+∞, (15) where the constant 𝑑_𝑐 is negative and 𝑏_𝑐 is positive.

We next consider the following version of the KPP system: 𝜔^′′−𝑐𝜔^′+()𝜔()𝜔()()1−𝜆1−𝜔=0,−∞=0,𝜔+∞=1. (16) According to Lemma 2, for every 𝑐≥2√1−𝜆 system, (16) has a unique (up to a translation of the origin) monotone solution 𝑢(𝜉), 𝜉∈ℝ. Now, fix this 𝑢(𝜉) and consider the equation −𝑐𝑣_𝜉+𝜆𝑢(1−𝐾𝑣)=0. (17)

For each fixed 𝑐≥2√1−𝜆 and the corresponding 𝑢(𝜉), (17) has a solution 𝑣(𝜉)=1𝐾1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞}𝑢_{(𝑠)𝑑𝑠}. (18)

We next compare 𝑢(𝜉) with 𝑣(𝜉) for 𝜉∈ℝ.

Lemma 3. There exists a 𝜁₁≥0 such that if 20<𝜆≤2+𝐾1+√2, (19) one has 1𝐾𝑢𝜉+𝜁₁≥𝑣(𝜉),𝜉∈ℝ. (20)

Proof. According to Lemma 2, the wave solution 𝑢(𝜉) to (16) has the following asymptotic behaviors.
For 𝑐>2√1−𝜆, 𝑢(𝜉)=𝑎_𝜔𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜉}𝑢(𝜉)as𝜉⟶−∞,=𝑏−𝑏_𝜔𝑒^{((𝑐−𝑐2+4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4(1−𝜆))/2)𝜉}as𝜉⟶+∞, (21)(22) and 𝑎_𝜔, 𝑏_𝜔 are positive constants.
For 𝑐=2√1−𝜆, we have 𝑢(𝜉)=𝑑_𝑐𝜉𝑒^{√1−𝜆𝜉}+𝑜𝜉𝑒^{√1−𝜆𝜉}𝑢(𝜉)as𝜉⟶−∞,=𝑏−𝑏_𝑐𝑒^{√(1−√2)1−𝜆𝜉}+𝑜𝑒^{√(1−√2)1−𝜆𝜉}as𝜉⟶+∞, (23)(24) where the constant 𝑑_𝑐 is negative and 𝑏_𝑐 is positive.
We now study the asymptotics of the function 𝑣(𝜉). Formulas (21) and (23) imply that ^𝜉_−∞𝑢(𝑠)𝑑𝑠⟶0as𝜉⟶−∞. (25) We can then expand 𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞}𝑢_{(𝑠)𝑑𝑠}𝜆=1−𝑐𝐾^𝜉_−∞𝑢(𝑠)𝑑𝑠+𝑜^𝜉_−∞𝑢(𝑠)𝑑𝑠². (26)
A further expanding of (26) for 𝜉→−∞ and for 𝑐>2√1−𝜆 yields 1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞}𝑢_{(𝑠)𝑑𝑠}=2𝜆𝑎_𝜔𝑐−𝑐²()𝑐𝑒−41−𝜆^{((𝑐−𝑐2−4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜉}, (27) and for 𝑐=2√1−𝜆, 1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞}𝑢_{(𝑠)𝑑𝑠}=𝜆2√𝑑1−𝜆^𝑐𝑑𝜉−^𝑐𝑒1−𝜆^{√1−𝜆𝜉}+𝑜𝜉𝑒^{√1−𝜆𝜉}. (28)
As for 𝜉>0 sufficiently large, we have lim_𝜉→+∞∫^𝜉_−∞𝑢(𝑠)𝑑𝑠𝜉=lim_𝜉→+∞𝑢(𝜉)=1; (29) therefore, 𝑣(𝜉)=𝐾1−𝑒^{−(𝜆/𝑐)𝐾𝜉}+𝑜𝑒^{−(𝜆/𝑐)𝐾𝜉}as𝜉⟶+∞. (30)
We next show −𝜆𝑐𝐾≥𝑐−𝑐²+4(1−𝜆)2, (31) or equivalently 2𝑐(1−𝜆)𝑐+𝑐²+4(1−𝜆)≥𝜆𝐾. (32)
Setting 𝑔(𝑐)=2(1−𝜆)/1+1+4(1−𝜆)/𝑐², then it is easy to see that 𝑔(𝑐) increases as 𝑐 does. Hence, 𝑔(𝑐)≥𝑔2√=2()1−𝜆1−𝜆1+√2. (33)
We therefore require 0<𝜆≤2/(2+𝐾(1+√2)) to have (31).
We now shift 𝑢(𝜉). Since (16) is shifting invariant, 𝑢(𝜉+𝜁),   𝜉∈ℝ is also a solution for any 𝜁∈ℝ. It then follows from (21) for 𝑐>√1−𝜆, 𝑢(𝜉+𝜁)=𝑎_𝜔𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜁}𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜉}as𝜉⟶−∞, (34) and for 𝑐=√1−𝜆, 𝑢(𝜉+𝜁)=𝑑_𝑐(𝜉+𝜁)𝑒^{√1−𝜆𝜁}𝑒^{√1−𝜆𝜉},+𝑜𝜉𝑒^{√1−𝜆𝜉}as𝜉⟶−∞. (35)
If we choose 𝜁>0 sufficiently large, the positiveness of ((𝑐−𝑐²−4(1−𝜆))/2)𝜁 and √1−𝜆𝜁 implies that if 𝑐=2√1−𝜆, 𝑑_𝑐(𝜉+𝜁)𝑒^{√1−𝜆𝜁}>𝜆2√𝑑1−𝜆^𝑐𝑑𝜉−^𝑐1−𝜆 (36) and if 𝑐>2√1−𝜆, 𝑎_𝜔𝑒^{((𝑐−𝑐2−4(1−𝜆))/2)𝜁}>2𝜆𝑎_𝜔𝑐−𝑐²()𝑐.−41−𝜆 (37)
It then follows from (31), (34), and (35) that there exists a 𝑁>0 sufficiently large, 1𝐾𝑢(𝜉+𝜁)≥𝑣(𝜉)for𝜉∈(−∞,−𝑁]∪[𝑁,+∞), (38) and for 𝜉∈[−𝑁,𝑁], since 𝑢 and 𝑣 are both monotonically increasing on ℝ we can further shift 𝑢(𝜉+𝜁) to the left at most 2𝑁 units to have (1/𝐾)𝑢(𝜉+𝜁)≥𝑣(𝜉), 𝜉∈ℝ. Hence there exists a finite 𝜁₁≥0 such that the conclusion of the Lemma holds.

Now, we write 𝑢(𝜉)=𝑢(𝜉+𝜁₀),  𝜉∈ℝ and let 𝑣(𝜉) be defined in (18). We remark here that the computation of 𝑣(𝜉) still uses 𝑢(𝜉).

Lemma 4. Assume the conditions in Lemma 3, then (𝑢,𝑣)(𝜉), 𝜉∈ℝ defines an upper solution for (10) and (9).

Proof. We can easily verify that (𝑢,𝑣)(𝜉) satisfies the boundary conditions (12).
For the 𝑢 component, we have 𝑢^′′−𝑐𝑢^′+𝑢(1−𝜆−𝑢+𝜆𝐾𝑣)=𝑢[1−𝜆−𝑢+𝜆𝐾𝑣−(1−𝜆)(1−𝑢)]=−𝜆𝐾𝑢1𝐾𝑢−𝑣≤0. (39) The last inequality follows from the previous lemma.
As for the 𝑣 component, for each 𝑢, we have −𝑐𝑣_𝜉+𝜆𝑢(1−𝐾𝑣)=0. (40)

We next set up the lower solution for (10) and (9).

For a fixed 𝑙>0, we consider another version of the KPP system: 𝑤^′′−𝑐𝑤^′+(1−𝜆)𝑤1−1−𝜆+𝑙𝑤𝑤()()=1−𝜆=0,−∞=0,𝑤+∞1−𝜆1−𝜆+𝑙<1. (41) Then, for any 𝑐≥2√1−𝜆, (41) has correspondingly a unique wave solution 𝑢̆(𝜉), 𝜉∈ℝ.

We define 𝑣(𝜉)=1𝐾1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞̆𝑢(𝑠)𝑑𝑠}. (42)

The next lemma gives the relation between 𝑢̆(𝜉) and 𝑣(𝜉), 𝜉∈ℝ.

Lemma 5. There exists a 𝜁₁≥0 such that 1𝐾𝑢̆𝜉−𝜁₁≤𝑣(𝜉),𝜉∈ℝ. (43)

Proof. The proof is similar to that of Lemma 3. Noting as 𝜉→+∞,  𝑢̆(𝜉)→(1/𝐾)((1−𝜆)/(1−𝜆+𝑙))<1/𝐾. Hence, we do not need condition (19) here.

We denote 𝑢(𝜉)=𝑢̆(𝜉−𝜁₁),  𝜉∈ℝ. Then, we have the following.

Lemma 6. Such defined (𝑢,𝑣)(𝜉), 𝜉∈ℝ consists of a lower solution for (10) and (9).

Proof. One the boundary, we have 𝑢,𝑣()=(),𝑢−∞0,0,𝑣(+∞)=1−𝜆,11−𝜆+𝑙𝐾≤11,𝐾 (44) and for the 𝑢 component, 𝑢^′′−𝑐𝑢^′+𝑢1−𝜆−𝑢+𝜆𝐾𝑣=𝑢^′′−𝑐𝑢^′+(1−𝜆)𝑢1−1−𝜆+𝑙𝑢1−𝜆−(1−𝜆)𝑢1−1−𝜆+𝑙𝑢1−𝜆+𝑢1−𝜆−𝑢+𝜆𝐾𝑣=𝑢1−𝜆−𝑢+𝜆𝐾𝑣−(1−𝜆)+(1−𝜆+𝑙)𝑢≥0 (45) due to Lemma 5.
Noting that 𝑣(𝜉) solves the equation −𝑐𝑣_𝜉+𝜆𝑢̆1−𝐾𝑣=0, (46) it satisfies the inequality trivially.

Lemma 7. The upper and lower solutions are ordered (𝑢,𝑣)(𝜉)≥𝑢,𝑣(𝜉),𝜉∈ℝ. (47)

Proof. For each fixed 𝑐≥2√1−𝜆, if 𝑢̆(𝜉) solves the system (41), then the function 𝑢(𝜉)=((1−𝜆+𝑙)/(1−𝜆))𝑢̆(𝜉) solves (16). Hence, it follows that 𝑢(𝜉)>𝑢̆(𝜉), and then 𝑢(𝜉)>𝑢(𝜉) for all 𝜉∈ℝ.
By definition of 𝑣(𝜉) and 𝑣(𝜉), we have 𝑣(𝜉)=1𝐾1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞}𝑢_{(𝑠)𝑑𝑠}=1𝐾1−𝑒^{−(𝜆/𝑐)𝐾((1−𝜆+𝑙)/(1−𝜆))∫𝜉−∞̆𝑢(𝑠)𝑑𝑠}>1𝐾1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞̆𝑢(𝑠)𝑑𝑠}=𝑣(𝜉). (48)
Hence, the conclusion of the Lemma holds.

Theorem 8. Let the parameters satisfy (19), then for each 𝑐≥2√1−𝜆, system (10) and (9) have a unique (up to a translation of the origin) strictly monotonically increasing traveling wave solution, while for 0<𝑐<2√1−𝜆, there is no monotonic traveling wave. The traveling wave solution has the following asymptotic behaviors.
For 𝑐=2√1−𝜆, 𝑢𝑣(𝜉)=𝑐₁₁𝜉𝑐₁₂𝜉𝑒^{√1−𝜆𝜉}+𝑜𝑒^{√1−𝜆𝜉}𝑢𝑣(𝜉)=11,𝑎𝑠𝜉⟶−∞,𝐾−𝑐₂₁𝑒^{√−(𝜆𝐾/21−𝜆)𝜉}𝑐₂₂𝑒^{√−(𝜆𝐾/21−𝜆)𝜉}+𝑜𝑒^{√−(𝜆𝐾/21−𝜆)𝜉}𝑜𝑒^{√−(𝜆𝐾/21−𝜆)𝜉},𝑎𝑠𝜉⟶+∞, (49)(50) and for 𝑐>2√1−𝜆, 𝑢𝑣(𝜉)=𝑑₁₁𝜉𝑑₁₂𝜉𝑒^{((𝑐−𝑐2+4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4(1−𝜆))/2)𝜉}𝑢𝑣𝜉)=11,𝑎𝑠𝜉⟶−∞,(𝐾−𝑑₂₁𝑒^{−(𝜆𝐾/𝑐)𝜉}𝑑₂₂𝑒^{−(𝜆𝐾/𝑐)𝜉}+𝑜𝑒^{−(𝜆𝐾/𝑐)𝜉}𝑜𝑒^{−(𝜆𝐾/𝑐)𝜉}𝑎𝑠𝜉⟶+∞, (51)(52) where 𝑐₁₁, 𝑐₁₂, 𝑐₂₁, 𝑐₂₂, 𝑑₂₁, and 𝑑₂₂>0 and 𝑑₁₁,𝑑₁₂<0.

Proof. Noting that between the upper and lower solutions, there is no equilibrium other than (0,0) and (1,1/𝐾) of system (10) and (9). Hence, the monotone iteration scheme developed in [10] is still applicable. Such monotone iteration scheme reduces the existence of the traveling wave solutions to that of the ordered upper and lower solution pairs, and the existence of the traveling waves then follows by Lemmas 6, 4, and 7, and by [10], so the obtained traveling wave solutions are nondecreasing, while for 𝑐<2√1−𝜆, it is easy to verify, by analyzing the equilibrium (0,0), that the nontrivial bounded solutions of (10) are oscillatory.
We next show that the wave solutions are strictly monotonically increasing on ℝ.
For any fixed 𝑐≥√1−𝜆, let (𝑢_𝑐,𝑣_𝑐)(𝜉) be the corresponding traveling wave solution and (𝑤₁(𝜉),𝑤₂(𝜉)) be its derivative. Then, (𝑤₁(𝜉),𝑤₂(𝜉))≥0 for 𝜉∈ℝ, and (𝑤₁(𝜉),𝑤₂(𝜉)) satisfies the following systems: 𝑤_1,𝜉𝜉−𝑐𝑤_1,𝜉+1−𝜆−2𝑢_𝑐+𝜆𝐾𝑣_𝑐𝑤₁+𝜆𝐾𝑢_𝑐𝑤₂=0,−𝑐𝑤_2,𝜉+𝜆1−𝐾𝑣_𝑐𝑤₁−𝜆𝐾𝑢_𝑐𝑤₂𝑤=0,₁,𝑤₂(±∞)=0. (53) It then follows that 𝑤_1,𝜉𝜉−𝑐𝑤_1,𝜉+1−𝜆−2𝑢_𝑐+𝜆𝐾𝑣_𝑐𝑤₁≤0,−𝑐𝑤_2,𝜉+𝜆1−𝐾𝑣_𝑐𝑤₁−𝜆𝐾𝑢_𝑐𝑤₂𝑤=0,₁,𝑤₂(±∞)=0. (54)
Applying the maximum principle to the first inequality of (54), we immediately conclude that 𝑤₁(𝜉)>0 for 𝜉∈ℝ. Thu, 𝑢_𝑐(𝜉) is strictly monotonically increasing.
The strict monotonicity of 𝑣_𝑐(𝜉) comes from (10). Since 𝑢_𝑐(𝜉)>0 for all 𝜉∈ℝ, and for such 𝑢_𝑐(𝜉), we have 𝑣_𝑐(𝜉)=1𝐾1−𝑒^{−(𝜆/𝑐)𝐾∫𝜉−∞𝑢𝑐(𝑠)𝑑𝑠}, (55) then it follows that 𝑤₂(𝜉)=𝑣^′_𝑐(𝜉)>0, 𝜉∈ℝ. This shows that the wave solution (𝑢_𝑐,𝑣_𝑐) is strictly monotonically increasing.
We then derive the asymptotics of the wave solutions at ±∞. Noting that the upper and lower solutions have the same exponential decay rate at −∞, (49) and (51) come directly from comparison.
We next study the asymptotics of the function (𝑤₁,𝑤₂)(𝜉) at +∞, recalling that (𝑤₁,𝑤₂)(𝜉)=(𝑢_𝑐,𝑣_𝑐)^′(𝜉) and that it satisfies the system (53). Since this system is hyperbolic at +∞, (𝑤₁,𝑤₂) approaches (0,0) exponentially. We will derive the exact exponential rate.
The limit equation at +∞ of system (53) is 𝑤⁺_1,𝜉𝜉−𝑐𝑤⁺_1,𝜉−𝑤⁺₁+𝜆𝐾𝑤⁺₂=0,−𝑐𝑤⁺_2,𝜉−𝜆𝐾𝑢_𝑐𝑤⁺₂=0. (56)
Since the second equation is decoupled from the system, we immediately have 𝑤⁺₂(𝜉)=𝐴𝑒^{−(𝜆𝐾/𝑐)𝜉}. (57) Plugging the above into the first equation yields a bounded solution (up to the first order approximation) of the form 𝑤⁺₁(𝜉)=𝐴₁𝑒^{−(𝜆𝐾/𝑐)𝜉}+𝐴₂𝑒^{√((𝑐−𝑐+4)/2)𝜉}. (58)
By roughness of exponential dichotomy [15], we have 𝑤₁(𝜉)𝑤₂(𝜉)=𝐴₁𝑒^{−(𝜆𝐾/𝑐)𝜉}𝐴₂𝑒^𝜇𝜉+𝑜𝑒^{−(𝜆𝐾/𝑐)𝜉}𝑜𝑒^𝜇𝜉, (59) where 𝜇 is either −𝜆𝐾/𝑐 or (𝑐−√𝑐+4)/2.
Integrating the above from 𝜉₀ to +∞ and comparing the decay rates of (𝑢_𝑐,𝑣_𝑐)(𝜉) with that of the upper solution (𝑢,𝑣)(𝜉), we have (50) and (52).
On the uniqueness of the traveling wave solution for every 𝑐≥2√1−𝜆, we only prove the conclusion for traveling wave solutions with asymptotic rates given in (51) and (52) since the other case can be proved similarly. Let 𝑈₁(𝜉)=(𝑢₁,𝑣₁)(𝜉) and 𝑈₂(𝜉)=(𝑢₂,𝑣₂)(𝜉) be two traveling wave solutions of system (10) and (9) with the same speed 𝑐>2√1−𝜆. There exist positive constants 𝐴_𝑖𝑗, 𝐵_𝑖𝑗,  𝑖,𝑗=1,2 and a large number 𝑁>0 such that for 𝜉<−𝑁, 𝑈₁(𝜉)=𝐴₁₁𝐴₁₂𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜉}𝑈₂(𝜉)=𝐴₂₁𝐴₂₂𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜉} (60)(61) and for 𝜉>𝑁, 𝑈₁(𝜉)=1−𝐵₁₁𝑒^{−(𝜆/𝑐)𝐾𝜉}1𝐾−𝐵₁₂𝑒^{−(𝜆/𝑐)𝐾𝜉}+𝑜𝑒^{−(𝜆/𝑐)𝐾𝜉}𝑒^{−(𝜆/𝑐)𝐾𝜉}𝑈₂,(𝜉)=1−𝐵₂₁𝑒^{−(𝜆/𝑐)𝐾𝜉}1𝐾−𝐵₂₂𝑒^{−(𝜆/𝑐)𝐾𝜉}+𝑜𝑒^{−(𝜆/𝑐)𝐾𝜉}𝑒^{−(𝜆/𝑐)𝐾𝜉}. (62)(63) The traveling wave solutions of system (10)-(9) are translation invariant; thus, for any 𝜃>0, 𝑈^𝜃₁(𝜉)∶=𝑈₁(𝜉+𝜃) is also a traveling wave solution of (10)-(9). By (60) and (62), the solution 𝑈₁(𝜉+𝜃) has the asymptotics 𝑈^𝜃₁(𝜉)=𝐴₁₁𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜃}𝐴₁₂𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜃}𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜉}+𝑜𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜉} (64) for 𝜉≤−𝑁 and 𝑈^𝜃₁(𝜉)=1−𝐵₁₁𝑒^{−(𝜆/𝑐)𝐾𝜃}𝑒^{−(𝜆/𝑐)𝐾𝜉}1𝐾−𝐵₁₂𝑒^{−(𝜆/𝑐)𝐾𝜃}𝑒^{−(𝜆/𝑐)𝐾𝜉}+𝑜𝑒^{−(𝜆/𝑐)𝐾𝜉}𝑒^{−(𝜆/𝑐)𝐾𝜉} (65) for 𝜉≥𝑁.
Choosing 𝜃>0 large enough such that 𝐴₁₁𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜃}>𝐴₂₁,𝐴₁₂𝑒^{((𝑐+𝑐2−4(1−𝜆))/2)𝜃}>𝐴₂₂,𝐵₁₁𝑒^{−(𝜆/𝑐)𝐾𝜃}<𝐵₂₁,𝐵₁₂𝑒^{−(𝜆/𝑐)𝐾𝜃}<𝐵₂₂, (66) then one has for 𝜉∈(−∞,−𝑁]∪[𝑁,+∞), 𝑈^𝜃₁(𝜉)>𝑈₂(𝜉). (67)
We now consider system (10) on [−𝑁,+𝑁]. There are two possibilities.
Case 1. Suppose that we already have 𝑈^𝜃₁(𝜉)≥𝑈₂(𝜉) on [−𝑁,+𝑁], then the function 𝑊(𝜉)=(𝑤₁(𝜉),𝑤₂(𝜉))^𝑇∶=𝑈^𝜃₁(𝜉)−𝑈₂(𝜉)≥0 and it satisfies for some 𝜁_𝑖∈(0,1), 𝑖=1,2, 𝑤₁^′′0𝑤−𝑐^′1𝑤^′₂𝑤+𝑀¹𝑤₂𝑊()()(),=0,−𝑁>0,𝑊+𝑁>0.𝜉∈−𝑁,𝑁 (68) where the matrix 𝑀 is given by 𝑀𝑤₁,𝑤₂=1−𝜆−2𝑢₂+𝜁₁𝑤₁+𝜆𝐾𝑣₂+𝜁₂𝑤₂,𝜆𝐾𝑢₂+𝜁₁𝑤₁𝜆1−𝐾𝑣₂+𝜁₂𝑤₂,−𝜆𝐾𝑢₂+𝜁₂𝑤₁. (69)
Since 𝑤₁(𝜉)≥0, 𝜉∈[−𝑁,𝑁] and 𝜆𝐾(𝑢₂+𝜁₁𝑤₁)≥0, then we have on 𝜉∈[−𝑁,𝑁], 𝑤₁^′′−𝑐𝑤^′₁+1−𝜆−2𝑢₂+𝜁₁𝑤₁+𝜆𝐾𝑣₂+𝜁₂𝑤₂𝑤₁+𝜆𝐾𝑢₂+𝜁₂𝑤₁𝑤₂𝑤=0,₁(−𝑁)>0,𝑤₁(𝑁)>0. (70)
The maximum principle then implies that 𝑤₁(𝜉)>0 on [−𝑁,𝑁]. We then move to the second equation of (68). We have −𝑐𝑤^′₂−𝜆𝐾𝑢₂+𝜁₂𝑤₁𝑤₂=−𝜆1−𝐾𝑣₂+𝜁₂𝑤₂𝑤₁[],𝑤<0,𝜉∈−𝑁,𝑁₂(−𝑁)>0,𝑤₂(𝑁)>0. (71)
The strict inequality comes from the fact that 𝑣₂(𝜉)≤𝑣₂(𝜉)+𝜁₂𝑤₂(𝜉)≤𝑣^𝜃₁<1/𝐾 for 𝜉∈[−𝑁,𝑁]. It then follows that 𝑤₂(𝜉)>0 for 𝜉∈[−𝑁,𝑁]. For if there is a 𝜉∈(−𝑁,𝑁) such that 𝑤₂(𝜉)=0, then 𝑤₂ takes local minimum at 𝜉 and the left hand side of the first inequality of (71) is zero at 𝜉. We then have a contradiction.
Case 2. We may suppose that there is some point in (−𝑁,𝑁) such that one of the components, say the 𝑗th component, satisfies (𝑈^𝜃₁(𝜉))_𝑗<(𝑈₂(𝜉))_𝑗 at that point, 𝑗=1 or 2. We then increase 𝜃, that is, shift 𝑈^𝜃₁(𝜉) further left, so that 𝑈^𝜃₁(−𝑁)>𝑈₂(−𝑁),  𝑈^𝜃₁(𝑁)>𝑈₂(𝑁). By the monotonicity of 𝑈^𝜃₁ and 𝑈₂, we can find a 𝜃∈(0,2𝑁) such that in the interval (−𝑁,𝑁) we have 𝑈^𝜃₁(𝜉+𝜃)>𝑈₂(𝜉). Shifting 𝑈^𝜃₁(𝜉+𝜃) back until one component of 𝑈^𝜃₁(𝜉+𝜃) first touches its counterpart of 𝑈₂(𝜉) at some point 𝜉∈[−𝑁,𝑁], we then return back to Case 1 again, where it has been shown that this is impossible. Therefore, we must have 𝑈^𝜃₁(𝜉)>𝑈₂(𝜉) (72) for all 𝜉∈ℝ, where 𝜃 is the one chosen by means of (66) as described above.
Now, decrease 𝜃 until one of the following situations happens.(1)There exists a 𝜃≥0, such that 𝑈^𝜃₁(𝜉)≡𝑈₂(𝜉). In this case, we have finished the proof.(2)There exists a 𝜃≥0 and 𝜉₁∈ℝ, such that one of the components of 𝑈^𝜃 and 𝑈₂ are equal there; for all 𝜉∈ℝ, we have 𝑈^𝜃₁(𝜉)≥𝑈₂(𝜉). On applying the maximum principle on ℝ and using the same argument as we did for Case 1, we see that this is impossible.
Consequently, in either situation, there exists a 𝜃≥0, such that 𝑈^𝜃₁(𝜉)≡𝑈₂(𝜉) (73) for all 𝜉∈ℝ.