Dynamics of a Three Species Competition Model

We investigate the dynamics of a three species competition model, in which all species have the same population dynamics but distinct dispersal strategies. Gejji et al. [15] introduced a general dispersal strategy for two species, termed as an ideal free pair in this paper, which can result in the ideal free distributions of two competing species at equilibrium. We show that if one of the three species adopts a dispersal strategy which produces the ideal free distribution, then none of the other two species can persist if they do not form an ideal free pair. We also show that if two species form an ideal free pair, then the third species in general can not invade. When none of the three species is adopting a dispersal strategy which can produce the ideal free distribution, we find some class of resource functions such that three species competing for the same resource can be ecologically permanent by using distinct dispersal strategies.


Introduction.
Understanding the dynamics of interacting species has always been an important subject in population dynamics. For systems consisting of mul tiple competing species, of great interest are issues that concern competitive exclu sion and the coexistence of species. In recent years many studies have considered the role of spatial movement of organisms on the persistence of interacting species [6,26,27,30,32]. In this article we shall address the following question of Chris Cosner [9]: in a spatially heterogeneous environment, can three competing species with the same population dynamics coexist, via different dispersal strategies? In order to give some context for Cosner's question, we provide a brief review of previ ous works on two competing species, starting with the work of Dockery et al. [12], in which they considered the following model for two competing species: Here u(x, t) and υ(x, t) account for the densities of two competing species at location x and time t, μ, v correspond to their random diffusion rates, and Δ := ΣNi= 1 a 2 / a x 2 i denotes the Laplace operator in the Euclidean space RN . The function m(x) represents the intrinsic growth rate of both species at location x and is always assumed to be non-constant to reflect the spatial heterogeneity of the habitat (e.g., hetero geneous spatial distribution of resources and predation rates). The habitat Ω is a bounded domain in RN with smooth boundary aΩ. The vector n denotes the outward unit normal vector on aΩ, and the boundary conditions in (1.1) mean that there is no net flux of population across the boundary.
Dockery et al. [12] showed that if both species disperse by random diffusion, then the slower diffusing species will drive the faster diffusing species to extinction. In terms of the persistence theory, this result implies that in a spatially varying but temporally constant environment, two competing species with the same popu lation dynamics but different random diffusion rates cannot coexist. Hastings [19] suggested that environmental cues can have important effect on the evolution of dis persal strategy for species. Following the work of Belgacem and Cosner [3], Cantrell et al. [7] extended model (1.1) by adding an advection term for species u as follows: where α > 0 is the advection rate of species u, which measures the tendency of the species u to move upward along the gradient of rn. The following result is proved in [7]: Theorem 1.1. [7] Suppose that m ∈ C2(Ω), lΏm > 0, the set of critical points of m has Lebesque measure zero, and m has at least one isolated global maximum. Then for every μ > 0 and v > 0, there exists some positive constant a* such that for a > a*, system (1.2) has at least one stable positive steady state. Theorem 1.1 says that as long as species u has sufficiently large advection, both species can persist within the habitat. Cantrell et al. [7] suggested that such a coexistence result is possible because species u concentrates primarily on the local maxima of m, leaving enough resources in other locations for the species υ to utilize. Chen and Lou [10] demonstrated that for appropriate m with a unique local maximum in Ω, species u with large advection is concentrated at this maximum as a Gaussian distribution. When m has multiple local maxima, Lam and Ni [24] and Lam [22,23] recently completely determined the profiles of all positive steady states of (1.2). These works illustrate a general mechanism for the coexistence of two competing species with the same population dynamics but different dispersal strategies.
In another closely related direction, Cantrell et al. [8] observed that the influence of spatial resource variability on the competition of species is linked to the fact that diffusion generally produces a mismatch between population density and the quality of the environment. Is it possible to find dispersal mechanisms that can produce a perfect match of the population density with the environment? To this end, they generalized model (1.2)

as
Here P(x), Q(x) ∈ C2(Ω) provide advective directions for the respective species as well as regulate their speeds in such directions. This generalization allows for the possibility that populations can "match environmental quality perfectly". In particular, consider the single species equation for u in (1.3) (i.e. set v = 0), if P = ln m, then u* = m is always a positive steady state. Note that the net flux for species u satisfies Vu* -u*V(ln m) = 0 in Ω and the fitness of species u is equilibrated throughout Ω: m∕u* ≡ 1. A population exhibiting such a spatial distribution is said to have an ideal free distribution [13]. That is, the density of species u at any location x ∈ Ω is proportional to the habitat quality m(x). In this context, we shall refer P = ln m as an ideal free dispersal strategy. Furthermore, Cantrell et al. [8] demonstrated that selection favors this ideal free dispersal strategy as it can beat any other "nearby" strategy. In [8] they conjectured that the ideal free dispersal strategy should be a global evolutionary stable strategy. Averill et al. [1] recently proved this conjecture in the following result.

Theorem 1.2. [1]
Suppose that m is a positive, non-constant function and m ∈ C2(Ω). If P = ln m and Q -In m is non-constant, then (m, 0) is a globally asymp totically stable steady state of (1.3) among initial data that are nonnegative and not identically zero. Theorem 1.2 says that if species u plays the ideal free strategy and species v does not play an ideal free dispersal strategy, then species u always drives v to extinction. This means that if one of P -ln m and Q -ln m is a constant function, then we cannot expect the coexistence of these two species. Is coexistence possible if neither Pln m nor Q -ln m is constant? To address this question, we consider the case that P = ln m + αR and Q = ln m + βR, where R(x) ∈ C2(Ω) and α,β ∈ R. Note that α = 0 and β = 0 correspond to ideal free dispersal strategies for species u and υ, respectively. With such assumptions on P and Q, Cantrell et al. [8] showed that as long as αβ < 0, coexistence is always possible provided that μ = v, Ω is an interval and Rx ≠ 0 in Ω. Recently Averill et al. [1]  Note that in Theorem 1.3 neither species plays the ideal free dispersal strategy because α ≠ 0 and β ≠ 0. The spirit of Theorem 1.3 is the same as that of Theorem 1.1. Consider Fig. 1. We call species υ in Fig. 1 a specialist as it pursues resources near the maxima of the resource function. In contrast, we call species u in Fig. 1 a generalist since it also makes use of resources away from such maxima. Theorem 1.3 demonstrates that generalist and specialist can persist together.
Since lΏ m2R.∕lΏ m2 < maxΏ R, we see that x0 in assumption (Al) can not be a global maximum of R. This implies that any function R satisfying (A1) has at least two local maxima.

Theorem 1.5. [1]
Suppose that R satisfies assumption (Al) and all critical points of R are non-degenerate. Assume P(x) = ln m + αR and Q(x) = ln m + βR. Then there exists some α0 > 0 such that for every a ∈ (O,α0), we can find some μ0 > 0 such that if μ > μ0, then given any v> 0, both (u*,0) and (0,υ*) are unstable for sufficiently large β and system (1.3) has at least one stable positive steady state. Theorem 1.5 demonstrates that for this new range of a and β both species act as specialists and they both can persist. While this seems to contradict the principle of competitive exclusion, as both species tend to purse locally the most favorable resources, Averill et al. [1] proposed that "if resource functions have two or more local maxima, the resident species at equilibrium may undermatch its resource at some local maximum of the resource, which makes it vulnerable to invasion by other species near such local maxima." Numerical simulations in [15] have indicated that this in effect leads to the species with the larger advection concentrating at some local (but not the global) maximum while the species with less advection concentrates at the global maximum. Biologically, this can be regarded as niche differentiation or niche separation as each species carves out a niche near different maximum of the resource. It seems a bit mysterious that it is the species with less advection, not the one with the larger advection, which concentrates at the global maximum of m.
To summarize the parameter ranges of α and β for the coexistence of two species, we see two possibilities: 1. for any resource J? ∈ C2(Ω), when aβ < 0 (generalist vs. specialist) 2. for R satisfying (Al), with α positive small and β sufficiently large (specialist vs. specialist). It is an open problem whether two generalists can coexist with each other.
2. Main results. We now move into the three species realm. Can we establish results for three competing species similar to those for two species? For example, can two specialists and one generalist persist? If one of the species adopting ideal free dispersal strategy while other species do not, do we expect to see competitive exclusion? To address these questions, one serious mathematical difficulty immedi ately emerges. It is well known that systems of three competing species, unlike the two species competition model, are not monotone dynamical systems. To overcome this difficulty, we rely on two different methods, namely, the Lyapunov functional method (for competition exclusion) and "practical persistence" (for permanence) as described in [5].
We consider the following three species model: where species w has diffusion rate y > 0, and L(x) ∈ C2(Ω). First, we aim to generalize Theorem 1.2 to (2.1). In view of Theorem 1.2, it is tempting to propose the following: Conjecture. If P -ln m is constant, Qln m is non-constant, and Lln m is non-constant, the steady state (m, 0,0) of system (2.1) is globally asymptotically stable for initial data that are non-negative and not identically zero.
It turns out that this conjecture is false. To see this, assume that P = ln m. Let r be any constant in (0,1) and let Q be any function satisfying (1r)to > eQ, and Qln to is non-constant in Ω. Set L = ln[(l -r)m -eQ], i.e., (1 -r)to = eQ + eL in Ω. Then (u,v,w) = (rm,eQ,eL) is always a positive steady state of (2.1). In particular, the steady state (m, 0,0) of system (2.1) cannot be globally stable. This example implies that the conjecture needs to be modified. In this connection, we first define an ideal free pair for species υ and w as follows: Definition 2.1. We say that Q and L form an ideal free pair if there exist nonnegative constants τ and η such that τeQ(x) > + ηeL(x) = m(x) in Ω.
Note that if Q and L form an ideal fee pair, then (u, v, w) = (0, τeQ(x), neL(x) is a non-negative and non-trivial steady state of (2.1) such that m-u-v-w ≡ 0 in Ω. Furthermore, the net fluxes for both species v and w at this equilibrium are equal to zero. In other words, a consequence of an ideal free pair for two competing species is that the spatial distributions of both species at equilibrium are ideal free; See [15]. If one of the two coefficients τ and η is equal to zero, say τ = 0, then Lln m is equal to some constant. Hence, an ideal free pair is a natural generalization of the ideal free distribution from a single species to two species.

Theorem 2.2.
Suppose that m is positive in Ω and non-constant, Pln m is constant, and Q and L do not form an ideal free pair. Then u( Theorem 2.2 says that the species playing the ideal free dispersal strategy (i.e., P = ln m up to a constant) will be the sole winner as long as the other two species do not form an ideal free pair. Continuing along this line of thought, we ask: suppose that two species adopt an ideal free pair dispersal strategy, can a third species invade? The answer in general is no, as shown by the following result.

Theorem 2.3. Suppose that m is positive and non-constant, and P and Q form an ideal free pair such that τep(x) + neQ(x) = m(x)
in Ω for some positive constants τ,η. If neither P and L nor Q and L form an ideal free pair, then Biologically, Theorem 2.3 implies that species w will be driven to extinction by species u and v.
Summarizing these results for three species, we see that under the hypotheses of Theorems 2.2 and 2.3, three species cannot coexist. For two species we see that one generalist and one specialist can always coexist, and for some non-monotone resource functions, two specialists can also coexist. Is it possible for two specialists and one generalist to coexist? Interestingly, we can show that this is possible. To this end, we consider the following three species model: Here we only study the case that the third competitor w moves via random diffusion. For species u and v, we assume that R = ln m, so that P = ln m + αR and Q = ln m + βR become P = (1 + α)hiTO and Q = (1 + β) ln m, respectively. Replacing 1 + α and 1 + β with a and β, respectively, we obtain the advective strategies, P = αln m and Q = β ln m.
As R is now a function of to, our goal is to find resource functions to and move ment parameters (μ, α, v, β, y) such that the permanence of three species is possible. In [6], Cantrell and Cosner describe permanence as "a qualitative criterion for ad dressing the qualitative issue of whether a model for interacting biological species predicts the coexistence of all the species in question." Practically, they say that "permanence in a model system for the densities of a collection of interacting species means the system possesses both an asymptotic 'ceiling' and a positive asymptotic 'floor' on the densities of all the species in question, the 'heights' of which are in dependent of the initial state of the system so long as each component is positive" [6]. Working from such a description, we utilize the following definition of ecological permanence as presented in [6]. with nonnegative and not identically zero initial data, then there is a T0 > 0 which depends only on the initial condition such that k < u(x,t) ≤ K, k < υ(x,t) < K, and k ≤ w(x, t) ≤ K for all x ∈ Ω and all t ≥ T0.
To establish the permanence of system (2.2), we rely on the following crucial assumption on m: (A2) There exists some x0 ∈ Ω such that x0 is local maximum of m(x) and We shall give some intuitive interpretation of (2.3) after the statement of the following result on the permanence of three competing species: Theorem 2.5. Suppose that m > 0, to ∈ C2(Ω), all critical points of m are nondegenerate, and it satisfies (A2). There exists some sufficiently large constant μ such that for any μ > μ, there exists some constant a. > 1 and close to 1 such that for every 1 < α < a and v, y > 0, there exists some sufficiently large constant β such that if β > β, system (2.2) is ecologically permanent. Furthermore, system (2.2) has at least one positive steady state.
From Theorem 1.3 we see that if α > 1, system (2.2) always has a steady state of the form (u, 0, w), where u and w are positive in Ω. In order for the species υ to invade when rare, since species υ has a strong tendency to concentrate at the local maxima of to, it is crucial that the growth rate of υ, given by muw, is strictly positive at some local maximum of to. If the inequalities in (2.3) are violated at each local maximum of to, it might cause muw to be negative at each local maximum of to and thus prevent the species υ to invade when rare. Biologically, species u and v both have an established niche as u concentrates near the global maximum of to and υ concentrates near some local maximum of to. Species w, on the other hand, has a more evenly spread distribution. In short, we may say that u pursues the "best" resource, v pursues the "second best" resource, and w goes after the "rest" of the resource (see Figure 2).
The rest of the paper is organized as follows: In Sect. 3 we establish Theorems 2.2 and 2.3. Sections 4-6 are devoted to the proofs of the uniform lower bounds of three species, respectively, among which the lower bound of species v is the most technical and is thus postponed to Sect. 6. The proof of Theorem 2.5 will be completed in Sect. 7. 3. Competitive exclusion. We first briefly discuss the well-posedness of the reaction-diffusion-advection model (2.1). We rewrite (2.1) as where u = ue-p, v = ve-Q and w = we-L in Ω. By the maximum principle for parabolic equations [28], if the initial data (u(x, 0), υ(x, 0), w(x, 0)) of (2.1) are nonnegative and not identically zero, then u(x, t),v(x, t),w(x, t) > 0 for every a: ∈ Ω and t > 0. Hence, u(x, t), υ(x, t),w(x, t) > 0 for every x ∈ Ω and t > 0. Similarly by the maximum principle we can show a priori that u, v, and w are uniformly bounded FIGURE 2. Illustration of Theorem 2.5: Graphs of m = 3e -50 (x -2)2 + 1.7e-40(x-.8)2 + .2 (black) and three species at equilibrium: u (red), v (green), w (blue), μ = 1000, v = 10, α = 4, β = 80, y = .05.
in L∞(Ω) norm for all t > 0. By the regularity theory of parabolic partial differential equations [14], u,υ, and w exist for all time and belong to C2,1(Ω × (0,∞)). Using the theory of analytic semi-groups and parabolic partial differential equations, we can recast system (2.1) as a dynamical system ∏[(u0, υ0. w0), t] defined on the space

LaSalle's invariance principle.
Let (B,p) be a complete metric space and {S(t),t ≥ 0} : B → B be a dynamical system on B. For any x ϵ B, the positive trajectory of x is defined as 7(x) := {S,(t)x|t ≥ 0}. We say that x is a ω-limit point of y(x0) if there exist tn > 0, tn → ∞ as n → ∞ such that limn→∞ p(S(tn)x0, x) = 0. We say that V is a Lyapunov functional on a set G in B if V is continuous in G (the closure of G) and for every x ∈ G, We will make use of the following version of LaSalle's invariance principle for infinite dimensional systems [17].

Proof of Theorem 2.2. Set
We first establish some a priori estimates for positive solutions of (2.1).
Proof. Integrating the equations of u, υ, and w in Ω and then summing up the results, we get Next, Integrating by parts, we see that Hence, we have our result.
Next we show that u has a positive uniform lower bound in Ω.
Similarly, we have Because (P, L) is not an ideal free pair, as above, we see that there are two possi bilities: (iii) n -n1 < 0 or (iv) nt1 -η1t < 0 and n -n1 > 0. In summary, we must rule out four cases: Case 1: (i) and (iii) hold, τ -τ1 < 0 and n -n1 <0. From (3.17) we see this is not possible.

Lower bound for species w. The goal of this section is to establish
Let u* denote the unique positive solution of The existence and uniqueness of u* is well-known as we are assuming m > 0 on Ω [6]. Clearly, when a = 1, then u* = m. The following result illustrates some interesting properties of u* when α ≠ 1. We first claim that if α ≠ 1, It suffices to show that u*∕mα is non-constant. We argue by contradiction. Suppose that u*∕mα is constant, then from (4.2), we see that u* = m. This implies that m∕mα is a constant. Since a ≠ 1 we must have that m is constant, which is a contradiction.
To complete the proof, we consider two cases:  Proof. See [6] for details. □ Similarly, we have the following result.

Lemma 4.4. Let v be a positive solution of
Then v(x,t) → υ*(x) uniformly as t → ∞, where v* is the unique positive steady state of (4.6).
Using Lemmas 4.3, 4.4, and comparison of solutions, we state the following in equalities. where β > 0 will be chosen larger later. By Lemma 4.2, if we assume a > 1, then ∫ω u* < ∫ω m. By Theorem 3.5 in [7] we see that as β -> ∞, ∫Ώ v* → 0. Hence, we can choose β > β1 such that We claim w → w* uniformly as t → ∞, where w* is the unique positive solution of the equation Since ∫Ώ(mu* -v* -2∕β) > 0, the zero steady state of (4.8) is unstable, thus w* exists for all y, and w → w* uniformly as t → ∞ [6]. By Corollary 1, w is super-solution of (4.7), that is, for all t ≥ T(β), w(x, t) ≥ w(x,t) on Ω. Finally, because w → w* uniformly, we can choose δ1 = minΏ w* to conclude that for all x ∈ Ω, lim inft→∞ w(x, t) ≥ δ1 > 0.  (2.2). Then for any μ, v, and y > 0 and for any a > 1, there exists a β2 such that for all β > β2 , there is a δ2 > 0 such that for all x ∈ Ω, lim inft→∞ u(x, t) ≥ δ2 Proof. To show this, we consider the following equations of u and w (coming from (2.2)):

By Corollary 1, v(x,t) ≤ υ*(x) + 1/β for t ≥ T(β). Hence we see that
Thus we are led to consider the following system:

Put y = [u∕mα). Then (5.3) becomes
We first note that (5.4) is a strongly monotone dynamical system (by Theorem 1.20 in [6] and the strong maximum principle). We claim that the semi-trivial steady state (0,w*) of (5.4) is unstable. Upon showing this, by the monotone dynamical system theory [20,31], any solution with nonnegative initial data will either be attracted to (y*,0), where y* > 0 in Ω, or an order interval bounded above and below by positive equilibria of (5.4). In any case, there exists a δ > 0 such that for all x ∈ Ω and t > T2, where T2 depends on the initial conditions and β, y(x, t) ≥ δ > 0. Since y = u∕mα and infΏ mα > 0, there exists a δ2 >0 such that for all x ∈ Ω and t > T2, u(x,t) > δ2 . Since u(x,t) ≥ u(x,t) on Ω and for t > T2, we see that lim inft→∞ u(x, t) ≥ δ2 > 0• To show that (0, w*) is unstable, we consider the following eigenvalue problem: where λ1 denotes the smallest eigenvalue of (5.5) and ψ is a positive eigenfunction associated to λ1. Dividing (5.5) by ψ and integrating the resulting equation over Ω, we obtain Also note that w* satisfies If we multiply (5.7) by (w*)α-1 and integrate the resulting equation over Ω we have that Combining (5.6) and (5.8), we get Since 7 > 0, α > 1, and w* > 0 on Ω, we notice that where equality holds if and only if both w* and ψ are constant functions. Now notice that [mα -(w*)α](m -w*) ≥ 0 in Ω with equality if and only if m = w*. Hence ∫Ώ[mα -(w*)α](mw*) = 0 if and only if m ≡ w* in Ω. We claim that ∫Ώ[mα -(w*)α](m -w*) > 0. To see this, suppose that m ≡ w* in Ω. Then m must satisfy equation (5.7) and by the maximum principle, m ≡ constant in Ω. This is a contradiction and thus ∫Ώ[mα -(w*)α](m -w*) > 0. By Theorem 3.5 in [7] we know that as β → ∞, ∫Ώv* → 0. Hence for large enough β, From (5.11) we have that λ1 < 0, proving that (0,w*) is unstable. □ 6. Lower bound for species υ. This is the most technical part of the paper. As we mentioned earlier, we work with non-monotone functions m that satisfy assumption (A2), looking for parameter values that cause species u to concentrate at the global maximum of m, species υ to concentrate at the local maximum of m at xo, and species w to pursue resources away from these maxima. However, for v to be able to concentrate at some x0 and persist, its growth rate (2.2) near x0 needs to be positive. That is, we need m(x) -u(x)w(x) > 0 in a neighborhood of x0, where (u, 0,w) is a steady state of the three species model (2.2) with u > 0 and w > 0 in Ω. So we first seek to understand the structure of the solution set (u,w) as we vary the advection parameter α near α = 1.
6.1. Structure of positive steady states for two species model. Consider the following two species model (this is steady state problem for system (2.2) with υ = 0 and α = 1 + ϵ): We note that for a > 1, there exists at least one steady state (u, w) of system (6.1) where both u and w are positive in Ω (see Theorem 1.4 in [1]). We also know that when α = 1 that (m, 0) is a solution of (6.1). Essentially, we will show that for α slightly larger than 1, system (6.1) has a unique branch of positive steady states bifurcating from (m, 0) at α = 1.
Proof. To see this, recall that φ satisfies (6.5). If we multiply (6.5) by mφ and integrate the result in Ω, we have So, we have that where strict inequality holds as φ is nonconstant. □ By Lemmas 6.4 and 6.5 we see that both the numerator and the denominator of (6.14) must be negative. Hence, λ'(0) > 0 and for both s and e small, Noticing that u = m1+ϵ(u + uϵ), we have demonstrated that we can parameterize the positive solution (u, w) of system (6.1) in terms of ϵ as follows: Theorem 6.6. Let (u,w) be a positive solution pair of system (6.1). Then for sufficiently small e, where Ʌ = l∕λ'(0). The next two results establish the fact that for a slightly larger than 1, the only positive solutions (u, w) of (6.1) are on the solution branch bifurcating from (m, 0) as described by Theorem 6.1. This completes the global picture for positive solutions of (6.1) for α slightly larger than 1. In fact, (6.1) has no positive solutions for α ≤1 and close to 1. Lemma 6.7. Consider a positive solution (u,w) to (6.1). Then limϵ→0+(u, w) = (m, 0).
Proof By elliptic regularity and the Sobolev embedding theorem, for 0 < e << 1, (u,w) is uniformly bounded in C2,τ(Ω) for some τ ∈ (0,1) [16]. If we let e → 0+, passing to a subsequence if necessary, then by the Ascoli-Arzela lemma, we see that (u,w) converges to (u,w) in C2(Ω), where (u,w) satisfies Suppose that (u,w) = (0,0). Set u = u∕║it║∞. By elliptic regularity and the equation of u we see that u → u1 in C2(Ω) for some u1 ≥ 0 in Ω, which satisfies ║u1║∞ = 1 and Integrating both sides of equation (6.18) over Ω and using the boundary condi tion, we see that But this is a contradiction since m > 0 and u1 ≥ 0, u1 ≡ on Ω. Now suppose (u, w) = (0, w*). Again we set u = u∕║u║∞ and see that by elliptic regularity and the equation of u we see that u → u1 in C2(Ω) for some u1 ≥ 0 in Ω, which satisfies ║u1║∞ = 1 and Since u1 ≥ 0,u1 ≠ 0, we see 0 is the principal eigenvalue for the eigenvalue problem But this contradicts the result in Theorem 2 of [1] which says that the above ei genvalue problem has a negative principal eigenvalue. Hence, we must have that (u,w) = (m, 0). □ Lemma 6.8. There exists ϵ0 > 0 such that for all e with 0 < ϵ < ∈0, (u*,w*) is the unique steady state of (6.1) and is linearly stable.
Using this and the fact that s = eΛ, where Λ = l∕λ'(0) we can write u* = (m + ϵm ln m)(Λϵφ+l + eu'0) + O(ϵ2) and w* = eΛ + O(e2). Substituting these expansions into the second equation of (6.22) we obtain the following equation in Ω Thus if we integrate both sides of (6.24), using the boundary condition and the definition of Λ, Because n1 > 0, we conclude that for sufficiently small positive e, η > 0. □ 6.2. Bounds on solutions of the three species system. Given a solution (u(x, t), υ(x,t),w(x, t)) of system (2.2), we aim to establish upper bounds on u(x, t) and w(x,t) in Ω x (T, ∞) for some T which depends on the nonnegative, not iden tically zero initial data of the solution (u, υ,w). The main result of this section is Theorem 6.9. Let (u,v,w) be any positive solution of (2.2) with α = l + ϵ. Assume that the set {x ∈ Ω : |▼m(x)| = 0} has Lebesgue measure zero. Then there exists an ∈0 > 0 such that for every 0 < ϵ < ϵ0, there exists Γ such that for all β > Γ, there exists a T > 0 such that u(x,t) ≤ u*(x) + 1∕β and w(x,t) ≤ w*(x) + ∕β on Ω x (T, ∞), where (u*,w*) is the unique positive steady state of (6.1).
To establish this result, we make use of appropriate "sub/super" systems as follows.