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

Definition 1. A C²(ℝ) × C¹(ℝ) function $$, ξ ∈ ℝ is an upper solution of (10) and (9) if it satisfies

and the boundary conditions

We can similarly define the lower solution $$, ξ ∈ ℝ by reversing the inequalities (11) and (12).

Lemma 2. Corresponding to every $$, 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 $$, one has

where $$ and $$ are positive constants.

For the wave with critical speed $$, one has

where the constant $$ is negative and $$ is positive.

Lemma 3. There exists a ζ₁ ≥ 0 such that if

one has

Proof. According to Lemma 2, the wave solution $$ to (16) has the following asymptotic behaviors.

For $$,

and a_ω, b_ω are positive constants.

For $$, we have

where the constant d_c is negative and b_c 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 ξ → −∞ and for $$ yields

and for $$,

As for ξ > 0 sufficiently large, we have

therefore,

We next show

or equivalently

Setting $$, then it is easy to see that g(c) increases as c does. Hence,

We therefore require $$ to have (31).

We now shift $$. Since (16) is shifting invariant, $$,   ξ ∈ ℝ is also a solution for any ζ ∈ ℝ. It then follows from (21) for $$,

and for $$,

If we choose ζ > 0 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 ξ ∈ [−N, N], since $$ and $$ are both monotonically increasing on ℝ we can further shift $$ to the left at most 2N units to have $$, ξ ∈ ℝ. Hence there exists a finite ζ₁ ≥ 0 such that the conclusion of the Lemma holds.

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

Proof. We can easily verify that $$ satisfies the boundary conditions (12).

For the u component, we have

The last inequality follows from the previous lemma.

As for the v component, for each $$, we have

Lemma 5. There exists a ζ₁ ≥ 0 such that

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

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

Proof. One the boundary, we have

and for the u 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 ξ ∈ ℝ.

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 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 $$, 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 $$, 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:

It then follows that

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

then it follows that $$, 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 $$ 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

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 μ is either −λK/c or $$.

Integrating the above from ξ₀ to +∞ and comparing the decay rates of (u_c, v_c)(ξ) with that of the upper solution $$, we have (50) and (52).

On the uniqueness of the traveling wave solution for every $$, we only prove the conclusion for traveling wave solutions with asymptotic rates given in (51) and (52) since the other case can be proved similarly. Let U₁(ξ) = (u₁, v₁)(ξ) and U₂(ξ) = (u₂, v₂)(ξ) be two traveling wave solutions of system (10) and (9) with the same speed $$. There exist positive constants A_ij, B_ij,  i, j = 1,2 and a large number N > 0 such that for ξ < −N,

and for ξ > N, The traveling wave solutions of system (10)-(9) are translation invariant; thus, for any θ > 0, $$ is also a traveling wave solution of (10)-(9). By (60) and (62), the solution U₁(ξ + θ) has the asymptotics for ξ ≤ −N and for ξ ≥ N.

Choosing θ > 0 large enough such that

then one has for ξ ∈ (−∞, −N]∪[N, +∞),

We now consider system (10) on [−N, +N]. There are two possibilities.

Case 1. Suppose that we already have $$ on [−N, +N], then the function $$ and it satisfies for some ζ_i ∈ (0,1), i = 1,2,

where the matrix M is given by

Since w₁(ξ) ≥ 0, ξ ∈ [−N, N] and λK(u₂ + ζ₁w₁) ≥ 0, then we have on ξ ∈ [−N, N],

The maximum principle then implies that w₁(ξ) > 0 on [−N, N]. We then move to the second equation of (68). We have

The strict inequality comes from the fact that $$ for ξ ∈ [−N, N]. It then follows that w₂(ξ) > 0 for ξ ∈ [−N, N]. For if there is a $$ such that $$, then w₂ takes local minimum at $$ 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 (−N, N) such that one of the components, say the jth component, satisfies $$ at that point, j = 1 or 2. We then increase θ, that is, shift $$ further left, so that $$,  $$. By the monotonicity of $$ and U₂, we can find a $$ such that in the interval (−N, N) we have $$. Shifting $$ back until one component of $$ first touches its counterpart of U₂(ξ) at some point $$, we then return back to Case 1 again, where it has been shown that this is impossible. Therefore, we must have

for all 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 $$, such that $$. In this case, we have finished the proof.

• (2)

  There exists a $$ and ξ₁ ∈ ℝ, such that one of the components of $$ and U₂ are equal there; for all ξ ∈ ℝ, we have $$. 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 $$, such that

for all ξ ∈ ℝ.