Analysis of a model arising from invasion by precursor and differentiated cells

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 equals a certain number, and below which there is no monotonic wave solutions. We also investigate the monotonicity, uniqueness and asymptotics of the waves.


introduction
In [8], the following coupled partial differential equation system was proposed to study the invasion by precursor and differentiated cells: , 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 k 1 > 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 k 1 . The cell population density v is limited by its carrying capacity k 2 and has a max. differentation rate β > 0. The model assumes that the differentiated cells do not have mobility.
Letting (see [8] and dropping the hat notation for convenience, system (1.1) is changed into where λ = β α and K = k1 k2 . System (1.1) or (1.2) belongs to reaction diffusion systems of degenerate type, and such systems have attracted much attention in the research such as epidemics and wound healing [9,10,3]. However, system (1.2) differs from the above systems in the appearance of degenerate reaction terms. In fact, u = 0 coupling with any v = constant consist of a constant solution of (1.2). This resembles the combustion wave equation considered in [5], however our method in proving the existence of the fronts of (1.2) differs from theirs.
If the parameters satisfy then system (1.2) 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 (1.2) solves with boundary conditions: For the notional convenience we further set and dropping the bar on u to have Numerical investigations [8] strongly suggest that system (1.8) and (1.7) admit traveling wave solutions for ν = 0 and ν = 1. When the differentiated cell density does not affect the proliferation of the the percursor cells, we have ν = 0; and when the the total cell population contributes to the proliferation carrying capacity, we have ν = 1. Numerically, however when ν = 1, (1.6) may have non-monotone traveling wave solutions and requires a different treatment. Hence in this paper we only study the wave solutions for ν = 0. The system (1.6) in this case can be further reduced to The computations in [8] shows 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, we can use the monotone iteration scheme developed in [4] for the existence proof. Such method reduces the existence of the wave solutions to that of the ordered upper and lower solution pairs for (1.8) and (1.7). The upper and lower solutions in this paper come straightly from two KPP type equations, which are so constructed that they have the same decay rate at −∞. Such information is also relevent 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 [1]) 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 [6].

the main result.
In this section we will use monotone iteration method to set up the upper and lower solutions for system (1.8) and (1.7).
and the boundary conditions We can similarly define the lower solution (u, v)(ξ), ξ ∈ R by reversing the inequalities (2.1) and (2.2).
The following known result ( [7]) is needed in the construction of the upper and lower solutions: Consider the following form of the KPP equation: Corresponding to every c ≥ 2 √ā , system (2.3) 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 non-critical speed c > 2 √ā , we have whereā ω andb ω are positive constants. For the wave with critical speed c = 2 √ā , we have where the constantd c is negative,b c is positive.
Lemma 3. There exists a ζ 1 ≥ 0 such that if Proof. According to Lemma 2, the wave solutionũ(ξ) to (2.8) has the following asymptotic behaviors: and a ω , b ω are positive constants.
For the u component we havē The last inequality follows from the previous Lemma.
We next set up the lower solution for (1.8) and (1.7). For a fixed l > 0 we consider another version of the KPP system: Then for any c ≥ 2 √ 1 − λ, (2.24) has correspondingly a unique wave solutionȗ(ξ), ξ ∈ R.
By the definition ofv(ξ) and v(ξ), we havē Hence the conclusion of the Lemma holds.
Proof. Noting that between the upper and lower solutions, there is no equilibrium other than (0, 0) and (1, 1 K ) of system (1.8) and (1.7). Hence the monotone iteration scheme developed in [4] 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, the existence of the traveling waves then follows by Lemma 6, Lemma 4 and Lemma 7, and by [4], such 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 (1.8) are oscillatory. We next show that the wave solutions are strictly monotonically increasing on R.
The strict monotonicity of v c (ξ) comes from (1.8). Since u c (ξ) > 0 for all ξ ∈ R, and for such u c (ξ) we have 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 −∞, (2.27) and (2.29) come directly from comparison.
The limit equation at +∞ of system (2.31) is −cw + 2,ξ − λKu c w + 2 = 0 Since the second equation is decoupled from the system, we immediately have Plugging the above into the first equation yields a bounded solution (up to the first order approximation) of the form By roughness of exponential dichotomy [?], we have Integrating the above from ξ 0 to +∞, and comparing the decay rates of (u c , v c )(ξ) with that of the upper solution (ū,v)(ξ), we have (2.28) and (2.30).
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 (2.29) and (2.30) since the other case can be proved similarly. Let U 1 (ξ) = (u 1 , v 1 )(ξ) and U 2 (ξ) = (u 2 , v 2 )(ξ) be two traveling wave solutions of system (1.8) and (1.7) 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 ,