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International Journal of Differential Equations
Volume 2013 (2013), Article ID 341473, 7 pages
http://dx.doi.org/10.1155/2013/341473
Research Article

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

Department of Mathematics, University of North Carolina Wilmington, Wilmington, NC 28403, USA

Received 9 April 2013; Accepted 7 August 2013

Academic Editor: Zhi-Qiang Wang

Copyright © 2013 Xiaojie Hou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: where denotes the population densities of the precursor cells. The constant is the diffusion rate of the cell , which has proliferation rate , and is the carrying capacity of . The parameter measures the relative contribution that the differentiated cell with population density makes to the carrying capacity . The cell population density is limited by its carrying capacity and has a maximum differentiation rate . The model assumes that the differentiated cells do not have mobility.

By letting (see [1]) and dropping the hat notation for convenience, system (1) is changed into where and .

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 [24] as well as combustion and calcium wave problems [58]. However, system (3) differs from the above systems in the appearance of degenerate reaction terms. In fact, coupling with any 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 then system (3) admits an additional equilibrium: representing the state that the spatial domain is successfully invaded. We also separate the equilibrium from the rest of the line of equilibria, . The unstable equilibrium represents the state before the invasion.

We are interested in the existence of the wave solutions connecting with as time and space evolve from to . Setting , , , a traveling wave solution to (3) solves with boundary conditions: For the notational convenience, we further set and drop the bar on to have

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

The computations in [1] show that the wave may exist for , 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 function , is an upper solution of (10) and (9) if it satisfies and the boundary conditions
We can similarly define the lower solution , 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: where and on the open interval with , , and .

Lemma 2. Corresponding to every , system (13) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution for . The traveling wave solution has the following asymptotic behaviors.
For the wave solution with noncritical speed , one has where and are positive constants.
For the wave with critical speed , one has where the constant is negative and is positive.

We next consider the following version of the KPP system: According to Lemma 2, for every system, (16) has a unique (up to a translation of the origin) monotone solution , . Now, fix this and consider the equation

For each fixed and the corresponding , (17) has a solution

We next compare with for .

Lemma 3. There exists a such that if one has

Proof. According to Lemma 2, the wave solution to (16) has the following asymptotic behaviors.
For , and , are positive constants.
For , we have where the constant is negative and is positive.
We now study the asymptotics of the function . Formulas (21) and (23) imply that We can then expand
A further expanding of (26) for and for yields and for ,
As for sufficiently large, we have therefore,
We next show or equivalently
Setting , then it is easy to see that increases as does. Hence,
We therefore require to have (31).
We now shift . Since (16) is shifting invariant, ,    is also a solution for any . It then follows from (21) for , and for ,
If we choose sufficiently large, the positiveness of and implies that if , and if ,
It then follows from (31), (34), and (35) that there exists a sufficiently large, and for , since and are both monotonically increasing on we can further shift to the left at most units to have , . Hence there exists a finite such that the conclusion of the Lemma holds.

Now, we write ,   and let be defined in (18). We remark here that the computation of still uses .

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

Proof. We can easily verify that satisfies the boundary conditions (12).
For the component, we have The last inequality follows from the previous lemma.
As for the component, for each , we have

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

For a fixed , we consider another version of the KPP system: Then, for any , (41) has correspondingly a unique wave solution , .

We define

The next lemma gives the relation between and , .

Lemma 5. There exists a such that

Proof. The proof is similar to that of Lemma 3. Noting as ,  . Hence, we do not need condition (19) here.

We denote ,  . Then, we have the following.

Lemma 6. Such defined , consists of a lower solution for (10) and (9).

Proof. One the boundary, we have and for the component, due to Lemma 5.
Noting that solves the equation it satisfies the inequality trivially.

Lemma 7. The upper and lower solutions are ordered

Proof. For each fixed , if solves the system (41), then the function solves (16). Hence, it follows that , and then for all .
By definition of and , we have
Hence, the conclusion of the Lemma holds.

Theorem 8. Let the parameters satisfy (19), then for each , system (10) and (9) have a unique (up to a translation of the origin) strictly monotonically increasing traveling wave solution, while for , there is no monotonic traveling wave. The traveling wave solution has the following asymptotic behaviors.
For , and for , where , , , , , and and .

Proof. Noting that between the upper and lower solutions, there is no equilibrium other than and 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 , it is easy to verify, by analyzing the equilibrium , that the nontrivial bounded solutions of (10) are oscillatory.
We next show that the wave solutions are strictly monotonically increasing on .
For any fixed , let be the corresponding traveling wave solution and be its derivative. Then, for , and satisfies the following systems: It then follows that
Applying the maximum principle to the first inequality of (54), we immediately conclude that for . Thu, is strictly monotonically increasing.
The strict monotonicity of comes from (10). Since for all , and for such , we have then it follows that , . This shows that the wave solution 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 at , recalling that and that it satisfies the system (53). Since this system is hyperbolic at , approaches exponentially. We will derive the exact exponential rate.
The limit equation at of system (53) is
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 [15], we have where is either or .
Integrating the above from to and comparing the decay rates of with that of the upper solution , we have (50) and (52).
On the uniqueness of the traveling wave solution for every , 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 and be two traveling wave solutions of system (10) and (9) with the same speed . There exist positive constants , ,   and a large number such that for , and for , The traveling wave solutions of system (10)-(9) are translation invariant; thus, for any , is also a traveling wave solution of (10)-(9). By (60) and (62), the solution has the asymptotics for and for .
Choosing large enough such that then one has for ,
We now consider system (10) on . There are two possibilities.
Case 1. Suppose that we already have on , then the function and it satisfies for some , , where the matrix is given by
Since , and , then we have on ,
The maximum principle then implies that on . We then move to the second equation of (68). We have
The strict inequality comes from the fact that for . It then follows that for . For if there is a such that , then takes local minimum at and the left hand side of the first inequality of (71) is zero at . We then have a contradiction.
Case 2. We may suppose that there is some point in such that one of the components, say the th component, satisfies at that point, or . We then increase , that is, shift further left, so that ,  . By the monotonicity of and , we can find a such that in the interval we have . Shifting back until one component of first touches its counterpart of at some point , we then return back to Case 1 again, where it has been shown that this is impossible. Therefore, we must have for all , where is the one chosen by means of (66) as described above.
Now, decrease until one of the following situations happens.(1)There exists a , such that . In this case, we have finished the proof.(2)There exists a and , such that one of the components of and are equal there; for all , we have . On applying the maximum principle on and using the same argument as we did for Case 1, we see that this is impossible.
Consequently, in either situation, there exists a , such that for all .

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