Spreading speeds and traveling waves for non-cooperative integro-difference systems

The development of macroscopic descriptions for the joint dynamics and behavior of large heterogeneous ensembles subject to ecological forces like dispersal remains a central challenge for mathematicians and biological scientists alike. Over the past century, specific attention has been directed to the role played by dispersal in shaping plant communities, or on the dynamics of marine open-ocean and intertidal systems, or on biological invasions, or on the spread of disease, to name a few. Mathematicians and theoreticians, starting with the efforts of researchers that include Aronson, Fisher, Kolmogorov, Levin, Okubo, Skellam, Slobodkin, Weinberger and many others, set the foundation of a fertile area of research at the interface of ecology, mathematics, population biology and evolutionary biology. Integrodifference systems, the subject of this manuscript, arise naturally in the study of the spatial dispersal of organisms whose local population dynamics are prescribed by models with discrete generations. The brunt of the mathematical research has focused on the the study of existence traveling wave solutions and characterizations of the spreading speed particularly, in the context of cooperative systems. In this paper, we characterize the spreading speed for a large class of non cooperative systems, all formulated in terms of integrodifference equations, by the convergence of initial data to wave solutions. In this setting, the spreading speed is characterized as the slowest speed of a family of non-constant traveling wave solutions. Our results are applied to a specific non-cooperative competition system in detail.

tion dynamics governed by Hassell and Comins' non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This article is dedicated to Simon A. Levin whose contributions to the fields of ecology, evolutionary biology, and the environmental sciences have driven and inspired the research of generations of mathematicians, mathematical biologists, and life and social scientists, around the world for over four decades.

Introduction
Finding and developing macroscopic descriptions for the dynamics and behavior of heterogeneous large ensembles of individuals subject to ecological forces like dispersal continues to provide challenges and opportunities for mathematical and biological scientists. Over the past century, particular attention has been placed on the study of the role played by dispersal in shaping plant communities, in helping understand biological invasions, in assisting in the quantification and control of the spread of infectious disease, or in disentangling the dynamics of marine open-ocean and intertidal systems, to name but a few examples. The work of pioneers like Aronson [1,2], Fisher [7], Hadeler [8][9][10], Kolmogorov [15], Levin [18], Okubo [19], Skellam [29], Slobodkin [14], Weinberger [33] and the subsequent cadre of distinguished mathematicians and theoreticians across the world who have worked at this interface, set not only the foundation of an important and fertile area of interdisciplinary research (ecology, mathematics, and evolutionary biology) but in the process it has inspired novel mathematical research while being re-energized by unsolved questions in emerging fields like urban ecology and sustainability and the challenges and opportunities posed by the growing body of research on the co-evolving dynamics of socio-biological systems [5] [17].
The theme of finding mathematical macroscopic descriptions for the spatial dynamics of heterogeneous large-ensembles of populations was set in "motion" by the fundamental ecological contributions of Skellam (1951) [29], Kierstad and Slobodkin (1953) [14], Levin and Paine (1974) [18], Okubo (1980) [19], and others. The study of integro-difference equations dispersal models in the mathematical literature has its origins in the study of the coupled spatial dynamics of organisms with discrete primarily non-overlapping (but see [30]) local dynamics with dispersal processes modeled via re-distribution kernels [1,2].
Early models for the dispersal of invasive species used nonlinear reactiondiffusion equations, with the prototype provided by Fisher's Equation [7]. The primary motivation or emphasis have always been on the characterization of the speed of propagation of species invading unoccupied habitats. The seminal contributions of Fisher [7] and Kolmogorov, Petrowski, and Piscounov [15], Aronson and Weinberger [1,2] jointly handled the mathematical challenges posed by their efforts to classify rigorously the concept of speed of propagation for quite general continuous in space and discrete time, integro-difference equations. Weinberger [33] and Lui [24] research expanded the mathematical foundation for the theory of spreading speeds and traveling waves, through their analysis of traveling waves via the convergence of initial data to wave solutions, in the context of cooperative operators. Recently, Weinberger, Lewis and Li made additional contributions [31,20,21,32]. The mathematical analyses of integro-difference spatially explicit systems enhances the understanding of the dynamics of introduced species like weeds or pests in terrestrial systems, or the study of the impact of dominant alien species in freshwaters while generating additional challenges and opportunities to mathematicians, whose interests, are driven by the study of challenging dynamical systems.
The pervasiveness of overcompensation in biological systems implies that integro-difference equations models are in general non-cooperative, and therefore, existing theoretical work has yet to address effectively the mathematical consequences of non-cooperative local dynamics on dispersal. In other words, the incorporation of biological forces/mechanisms that drive population overcompensation leads to mathematical models whose dynamics have yet to be satisfactorily teased out in the context of relevant biological settings. Deep mathematical challenges remain [16]. The research in this manuscript does not start in the vacuum since relevant mathematical work for non-cooperative systems has been carried out by several researchers. Thieme [27] showed, in the context of a general model with non-monotone growth functions, that the asymptotic spreading speed could still be obtained with the aid of carefully constructed monotone functions. Hsu and Zhao [13] and Li, Lewis and Weinberger [22] just extended the theory of spreading speeds in the context of nonmonotone integro-difference equations. Their extensions relied on two methods: the construction of two monotone operators (with appropriate properties) and the application of fixed point theorems in Banach spaces-an approach also used in Ma [26] and Wang [28] to establish the existence of traveling wave solutions of reaction-diffusion equations. The results in this manuscript on the speed of propagation for non-cooperative systems in the context of integrodifference equations rely on the spreading results for monotone systems in Weinberger et al. [31].
We highlight our results in the context of a two-dimensional nonlinear discrete system describing the local nonlinear dynamics of two competing species with discrete reproduction cycles [11]. The model focuses on the growth and spread of these competing species with their population densities at generation n and spatial location x being tracked by the state variables X n (x) and Y n (x), respectively. The system is a natural extension of the classical single population "scramble" competition model of Ricker [3]. Specifically, the nonspatial interference-competition model of Hassell and Comins is given by the following system of coupled nonlinear difference equations: where r 1 , r 2 , σ 1 , σ 2 are all positive constants. The possibility that individuals in the above two populations may disperse to different sites is modeled with a redistribution kernel k i (y). Hence, a discrete-time model, where individuals interact locally according to Model (1), can be naturally formulated via a system of coupled nonlinear integrodifference equations. Hence, we have that where the dispersal of the i-species is modeled by a redistribution kernel k i , i = 1, 2 that depends just on the signed distance x − y, connecting the "birth" y location and the "settlement" location x. In other words, k i (y) is a homogenous "probability" kernel that satisfies ∞ −∞ k i (y)dy = 1. Since the above system is non-cooperative in general, it is in such a context that new results will be formulated, and illustrated but first, we introduce the notation that will be used for the explicit mathematical formulation of the dynamics of two-interacting, dispersing, and competing populations. Consequently, β, β ± , F, F ± , r, u, v are used to denote vectors in R N or N -vector valued functions while x, y, ξ are used to denote variables in R. The use of u = (u i ) and v = (v i ) ∈ R N allow us to define u ≥ v whenever u i ≥ v i for all i; and u ≫ v whenever u i > v i for all i. We further define for any where C(R, R) is the set of all continuous functions from R to R. Our focus will be on the set C β + , β + ≫ 0.
Specifically, we consider the system of integro-difference equations where u n = (u i n ) ∈ C β + , F (u) = (f i (u)); u n (x) is the density of individuals at point x and time/generation n; F (u) is the density-dependent fecundity (local growth rate); and k i (x − y) (dispersal kernel) models the dispersal of u, assumed to depend only on the signed distance x−y between the location of "birth" y and the "settlement" or "landing" location x. As noted before, k i (x − y) can be viewed as a probability kernel since ∞ −∞ k i (x)dx = 1. The notation Q[F (u n )] is slightly different from these used in [13,22,24,31] and hence those wishing to compare results must account for this. Specifically, no F can be found in the standard literature notation.
Here, F has been included to carry out the proofs involving non-monotone systems effectively.
The integro-difference system (3) models the reproduction and dispersal of a time-synchronized species where all individuals first undergo reproduction, then redistribute their offspring, and then proceed to reproduce again. The goal is to carry out the characterization of the spreading speed in a system involving a rather general non-cooperative system (3) as the slowest speed of a family of non-constant traveling wave solutions of (3).

Non-cooperative Systems' Results
The focus is on the characterization of the speeds of propagation for (3) when the system is non-cooperative. As it is typical in mathematics, we make use of prior results established results for cooperative systems ( [31]). The existence of two additional monotone operators F ± with the properties that the first lies above F and the second below F is required by our method of proof. The use of this approach is motivated by the work on non-monotone equations carried out in [27,13,22,26,35,28]. Specifically, we observe that F ± can be "constructed" via piecewise functions made up of "pieces' of F and the incorporation of appropriate constants. If F happens to be monotone, then F ± = F . We introduce additional technical assumptions below. The assumptions are critical because the feasibility of the mathematical analysis depends on whether or not the components of our problem meet them: (H1) For i = 1, ..., N , k i (τ ) ≥ 0 is integrable on R, k i (τ ) = k i (−τ ), τ ∈ R, and R k i (τ )dτ = 1, R k i (τ )e λτ dτ < +∞, for all λ > 0. (H2) (i) Given that F : [0, β + ] → [0, β + ] is a continuous, twice piecewise continuous differentiable function with it is assumed that there exist continuous, twice piecewise continuous (ii) F (0) = 0, F (β) = β and there is no other positive equilibrium of Q[F ] between 0 and β (that is, there is no constant v = β such that F (v) = v, 0 ≪ v ≤ β). F ± (0) = 0, F ± (β ± ) = β ± and there is no other positive equilibrium of Q[F ± ] between 0 and β ± . F has a finite number of equilibria in [0, β + ]. (iii) F ± are nondecreasing functions on [0, β + ] and F ± (u) and F (u) have the same Jacobian at 0.
Assumptions (H1-H2) do not suffice if the goal is to characterize the speeds of propagation for (3). The assumption (H3), which includes the requirement that the operator grows less than its linearization along the particular function ν µ e −µx , is essential and implies that the operator Q does not display an Allee effect for this particular function (see [31]). Assumption (H3), explicitly formulated below, is satisfied by several biological systems of interest this will be highlighted in our example. Assumption (H3), therefore does not severely handicap the usefulness of the results in this manuscript.
The need for Frobenius' theorem stating that any nonzero irreducible matrix with nonnegative entries has a unique principal positive eigenvalue with a corresponding principal eigenvector "made up" of strictly positive coordinates is implicit in Assumption (H3). The formulation of (H3) depends on the concept of irreducibility. A matrix is irreducible if it is not similar to a lower triangular matrix with two blocks via a permutation (See [12,31]). By reordering the coordinates, one can put any matrix into a block lower triangular form, then we say that the matrix is in Frobenius form if all the diagonal blocks are irreducible (an irreducible matrix consists of the single diagonal block which is the matrix itself). Here we use the definition of Frobenius form in Weinberger et al. [31]. Following the approach in [31], that for each µ > 0, the N × N matrix B µ that results from the linearization of (3) at 0, namely where b i,j µ is the (i, j) entry of the matrix, is in Frobenius form ( [31]). In other words, it is assume that the required reordering has been done for B µ ( [31]). If we now let λ(µ) denote the principal eigenvalue of the first diagonal blocks, we reach the formulation required by Assumption (H3): (H3) (i) Assume that B µ is in Frobenius form and that the principal eigenvalue, λ(µ), of the first diagonal block is strictly larger than the principal eigenvalues of other diagonal blocks. Further, let's assume that B µ has a positive eigenvector ν µ = (ν i µ ) ≫ 0 corresponding to λ(µ) with the additional requirement that λ(0) > 1.
(iii) For every sufficiently larger positive integer k, there is a small constant vector ω = (ω i ) ≫ 0 such that It follows from (H1) that λ(µ) is an even function. In fact, it was shown by Lui [24] that ln λ(µ) is a convex function and therefore, ln λ(µ) achieves its minimum at µ = 0 and, therefore the assumption that λ(0) > 1 implies that ln λ(µ) > 0. The statement in Proposition 1 below which is critical to the rest of analysis that leads to the main result and it involves the following function of the largest principal eigenvalue λ(µ) Part (5) of Proposition 1 highlights the use of this function in the construction of lower solutions and estimates of the traveling wave solutions.
Parts (1)-(4) of Proposition 1 are essentially due to Lui [24]. However, Lui's results only guarantee that c * ≥ 0. The proof of the strict inequality, that is, that c * > 0, is found in the Appendix. Since λ(µ) is a simple root of the characteristic equation of an irreducible block, it can be shown that λ(µ) is twice continuously differentiable on R. Part (5) is a direct consequence of the results stated in Parts (1)-(4).
A traveling wave solution u n of (3) is defined as a solution of the form u n (ξ) = u(ξ−cn), u ∈ C(R, R N ). The theorems that guarantee the existence of traveling wave solutions for cooperative systems have been already established (e.g. [21]). It also has been established that the asymptotic spreading speed, for such systems, can be characterized as the speed of the slowest non-constant traveling wave solution for monotone operators [21] and for scalar equations [34,33,13,22].
We start with the statement of Theorem 1, the main theorem, which generalizes results previously established for cooperative systems to non-cooperative systems. Some parts of Theorem 1 such as the asymptotic behavior of traveling waves are new even for cooperative systems. The new information about cooperative systems are, in fact, required to be able to carry out the proofs of the results for non-cooperative systems. The details associated with the proof of the main result are collected in a series of lemmas and theorems all collected in the following sections.
The two major new contributions in this paper are: 1) for a large class of non-monotone systems (3), the question of the existence of the minimum speed of propagation is settled (Theorem 1(i-ii)) and this speed is characterize as the speed of the slowest non-constant traveling wave solution (Theorem 1(iii-v)); and 2) in the case of competition model, a direct application of the main theorem helps identify simple and meaningful conditions needed for the existence of traveling waves with the minimum speed of propagation (Theorem 3). That is, what is required to guarantee the success of a biological invasion. It is worth re-iterating that the application of the results in this manuscript to the study of relevant monotone operators case [21,33] does give additional information. In fact, these results help explicitly characterize the asymptotic behavior of traveling waves via the careful analysis of eigenvalues and upper-lower solutions. This analysis was not done before [21,33] most likely because the focus was exclusively in establishing the existence of traveling waves. The results and analysis for the n-dimensional case is typically harder. Our approach works because the analysis of the n dimensional case is closely related to structure of the eigenvalues and corresponding eigenvectors, an analysis that is embedded in our study of the relevant monotone operators.
The following theorem summarizes the main results.
Theorem 1 Assume (H1) − (H3) hold. Then the following statements are valid: (i) For any u 0 ∈ C β with compact support and 0 ≤ u 0 ≪ β, the solution u n of (3) satisfies lim n→∞ sup |x|≥nc u n (x) = 0, for c > c * (ii) For any strictly positive vector ω ∈ R N , there is a positive R ω with the property that if u 0 ∈ C β and u 0 ≥ ω on an interval of length 2R ω , then the solution u n (x) of (3) satisfies If, in addition, F is non-decreasing on C β , then u is non-increasing on R.

Remark 2
The assumption that F has a finite number of equilibria in [0, β + ] is only used in the proof of Theorem 1 (iv), and can be further relaxed. In fact, as long as for some component i and a sufficiently small positive number δ, u = (u i ) ≥ 0 with u i = δ are not equilibria of F , the conclusion is still valid from the proof.
We shall establish Theorem 1 in Sections 3 and 4.

Spreading Speeds
Our results on the speed of propagation for non-cooperative systems make use of Theorem 2 below which collects the properties of the spreading speed c * for monotone systems as established in Weinberger, Lewis and Li [31]. Theorem 2 extends the related spreading results in Lui [24] to systems of monotone recursive operators with more than two equilibria. The operator at the center of this manuscript may support more than two equilibria with one lying at the boundary as in [31] (see Section 5).
Further, for u = (u i ) ∈ C β + and x ∈ R, we have and therefore We are now able to establish Part (i) and (ii) of Theorem 1 by following essentially the proof for the scalar cases found in [13,22].
Proof of Parts (i) and (ii) of Theorem 1. Part (i). For a given u 0 ∈ C β with compact support, let u n be the n-th iteration of Q[F ] starting from u 0 and let u + n be the n-th iteration of Q[F + ] starting from u 0 . By (H2), we have Thus for any c > c * , it follows from Theorem 2 (i) that Theorem 2 (ii) states that for any strictly positive constant ω, there is a positive R ω (choose the larger one between the R ω for F + and the R ω for F − ) with the property that if u 0 ≥ ω on an interval of length 2R ω . Hence, it follows that the solutions u ± n (x) satisfy lim inf t→∞ inf |x|≤tc u ± (x) = β ± , for 0 < c < c * .
Thus for any c < c * , it follows from Theorem 2 (ii) that and consequently, that 4 Characterization of c * as the slowest speeds of traveling waves A non-constant solution of (3) is a traveling wave of speed c provided that it has the form u n (x) = u(x − cn), where u ∈ C(R, R N ) and, of course, if it satisfies (3). By substituting this form into (3), it follows that u(ξ) must satisfy the following system of equations.
In this section we complete the proof of Theorem 1 (iii), (iv) and (v), that is, the portion of our main result that characterizes the spread speed c * as the speed of the slowest member of a family of non-constant traveling wave solutions. This is an extension of prior results for monotone operators [21] and for scalar equations [34,33,13,22].

Upper and lower solutions
In this subsection, we shall verify that φ + and φ − defined below are the upper and lower solutions of (6) respectively. These solutions are only continuous on R. Upper and lower solutions of this type have been frequently used in the literature (see Diekmann [6], Weinberger [34], Lui [24], Weinberger, Lewis and Li [31], Rass and Radcliffe [25], Weng and Zhao [36], more recently by Ma [26] and Wang [28]). In particular, the explicit use of upper vector-valued solutions can be traced to the work in [24,31,25,36]; for lower vector-valued solutions, in the context of multi-type epidemic models, to the work in [25]; and in [36] in the context of multi-type SIS epidemic models. Our construction of φ + and φ − , the upper and lower solutions of (6), is motivated by the research in these references.
Our verification of the lower and upper solutions for n-dimensional systems is new and different from the above mentioned references. The details follow below.
Let c > c * , 1 < γ < 2, q > 1 and recall the definitions of Λ c and γΛ c as utilized in Proposition 1. The corresponding positive eigenvectors ν Λc and ν γΛc of B µ for the eigenvalues λ µ when µ = Λ c , γΛ c can therefore be identified. Define We verify in the two lemmas below that φ + and φ − are upper and lower solutions of (6) respectively. It is assumed that Lemma 1 is valid when F is monotone. In this case, F ± = F, β ± = β.
Thus, for ξ ∈ R, in view of (4), (H3), Proposition 1, we obtain that On the other hand, since This completes the proof of Lemma 1.
In order to verify the lower solution, the following estimate for F is needed. For N = 1, 2, Lemma 2 can be found in [28]. (H1 − H2) hold. There exist positive constants D i , i = 1, ..., N such that

Lemma 2 Assume
Proof In a sufficiently small neighborhood of the origin, since F is twice continuously differentiable. From the Taylor's Theorem for multi-variable functions (the big Oh notation version), for u sufficiently small.
There exist small ǫ > 0 and D ′ i > 0 such that for n j=1 (u j ) 2 < ǫ For u ∈ [0, β] and n j=1 (u j ) 2 ≥ ǫ, noting that f i (u), N j=1 ∂ j f i (0)u j are bounded, we always choose a sufficiently large constant D ′′ i > 0 such that Thus if we let D i = max{D ′ i , D ′′ i }, then Lemma 2 is proved.
Lemma 3 Assume (H1) − (H3) hold. For any c > c * if q (which is independent of ξ) and that it is sufficiently large, φ − is a lower solution of Q c [F ].
That is It is easy to see that For ξ ∈ R, in view of Lemma 2, we have, for ξ ∈ R, i = 1, ..., N , where For ξ ≥ min i ξ * i , e (γ−2)Λcξ is bounded above. Finally, from (12) and the fact that Φ(γΛ c ) < c, we conclude that there exists q > 0, which is independent of ξ, such that, for ξ ≥ ξ * And This completes the proof.

Proof of Theorem 1 (iii) with monotonicity of F
Theorems that guarantee the existence of traveling wave solutions for cooperative systems have been established (e.g. [21,33]). In this section, it is assumed that F is non-decreasing on [0, β] and from this assumption, we proceed to establish Theorem 1.
As we state in Section 2, even for the case of monotone operators, the results and analysis in this manuscript are different from [21,33]. Here, we are able to characterize explicitly the asymptotic behavior of traveling waves through a careful analysis of eigenvalues and upper-lower solutions ( an analysis not provided in [21,33]). As we shall see, the analysis of the asymptotic behavior of traveling wave solutions for monotone operator enable us also to prove the existence of traveling wave solutions for non monotone operators.
In order to complete the last step, we need to make use of the following Banach space where C(R) denotes the set of all continuous functions on R, and where ρ is a positive constant such that ρ < Λ c . It follows that φ + ∈ B ρ and φ − ∈ B ρ . Finally, the following set is required (domain of the operator of interest): It is clear that A ρ ⊆ C β . By the standard procedure (see [26,13,28]), it can be shown that Q c [F ] is a continuous map of the bounded set A ρ into a compact set.
Lemma 4 Assume (H1) − (H3) hold. Then Q c [F ] : A ρ → A ρ is continuous with the weighted norm . ρ and relatively compact in B ρ . Now we are in a position to prove Theorem 1 when F is monotone. Define the following iteration From Lemmas 1, 3, and the fact that F is non-decreasing, u n is non-increasing on R, it follows that .., N. By Lemma 4 and monotonicity of (u n ), there is u ∈ A ρ such that lim n→∞ u n − u ρ = 0. Lemma 4 implies that Q[u] = u. Furthermore, u is non-increasing. It is clear that lim ξ→∞ u i (ξ) = 0, i = 1, ..., N . Assume that lim ξ→−∞ u i (ξ) = k i , i = 1, ..., Nk i > 0, i = 1, ..., N because of u ∈ A ρ . Applying the dominated convergence theorem, we getk i = f i (k). By (H2),k = β. Finally, note that This completes the proof of Theorem 1 when F is monotone.

Proof of Theorem 1 (iii)
We proceed to characterize traveling wave solutions when the assumption that F is monotone is dropped. Our treatment is different even for the scalar case (N = 1). The key mathematical ideas used can be found in the literature albeit there are differences. Our use of the Schauder Fixed Point Theorem and the construction of the bounded set D ρ are different from those found in [13,22]. As in Section 4, both Q c [F + ] and Q c [F − ] are monotone. Note that F, F + , F − have the same linearization at the origin. In view of the results in Section 4, there exists a non-increasing fixed point Therefore, Q c [F ] : D ρ → D ρ . Note that the proof of Lemmas 4 does not need the monotonicity of F − . In the same way as in Lemmas 4, we can show that Q c [F − ] : D ρ → B ρ is continuous and maps bounded sets into compact sets. Therefore, the Schauder Fixed Point Theorem shows that the operator Q c [F ] has a fixed point u in D ρ , which is a traveling wave solution of (3) for c > c * .

Proof of Theorem 1 (iv)
Proof The proof in this subsection follows the approach found in [4,13]. We make use of the results in Theorem 1 (iii). Hence, for each m ∈ N, we choose c m > c * such that lim m→∞ c m = c * . According to Theorem 1 (iii), for each c m there is a traveling wave solution u m = (u i m ) of (3) such that By the standard procedure (see [26,13,28]), (u m ) is equicontinuous and uniformly bounded on R. Hence, the Ascoli's theorem implies that there is a vector valued continuous function u = (u i ) on R and subsequence (u m k ) of (u m ), such that lim k→∞ u m k (ξ) = u(ξ) uniformly in ξ on any compact interval of R. Further, the use of the dominated convergence theorem guarantees that we have Because of the translation invariance of u m , we always can assume that the first component u 1 m (0) equals to a sufficiently small positive number σ > 0 for all m. Since there is only a finite number of equilibria, we can choose σ in such a way that it is not the first component of any nontrivial equilibrium. Consequently u is a nonconstant traveling solution of (3) for c = c * .

Minimum speeds and traveling waves for a competition model
Hassell and Comins' model of the growth and spread of two population densities at time n and location x under an interference competition regime is used to highlight the applicability of the results in this manuscript. We makes use of the local analysis results of their model reported in [11]. The addition of the possibility of dispersal via the re-distribution kernel k i (x − y) leads to (2). If their two densities are denoted by X n (x) and Y n (x) then their model is given locally by a set of nonlinear coupled difference equations (1) with the addition of dispersal leading to (2). The following results highlight the contributions that the main theorem makes towards increasing our understanding of the role of dispersal, in the context of local competitive systems. Li [23] also investigated the minimum speed of (2). Model (2) can support four constant equilibria: the unpopulated state (0, 0); the second-species monoculture state (0, r 2 ); the first monculture state (r 1 , 0); and ( r1−σ1r2 1−σ1σ2 , r2−σ2r1 1−σ1σ2 ). The change of variables p = X, q = r 2 −Y allows to convert system (2) into the following coupled system of integro-difference equations p n+1 (x) = R k 1 (x − y)f (p n (y), q n (y))dy q n+1 (x) = R k 2 (x − y)g(p n (y), q n (y))dy. (16) where f (p, q) = h(p)e r1−σ1r2+σ1q g(p, q) = r 2 − r 2 − q e q−σ2p h(p) = pe −p It is clear that (2) and (16) are not monotone systems. A straightforward calculation shows that (16) has four equilibria (0, 0), (0, r 2 ), (r 1 , r 2 ) and In fact, under the conditions of Theorem 3, we show in Appendix that there are no positive equilibrium of (16) between (0, 0) and (r 1 , r 2 ). Theorem 1 is used to guarantee the existence of a spreading speed and traveling wave solutions of the nonmonotone system (16) with its accompanying results on the speed of propagation.. We summarize the results obtained in the context of this example in Theorem 3. Its proof is outlined in the Appendix.
Then the conclusions of Theorem 1 hold for (16).
The biological interpretation of the conditions in Theorem 3 in the context of our application are straightforward. For an invasion to be successful, the overall dispersal of the invader (X) is relatively larger than the overall dispersal of the out-competed resident (Y). Further competition favors the invader whenever σ 1 is sufficiently small (invader less affected by competition) and σ 2 is sufficiently large (a relatively fragile resident, that is, more susceptible to interference competition). Under these conditions, there are traveling wave solutions of (16) "loosely" connecting its two equilibria (0, 0) and (r 1 , r 2 ). Equivalently, there are traveling wave solutions of (2) "loosely" connecting its two boundary states (0, r 2 ) and (r 1 , 0). Here the term "loosely" means the traveling waves may oscillate around the equilibria since they are not necessarily monotone. For specific k i , the exact value of c * can be computed and compared to experimental data as it has been done by Kot, Lewis, others and their collaborators.

Conclusions
integro-difference systems arise naturally in the study of the dispersal of populations, including interacting populations, composed of organisms that reproduce locally via discrete generations and compete for resources, before dispersing . The brunt of the mathematical research has focused on the the study of the existence of traveling wave solutions and characterizations of the spreading speed in the context of cooperative systems. In this paper, we characterize the spreading speed for a large class of non cooperative systems, formulated in terms of integro-difference equations, via the convergence of initial data to wave solutions. The spreading speed is characterized as the slowest speed of a family of non-constant traveling wave solutions. The results are applied to the non-cooperative competitive system proposed by Hassell and Comins (1976) [11]. We are in the process of applying these results to additional ecological and epidemiological systems where the local dynamics are naturally non-cooperative with the hope that increasing our understanding of the role of dispersal in communities where the local dynamics are richer, more realistic, than those previously supported by the mathematical theory.