BIFURCATION DIAGRAMS OF POSITIVE SOLUTIONS FOR ONE-DIMENSIONAL MINKOWSKI-CURVATURE PROBLEM AND ITS APPLICATIONS

. In this paper, we study the classiﬁcation and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem where λ > 0 is a bifurcation parameter, L > 0 is an evolution parameter, f ∈ C [0 , ∞ ) ∩ C 2 (0 , ∞ ) and there exists β > 0 such that ( β − z ) f ( z ) > 0 for z (cid:54) = β . In particular, we ﬁnd that the bifurcation curve S L is monotone increasing for all L > 0 when f ( u ) /u is of Logistic type , and is either ⊂ -shaped or S-shaped for large L > 0 when f ( u ) /u is of weak Allee eﬀect type . Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.

1. Introduction. In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem − u / 1 − u 2 = λf (u), in (−L, L) , u(−L) = u(L) = 0, (1) where λ > 0 is a bifurcation parameter, L > 0 is an evolution parameter, f ∈ C[0, ∞) ∩ C 2 (0, ∞) and there exists β > 0 such that (β − z) f (z) > 0 for z = β. It is well-known that studying the exact shape of bifurcation curve S L of (1) is equivalent to studying of the exact multiplicity of positive solutions of problem (1) where is a positive solution of (1)} .
Notice that the problem (1) plays an important role in certain fundamental issues in differential geometry and in the special theory of relativity, see for example [3,5]. We refer the readers, for motivations and results, to [1] and the references cited therein.
The solutions of (3) are the steady state solutions of a reaction-diffusion population model in one space dimension. A typical form of reaction-diffusion population model equation is where u(x, t) is the population density, d > 0 is the diffusion constant and f (u)/u is the growth rate per capita. We refer to the work of McCabe, Leach and Needham [14], Shi and Shivaji [17], Wang and Kot [19], and Xin [20] and the references therein. In (3), the constant L can be scaled out so one defines the bifurcation curveS of (3) on the (λ, u ∞ )-plane bȳ S ≡ {(λ, u λ ∞ ) : λ > 0 and u λ is a positive solution of (3)} .
Next, we give some terminologies related to the shapes of bifurcation curves S L on the (λ, u ∞ )-plane (while similar terminologies forS also holds).
Monotone increasing: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S L is monotone increasing if S L is a continuous curve and for each pair of points (λ 1 , u λ1 ∞ ) and (λ 2 , u λ2 ∞ ) of S L , u λ1 ∞ < u λ2 ∞ implies λ 1 ≤ λ 2 , see Figure 1(i). ⊂-shaped: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S L is ⊂shaped if S L is a continuous curve, initially continues to the left and eventually continues to the right, see Figure 1(ii). S-shaped:: We say that, on the (λ, u ∞ )-plane, the bifurcation curve S L is S-shaped if S L is a continuous curve, initially continues to the right (or starts from (0, 0)), eventually continues to the right and has a turning point which turns to the left, see Figure 1(ii). There are many references in studying the bifurcation curveS of (3), cf. [2,10,11,18,12,15,16,21,22]. For instance, Ouyang and Shi [15] obtained the bifurcation diagrams for the problem (3) with nonlinearity f (u) = u p − q q , 1 < p < q.
They proved that the corresponding bifurcation curveS is either monotone increasing or ⊂-shaped. Lee et al. [12] studied the problem (3) with nonlinearity This model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. They proved that the corresponding bifurcation curveS is S-shaped for c ∈ ( 8 √ 27 , 2) and large K > 0. Huang and Wang [11, Theorem 2.1] studied the problem (3) with nonlinearity They proved that there exists a 0 > 0 such that corresponding bifurcation curveS is S-shaped for 0 < a < a 0 and monotone increasing for a ≥ a 0 .
We prove that the bifurcation curve S L is monotone increasing for all L > 0 when g is of Logistic type, and is either ⊂-shaped or S-shaped for large L > 0 when g is of weak Allee effect type, see Theorems 2.1, 2.2 and Proposition 1. As applications, we use these results to obtain the bifurcation diagrams for the problem (1), (5), the problem (1), (6) and the problem (1) with f (u) = au p − bu q + cu, q ≥ 1, q > p > 0, a, b > 0 and c ≥ 0, see Theorems 3.1-3.3 stated below. Obviously, (4) is the special case of (7). Throughout this paper, we let Then there are seven possibilities: The paper is organized as follows. Section 2 contains statements of main results. Section 3 contains three important applications. Section 4 contains preparatory lemmas. Finally, section 4 contains the proofs of the main results. Main results. In this section, we present our main results. First, we classify the bifurcation curves S L of (1) for L > 0.
Furthermore, (i) if one of (C1), (C3) and (C5) holds, then S L is ⊂-shaped for all L > 0; (ii) if one of (C2), (C4) and (C6) holds, then S L is either monotone increasing or S-shaped for L > 0. Moreover, S L is S-shaped for large L > 0 if the bifurcation curveS of (3) is not monotone increasing. (iii) if (C7) holds, then S L is ⊂-shaped for L >L, and is either monotone increasing or S-shaped forL > L > 0 where In Theorem 2.1, we find that the shape of bifurcation curve S L may change with varying L > 0. In order to determine the evolution of bifurcation curve S L with varying L > 0, we need the following assumptions (H1)-(H3): (H1) [f (u)/u] < 0 on (0, β) (f (u)/u is of logistic type).
(ii) The shape of bifurcation curveS of (3) has been widely studied. Thus we can apply these results and techniques to determine the monotonicity of the bifurcation curveS of (3), see Proposition 1. Remark 2. Assume that f (u)/u is of weak Allee effect type. By Theorems 2.1, 2.3 and Proposition 1(ii), we find that the bifurcation curve S L is ⊂-shaped for large L > 0 when (C1), (C3), (C5) or (C7) holds, and is S-shaped for large L > 0 when (C2), (C4) or (C6) holds.
(i) If the bifurcation curve SL of (1) is monotone increasing, then the bifurcation curve S L of (1) is ⊂-shaped for L >L and monotone increasing for 0 < L ≤ L. (ii) If the bifurcation curve SL of (1) is not monotone increasing, then there exists L ∈ (0,L) such that the bifurcation curve S L of (1) is ⊂-shaped for L >L, S-shaped forL < L <L and monotone increasing for 0 < L ≤L.
The proof is complete.

Proofs of theorems.
Proof of Theorem 2.1. Let L > 0 be given. By Lemma 4.6(i)(iii), the bifurcation curve S L of (1) is continuous in (λ, u λ ∞ )-plane, starts from the point (κ, 0) and goes to (∞, m L,β ) for L > 0. Next, we divide the remainder proofs of Theorem 2.1 into the following three steps.
The proof is complete.
By Theorem 2.1, the bifurcation curve S L is S-shaped for L >L and monotone increasing for 0 < L ≤L. Assume that one of the following conditions holds: ((C2) and (H2)), (C4) or (C6).
Since (H3) holds, and by Theorem 2.2(ii), we see that S L is monotone increasing for small L > 0. SoL > 0. The proof is complete.
(39) By Theorem 2.1, the bifurcation curve S L of (7) starts from (κ, 0) and goes to (∞, m L,β ) for L > 0 where κ is defined by (9). Let g(u) = f (u)/u = au p−1 − bu q−1 + c. Then we have that Next, we divide this proof into the following six steps.
Step 1. We prove statement (i). Assume that c = 0. Then we consider two cases.
Case 1. Assume that 0 < p ≤ 1. Since q > p, and by (40), we see that g (u) < 0 for u > 0. So by Theorem 2.2(i), the bifurcation curve S L is monotone increasing for all L > 0.
Case 2. Assume that p > 1. Since (C1) holds by (39), and by Theorem 2.1(i), the bifurcation curve S L is ⊂-shaped for all L > 0. Thus statement (i) holds by Cases 1 and 2.
Step 2. We prove statement (ii)(a). Assume that 0 < p ≤ 1. Since q > p, and by (40), we see that g (u) < 0 for u > 0. So by Theorem 2.2(i), the bifurcation curve S L is monotone increasing for all L > 0. Then statement (a) holds.
The proof is complete.