COMPLETE CLASSIFICATION OF BIFURCATION CURVES FOR A MULTIPARAMETER DIFFUSIVE LOGISTIC PROBLEM WITH GENERALIZED HOLLING TYPE-IV FUNCTIONAL RESPONSE

. We study exact multiplicity and bifurcation curves of positive solutions for the diﬀusive logistic problem with generalized Holling type-IV functional response

In population dynamics, a functional response of the predator to the prey density depends on the change in the density of prey susceptible to each predator per unit time. The simplest response function is where k, a > 0, which is called Holling type-I function in [6]. Michaelis and Menten proposed the response function in the studying enzymatic reactions, where a, c > 0, which is called Holling type-II function in [6]. Another class of response function is where a, c > 0, it is known as a Holling type-III function. Wang and Yeh [24,26] studied a multiparameter diffusive logistic problem with Holling type-III functional response. Note that p 1 (u), p 2 (u), p 3 (u) are monotonic on (0, ∞). Sokol and Howell [19] proposed a non-monotonic response function which is called the simplified Holling type-IV function. This casse has been extensively studied by many authors, see Baek [1], Li and Xiao [9], Lian and Xu [10], Qolizadeh Amirabad et al. [16], Ruan and Xiao [17], and Yeh [25]. Another class of non-monotonic response functions is the generalized Holling type-IV function Collings [4] used the response functionp 4 (u) in a mite predator-prey interaction model for b ≥ √ a. The generalized Holling type-IV function has been studied by Huang and Xiao [7], Liu and Huang [11], and Upadhyay et al. [21].
The idea of using diffusion to study population dynamics was introduced by Skellam [18] in the early 1950s. Since then, reaction-diffusion equations have been widely used for the formation of spatial population patterns and the description of the effects of organisms' spatial dispersal in population dynamics; see Britton [2], Cantrell and Cosner [3], Fife [5], Murray [14], and Okubo [15]. Sounvoravong et al. [20] studied a reaction-diffusion system for a SIRS epidemic model. Problem (1.1) is motivated by the reaction-diffusion population model where T > 0, D > 0 is the diffusion constant, N is the prey population density, r N is the intrinsic growth rate of the prey population, K N is the carrying capacity, and a, c > 0, b ≥ 0. The second term on the right-hand side of (1.3) is a logistic term. The third term on the right-hand side of (1.3) gives the rate of consumption of prey by predators, which is called the predation term; see Ludwig et al. [12,13]. We consider the problem (1.3) with Then problem (1.3) takes the form Assume that a habitat −L/2 ≤x ≤ L/2 is surrounded by a totally hostile, outer environment. That is, equation (1.4) holds in the strip |x| < L/2 and w(−L/2,t) = w(L/2,t) = 0,t > 0. (1.5) Let v(x, t) = w(x,t) with x = 2x/L, t = 4t/L 2 , and let Then problem (1.4), (1.5) takes the form  For m = 0, problem (1.1) takes the form Applying the quadrature method (time-map method), Yeh [25] proved that either r ≤ η 1 q and (q, r) lies above the curve , drawn on the first quadrant of (q, r)-parameter plane.
In this article we study exact multiplicity of positive solutions and shapes of bifurcation curves of (1.1) for parameters m ≥ 1 and q, r > 0. We first find the number of positive zeros of growth rate per capita Then we give a classification of g(u) on the first quadrant of (q, r)-parameter plane according to the monotonicity of g(u). We divide the first quadrant of (q, r)parameter plane into the disjoint union of three curves Γ 1 , Γ 2 , Γ 3 and five regions R 1 , R 2 , R 3 , R 4 , R 5 defined as follows: : 0 < r < mq and r ≥ 1}, R 2 = (q, r) : 0 < r < mq and (q, r) lies between the curve Γ 1 and the line r = 1 , R 3 = (q, r) : 0 < r < mq and (q, r) lies below the curves Γ 1 and Γ 3 , It is well known that the curve Γ 1 is continuous and strictly decreasing on the (q, r)plane. Note that, for (q(a), r(a)) ∈ Γ 1 , lim a→0 + q(a) = 1/m and lim a→0 + r(a) = 1, lim a→∞ q(a) = ∞ and lim a→∞ r(a) = 0. Therefore, we write on curve Γ 1 , the function r 1 (q) with (q, r 1 (q)) ∈ Γ 1 for q > 1/m. We define the bifurcation curve of (1.1) S = {(λ, u λ ∞ ) : λ > 0 and u λ is a positive solution of (1.1)}.

Proofs of main results
For f (u) = ug(u) from the analysis of g(u) in Section 1, we obtain the following result. (i) if (q, r) ∈ R 1 , then the time map is See Laetsch [8] for the derivation of the time map formula. So positive solutions u λ of (1.1) correspond to u λ ∞ = α and T (α) = √ λ. Thus, studying the exact number of positive solutions of (1.1) is equivalent to studying the number of roots of the equation T (α) = √ λ. We define with m ≥ 1, q, r > 0. If (q, r) ∈ R 1 ∪R 2 , then there exists a positive number B ∈ (0, β) such that Proof. For m ≥ 1 and q, r > 0, by (3.3), we have We compute that    In addition, if 0 < r/q ≤ m, we have for u large enough. By  (ii) For (q, r) ∈R 2 , we know that 0 < r/q < m and In addition, T (α) has exactly one critical point, a minimum, on (0, β).
Proof of Theorem 2.1. By (3.1), f (0) = r − 1 and Lemma 3.3, the results in (2.1) hold, and the bifurcation curveS of (1.1) is a C-shaped curve on the (λ, u ∞ )plane. More precisely,S consists of a continuous curve with exactly one turning point, (λ * , u λ * ∞ ) such that 0 < λ * <λ ≤ ∞ and 0 < u λ * ∞ < β, where the curve turns to the right. So we obtain immediately the exact multiplicity result and ordering results of the solutions in parts (i)-(iii). The proofs of results in (2.2) and (2.3) are easy but tedious; we omit them.
Proof of Theorem 2.2. By (3.2) and Lemma 3.4, the results in (2.4) hold, and the bifurcation curveS of (1.1) is a C-shaped curve on the (λ, u ∞ )-plane. More precisely,S consists of a continuous curve with exactly one turning point, (λ * , u λ * ∞ ) such that 0 < λ * < ∞ and γ < u λ * ∞ < β, where the curve turns to the right. So we obtain immediately the exact multiplicity result and ordering results of the solutions in parts (i)-(iii). The proofs of results in (2.5) are easy but tedious; we omit them.