S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-III functional response

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response \begin{document}${\left\{ {\begin{array}{*{20}{l}} {{u^{\prime \prime }}(x) + \lambda \left[ {ru(1 - \frac{u}{q}) - \frac{{{u^p}}}{{1 + {u^p}}}\% } \right] = 0{\text{,}} - {\text{1}} where u is the population density of the species, p > 1, q , r are two positive dimensionless parameters, and λ > 0 is a bifurcation parameter. For fixed p > 1, assume that q , r satisfy one of the following conditions: (ⅰ) r ≤ η 1, p * q and ( q , r ) lies above the curve \begin{document}$\begin{array}{l}{\Gamma _1} = \{ (q,r):q(a) = \frac{{a[2{a^p} - (p - 2)]}}{{{a^p} - (p - 1)}}{\rm{, }}\\\quad \quad \quad \quad \quad r(a) = \frac{{{a^{p - 1}}[2{a^p} - (p - 2)]}}{{{{({a^p} + 1)}^2}}}{\rm{, }}\sqrt[p]{{p - 1}}\% (ⅱ) r ≤ η 2, p * q and ( q , r ) lies on or below the curve Γ1, where η 1, p * and η 2, p * are two positive constants, and $C_{p}^{*}={{\left(\frac{{{p}^{2}}+3p-4+p\sqrt{%{{p}^{2}}+6p-7}}{4} \right)}^{1/p}}$. Then on the ( λ , || u || ∞ )-plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of q , r and λ .

(ii) r ≤ η * 2,p q and (q, r) lies on or below the curve Γ 1 , where η * 1,p and η * 2,p are two positive constants, and C * p = p 2 +3p−4+p √ p 2 +6p−7 4 1/p . Then on the (λ, ||u||∞)-plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of q, r and λ.
1. Introduction. We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter diffusive logistic problem with Holling type-III functional response u (x) + λf (u) = 0, − 1 < x < 1, where u is the population density of the species, f (u) = ug(u) is the growth rate, where N (T ) is the vegetation biomass, G(N ) is the growth rate of vegetation in absence of grazing, H is the herbivore population density, and c(N ) is the per capita consumption rate of vegetation by the herbivore. For problem (3), if G(N ) is given by the logistic function, and c(N ) is the Holling type III function, then (3) takes the form where p > 1 and A, B, r N , K N > 0, cf. [2, p. 37]. The Holling type III functional response was also considered in Sugie et al. [15] and Sugie and Katagama [14]. They studied the existence of stable limit cycle and global asymptotic stability for a predator-prey system In addition, the Holling type III functional response has also appeared in the dynamics of lake eutrophication where N (T ) is the level of nutrients suspended in phytoplankton causing turbidity, a is the nutrient loading, b is the nutrient removal rate, and B is the rate of internal nutrient recycling, see Carpenter et al. [1] and Scheffer et al. [12].
Adding a diffusion term to (4), we consider a reaction-diffusion model governed by the equation in spatial one dimension, where D > 0 is the diffusion coefficient. Under some assumptions and by using some transformations, we convert (5) into where p > 1 and q, r, λ > 0, see Wang and Yeh [16, p. 815] for p = 2. Let u(x) denote a positive steady-state population density of (6). Then u(x) satisfies problem (1).
For p = 2, problem (1) takes the form Applying the quadrature method (time-map method), Ludwig et al. [6] showed that the rough bifurcation curve goes from a monotone curve with a unique small steady state, to a broken S-shaped curve, to an S-shaped curve, and finally a monotone curve with a unique large steady state, when r increases from 0 + to a large value. Note that the results of evolutionary bifurcation curves in Ludwig et al. [6] are not exact, and it was only shown that the equation has at least three positive solutions but not exactly three. Recently, Wang and Yeh [16, Theorems 2.1-2.3] gave a partial answer of this conjecture in Ludwig et al. [6]. Assume that either r ≤ ρ 1 q and (q, r) lies above the curve or r ≤ ρ 2 q for some constants ρ 1 ≈ 0.0939 and ρ 2 ≈ 0.0766. Then on the (λ, ||u|| ∞ )plane, they gave a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Their results settled rigorously a long-standing open problem in Ludwig et al. [6]. Note that, for p = 1, the function h(u) = u/(1 + u) is called a Holling type-II function. A n-dimensional Dirichlet problem of (1) with p = 1 was considered by Korman and Shi [3]. They obtained two qualitatively different bifurcation curves: a ⊂-shaped curve and a monotone increasing curve, see [3,Theorem 3.1].
We define the bifurcation curve of (1) S = {(λ, u λ ∞ ) : λ > 0 and u λ is a positive solution of (1)} . In this paper we mainly study exact multiplicity of positive solutions and shapes of bifurcation curvesS of (1) for parameters q, r > 0 and p > 1. We first determine numbers of positive zeros of g(u) = r(1 − u q ) − u p−1 1+u p . We then give a classification of growth rate per capita g(u) on the first quadrant of (q, r)-parameter plane according to the monotonicity of g(u).
2. Main results. The main results in this paper are Theorems 2.1-2.3. We first define two positive numbers η 1,p and η 2,p for p > 1 as follows: (i) Let η 1,p be the unique positive intersection value of the two curves η = I(u) and η = K(u) for u > 0, where functions and where function See Figs. 3 and 6.
For p > 1 and 0 < η < m p , let B 1,p (η) be the smallest positive root of I(u) = η and C 2,p (η) be the largest positive root of J(u) = η, see the proof of Lemma 3.2 in Section 3 below. We also define two positive numbers η * 1,p and η * 2,p for p > 1 as follows: (i) Let (16) and see Lemma 3.6. Let u λ be a positive solution of (1) with α ≡ u λ ∞ > 0.
with p > 1, by analyses for g(u) in Section 1, we obtain the following Lemma 3.1.
Let F (u) ≡ u 0 f (t)dt, and u λ be a positive solution of (1) with α ≡ u λ ∞ > 0. The time map formula which we apply to study problem (1), takes the form as follows: (i) if (q, r) ∈ R 1 , then the time map (ii) if (q, r) ∈R 2 , then the time map (iii) if (q, r) ∈ Γ 1 , then the time map see Laetsch [4] for the derivation of the time map formula T (α) for problem (1). So positive solutions u λ of (1) correspond to u λ ∞ = α and T (α) = √ λ. Thus, studying the exact number of positive solutions of (1) is equivalent to studying the number of roots of the equation T (α) = √ λ. We define and We also compute and and H(B 2,p ) ≤ 0.
Thus we obtain that We compute that Thus we obtain that, if 1 < p ≤ 2, and In addition, it is clear that lim u→∞ I(u) = 0, and hence max u∈(0,∞) I(u) = I(B * p ).

By (12) and (24), it is clear that u =ũ is a positive zero of H(u) if and only if
and henceK Thus we obtain that, if 1 < p ≤ 2, . SinceK(0) = 0 and lim u→∞K (u) = −∞, we obtain that there exist two positive In addition, it is clear that and lim u→∞ K(u) = 0, and hence there exists a positive number and max (Note that the case of (34) contradicts to lim u→0 + K(u) = lim u→∞ K(u) = 0.) By the above analysis for K(u), for 0 < r < K(D * p )q, we obtain that the equation K(u) = r/q has exactly two positive roots, say, D 1,p , D 2,p withD p < D 1,p < D * p < D 2,p , see Fig. 5. Thus, for 0 < r < K(D * p )q, has two positive zeros D 1,p , D 2,p such that We compute that and by (30) and (33). Thus, if 1 < p ≤ 2, and if p > 2, It is easy to check that B * p < C * p and hence by (39) and (41). In addition, we know that C * p < D * p and hence J(C * p ) = max by (40) and (42). So we obtain . In addition, we also obtain  , ∞). J(u) < 0 on (0, p √ p − 1) and J(u) has exactly one critical point, a local maximum, at C * p , on ( p √ p − 1, ∞). K(u) < 0 on (0,D p ) and K(u) has exactly one critical point, a local maximum, at D * p , on (D p , ∞). are C * p and D * p respectively. We define the largest positive zero of function I(u) − K(u) is ξ p . Thus we obtain C * p < ξ p < D * p by C * p < D * p and property (5). Hence by (10) and property (1). See Fig. 5 for graphs of functions I(u), J(u), and K(u) on (0, ∞).
The proof of Lemma 3.2 is complete.
The proof of Theorem 2.1 is now complete. To prove Theorems 2.2-2.3, we need Lemma 3.1 and the following Lemmas 3.5-3.8.
The proofs of Theorems 2.2-2.3 follows by Lemmas 3.7-3.8 and the fact that λ < λ * which see the proof of Theorem 2.1, and hence we omit it.