ON THE EFFECTS OF THE EXTERIOR MATRIX HOSTILITY AND A U-SHAPED DENSITY DEPENDENT DISPERSAL ON A DIFFUSIVE LOGISTIC GROWTH MODEL

. We study positive solutions to a steady state reaction diﬀusion equation arising in population dynamics, namely, (cid:40) where Ω is a bounded domain in R N ; N > 1 with smooth boundary ∂ Ω or Ω = (0 , 1), ∂u∂η is the outward normal derivative of u on ∂ Ω, λ is a domain scaling parameter, γ is a measure of the exterior matrix (Ω c ) hostility, and A ∈ (0 , 1) and (cid:15) > 0 are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for u < A and increasing for u > A . We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee eﬀect for certain range of λ . When Ω = (0 , 1) we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter γ varies. Our results indicate that when γ is large there is no Allee eﬀect for any λ . We employ a method of sub-supersolutions to obtain existence and multiplicity results when N > 1, and the quadrature method to study the case N = 1.

Habitat Ω and the exterior matrix Ω c .
1. Introduction. We study the steady state reaction diffusion model governed by a logistic growth and given by: where Ω is a bounded region in R N ; N > 1 with smooth boundary ∂Ω or Ω = (0, 1), ∂u ∂η is the outward normal derivative of u on ∂Ω, λ is a domain scaling parameter, γ is a measure of the exterior matrix (Ω c ) hostility, and α(u) is the probability of the population staying in the habitat Ω when it reaches the boundary. See [3], [6], [11] and [14] for the discussion on the derivation of the model. In this paper, we will focus on a case when the dispersal 1 − α(u) is U-shaped, that is when it is decreasing for lower densities and increasing for higher densities. In particular, we will discuss the case when α is of the form: is as shown in Figure 2 Note that the minimum dispersal is /[1 + ]. In [6] the authors studied the case when = 0. However, ecologists have noted that in many cases the minimum dispersal on the boundary never becomes zero ( > 0). In this paper we focus on the case when > 0, and establish non-existence, existence, uniqueness and multiplicity results for the model: It turns out that the analysis in [6] for the = 0 does not extend to this case and we require new ideas to study (1.4). Let D > 0 be a constant and E 1 (γ, D) be the principal eigenvalue of −∆v = Ev; x ∈ Ω ∂v ∂η + γ √ EDv = 0; x ∈ ∂Ω.
(1.5) Figure 2. An example that illustrates U-shaped density dependent dispersal (1 − α(u)) on the boundary. Note that the existence of E 1 (γ, D) > 0 follows from [13] where the authors study the eigenvalue problem: for any κ ∈ R. They prove for each κ, the principal eigenvalue B(κ) exists, and the eigencurve B(κ) is Lipschitz continuous, strictly decreasing, and concave. Further, In the case of (1.5), treating κ = −γD √ E (or E = κ 2 D 2 γ 2 ), we see that the principal eigenvalue E 1 (γ, D) of (1.5) is given by E 1 (γ, D) = C where (−γD √ C, C) with C > 0 is the point of intersection of the curves B(κ) and κ 2 D 2 γ 2 as shown in Figure  3.
We establish the following results: There is no positive solution of (1.4) for λ ∈ (0, E 1 (γ, )].  Next we recall that for γ > 0 fixed, the boundary value problem: has a positive solution w λ for λ > 0 such that A < w λ (x) ≤ 1 for x ∈ Ω, and this solution is unique (see [6]). We also note that w λ is continuous with respect to λ and E 1 (γ, [7]). Let w * λ := min x∈∂Ω w λ (x) and δ γ := min Then we establish the following multiplicity result which ensures a prediction of an Allee effect in the model. Theorem 1.3. Let γ > 0, * γ := min{δ γ , A 2 } and Γ := {u ∈ C 2 (Ω)∩C 1 (Ω) | u(x) ∈ [A, 1] for x ∈ Ω}. For each ∈ (0, * γ ), there exists λ * > 0 such that if λ ∈ (λ * , E 1 (γ, A 2 + )) then (1.4) has at least two positive solutions u * and u * such that u * ∈ Γ and u * ∈ Γ. In particular, in Γ, (1.4) has a unique solution and this solution is u * . Remark 1. Note that the time dependent problem related to (1.4) is of the form: is the solution of (1.8). In addition, if there existsδ > 0 such that when u 0 − u ∞ <δ, v(t, .) − u ∞ −→ 0 as t −→ ∞, then u is called asymptotically stable. The solution u is called unstable if it is not stable. We note that the solution u * ∈ Γ in Theorem 1.3 is asymptotically stable (see Theorem 6.7 of Chapter 5 in [12]). We also note that the trivial solution of (1.4) is asymptotically stable for λ < E 1 (γ, A 2 + ) and unstable for λ > E 1 (γ, A 2 + ) following the proof of Theorem 1.1 in [9]. In particular, when λ ∈ (λ * , E 1 (γ, Hence there is an Allee effect for λ ∈ (λ * , E 1 (γ, A 2 + )). See also [2] where the authors show existence of an Allee effect in a logistic model but with negative density dependent emigration. Note that with Dirichlet boundary condition, an Allee effect does not occur for a logistic growth model. For more details on the discussion of an Allee effect, see [1] and [15].
Next we consider the case when Ω = (0, 1). In this case (1.4) reduces to the two-point boundary value problem: We note that if u is a positive solution of (1.9) then u has a unique interior maximum, say at t 0 , and the solution is symmetric about t 0 (see [8]). We establish: Theorem 1.4. For λ > 0, (1.9) has a positive solution u such that u(t 0 ) = u ∞ = ρ, u(0) = q 1 , u(1) = q 2 , with 0 < q 1 , q 2 < ρ if and only if λ, ρ, q 1 and q 2 satisfy: We will use Theorem 1.4, namely, equations (1.10) and (1.11) to obtain numerical bifurcation diagrams via Mathematica.
Next we establish conditions that ensures the symmetry of positive solutions of (1.9).
3 then all positive solutions of (1.9) are symmetric.
In Section 2, we state some preliminaries. In Section 3, we present the proofs of Theorems 1.1 -1.3. The proof of Theorem 1.4 will be discussed in Section 4. Finally, the proofs of Theorems 1.5 -1.6 and the numerical bifurcation results will be presented in Section 5.

Preliminaries.
In this section, we present definitions of a subsolution and a supersolution of (1.4). We also provide a sub-supersolution theorem and a three solution theorem that we will use to prove our existence and multiplicity results.
A strict subsolution of (1.4) is a subsolution which is not a solution. A strict supersolution of (1.4) is a supersolution which is not a solution.
Lemma 2.1. Let ψ and Z be a subsolution and a supersolution of (1.4) respectively such that ψ ≤ Z.
Lemma 2.2. Let u 1 and u 2 be a subsolution and a supersolution of (1.4) respectively such that u 1 ≤ u 2 . Let u 2 and u 1 be a strict subsolution and a strict supersolution of (1.4) respectively such that u 2 , u 1 ∈ [u 1 , u 2 ] and u 2 ≤ u 1 . Then (1.4) has at least three solutions u 1 , u 2 and u 3 where 3. Proofs of Theorems 1.1 -1.3. In this section, we will provide proofs of Theorems 1.1 -1.3. First, we present the proof of Theorem 1.1.
Proof of Theorem 1.1. Let λ ≤ E 1 (γ, ). Assume to the contrary that (1.4) has a positive solution u. Then there exist a unique λ ≤ such that λ is the principal eigenvalue of the boundary value problem: and equality holds if and only if λ = E 1 (γ, ). This easily follows from the behavior of κ 2 γ 2 2 as varies (see Figure 6). See also [7]. Let e > 0 be the corresponding normalized eigenfunction for the principal eigenvalue λ in (3.1). Then we have However, by the Green's second identity we have This is a contradiction since λ ≤ . Hence the proof is complete. Next, we provide a proof of Theorem 1.2.
Finally, a proof of Theorem 1.3 is provided.
Since µ λ > 0, choosing β λ ≈ 0, it follows that φ 2 is a strict supersolution for (1.4). We next construct a strict subsolution for (1.4). For each ∈ (0, * γ ), there exists λ * < E 1 (γ, w λ on ∂Ω and hence ψ 2 is a strict subsolution. We note that ψ 2 ∞ > A and φ 2 ∞ < A. By Lemma 2.2, we obtain solutions u, u * and u * such that u ∈ [ψ 1 , . Clearly u * and u * are positive solutions. Further, u * ∈ Γ since u * ≥ ψ 2 > A on Ω. Next in Γ, we show that (1.4) has a unique positive solution. Assume to the contrary that in Γ there exist two distinct positive solutions u and v. Without loss of generality, we assume u ≤ v since φ 1 ≡ 1 is a global supersolution. Therefore we have However, by the Green's second identity we have which is a contradiction. Hence in Γ, there exists a unique positive solution, which is u * , and u * is a positive solution which does not belong to Γ. Hence the proof is complete.

4.
Proof of Theorem 1.4. In this section, we provide a proof of Theorem 1.4.

5.
Proofs of Theorems 1.5 -1.6 and numerical bifurcation results. In this section, we will provide proofs of Theorems 1.5 -1.6 and we will present our numerical bifurcation results. First, we present the proof of Theorem 1.5.
Proof of Theorem 1.5. Let u be a positive solution such that u(0) = q 1 and u(1) = q 2 . Assume t 0 < 1 2 . Since u is symmetric about t 0 and u is concave, q 1 > q 2 and hence |u (0)| < |u (1)|. By the boundary conditions we have This is a contradiction. Similar contradiction can be obtained when t 0 > 1 2 . Hence the solution is symmetric if > A 2 3 . Next, we provide the proof of Theorem 1.6.
Proof of Theorem 1.6. Let u be a positive solution such that u(0) = q 1 and u(1) = q 2 . To show that the solution is symmetric, we need to show q 1 = q 2 . By Theorem 1.4, this follows by showing that for any fixed ρ ∈ (0, 1), It is easy to see that lim q→0 + H(q) = ∞ and H(ρ) = 0. Further, we have Thus we obtain lim q→0 + H (q) = lim q→ρ H (q) = −∞. This implies (5.1) has only one solution q ∈ (0, ρ) for γ 1 or γ ≈ 0 (see Figure 7 for an illustration). Hence the proof is complete. Finally, we present some bifurcation curves for a couple of parameter selections.
Here we briefly explain how we obtain numerical bifurcation diagrams. Let γ > 0 be fixed and let x i = i n+1 ; i = 1, ..., n for some n ≥ 1. Letting ρ = x 1 , we numerically solve the equation (1.11) for q 1 and q 2 using the FindRoot command in Mathematica. The values of q 1 , q 2 and ρ are substituted into (1.10) to find the corresponding value of λ. Repeating this procedure for ρ = x i , i = 2, ..., n, we obtain (λ, ρ) points for the bifurcation diagram. Example 1. Let = 0.1 and A = 0.5. We note that by Theorem 1.5, every positive solutions of (1.9) is symmetric. Here we provide bifurcation curves numerically generated via Mathematica for various γ values. See Figure 8 consisting of 6 bifurcation curves, the first five are in the ascending order of γ from left to right and the last one is the bifurcation curve with Dirichlet boundary condition. We note that as γ increases the bifurcation diagrams shift to right. In particular, the Allee effect is lost when γ > 6.7.
Example 2. Here we present an example where we get both symmetric and nonsymmetric solutions of (1.9) for certain values of γ, when = 0.01 and A = 0.8. We observe that solutions are symmetric for γ = 1, γ = 23 and γ = 25 (see (a), (f ) and (g) in Figure 9). We also find that for some γ values, (5.1) has three distinct q-values, say q 1 , q 2 and q 3 , for a certain range of ρ values. This implies that there exist three symmetric solutions such that u ∞ = ρ and u(0) = u(1) = q i for i = 1, 2, 3 and six non-symmetric solutions such that u ∞ = ρ, u(0) = q i and u(1) = q j for i, j = 1, 2, 3 and i = j (Note: In general, if (5.1) has n q−value solutions then there are n 2 total solutions). See (c), (d) and (e) in Figure 9 for bifurcation diagrams when γ = 6, γ = 10 and γ = 16, respectively. Here the bifurcation curves for symmetric solutions are in red and the bifurcation curves for non-symmetric solutions are in green (Note: green points represent two solutions each while red represent only one solution each). Note that (h) in Figure 9 is the bifurcation curve with Dirichlet boundary condition i.e., the boundary condition is u(0) = 0 = u(1). We observe that bifurcation curves of (1.9) approaches the bifurcation curve with Dirichlet boundary condition when γ −→ ∞. However, for a fixed γ > 3, we observe that there always exists a range of λ in which there exists at least three solutions. 6. Appendix. In this section, we recall important results from [7], namely, Lemmas 4.3 and 4.4 in the Appendix. For the convenience of the reader we also provide the proofs of these results here.