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1 SIAM J. MATH. ANAL. Vol. 39, No., pp. 3 5 c 7 Society for Industrial and Applied Mathematics TRAVELING WAVEFRONTS IN A DELAYED FOOD-LIMITED POPULATION MODEL CHUNHUA OU AND JIANHONG WU Abstract. In this paper we develop a new method to establish the existence of traveling wavefronts for a food-limited population model with nonmonotone delayed nonlocal effects. Our approach is based on a combination of perturbation methods, the Fredholm theory, and the Banach fixed point theorem. We also develop and theoretically justify Canosa s asymptotic method for the wavefronts with large wave speeds. Numerical simulations are provided to show that there exists a prominent hump when the delay is large. Key words. traveling wave fronts, nonmonotone, nonlocal, food-limited AMS subject classifications. 35K55, 35K57, 35R, 9D5 DOI..37/5638. Introduction. There has been some success in establishing the existence of traveling wavefronts for the reaction-diffusion equation with nonlocal delayed nonlinearity. When the nonlinearity is monotone, the existence of traveling wavefronts can be obtained by extension of the methods of the super/subsolution pair [], [7], [9], homotopy [3], and Leray-Schauder degree [8]. Unfortunately, when the delayed nonlinearity is no longer monotone, very little has been achieved (except for the work in [9]). While one suspects that the method developed by Wu and Zou [9] and based on a nonstandard ordering could be applicable, the construction of a supersolution and subsolution pair is nontrivial, and it is almost as difficult as solving the original given equations. In this paper we develop a new approach to establish the existence of traveling wavefronts in the case when the delayed nonlinearity is nonmonotone. We shall demonstrate this approach by considering the following food-limited reaction-diffusion equation u (.) t (t, x) u (f u)(t, x) = (t, x)+u(t, x) x +γ(f u)(t, x), where the parameter γ>, and the spatiotemporal convolution f u is defined by (.) f u = t f(t, s, x, y)u(s, y)dyds, with the kernel f(t, s, x, y) satisfying the normaliation condition t f(t, s, x, y)dyds =. Received by the editors August, 5; accepted for publication (in revised form) December 7, 6; published electronically April 3, 7. This work was partially supported by the Canada Research Chairs Program, by the NCE Center of Mathematics for Information Technology and Complex Systems, and by the Natural Sciences and Engineering Research Council of Canada. Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J P3, Canada. Current address: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John s, Canada (ou@math.mun.ca). Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J P3, Canada (wujh@mathstat.yorku.ca). 3

2 4 CHUNHUA OU AND JIANHONG WU The simplest version of (.) without diffusion is the following ODE: (.3) du dt = ru(t) K u(t) K + γu(t), where r, K, and γ are positive constants. This equation was first proposed by Smith [5] as a mathematical model for population of Daphnia (water flea), and a derivation of this equation is given in [3]. The equation can also be used to study the effects of environmental toxicants on aquatic populations [6]. The delayed food-limited model (.4) du dt = ru(t) K u(t τ) K + γu(t τ), τ >, has been studied recently by several authors; see [3], [7], [7], [8], and [8]. It seems that the best result for the local stability of the positive equilibrium u = K is given in [7]. For the first time, the global stability of the positive equilibrium was established in [8]; see also [8] for further generaliations. Equation (.3) incorporating spatial dispersal was investigated by Feng and Lu []. They considered both the reaction-diffusion equation without time delay (.5) u t Au(t, x) =r(x)u(t, x) K(x) u(t, x) K(x)+γ(x)u(t, x) and the corresponding time-delay model u K(x) au(t, x) bu(t τ,x) (.6) Au(t, x) =r(x)u(t, x) t K(x)+aγ(x)u(t, x)+bγ(x)u(t τ,x), where x =(x,x,...,x n ) Ω R n, with Ω bounded and the operator A, given by A = n a ij (x) + x i x j i,j= n j= β j (x) x j, is uniformly strongly elliptic and has coefficient functions that are uniformly Hölder continuous in Ω. Feng and Lu studied the above problems subject to general boundary conditions that include both the ero-dirichlet and ero-neumann cases, and they established a global convergence result for the nonero steady state. We are here concerned about the general case (.), and we first note that this includes various types of special cases by choosing the kernel function f. (i) If the kernel f is taken to be f(t, s, x, y) =δ(t s)δ(x y), (.) becomes the reaction-diffusion equation without delay u (.7) t (t, x) u u(t, x) = (t, x)+u(t, x) x +γu(t, x), which is a special case of (.5). (ii) If the kernel function f has a discrete time lag τ and spatial averaging, that is, f(t, s, x, y) = 4π(t s) e (x y) /4(t s) δ(t s τ),

3 then (.) becomes (.8) TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 5 u t = u x + u +γ e (x y) /4τ u(t τ,y)dy e (x y) /4τ u(t τ,y)dy. A derivation of this type of model, using probabilistic arguments, was given in [4]. In this model, the movement of individuals to their present positions from where they have been at previous times is accounted for by a spatial convolution with a kernel that spreads normally with a dependence on the delay. (iii) If f(t, s, x, y) =δ(x y)g(t s), where (.9) G(t) = τ e t/τ or G(t) = t τ e t/τ, (.) becomes a model of reaction diffusion equation with distributed delay: t G(t )u(, x)d (.) u t = u x + u +γ t. G(t )u(, x)d The parameter τ measures time delay and is comparable to the discrete delay τ in (.8). The two kernel functions G in (.9) are used frequently in the literature on delay differential equations. The first of the two functions G is sometimes called the weak generic kernel because it implies that the importance of events in the past decreases exponentially. The second kernel (the strong generic case) is different because it implies that a particular time in the past, namely, τ time units ago, is more important than any other since this kernel achieves its unique maximum when t = τ. This kernel can be viewed as a smoothed out version of the case G(t) =δ(t τ), which gives rise to the discrete delay model. (iv) If the kernel f is taken to be f(t, s, x, y) = 4π(t s) e (x y) /4(t s) G(t s), then (.) is a reaction diffusion equation with both distributed delay and spatial averaging. In the distributed delay case with G(t) = τ e t/τ, a formal asymptotic expansion of traveling wavefront to (.) when τ is small was found recently by Gourley and Chaplain by using the so-called linear chain techniques; see [4], [5]. But the convergence of this series or the proof of validity of this expansion has been absent. The central idea of this trick is to recast the traveling wave equation into a higher dimensional system of ODEs without delay. When τ is small, Fenichel s geometrical singular perturbation theory (see [] or part two of []) is applicable. As mentioned in [4], if G(t) = t τ e t/τ, linear chain techniques are still applicable, but the system of traveling wave profile equation is six-dimensional. While the trick remains to be effective theoretically, it will be much more difficult in practice. Apparently, it is well known that the drawback of this method is that it is applicable only for models with the special distributed delays. One cannot extend this technique to the discrete case. Another disadvantage of this approach is that if the unperturbed system (τ = )isa higher dimensional system, the construction of a traveling wavefront is extraordinarily difficult. As mentioned in section 3 of [4], traveling wave solutions to (.) in the discrete case are much more difficult to study than in the distributed case with specific kernels, because we are no longer able to recast the wave profile equation of (.8) into

4 6 CHUNHUA OU AND JIANHONG WU a nondelay equation, and thus Fenichel s geometrical singular perturbation theory cannot be directly used to find a heteroclinic connection in a finite dimensional manifold. Generally, it is well known that in this case the search for traveling wavefronts becomes a much more difficult task. The purpose of this paper is to develop a completely new approach suitable for all of the aforementioned cases for the existence of traveling wavefronts. The principal result can be stated as follows. Theorem.. For any fixed constant c, there exists a real number δ = δ(c) > so that for τ [,δ], (.) possesses a traveling wavefront u(t, x) =U(x ct) satisfying U() =and U() =. We should remark that our approach here can be developed to study more general equations including u t (t, x) u = (t, x)+u(t, x)h((f u)(t, x)), t with H( ) a decreasing function; we refer to [] for this development. Although the main result should be of interest, we wish to emphasie the novelty of the approach that we develop. This approach is based on a combination of the perturbation analysis, the Fredholm operator theory, and the fixed point theorems and is expected to find applications in other models as well. A detailed proof is given for case (ii) and is sketched to emphasie the key differences for cases (iii) and (iv). This approach does not work when the delay is not small. In the case where the delay is arbitrary, we develop in section 5 Faria, Huang, and Wu s perturbation method [9] for traveling wavefronts with large wave speeds. We shall provide both theoretical justifications and numerical simulations for this method.. The discrete-delay and spatial-averaging case. In the discrete-delay and spatial-averaging case, i.e., the case when the delayed term involves an evaluation of the dependence exactly time τ ago and a convolution in space to account for the movement of individuals to their present positions from their past positions at previous times, (.) becomes ( u (.) t = u + u(t, x) / ) e (x y) /(4τ) u(t τ,y)dy x +γ /. e (x y) /(4τ) u(t τ,y)dy Our intention here is to establish the existence of traveling waves to (.) connecting the two uniform steady states u = and u =. For this purpose, we first show the existence of such wavefronts when the delay τ is ero. Letting τ +, we arrive at the following nondelay version of the food-limited model: (.) u t = u x + u u +γu, which is actually a modified version of the well-known Fisher s equation. Obviously, (.) has two uniform steady-state solutions u = and u =. Considering the traveling wavefront form by setting u(t, x) =U () =U (x ct) in (.), we obtain the following second-order ODE for U (): (.3) U + cu + U U +γu =

5 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 7 or equivalently the following first-order coupled system (.4) { U = V, V = cv U U +γu. The existence of traveling wavefronts to (.4) can be established by using the standard phase-plane techniques. Here we present only the result below. Theorem.. If c, then in the (U,V ) phase plane, a heteroclinic connection exists between the critical points (U,V )=(, ) and (, ). Furthermore, the traveling front U () is strictly monotonically decreasing. It is easy to see that the equilibrium (, ) is a saddle and the origin (, ) is a stable node. To obtain an extension when τ>, we need the following estimate on the derivative of the wave profile U. Theorem.. Let U be a traveling wavefront solution to (.3). Then (.5) γ <U () for all (, ). Proof. Since U is strictly monotonically decreasing, it is obvious that U (). It remains to prove that U () > /( γ). Note that (.3) can be rewritten as (.6) U + cu ( γ U + + ) U =. γ +γu Let λ = c c +4/γ Then it follows from (.6) that <, λ = c + c +4/γ U () = ( + γ ) [ e λ( s) U ds + λ λ +γu and hence using the fact that <U <, we have U () = ( + γ ) [λ e λ( s) U ds + λ λ λ +γu >. ] e λ( s) U ds, +γu > ( + γ )λ e λ( s) U ds λ λ +γu ( + γ )λ ( ) U max e λ( s) ds λ λ +γu ( + γ = ) ( + γ) c +4/γ γ. ] e λ( s) U ds +γu The proof is complete. Now we are in a position to establish the existence of traveling wavefronts to (.). We will show that the traveling fronts to (.) can be approximated by the corresponding wavefronts U () of (.3) when τ is small. First, we introduce some

6 8 CHUNHUA OU AND JIANHONG WU notations. Let C(R, R) be the Banach space of continuous and bounded functions from R to R equipped with the standard norm φ C = sup{ φ(t),t R}. Let C = C (R, R) ={φ C : φ C}, C = {φ C : φ C}, C = {φ C : lim t ± φ = }, and C = {φ C : φ C }, where the corresponding norms are defined by and φ C = φ C, φ C = φ C = φ C + φ C, φ C = φ C + φ C + φ C. Set u(t, x) =U() =U(x ct) in (.). Then U() satisfies the profile equation (.7) cu = U + U H(U)() +γh(u)(), where H(U)() = e y /(4τ) U( y + cτ)dy. We suppose that U can be approximated by U and hence assume that U = U + W. Then an equation for W is given by (.8) cw = W +(U + W ) H(U + W )() +γh(u + W )() U U () +γu (). Applying Taylor s expansions to (U + W ) H(U + W )() +γh(u + W )() yields (U + W ) H(U + W )() +γh(u + W )() = U H(U ) +γh(u ) +W H(U ) +γh(u ) ( + γ)u ( + γh(u )) H(W ) (.9) +R (,τ,w), where R (,τ,w) is the remainder (higher order terms) of this expansion, and for the time being we write it as (.) R (,τ,w) =(U + W ) H(U + W )() +γh(u + W )() H(U ) U +γh(u ) W H(U ) +γh(u ) + ( + γ)u H(W ). ( + γh(u )) Let g(x) =x x +γx. Then we have g (x) = x ( + γ)x +γx ( + γx).

7 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 9 Therefore, by (.9), (.8) becomes (.) cw = W + g (U ())W () + R (,τ,w)+r (,τ)+r 3 (,τ,w), where and R (,τ) =U H(U ) +γh(u ) g(u ), R 3 (,τ,w) = W H(U ) +γh(u ) ( + γ)u ( + γh(u )) H(W ) g (U ())W (). Next we transform (.) into an integral equation as follows. Recall that the equation has the characteristic equation that has two real roots λ = c c +4 W + cw W = λ + cλ = <, λ = c + c +4 Thus (.) is equivalent to the following integral equation: eλ( s) [( + g (U (s)))w (s)+r + R + R 3 ] ds (.) W = λ λ +. e λ( s) [( + g (U (s)))w (s)+r + R + R 3 ] ds We will study the existence of a solution W C to (.). For this purpose, we define a linear operator L : C C by L(W )() =W eλ( s) ( + g (U (s)))w (s)ds λ λ e λ( s) ( + g (U (s)))w (s)ds. λ λ It is obvious that L(W ) C if W C. In order to verify the existence of a solution W C to (.), we need to establish some estimates for the terms in the right-hand side of (.) when W C. We have the following. Lemma.3. For each δ>, there is a σ> such that (.3) R (,τ,φ) R (,τ,ϕ) C δ φ ϕ C >. and e λ( s) R (s, τ, φ) R (s, τ, ϕ) ds + δ φ ϕ C (.4) e λ( s) R (s, τ, φ) R (s, τ, ϕ) ds

8 CHUNHUA OU AND JIANHONG WU for all finite τ and all φ, ϕ B(σ), where B(σ) is the ball in C with radius σ and the center at the origin. Proof. Since R is the remainder of Taylor s expansion and Hφ φ for any φ C, wehave (.5) R (,τ,φ) = O( φ C ) as φ C, uniformly for all finite τ. Obviously R (,τ,φ), (R ) φ (,τ,φ) (the derivative of R with respect to φ), and (R ) φφ (,τ,φ) (the second derivative of R with respect to φ) are continuous for φ in a neighborhood of the origin in C, and τ [,τ ], where τ is a positive number. Therefore, (.3) and (.4) follow from (.5). Lemma.4. As τ, we have e λ( s) R (s, τ)ds + e λ( s) R (s, τ)ds = O( τ) uniformly for all (, ). Proof. Since H(U ) R (,τ) =U +γh(u ) g(u ) H(U ) = U +γh(u ) U U, +γu we need to show only that when τ is small, the following: (.6) and (.7) hold. In fact, we have = = = O( τ). e λ( s) H(U )(s) U (s) ds = O( τ) e λ( s) H(U )(s) U (s) ds = O( τ) e λ( s) H(U )(s) U (s) ds e λ( s) e y /4τ U (s y + cτ)dy U (s) ds e λ( s) e y /4τ [U (s y + cτ) U (s)] dy ds e λ( s) e y /4τ ( y + cτ)dyds U Similarly we can show (.7). Lemma.5. There exists an M > such that for all W C, the following inequality: (.8) e λ( s) R 3 (s, τ, W )ds + e λ( s) R 3 (s, τ, W )ds τm W C

9 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL holds. Furthermore for any two elements φ and ϕ in C, we have e λ( s) (R 3 (s, τ, φ) R 3 (s, τ, ϕ))ds + e λ( s) (R 3 (s, τ, φ) R 3 (s, τ, ϕ))ds = O( τ) φ ϕ C. (.9) Proof. We rewrite R 3 as ( H(U ) R 3 (,τ,w) = W +γh(u ) U ) +γu ( ( + γ)u H(W ) ( + γh(u )) ( + γ)u ) ( + γu ) (.) ( + γ)u ( + γu ) (H(W ) W ). Therefore, for the integrations of the first and the second lines on the right-hand side of (.), we have from (.6) the following estimates: ( e λ( s) H(U ) W +γh(u ) U ) (.) ds +γu = O( τ W ), and (.) ( ( + e λ( s) γ)u H(W ) ( + γh(u )) ( + γ)u ) ( + γu ) ds = O( τ W ), due to the fact H(W ) W. For the integration of the function in the last line of (.), if W C, by exchanging the order of integration and integration by parts, we have = = = e λ( s) ( + γ)u (H(W ) W ) ds ( + γu ) e λ( s) ( + γ)u ( + γu ) e λ( s) ( + γ)u ( + γu ) e y /(4τ) = ( + γ)u () ( + γu ()) = O( τ W ). (.3) e y /(4τ) y+cτ e y /(4τ) y+cτ e y /(4τ) (W (s y + cτ) W (s)) dyds e y /(4τ) y+cτ y+cτ W (s + )ddyds e λ( s) ( + γ)u ( + γu ) W (s + )dsddy W (s + ) W ( + )ddy To obtain the above result, we have used the fact that [ d e λ( s) ( + γ)u ] (s) ds ( + γu (s)) ds [ e λ( s) ( + γ)u ] ( + γu ) dsddy

10 CHUNHUA OU AND JIANHONG WU is uniformly bounded for all (, ). Therefore, from (.), (.), and (.3) it follows that there exists a constant M such that (.4) e λ( s) R 3 (s, τ, W )ds τm W C. Similarly, we can prove that there exists a constant M so that (.5) e λ( s) R 3 (s, τ, W )ds τm W C. Therefore, it follows from (.4) and (.5) that for any W C, we have e λ( s) R 3 (s, τ, W )ds + e λ( s) R 3 (s, τ, W )ds τm W C, with M = M j. j= Since C is dense in C, the inequality (.8) holds for all W C. Thus (.9) is satisfied due to the fact that R 3 (s, τ, W ) is a linear functional of W. We should mention that for each τ>and W C, we have R,R, and R 3 C and hence e λ( s) (R + R + R 3 )ds + e λ( s) [R + R + R 3 ] ds C. Now we are ready to prove our main result. Theorem.6. For any c, there exists a constant δ = δ(c) > so that for any τ [,δ], (.) possesses a traveling wavefront u(t, x) =U(x ct) satisfying U() =and U() =. Proof. Define an operator T :Ψ C C from the homogeneous part of (.) as follows: (.6) T Ψ() =cψ ()+Ψ ()+g (U ())Ψ(). The formal adjoint equation of T Ψ = is given by (.7) cφ ()+Φ ()+g (U ())Φ() =, R. We now divide our proof into five steps. Step. We claim that if Φ C is a solution of (.7) and Φ is C -smooth, then Φ=. Moreover, we have R(T )=C, where R(T ) is the range space of T. Indeed, when,u () and g (U ()). Then (.7) tends asymptotically to an equation with constant coefficients (.8) cφ ()+Φ ()+Φ() =. The corresponding characteristic equation of (.8) is (.9) λ cλ +=.

11 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 3 Both roots of (.9) have a positive real part as c, and thus we can conclude that any bounded solution to (.8) must be the ero solution. So as, any solution to (.7) other than the ero solution must grow exponentially for large. Then the only solution satisfying Φ(±) = is the ero solution. By the Fredholm theory (see Lemma 4. in []) we have that R(T )=C. Step. Let Θ C be given. We conclude that if Ψ is a bounded solution of T Ψ=Θ, then we have lim ± Ψ() =. In fact when, the equation (.3) cψ ()+Ψ ()+g (U ())Ψ() =Θ tends asymptotically to (.3) cψ ()+Ψ ()+Ψ() =. Note for (.3), the ω-limit set of every bounded solution is just the critical point Ψ=. Using Theorem.8 from [9], we find that every bounded solution of (.3) also satisfies lim Ψ() =. When, (.3) tends asymptotically to (.3) cψ ()+Ψ ()+g ()Ψ() =. Since g () =, the characteristic equation of (.3) has two eigenvalues: λ < and λ >. Thus every bounded solution of (.3) must satisfy lim Ψ() =. Inverting the time from to and using the result in [9] again, we know that any bounded solution to (.3) satisfies lim Ψ() =. Hence the claim of Step holds. Step 3. For a linear operator L : C C defined by eλ( s) ( + g (U (s)))w (s)ds L(W )() =W λ λ +, e λ( s) ( + g (U (s)))w (s)ds we want to prove that R(L) =C ; that is, for each Z C, we have a W C so that eλ( s) ( + g (U (s)))w (s)ds W λ λ + = Z(). e λ( s) ( + g (U (s))w (s))ds To see this, we assume that ξ() =W () Z() and obtain an equation for ξ as follows: eλ( s) ( + g (U (s)))ξ(s)ds ξ() = λ λ + e λ( s) ( + g (U (s)))ξ(s)ds eλ( s) ( + g (U (s)))z(s)ds + λ λ +. e λ( s) ( + g (U (s)))z(s)ds

12 4 CHUNHUA OU AND JIANHONG WU Differentiating both sides twice yields (.33) cξ ξ () g (U ())ξ() =(+g (U ()))Z(). Using the results that R(T )=C in Step and Z C, we obtain by Step that there exists a solution ξ() satisfying (.33) and ξ(±) =. Returning to the variable W, we have W = ξ + Z C. Step 4. Let N(L) be the null space of operator L. By Lemma 5. in [9], there is a subspace N (L) inc so that C = N (L) N(L); see also []. It is clear that N (L) is a Banach space. If we let S = L N (L) be the restriction of L to N (L), then S : N (L) C is one-to-one and onto. By the well-known Banach inverse operator theorem, we have that S : C N (L) isa linear bounded operator. Step 5. When L is restricted to N (L), (.) can be written as eλ( s) [R + R + R 3 ] ds S(W )() = λ λ +. e λ( s) [R + R + R 3 ] ds Since the norm S is independent of τ, it follows from Lemmas.3,.4, and.5 that there exist σ>,δ >, and <ρ< such that for all τ (,δ] and W, ϕ, ψ B(σ) N (L), and F (,W) ( W + σ) 3 F (,ϕ) F (,ψ) ρ ϕ ψ, where F (,W) = S λ λ eλ( s) [R + R + R 3 (τ, s, W )] ds + e λ( s) [R + R + R 3 (τ, s, W )] ds. It is easy to know that for any W B(σ) N (L), we have F (,W) ( W + σ) σ. 3 Hence F (,ϕ) is a uniform contractive mapping for W N (L) B(σ). By using the Banach contraction principle, it follows that for τ [,δ], equation (.) has a unique solution W N (L). Returning to the original variable, W + U is a heteroclinic connection between the two equilibria and. This completes the proof. 3. The distributed delay case. In this section we consider (.) with the kernel function f(t, s, x, y) =G(t s)δ(x y),

13 where TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 5 (3.) G(t) = τ e t/τ or G(t) = t τ e t/τ. We shall focus on the following equation (3.) u t = u x + u t t τ +γ t e (t )/τ u(, x)d t τ e (t )/τ u(, x)d since the corresponding analysis for the weak kernel G(t) = τ e t/τ is much easier. Instead of using the linear chain trick which is valid only for the kernels in (3.), we shall use the approach in section to prove rigorously that traveling fronts exist when τ is small. As in section, assume that u(t, x) =U(), = x ct, where c. Substituting u = U() into (3.), we have a wave equation for U cu = U + U +γ τ e /τ U( + c)d τ e /τ U( + c)d. Let U = U + W, where U is the traveling fronts for (.4). Then we have an equation for W of the form (3.3) cw = W U W +(U + W )h[u + W ] U, +γu where the functional h is defined by h[u]() = +γ Applying Taylor s expansion to h[u + W ], we have h[u + W ]() = = h[u ]() ( +γ Thus (3.3) becomes +γ ( + γ) τ e /τ U( + c)d τ e /τ U( + c)d. τ e /τ [U ( + c)+w ( + c]dyd τ e /τ [U ( + c)+w ( + c]dyd τ e /τ U ( + c)d ) cw = W + g (U ())W () (3.4) + R (,τ,w)+r (,τ)+r 3 (,τ,w), where R (,τ,w) =(U + W )h[u + W ]() U h[u ]() ( + γ) (3.5) + U ( +γ τ e /τ U ( + c)d ) h[u ]W, (3.6) R (,τ) =U h[u ]() U U +γu, τ e /τ W ( + c)d +. τ e /τ W ( + c)d

14 6 CHUNHUA OU AND JIANHONG WU and (3.7) R 3 (,τ,w) = ( + γ) U ( +γ τ e /τ W ( + c)d τ e /τ U ( + c)d ) +h[u ]W () g (U ())W (). To obtain the existence of the traveling fronts when τ is small, we need to prove that Lemmas.3,.4, and.5 hold. The proofs are quite similar to those in the discrete case, so we shall prove only Lemma.5 as an illustration and leave the proofs of Lemmas.3 and.4 to interested readers. Proof of Lemma.5 in the case of distributed delay. Note that g (U ()) = U +γu ( + γ)u ( + γu ). So where ( ) R 3 (,τ,w) = F (U )() τ e /τ W ( + c)d W () ( ) ( + γ) U ( +γ τ e /τ U ( + c)d ) ( + γ)u ( + γu ) W ( + h[u ] U ) W, +γu ( + γ)u F (U )() = ( +γ τ e /τ U ( + c)d ). We note that when τ is small, τ e /τ U ( + c)d U () =O(τ) holds uniformly for any (, ). Therefore, we have ( ) e λ( s) ( + γ) U ( +γ τ e /τ U (s + c)d ) ( + γ)u ( + γu ) W (s)ds = O(τ W ) and ( e λ( s) h[u ] U ) W (s)ds = O(τ W ). +γu We now prove that ( ) e λ( s) F (U )(s) τ e /τ (W (s + c) W (s)) d ds = O(τ W ). Using the fact that F (U )(s) =O()

15 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 7 for any s (, ), we need only prove that ( ) e λ( s) τ e /τ (W (s + c) W (s)) d ds = O(τ W ). Indeed, when W C, we have W (s + c) W (s) = c W (s + v)dv. Exchanging the order of integration and using the integration by parts, we obtain ( ) e λ( s) τ e /τ (W (s + c) W (s)) d ds ( c ) = τ e /τ e λ( s) W (s + v)dsdv d = O(τ W C ). Continuing the process as in section, we can show that Lemma.5 remains true, and so does the result in Theorem.6 for (3.). Similarly, we can prove that Theorem.6 is true if the kernel function is replaced by f(t, s, x, y) = τ e (t s) δ(x y). 4. The distributed delay and spatial-averaging case. In this section, we consider (.) with the distributed delay and spatial averaging. Namely, we study the following equation: (4.) u t = u + u(t, x) x + t τ e (t )/τ t τ e (t )/τ exp( (x y) /(4(t ))) 4π(t ) exp( (x y) /(4(t ))) 4π(t ) u(, y)dyd. u(, y)dyd As before, by a traveling wavefront, we mean a solution u(t, x) =U() =U( ct + x), where c> is the wave speed. Thus this specific kind of solution satisfies the following second-order ODE: (4.) cu = U + U H (U)() +γh (U)(), where H (U)() = τ e /τ e y /4 U( y + c)dyd. 4π We suppose that U can be approximated by U and hence assume that U = U + W. Then we obtain the equation for W as follows: (4.3) cw = W +(U + W ) H (U + W )() +γh (U + W )() U U () +γu ().

16 8 CHUNHUA OU AND JIANHONG WU Applying Taylor s expansions to (U + W ) H (U + W )() +γh (U + W )(), we have (U + W ) H (U + W )() +γh (U + W )() = U H (U ) +γh (U ) +W H (U ) +γh (U ) ( + γ) U ( + γh (U )) H (W ) (4.4) +R (,τ,w), where R (,τ,w) is the remainder (higher order terms) of this expansion, and this can be rewritten as R (,τ,w) =(U + W ) H (U + W )() +γh (U + W )() H (U ) U +γh (U ) W H (U ) +γh (U ) + ( + γ)u (4.5) ( + γh (U )) H (W ). Recall that g(x) =x x +γx and g (x) = x ( + γ)x +γx ( + γx). Therefore, in view of (4.4), (4.3) becomes (4.6) cw = W + g (U ())W () + R (,τ,w)+r (,τ)+r 3 (,τ,w), where and R (,τ) =U H (U ) +γh (U ) g(u ) R 3 (,τ,w) = W H (U ) +γh (U ) ( + γ)u ( + γh (U )) H (W ) g (U ())W (). As before, we transform (4.6) into the following integral equation eλ( s) [( + g (U (s)))w (s)+r + R + R 3 ] ds (4.7) W = λ λ +. e λ( s) [( + g (U (s)))w (s)+r + R + R 3 ] ds Now the above argument can be repeated to show that Theorem.6 holds for (4.). Similarly, we can prove Theorem.6 for (.) with the kernel function (4.8) f(t, s, x, y) = τ e (t s) 4π(t s) e (x y) /(4(t s)).

17 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 9 5. Traveling wavefronts with large wave speed. In section, we obtained a traveling wavefront for our model by assuming that the maturation time τ is small. In this section we utilie the idea of Canosa [6] to investigate the existence of traveling wavefront for (.) without the smallness requirement of τ. Although the method is originally a formal asymptotic analysis as the front speed approaches infinity, it is known that for Fisher s equation the method generates a solution that is accurate within a few percent of the true solution, even at the minimum speed. The method has also been applied to other reaction-diffusion equations, including coupled systems, with a very good accuracy; see [] and [4]. The main purpose here is to give a theoretical justification of the method for our food-limited model by showing the fact that when the wave speed tends to infinity, our traveling wavefront approaches a heteroclinic solution (the leading term of Canosa s expansions) of the original model without diffusion. The main idea of this section is from [9], except that we use some known results of global stability of the positive equilibrium instead of applying Smith and Thieme s order preserving semiflows theory [6]. Lineariing (.7) for U far ahead of the front, where U, gives cu () =U () U(). To ensure that we are studying ecologically realistic fronts that are positive for all values of, we assume, as in Fisher s equation, that the wave speed c. Following Canosa s approach, we introduce the small parameter and seek a solution of the form Equation (.7) becomes (5.) εg + G + G ε =/c 4 U() =G(ζ), +γ ζ = ε. e y /4τ G(ζ εy + τ)dy e y /4τ G(ζ εy + τ)dy =. When ε =, (5.) reduces to (5.) G G(ζ + τ) + G +γg(ζ + τ) =. For (5.), we have the following result concerning the heteroclinic orbit connecting the two equilibria G = and G =. Theorem 5.. Assume τ/( + γ) < 3. Then (5.) has a heteroclinic orbit g (ζ) connecting the two equilibria G =and G =. Proof. When ε =, we set g (ζ) =G( ζ) to invert (5.) into a delay differential equation (5.3) g g (ζ τ) = g +γg (ζ τ). By the result in [8] or [8], we know that the equilibrium g = is a global attractor as long as the initial value g (s) =φ(s), s [ τ,], satisfies φ() > and φ(s) for s [ τ,].

18 CHUNHUA OU AND JIANHONG WU Lineariing (5.3) around g =, we have (5.4) g = g. Therefore, the unstable space E u of the trivial solution in the usual phase space C τ = C([ τ,]; R) of continuous functions equipped with the sup-norm is spanned by χ(s) =e s,s [ τ,]. Let E s be the subspace in C τ so that C τ = E s E u ; then there exists ɛ > and a C -map w : E u E s, with w() = and Dw() = so that a local unstable manifold of g = is given by ɛχ + w(ɛχ) for ɛ ( ɛ,ɛ ). Choose ɛ > sufficiently small so that the operator norm Dw(ɛχ) <e τ for ɛ (,ɛ ). Then pick up ɛ (,ɛ ) and consider φ = ɛχ + w(ɛχ). We have φ(s) =ɛe s + w(ɛχ)(s) >ɛe s e τ ɛ χ ɛ(e τ e τ ). So the solution from the point φ on the local unstable manifold of g = is positive and tends to due to the global attractivity of the positive equilibrium g =. Returning to the original variable, we have an orbit G connecting the two equilibria G = and G =. This completes the proof. For (5.3), the positive equilibrium is a node, and all of the conditions in Theorem. in [9] are satisfied. Thus direct application of this result gives the following. Theorem 5.. Assume τ/( + γ) < 3. There is a constant c > such that for any c>c, (5.) has a traveling wave solution G(x ct) connecting the two equilibria and. When the wave speed c, the wave profile G(ξ) converges to a solution to (5.). Although the result is a consequence of Theorem. in [9], for the completeness of this paper and the convenience of readers, we outline the proof of this theorem as follows. For (5.), set Ḡ(ζ) =G( ζ). Then Ḡ satisfies the equation εḡ Ḡ + Ḡ +γ e y /4τ Ḡ(ζ + εy τ)dy e y /4τ Ḡ(ζ + εy τ)dy =. Now when ε is small, we use g to approximate the wavefront Ḡ = g + W. Then we have an equation for W Ḡ(ζ) in (5.). Let (5.5) W = εw + εg +(g + W ) h (g + W ) +γh (g + W ) g g (ζ τ) +γg (ζ τ), where the functional h is given by (5.6) h [U](ζ) = By means of Taylor s expansion, we have e /4τ U(ζ + ε τ)d. (5.7) (g + W ) h (g + W )(ζ) +γh (g + W )(ζ) = g h (g ) +γh (g ) +W h (g ) +γh (g ) ( + γ)g ( + γh (g )) h (W ) +R (ζ,τ,w),

19 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL where R (ζ,τ,w) is the remainder (higher order terms) of this expansion. Therefore by (5.7), (5.5) becomes (5.8) W = εw + P W ()+ R (ζ,τ,w)+r (ζ,τ)+r 3 (ζ,τ,w), where the linear operator P : C C is defined by and P W (ζ) = g (ζ τ) +γg (ζ τ) W (ζ) g ( + γ) W (ζ τ), ( + γg (ζ τ)) h (g ) R (ζ,τ)=g +γh (g ) g g (ζ τ) +γg (ζ τ) + εg, R 3 (ζ,τ,w)= W h (g ) +γh (g ) g ( + γ) ( + γh (g )) h (W ) W g (ζ τ) +γg (ζ τ) + g ( + γ) ( + γg (ζ τ)) h (W ). Now we prove that there exists a W C satisfying (5.8) when ε is small. Equation (5.8) can be transformed into an integral equation as follows. We first write (5.8) as (5.9) εw W W = W P W R R R 3. Since the equation has two real eros λ and λ, with (5.) λ = +4ε ε (5.9) is equivalent to the integral equation ελ λ = <, λ = + +4ε ε >, (5.) W (ζ) = ζ +4ε e λ(ζ t) [W (t)+p W (t)]dt + e λ(ζ t) [W (t)+p W (t)]dt +4ε ζ ζ + +4ε e λ(ζ t) [R + R + R 3 ]dt + e λ(ζ t) [R + R + R 3 ]dt. +4ε ζ It is easy to show that lim ε + λ =, lim λ =+. ε +

20 CHUNHUA OU AND JIANHONG WU Thus we have, from (5.), that (5.) ζ W (ζ) e (ζ t) [W (t)+p W (t)]dt ζ [ e λ (ζ t) ] = e (ζ t) [W (t)+p W (t)]dt +4ε + e λ(ζ t) [W (t)+p W (t)]dt +4ε ζ ζ + +4ε e λ(ζ t) [R + R + R 3 ]dt + e λ(ζ t) [R + R + R 3 ]dt. +4ε ζ For the right-hand side of (5.), in a similar manner as in section, we can prove that ζ [ e λ (ζ t) ] e (ζ t) [W (t)+p W (t)]dt = O( ε W C ), +4ε e λ(ζ t) [W (t)+p W (t)]dt = O( ε W C ), +4ε ζ ( ζ ) e λ(ζ t) R dt + e λ(ζ t) R dt = O( W ), +4ε ζ ( ζ ) e λ(ζ t) R dt + e λ(ζ t) R dt = O( ε), +4ε ζ and ( ζ +4ε e λ(ζ t) R 3 dt + ζ ) e λ(ζ t) R 3 dt = O( ε W C ). Let L be the linear operator defined by the left-hand side of (5.), namely, [LW ](ζ) =W (ζ) ζ e (ζ t) [W (t)+p W (t)]dt. It is obvious that if W C, then LW C. In order to use the argument in section to prove our result, we need to prove that R(L) =C, where R(L) is the range space of L; that is, for each u C, we need to show that equation LW = u or, equivalently, W (ζ) ζ e (ζ t) [W (t)+p W (t)]dt = u(ζ), ζ (, ) has a solution in C. For this purpose, we set w = W u. Upon substitution, we have an equation for w : (5.3) w = P w(ζ)+u(ζ)+p u(ζ). Define an operator T :C C by [Tw](ζ) =w (ζ) P w(ζ)

21 TRAVELING WAVEFRONTS IN A FOOD-LIMITED MODEL 3 and the formal adjoint equation of Tw =by (5.4) φ (t) = g (t τ) +γg (t τ) φ(t)+ g ( + γ) φ (t + τ),t (, ). ( + γg (t τ)) When t, (5.4) tends asymptotically to φ (t) = φ(t + τ). +γ When τ/( + γ) < π, it is easy to see that if φ is a bounded solution to (5.4), then φ =. From p. 7 of Chow, Lin, and Mallet-Paret [5], we see that T is Fredholm and R(T )=C. Therefore, (5.3) has a solution w C. From now on we can use the same argument as in sections 4 and 5 in [9] to verify that (5.) has a solution W C. 6. Summary and simulations. In this paper we have studied the existence of traveling wavefronts for the food-limited population model that involves nonmonotone delayed nonlocal response. The classical phase-plane approach or super/subsolution technique does not work for this type of model due to the lack of monotonicity. Hence, we develop a perturbation argument based on some analytical tools such as the contraction mapping principle and the Fredholm theory to establish the existence of traveling wavefronts. We consider three cases with spatiotemporal averaging when the delay is small. In the general case where the smallness condition on the delay is no longer required, we also developed Canosa s method to establish traveling wavefronts with large wave speeds. Our work shows how our perturbation analyses based on some analytical tools are particularly useful for models with small delay or for wavefronts with large wave speeds. We believe the smallness condition on τ and the largeness condition on the wave speed can be removed by a certain homotopy argument, and this remains to be a subject for future study. We should emphasie the difficulty caused by the nonmonotonicity of the delayed nonlocal response. In particular, we note that the traveling wavefronts obtained have prominent humps as the following two numerical simulations show. The first numerical simulation, reported in Figure, is for (.), carried out by using Matlab on a spatial domain L x L (for some L > ) with homogeneous Neumann boundary conditions at both ends. For initial data, we set a nonero steady state value at the left side and ero elsewhere for all t [ τ,]. The solution stabilies to a wavefront when time t goes on. For τ sufficiently small, the resulting traveling fronts appear to be strictly monotone. Increasing the value τ, we find that the monotonicity is lost and a prominent hump is exhibited. When γ = and τ =, the solutions at two different times are shown in Figure. The second numerical simulation is about (4.) with the kernel given by (4.8). Set t v(t, x) = τ e (t s) e (x y) /(4(t s)) u(s, y)dyds. 4π(t s) Then it is easy to recast (.) into the form u t = u xx + u v +γv v t = v xx + τ (u v).

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