NONLINEAR STABILITY OF TRAVELING WAVEFRONTS IN AN AGE-STRUCTURED REACTION-DIFFUSION POPULATION MODEL

The paper is devoted to the study of a time-delayed reactiondiffusion equation of age-structured single species population. Linear stability for this model was first presented by Gourley [4], when the time delay is small. Here, we extend the previous result to the nonlinear stability by using the technical weighted-energy method, when the initial perturbation around the wavefront decays to zero exponentially as x → −∞, but the initial perturbation can be arbitrarily large on other locations. The exponential convergent rate (in time) of the solution is obtained. Numerical simulations are carried out to confirm the theoretical results, and the traveling wavefronts with a large delay term in the model are reported.

1. Introduction.The population of a single species with age-structure is usually described as a time delayed reaction-diffusion equation where v(x, t) denotes the total population of mature species after the mature age τ > 0 at time t and location x.Here, d m > 0 is the diffusion rate of the mature species, d(v) > 0 is the death function, b(v) is the birth function, and ε > 0 is an impact factor of the death rate.Equation ( 1) can be derived from the Metz and Diekmann's dynamical population model [13]  where a denotes the age of the species and u(t, a, x) represents the density of the species with age a at a location x and in time t, d i (a) and γ(a) are the diffusion and death rates of the immatures.For the detailed derivation of equation (1), we refer the reader to [4,5,12,20].As shown in [4] (see also [1,2,3]), for simplicity, by choosing the birth and death functions of the matures as b(v(x, t − τ )) = αv(x, t − τ ) and d(v) = βv 2 , and let the death rates of the immatures γ(a) = γ be a constant which determines the impact factor of the death rate according to ε = e −γτ , and denoting the diffusion rate as d m = d, Eq.( 1) can be rewritten as It is noticed that two constant equilibria exist in equation ( 2); namely, Let c denote the speed of the wave solution, which depends on the maturation delay τ .In [3], Al-Omari and Gourley proved that for all speeds c exceeding a certain minimum value, equation ( 2) possesses a monotone traveling wavefront solution which connects the constant equilibria v − and v + .Later, by applying the weighted energy method as shown in [11], Gourley showed in [4] that the traveling wavefronts with speed c greater than the critical speed c 0 are linearly stable, if the initial perturbations around the wavefronts are exponentially decay in space as x → −∞.
In the present paper, we study equation (2) with the following initial value condition: v(x, s) = v 0 (x, s) → v ± , s ∈ [−τ, 0] as x → ±∞. ( The goal of the present work is to extend Gourley's linear stability of the wavefronts to the nonlinear stability.Although we require the initial perturbation around the wavefront to decay as x near −∞, the initial perturbations could be arbitrarily large in other locations.This is the so-called large initial perturbation problem for the stability, and the result is of particularly interest in mathematical and physical sciences.The method adopted here is based on the comparison principle together with the weighted L 2 -energy method, which was first applied by Lin and Mei [6] to prove the stability of traveling waves with large initial perturbations for the Nicholson's blowflies equation.For the wavefronts and their stabilities related to the other models, we refer to, for example, [8]- [12], [14], [16]- [25], and the references therein. Throughout the paper, C > 0 denotes a generic constant, while C i > 0 (i = 0, 1, 2, • • • ) represents a specific constant.Let I be an interval, typically I = R. L 2 (I) is the space of the square integrable functions on I, and H k (I) (k ≥ 0) is the Sobolev space of the L 2 -functions f (x) defined on the interval I whose derivatives w (I) represents the weighted L 2 -space with the weight w(x) > 0 and its norm is defined by is the weighted Sobolev space with the norm Let T > 0 and let B be a Banach space, we denote by C 0 ([0, T ]; B) the space of the B-valued continuous functions on [0, T ], and L 2 ([0, T ]; B) as the space of the Bvalued L 2 -functions on [0, T ].The corresponding spaces of the B-valued functions on [0, ∞) are defined similarly.
The paper is organized as follows.In Section 2, we introduce the traveling wavefronts and a weight function, the main result of this work, namely, the nonlinear stability of the traveling wavefronts is then presented.In Section 3, after having established the comparison principle and some key energy estimates in the weighted Sobolev's spaces, we prove the nonlinear stability.In Section 4, we report numerical simulations for several case studies, and the computed solutions confirm our theoretical results stated in Section 2.
As shown in [4], it is easily seen from Figure 1 that the critical wave speed c 0 is determined when the two curves F c (λ) and G c (λ) have a unique touched point.When c > c 0 , the two curves intersect at two points, says λ 1c and λ 2c .Let λ c be a point between λ 1c and λ 2c ; then Since λ = 2c/d is a nonzero root of the equation G c (λ) = 0, we have λ c < 2c/d.From the first graph of Figure 1, it is verified that G c 0 (λ) < F c 0 (λ) for all λ > 0 This implies c 0 < √ 4αde −γτ as mentioned in Proposition 1.It has been reported in [11,12] that when the wave is faster, namely, the wave speed is larger, then the wave is usually stable time-asymptotically.However, the more interesting and difficult case is to study the nonlinear stability of the slower waves, in particular, the wave with speed very close to the critical speed c 0 .Hence, in the present paper, we are interested in the waves with speed c satisfying As technically assumed in [4], we restrict the delay τ to be small, such that 4ατ e −γτ < cosh −1 (2); (10) then ( 9) is reduced to Thus, from λ c < 2c/d, we have Obviously, it holds Since lim ξ→∞ φ(ξ) = v + and φ(ξ) is increasing, there exists a number ξ 0 such that Let ṽ = v − φ be the linear perturbation around the wavefront φ; then ṽ satisfies For the linear stability of the traveling wavefronts, defining the weight function as Gourley [4] showed the following result.
) is the weight function given in (15), then the wavefront φ(x + ct) is linearly asymptotically stable in the sense that In this paper, by using the comparison principle together with the weighted energy method, we prove the following nonlinear stability of traveling wavefronts even when the initial perturbations are not small.Theorem 2.1 (Nonlinear Stability).Let the delay τ satisfy (10), for a given traveling wavefront φ(x + ct) with speed c satisfying (11), and the initial datum satisfying where w = w(x + cs) (f or s ∈ [−r, 0]) is the weight function given in (15), then the unique solution v(x, t) of the Cauchy problem (2) and (3) exists globally ) and it converges to the traveling wavefront φ(x + ct) time-asymptotically Remark 1.For the wave stability, we usually require the initial perturbation around the wave to be sufficiently small.However, such a restriction is not necessary in the present stability; namely, the initial perturbation ) can be large.This is the so-called large initial perturbation problem, and is quite significant and interesting in the studies of many problems in mathematical physics.
3. Proof of nonlinear stability.By using the energy method reported in [11], we can prove that equations (2) and (3) admit a unique global solution.The key step in the proof of the nonlinear stability is to establish the comparison principle and some energy estimates in the weighted space L 2 w (R).Now, we first demonstrate the positivity of the solution v(x, t) of the Cauchy problem (2) and (3), and then establish a comparison principle for the solution v(x, t).Consequently, the convergence of the solution to the wavefront is obtained.

Lemma 3.1 (Positivity). Let v(x, t) be a bounded solution of the Cauchy problem (2) and (3) with a nonnegative initial data
In fact, if we set v = we νt by choosing a large ν such that a(x, t) := ν + βv(x, t) ≥ 0 because of the boundedness of v(x, t), equation ( 18) is reduced to Applying the maximum principle (c.f.[15]), the above equation ensures that Consequently Combining ( 19) and ( 21), we prove By repeating this procedure, we have The proof is complete.
Lemma 3.2 (Comparison Principle).Let v(x, t) and v(x, t) be positive and bounded for (x, t) ∈ R × R + , and they satisfy Proof.Let From (23), it can be verified that where a 1 (x, t) := v(x, t) + v(x, t) > 0 is bounded.
We now prove the stability of the traveling wavefronts by the weighted energy method.Let Denote v + (x, t) and v − (x, t) as the corresponding solutions of equations ( 2) and (3) with respect to the above mentioned initial data v + 0 (x, τ ) and v − 0 (x, τ ); i.e., By the Comparison Principle, we have The convergence of the wave solutions with the initial data v + 0 (x, τ ), v − 0 (x, τ ), and v 0 (x, τ ) are discussed as follows.
Similarly, differentiating (36) with respect to ξ and multiplying by e 2µt w(ξ)z ξ (ξ, t), then by integrating the resultant equation with respect to (ξ, t) over R × [0, t], and applying (49), we obtain Combining ( 49) and (50), we prove that for the weight w(ξ) given in (15) (for the proof of the above Sobolev space embedding, we refer to Mei-Nishihara [9]), we obtain for all t ≥ 0. This implies (33), and proof is complete.
Case 3: The convergence of v(x, t) to φ(x + ct) Now we will prove Theorem 2.1, namely, the following convergence result.
Lemma 3.5.It holds Thanks to Lemmas 3.3 and 3.4, we have the following convergence results: Then, by using the Squeeze Theorem, we finally prove The proof is complete.
4. Numerical simulations.To investigate the stability of the traveling waves, we perform numerical simulations to confirm the theoretical results presented in Section 2.
The mathematical formulation given in equation ( 2) and ( 3) is a nonlinear timedelayed partial differential equation, and the nonlinearity is due to the term βv 2 .The computational results reported in this section are based on the following finitedifference approximation with a forward scheme for the time derivative and a central scheme for the spatial derivative: where ∆t and ∆x denote the step size in time and space, respectively.It is noted that the nonlinear term is calculated at the (n − 1)th time step; hence, it is an explicit scheme.The advantage of this simple scheme is that the resulting finitedifference equations are linear, and the solutions can easily be computed.The numerical scheme given in (54) is of first-order accurate in time and second-order accurate in space.We have also implemented an implicit numerical scheme in which the nonlinear term is computed at the nth time step, but the accuracy is the same as the above explicit scheme.For implicit formulation, an iterative method is required to solve the resulting nonlinear difference equations in each time step.However, we observe that the computed solutions are almost the same as those obtained from the explicit scheme.
In computation, the sizes of the time step and space step are chosen as ∆t = 0.01 and ∆x = 0.2, so that the condition to ensure a stable numerical computation ∆t/(∆x) 2 < 1/2 is satisfied.Although the original model assumes the spatial domain in (−∞, ∞), a finite computational domain (−L, L) is imposed.Here, we let L = 400 so that the computational domain is sufficiently large and no artificial numerical reflection is introduced in the computed solution.
In this section, we report the numerical simulations for six test cases.We first choose d = 1 and β = 1; other parameters and the initial data for each case study are listed in Table 1.The corresponding numerical solutions are displayed in Figures 4.2 to 4.7.
In the first group of the case studies, we focus on Cases 1, 2, and 3, in which a small delay τ = 1 is considered.It is noted that the condition 4ατ e −γτ < cosh −1 (2) is satisfied for each case, where cosh −1 (2) = 1.3710.For Case 1, we observe that starting with a discontinuous initial data, a rapid smoothing effect results in a short time.When t=10, Figure 4.2 illustrates that a stable traveling wave is established and it is moving in the negative x-direction as time increases.In Case 2, the initial datum is discontinuous with a big jump from 0 to 40, which causes the initial perturbation around the wavefront larger than 40 − 1 e 2 , because the monotone traveling wave is bounded between 0 and 1 e 2 .However, as shown in Figure 3, after time t = 50 the solution becomes a stable wavefront propagating from right to left.For Case 3, we test the continuous initial datum v 0 (x, s), and it has a maximum v 0 ( 5 200e−1 , s) = 50+ .From Figure 4, we observe that, the solution behaves as a stable wavefront moving from right to left after time t = 50.As shown in Figures 3 and 4, Cases 2 and 3 demonstrate numerically that the wavefronts are asymptotically stable even if the initial perturbations are really large.This confirms our theoretical stability result in Section 2. With the parameters chosen for Cases 1 through 3, we expect that the speeds of the traveling waves for Cases 1 and 3 are identical; and the wavefront will propagate faster for Case 2 than for Cases 1 and 3.The numerical simulations presented in Figures 4.2 through 4.4 indeed confirm our theoretical prediction.We observe that v(x, 50) and v(x, 100) are essentially the same for Case 1 and 3.   reduce the value of γ in Case 2 by 20, but we enlarge the delay τ twentyfold.For this case, the condition (10) is not satisfied.From the computed solutions given in Figure 4.7, we observe that a stable monotone traveling wave solution connecting two equilibria is obtained when the time is sufficiently large.The reported numerical simulations confirm the theoretical results presented in this work.In particular, we observe that stable traveling wavefronts are obtained even when the initial perturbation is not small.Moreover, the stable solution is not affected by a large delay term in which the condition (2.7) is not valid.

Figure 1 .
Figure 1.The graphs of F c (λ) and G c (λ) for c = c 0 and c > c 0 , respectively

Table 1 .
Case studies: Parameters and Initial data