On existence of wavefront solutions in mixed monotone reaction-diffusion systems

In this article, we give an existence-comparison 
theorem for wavefront solutions in a general class of 
reaction-diffusion systems. With mixed quasi-monotonicity and 
Lipschitz condition on the set bounded by coupled upper-lower 
solutions, the existence of wavefront solution is proven by applying 
the Schauder Fixed Point Theorem on a compact invariant set. Our 
main result is then applied to well-known examples: a 
ratio-dependent predator-prey model, a three-species food chain model of 
Lotka-Volterra type and a three-species competition model of 
Lotka-Volterra type. For each model, we establish conditions on the 
ecological parameters for the presence of wavefront solutions 
flowing towards the coexistent states through suitably constructed 
upper and lower solutions. Numerical simulations on those models are 
also demonstrated to illustrate our theoretical results.

1. Introduction. The existence, propagation, and asymptotic behavior of wavefront solutions of reaction-diffusion systems have been attracting the attention of researchers on mathematical models in biology, ecology, chemistry and physics. Among various approaches to the existence of wavefront solutions, the method of upper-lower solutions was widely applied to both single-equation and multi-equation systems. For the case of multi-equation systems, the existence of wavefront solutions were established for various two-dimensional competition models and threedimensional cooperative-competitive models, which can be transformed into a cooperative system through a simple substitution [6,9,10,19,21]. For those models with all reaction functions being quasimonotone nondecreasing, an ordered pair of upper and lower solutions were constructed through wavefront solutions of the classical K.P.P. equation ), and the wavefront solutions were generated by monotone iterations starting from the upper solution. The convergence of the iterative sequence also helped to obtain the monotonicity and asymptotic rates of the wavefront solutions in those models. However, it is not an easy task to extend this method to general reaction-diffusion systems with mixed quasi-monotone properties, including the two-species predator-prey models and three-dimensional competition models. "Note that if the dimension of the Lotka-Volterra competition system is greater than or equal to three, then one is not able to converse it into a cooperative system any more" [21]. Actually, it is impossible to transform most of the real-life models into quasi-monotone non-increasing systems.
Comparing with other methods for establishing the existence of wavefront solutions (for example, search for the generalized upper and lower solution pairs [3,24,30,31], geometric singular perturbation [2,8] and phase plane analysis [1,4,11,12,13,14,15,16,27,28,29,32]) construction of upper-lower solution pairs in the classical sense gives more straightforward results on the existence of wavefront solutions in terms of the the natural parameters, and also provides valuable information on the ultimate bounds and asymptotic rates of the solutions. In [20], an existence theorem was obtained (by the Schauder's Fixed Point Theorem) for the wavefront solutions in general reaction-diffusion systems with time delays. However, there is a gap on the proof for compactness of the operator F (in Lemma 3.5) because the application of Ascoli-Arzela Theorem is restricted to functional spaces on a compact domain but not for C(R, R n ). In addition, the definition of upper and lower solutions in (P1) (on page 398) requires that they have the same limits at both −∞ and +∞, which creates difficulty on construction of upper-lower solutions in complex models, therefore the general definition [25] should allow inequalities to hold on boundary conditions. Furthermore, the assumption on wavefront solutions having limit as 0 at −∞ is also unnecessarily restrictive since for many models there are traveling wavefront solutions connecting various equilibrium states.
In this article, we establish an existence-comparison theorem for wavefront solutions connecting two equilibrium states of a general reaction-diffusion system with mixed quasi-monotone properties. In Section 2, the upper-lower solutions are defined to satisfy mixed differential inequalities and their limits at ∓∞ also satisfy corresponding inequalities related to the equilibrium states. A complete proof of the existence-comparison theorem (Theorem 2.6) is given by applying Schauder's Fixed Point Theorem to a compact operator. We also apply our main theoretical result to three reaction-diffusion models with mixed quasi-monotone properties and give conditions for the existence of wavefront solutions through suitably constructed upper-lower solutions. In Section 3, we show the existence of wavefront solutions for a ratio-dependent predator-prey model flowing from the prey-only state (1, 0) to the coexistence state (u * , v * ). In Sections 4 and 5, we show the existence of wavefront solutions for two 3-species Lotka-Volterra models (food chain and competition) flowing from the trivial state (0, 0, 0) to the coexistence state (u * , v * , w * ). In our derivation of the wavefront solutions, the construction of the upper-lower solution pairs is based on the classical results on K.P.P. equation and relevant conditions on ecological parameters. For all models, we assume different diffusion rates for different population density functions, and find the minimum wave speed for wavefront solutions in terms of those diffusion rates and other ecological parameters. Finally, in Section 6, we demonstrate numerical simulations of wavefront solutions for the ratio-dependent predator-prey model and the food chain model of Lotka-Volterra type. Two numerical examples are illustrated in accordance with derived conditions for the existence of specific wavefront solutions. Through the numerical simulations, we observe that unlike the 2-dimensional competition models (transformable to a quasi-monotone nondecreasing system), the mixed quasi-monotone models no longer warrant the monotonicity of wavefront solutions on (−∞, +∞).
2. Existence-comparison theorem for wavefront solutions. We start with a general n-dimensional reaction-diffusion system for some u − , u + ∈ R n . We first define a pair of coupled upper and lower solutions for the wavefront system (2.2): Definition 2.1. Upper and lower solutions. A pair of bounded functionsw ≡ (w 1 , . . . ,w n ) andŵ ≡ (ŵ 1 , . . . ,ŵ n ) are coupled upper and lower solutions for (2.2) ifw ≥ŵ, where k i andk i are subsets of positive integers such that for each i = 1, . . . , n and lim t→−∞w (t), and lim t→∞ŵ (t) both exist with In this section, we show that under the following hypotheses for the reaction functions, there exists a wavefront solution (between the upper and lower solutions) for system (2.1). (H): of f between lim t→−∞ŵ (t) and lim t→−∞w (t) and u + is the only zero of f between lim t→∞ŵ (t) and lim t→∞w (t).
is monotone nondecreasing in [u] ki and is monotone nonincreasing in [u]k i . Also, there is β ≥ 0 such that f i (u) + βu i is nondecreasing in u i for each i. Without loss of generality, we assume that β > 0. 3. (Lipschitz condition) f satisfies the Lipschitz condition in any bounded set S of R n . That is, there is For the convenience of establishing the existence-comparison theorem, we look into a fixed point problem by writing the equation in (2.2) in the form We denote the Banach space and define the operator K on X by Kφ = ψ ≡ (ψ 1 , . . . , ψ n ) where (2.8) It can be verified that ψ i satisfies Further more, by l'Hôpital's rule provided that lim t→−∞ φ (t) and lim t→∞ φ (t) both exist. Since Thus a fixed point of K betweenŵ andw is a solution to (2.6). Now let Then for any φ ∈ S the function ψ = Kφ satisfies for i = 1, . . . , n. (2.9) The following lemma shows that S is an invariant set for K.
Proof. It suffices to show thatŵ i ≤ ψ i ≤w i for i = 1, . . . , n. Sinceŵ i ≤ φ i ≤w i in R for all i = 1, . . . , n, by the quasi-monotonicity of F i , it follows that Hence by (2.8) (2.10) LetF Thenŵ and by (2.3), for each i. Now, by (2.10), we can see that This completes the proof.
The next lemma gives the continuity of K on S.
Proof. Since f satisfies the Lipschitz condition given in (H), there is a constant L > 0 such that It follows that Let ζ ≡ (ζ 1 , . . . , ζ n ) and χ ≡ (χ 1 , . . . , χ n ) denote Kξ and Kη, respectively. By (2.8) and the above inequality, it follows that From (2.7) and the fact that Hence, by (2) if t < 0. Let M be maximum of the coefficients of |ξ − η| X on the left-hand sides of (2.11) and (2.12), it follows that |ζ − χ| X ≤ M |ξ − η| X . This proves the continuity of K on S.
Next, we show that there is a compact invariant set of K in X. Then for any φ ∈ S the function ψ = Kφ has the property Proof. By differentiation in (2.8), It can be shown through l'Hôpital's Rule that This proves the lemma.

Now let
Lemmas 2.2 and 2.4 imply thatŜ is invariant for K. We show thatŜ is compact in X.
Proof. Let (f n ) be a sequence of elements inŜ. We need to show that there is a convergent subsequence in X. Using a diagonal selection scheme, we can extract a subsequence (f n k ) that converges at each rational point in R. We show that this subsequence is Cauchy in X. Let ε > 0. There is an R > 0 such that |S| e −µR < ε/2 (2.14) where Let {r m } be the set of all rational numbers in the interval [−R, R]. By the convergence of (f n k ) on rational numbers, for each m there is N m ∈ N such that it follows that for any t ∈ U m , Choose a finite open cover of [−R, R] by U m1 , U m2 , . . . , U mj and let N = max N m1 , . . . , N mj .

It follows that
For any t ∈ [−R, R], by (2.14) This proves that (f n k ) is a Cauchy sequence in X.
Finally, we can give the following theorem for existence of wavefront solution (2.2) between the upper solutionw and the lower solutionŵ. Proof. This follows from Lemmas 2.2-2.5 and the Schauder Fixed Point Theorem.
3. Example 1: A ratio-dependent predator-prey model. In this section we look into the existence of traveling wavefronts in the following model [17] of ratiodependent interactions between a predator (v-species) and prey (u-species).
where a is the prey's intrinsic growth rate, e is the capturing rate, m is the half capturing saturation constant, f is the conversion rate, and d is the predator's death rate. The two-equation reaction-diffusion system in (3.1) is mixed quasi-monotone, and has the prey-only state (1, 0) and the coexistence state (u * , v * ) with The coexistence state is present under the following conditions: 1. d < f < ed e−ma when e > ma; 2. f > d when e ≤ ma. It is easily seen by the comparison argument that v(x, t) goes to extinction for f ≤ d. Based on the stability analysis done in [17] on the corresponding ordinary differential equation system, when f > d we have (1, 0) as a saddle point, and the Jacobian matrix J(u * , v * ) for (u * , v * ) has positive determinate while the trace of which indicates that (u * , v * ) is asymptotically stable for d/f close to 1. Throughout this section we assume that (H1): e > ma and d < f < min{ ed e−ma , 2d, d + e−ma m }.
We denote D = max{ D 1 , D 2 }, D = min{ D 1 , D 2 }. For each c ≥ 2D e−ma mD , we let Y be the unique (up to a translation of the origin) monotone increasing solution of the following K.P.P. equation: Also, for each c ≥ 2D e−ma mD (which already holds when c ≥ 2D e−ma mD ) and some 0 < l < 1 (to be determined later), let Z be the solution of the following K.P.P. equation: We next verify that the upper and lower solutions defined bỹ satisfy all the differential inequalities in Definition 3.1. It can be seen that: And, finally forV , we havê There is also an l 2 > 0 such that 0 ≤ e−ma m (l − Z) < for 0 < l < l 2 . Choosing l = min{ 1, l 1 , l 2 }, we now see thatV By the existence-comparison result in Theorem 2.6, we conclude that there exists a wavefront solution (U (ξ), V (ξ)) of system (3.4) for every c ≥ 2D e−ma mD . We are now ready to state the following theorem on the existence of the wavefront solutions for the ratio-dependent predator-prey model (3.1) connecting the prey-only state (1, 0) and the coexistence state (u * , v * ).

Example 2:
Three species food chain model of Lotka-Volterra type. In this section, we apply Theorem 2.6 to show the existence of wavefront solutions of a reaction-diffusion model for three-species food chain, which was studied in several earlier works [5,18,23]. In (4.1), the quantities u(x, t) v(x, t) and w(x, t) are scaled population densities of the three species (prey, predator and super-predator) in a food chain at t > 0 and x ∈ R, with corresponding intrinsic growth rates 1, r and s that satisfy 0 < s ≤ r ≤ 1. In each equation, the population is scaled such that the intra-species competition rate is 1, while the inter-species consumption coefficients a i and b i satisfy 0 < a i , b i < 1. This implies that the Lotka-Volterra model has the following trivial and semi-trivial steady states: (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), . Through linearization analysis on the corresponding ordinary differential system, we see that there exists a unique coexistence state (u * , v * , w * ) with while all the trivial and semi-trivial steady states are unstable. The following assumptions are made throughout this section: . In this section we look into the existence of wavefront solutions of (4.1) connecting the trivial steady state (0, 0, 0) and the coexistence state (u * , v * , w * ), in the form (u(x, t), v(x, t), w(x, t)) = (u(x + ct), v(x + ct), w(x + ct)) = (u(ξ), v(ξ), w(ξ)) where ξ ∈ R. We now consider the following equations with limiting conditions: In system (4.3), the reaction functions are mixed monotone and satisfy the Lipschitz condition in the region We next define the pair of upper and lower solutions for the wavefront system (4.3).
Definition 4.1. A pair of vector functions (Ũ (ξ),Ṽ (ξ),Ṽ (ξ)) and (Û (ξ),V (ξ),V (ξ)) in C 2 (R) × C 2 (R) × C 2 (R) are coupled upper and lower solutions of (4.3) if they satisfy the following differential inequalities with the limiting conditions Under the assumption , which ensures H2(a), the construction of upper and lower solutions for the wavefront system (4.2) is again in terms of the wavefront solution of the K.P.P. equations. Denote Let Y be the wavefront solution of the following K.P.P. equation: (4.7) By [27], for each c ≥ 2D/ √ D, there is a unique solution (up to a translation of the origin) Y of (4.7). Such solution satisfies Y (ξ) > 0 for ξ ∈ R. Set m = max{ a 1 (1 + a 2 ) Since m < 1, there exists an l such that 0 < l < 1 − m. Let Z be the wavefront solution of the following K.P.P. equation: (4.9) Also by [27], corresponding to every c ≥ 2D s D (which holds for c ≥ 2D/ there is a unique solution (up to a translation of the origin) Z of (4.9). Such solution satisfies Z (ξ) > 0 for ξ ∈ R. From the fact that 1 − l > m and both the K.P.P. systems (4.7) and (4.9) are translation invariant, by a suitable shift of the origin we will have 1−l l Z(ξ) ≥ mY (ξ) for all ξ ∈ R. For c ≥ 2D/ √ D, we define our upper and lower solutions as follows: We first verify the inequalities in (4.4) for the upper solution. It can be seen that We next show that the lower solution (Z, Z, Z) also satisfies the differential inequalities given in (4.4).

5.
Example 3: Three species competition model of Lotka-Volterra type. At last, we discuss the existence of wavefront solutions for a reaction-diffusion model for three species competition [7,18].
where the quantities u(x, t) v(x, t) and w(x, t) represent scaled population densities of the three competing species at t > 0 and x ∈ R, with corresponding intrinsic growth rates 1, r and s such that 0 < s ≤ r ≤ 1. In each equation, the population is scaled such that the intra-species competition rate is 1, while the inter-species consumption coefficients a i and b i satisfy 0 < a i , b i < 1. This implies that the Lotka-Volterra competition model (5.1) has the following trivial and semi-trivial steady states: (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), In the corresponding ordinary differential system, linearization analysis implies that all the trivial and semi-trivial states are unstable when In this case, there also exists a unique coexistence state (u * , v * , w * ) with Throughout this section we assume that conditions in H3(a) hold, and study the existence of wavefront solutions of (5.1) connecting the trivial steady state (0, 0, 0) and the coexistence state (u * , v * , w * ), in the form (u(x, t), v(x, t), w(x, t)) = (u(x + ct), v(x + ct), w(x + ct)) = (u(ξ), v(ξ), w(ξ)) where ξ ∈ R. The function (u(ξ), v(ξ), w(ξ)) needs to satisfy the following equations with limiting conditions: In system (5.3), the reaction functions are mixed monotone and satisfy the Lipschitz condition in the region We next define the upper-solution and lower-solution pair for the wavefront system (5.3).