Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 10, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu COMPLETE CLASSIFICATION OF BIFURCATION CURVES FOR A MULTIPARAMETER DIFFUSIVE LOGISTIC PROBLEM WITH GENERALIZED HOLLING TYPE-IV FUNCTIONAL RESPONSE JYUN-YUAN CIOU, TZUNG-SHIN YEH Abstract. We study exact multiplicity and bifurcation curves of positive so- lutions for the diffusive logistic problem with generalized Holling type-IV func- tional response u′′(x) + λ [ ru(1− u q )− u 1 +mu+ u2 ] = 0, −1 < x < 1, u(−1) = u(1) = 0, where the quantity in brackets is the growth rate function and λ > 0 is a bifurcation parameter. On the (λ, ‖u‖∞)-plane, we give a complete classifica- tion of two qualitatively different bifurcation curves: a C-shaped curve and a monotone increasing curve. 1. Introduction We study the exact multiplicity and bifurcation curves of positive solutions for the diffusive logistic problem with generalized Holling type-IV functional response, u′′(x) + λ [ ru(1− u q )− u 1 +mu+ u2 ] = 0, −1 < x < 1, u(−1) = u(1) = 0, (1.1) where the growth rate function is f(u) = ug(u) with g(u) = r(1− u q )− 1 1 +mu+ u2 , (1.2) m ≥ 1, q, r are two positive dimensionless parameters, and λ > 0 is a bifurcation parameter. 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 p1(u) = { au, 0 ≤ u < k/a, k, u ≥ k/a, 2010 Mathematics Subject Classification. 34B15, 34B18. Key words and phrases. Bifurcation curve; exact multiplicity; diffusive logistic problem; Holling type-IV functional response; time map. c©2021 Texas State University. Submitted April 5, 2019. Published February 23, 2021. 1 2 J.-Y. CIOU, T.-S. YEH EJDE-2021/10 where k, a > 0, which is called Holling type-I function in [6]. Michaelis and Menten proposed the response function p2(u) = cu a+ u , in the studying enzymatic reactions, where a, c > 0, which is called Holling type-II function in [6]. Another class of response function is p3(u) = cu2 a+ u2 , 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 p1(u), p2(u), p3(u) are monotonic on (0,∞). Sokol and Howell [19] proposed a non-monotonic response function p4(u) = cu a+ u2 , which is called the simplified Holling type-IV function. This casse has been exten- sively 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 p̄4(u) = cu a+ bu+ u2 , where a, c > 0 and b ≥ 0. The response function p̄4(u) satisfies p̄′4(u) > 0 on (0, √ a) and p̄′4(u) < 0 on ( √ a,∞). Collings [4] used the response function p̄4(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 ∂N ∂T = D ∂2N ∂X2 + rNN ( 1− N KN ) − cN a+ bN +N2 ,−L 2 √ D rN < X < L 2 √ D rN , (1.3) where T > 0, D > 0 is the diffusion constant, N is the prey population density, rN is the intrinsic growth rate of the prey population, KN 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 w = N√ a , t̃ = rNT, x̃ = √ rN D X, r = rNa c , q = KN√ a , m = b√ a . Then problem (1.3) takes the form ∂w ∂t̃ = ∂2w ∂x̃2 + w ( 1− w q ) − 1 r w 1 +mw + w2 , −L 2 < x̃ < L 2 , t̃ > 0. (1.4) EJDE-2021/10 BIFURCATION CURVES FOR A DIFFUSIVE LOGISTIC PROBLEM 3 Assume that a habitat −L/2 ≤ x̃ ≤ L/2 is surrounded by a totally hostile, outer environment. That is, equation (1.4) holds in the strip |x̃| < L/2 and w(−L/2, t̃) = w(L/2, t̃) = 0, t̃ > 0. (1.5) Let v(x, t) = w(x̃, t̃) with x = 2x̃/L, t = 4t̃/L2, and let λ = L2 4r . Then problem (1.4), (1.5) takes the form ∂v ∂t = ∂2v ∂x2 + λ [ rv(1− v q )− v 1 +mv + v2 ] , −1 < x < 1, t > 0, v(−1, t) = v(1, t) = 0, t > 0. (1.6) Let u(x) denote a positive steady-state population density of (1.6). Then u(x) satisfies (1.1). Figure 1. Classified bifurcation curves of (1.7), drawn on the first quadrant of (q, r)-parameter plane. For m = 0, problem (1.1) takes the form u′′(x) + λ [ ru(1− u q )− u 1 + u2 ] = 0, −1 < x < 1, u(−1) = u(1) = 0. (1.7) 4 J.-Y. CIOU, T.-S. YEH EJDE-2021/10 Applying the quadrature method (time-map method), Yeh [25] proved that either r ≤ η1q and (q, r) lies above the curve Γ = { (q, r) : q(a) = 1 + 3a2 2a , r(a) = 1 + 3a2 (1 + a2)2 , 0 < a < 1/ √ 3 } or r ≤ η2q for some constants η1 ≈ 0.618 and η2 ≈ 0.601. Then on the (λ, ‖u‖∞)- plane, he gave a classification of four qualitatively different bifurcation curves: an S- shaped curve, a broken S-shaped curve, a C-shaped curve and a monotone increasing curve, see [25, Theorems 2.1–2.4] and Figure 1. Figure 2. Classified graphs of growth rate per capita g(u) = r(1− u q )− 1 1+mu+u2 with m ≥ 1 on (0,∞), 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 g(u) = r(1− u q )− 1 1 +mu+ u2 . 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 Γ1, Γ2, Γ3 and five regions EJDE-2021/10 BIFURCATION CURVES FOR A DIFFUSIVE LOGISTIC PROBLEM 5 R1, R2, R3, R4, R5 defined as follows: Γ1 = { (q, r) : q(a) = 1 + 2ma+ 3a2 m+ 2a , r(a) = 1 + 2ma+ 3a2 (1 +ma+ a2)2 , 0 < a <∞ } , Γ2 = {(q, r) : r = mq > 1}, Γ3 = { (q, r) : 0 < r = mq ≤ 1 } , R1 = {(q, r) : 0 < r < mq and r ≥ 1}, R2 = { (q, r) : 0 < r < mq and (q, r) lies between the curve Γ1 and the line r = 1 } , R3 = { (q, r) : 0 < r < mq and (q, r) lies below the curves Γ1 and Γ3 } , R4 = {(q, r) : r > mq > 0 and r > 1}, R5 = {(q, r) : r > mq > 0 and r ≤ 1}. It is well known that the curve Γ1 is continuous and strictly decreasing on the (q, r)- plane. Note that, for (q(a), r(a)) ∈ Γ1, lima→0+ q(a) = 1/m and lima→0+ r(a) = 1, lima→∞ q(a) = ∞ and lima→∞ r(a) = 0. Therefore, we write on curve Γ1, the function r1(q) with (q, r1(q)) ∈ Γ1 for q > 1/m. Figure 3. (i),(ii),(iii) C-shaped bifurcation curves S̄ of (1.1). (iv) monotone increasing bifurcation curve S̄ of (1.1). We define the bifurcation curve of (1.1) S̄ = {(λ, ‖uλ‖∞) : λ > 0 and uλ is a positive solution of (1.1)}. By (1.2), g(0) = r− 1, g′(0) = m− r/q, and limu→∞ g(u) = −∞. According to the monotonicity of g(u), we give a classification of growth rate per capita g(u) on the first quadrant of (q, r)-parameter plane, see Figure 2. We have: (i) If (q, r) ∈ Γ1 ∪ Γ3 ∪ R3 ∪ R5, then g(u) ≤ 0 for all u ∈ [0,∞). Hence problem (1.1) has no positive solution for all λ > 0. 6 J.-Y. CIOU, T.-S. YEH EJDE-2021/10 Figure 4. Classified bifurcation curves of (1.1), drawn on the first quadrant of (q, r)-parameter plane. (ii) If (q, r) ∈ Γ2 ∪ R4, g(u) has exactly one positive zero at some β, g′(u) < 0 on (0, β), and g(u) ≤ 0 on [β,∞). Thus, the bifurcation curve S̄ of (1.1) is a monotone increasing curve since f(u) − uf ′(u) = −u2g′(u) > 0 on (0, β), see Figures 3(iv) and 4. (iii) If (q, r) ∈ R1, g(u) has exactly one positive zero at some β such that g(β) = 0, g(u) > 0 on (0, β), and g(u) < 0 on (β,∞). In addition, g(u) changes from increasing to decreasing on (0, β). In Theorem 2.1 stated below, we prove that the bifurcation curve S̄ of (1.1) has exactly one critical point, where the curve turns to the right on the (λ, ‖u‖∞)-plane when (q, r) ∈ R1, see Figures 3(ii)-(iii) and 4. (iv) If (q, r) ∈ R2, g(u) has exactly two positive zeros at some β1 < β such that g(β1) = g(β) = 0, g(u) < 0 on (0, β1)∪ (β,∞) and g(u) > 0 on (β1, β). In addition, g(u) changes from increasing to decreasing on (0, β). Thus, for each fixed q > 1/m, we have ∫ β 0 f(u)du { < 0 for r near r1(q)+, > 0 for r near 1−. In addition, for each fixed q > 1/m, d dr ∫ β 0 f(u)du = 1 6q β3 + 1 2r β2 1 +mβ + β2 > 0 EJDE-2021/10 BIFURCATION CURVES FOR A DIFFUSIVE LOGISTIC PROBLEM 7 because f(β) = 0. Hence for each fixed q > 1/m, there exists r̄1(q) ∈ (r1(q), 1) such that ∫ β 0 f(u)du < 0, for r1(q) < r < r̄1(q),∫ β 0 f(u)du = 0, for r = r̄1(q),∫ β 0 f(u)du > 0, for r̄1(q) < r < 1. We define the curve Γ̄1 = {(q, r) : q > 1 m and r = r̄1(q)} and regions R̂2 = { (q, r) : 0 < r < mq and (q, r) lies between the curve Γ̄1 and the line r = 1 } , R̄2 = { (q, r) : 0 < r < mq and (q, r) lies between curves Γ1 and Γ̄1 } . (So R2 = Γ̄1 ∪ R̄2 ∪ R̂2.) Notice that, for each fixed (q, r) ∈ R̂2, ∫ β 0 f(u)du > 0. Thus, for each fixed (q, r) ∈ R̂2, there exists a positive number γ ∈ (β1, β) satisfying∫ γ 0 f(u)du = 0 since f(u) < 0 on (0, β1) and f(u) > 0 on (β1, β). In Theorem 2.2 stated below, we prove that the bifurcation curve S̄ of (1.1) is a C-shaped curve on the (λ, ‖u‖∞)-plane when (q, r) ∈ R̂2, see Figures 3(i) and 4. In addition, we know that for (q, r) ∈ Γ̄1 ∪ R̄2, the problem (1.1) has no positive solution for all λ > 0 since ∫ β 0 f(u)du ≤ 0. 2. Main results Let uλ be a positive solution of (1.1) with α ≡ ‖uλ‖∞ > 0. Theorem 2.1 (See Figures 3(ii)-(iii) and 4). Consider (1.1). If (q, r) ∈ R1, then lim α→0+ λ(α) ≡ λ̂ = π2 4(r − 1) ∈ (0,∞], lim α→β− λ(α) =∞, (2.1) and the bifurcation curve S̄ of (1.1) is a C-shaped curve on the (λ, ‖u‖∞)-plane. More precisely, S̄ consists of a continuous curve with exactly one turning point, (λ∗, ‖uλ∗‖∞) such that 0 < λ∗ < λ̂ ≤ ∞ and 0 < ‖uλ∗‖∞ < β, where the curve turns to the right. Problem (1.1) has: (i) exactly two positive solutions uλ, vλ with uλ < vλ for λ∗ < λ < λ̂, (ii) exactly one positive solution uλ for λ = λ∗ and exactly one positive solution vλ for λ ≥ λ̂ (if r > 1), (iii) no positive solution for 0 < λ < λ∗. Furthermore, lim λ→∞ ‖uλ‖∞ = 0, if r = 1, lim λ→(λ̂)− ‖uλ‖∞ = 0, if r > 1, (2.2) and lim λ→∞ ‖vλ‖∞ = β. (2.3) 8 J.-Y. CIOU, T.-S. YEH EJDE-2021/10 Theorem 2.2 (See Figures 3(i) and 4). Consider (1.1). If (q, r) ∈ R̂2, then lim α→γ+ λ(α) = lim α→β− λ(α) =∞, (2.4) and the bifurcation curve S̄ of (1.1) is a C-shaped curve on the (λ, ‖u‖∞)-plane. More precisely, S̄ consists of a continuous curve with exactly one turning point, (λ∗, ‖uλ∗‖∞) such that 0 < λ∗ < ∞ and γ < ‖uλ∗‖∞ < β, where the curve turns to the right. Problem (1.1) has: (i) exactly two positive solutions uλ, vλ with uλ < vλ for λ∗ < λ <∞, (ii) exactly one positive solution uλ for λ = λ∗, (iii) no positive solution for 0 < λ < λ∗. Furthermore, lim λ→∞ ‖uλ‖∞ = γ, lim λ→∞ ‖vλ‖∞ = β. (2.5) 3. Proofs of main results For f(u) = ug(u) from the analysis of g(u) in Section 1, we obtain the following result. Lemma 3.1. Consider f(u) = ru(1− u q )− u 1 +mu+ u2 with m ≥ 1, q, r > 0. (i) If (q, r) ∈ R1, then there exists a positive number β such that f(0) = f(β) = 0, f(u) > 0 on (0, β), and f(u) < 0 on (β,∞). (ii) If (q, r) ∈ R̂2, then there exist two positive numbers β1 < β such that f(0) = f(β1) = f(β) = 0, f(u) < 0 on (0, β1) ∪ (β,∞), and f(u) > 0 on (β1, β). Also, there exists a positive number γ ∈ (β1, β) satisfying ∫ γ 0 f(u)du = 0. Let F (u) ≡ ∫ u 0 f(t)dt, and uλ be a positive solution of (1.1) with α ≡ ‖uλ‖∞ > 0. The time map formula for studying problem (1.1) takes the form: (i) if (q, r) ∈ R1, then the time map is T (α) ≡ 1√ 2 ∫ α 0 1 [F (α)− F (u)]1/2 du = √ λ for 0 < α < β; (3.1) (ii) if (q, r) ∈ R̂2, then the time map is T (α) ≡ 1√ 2 ∫ α 0 1 [F (α)− F (u)]1/2 du = √ λ for γ < α < β . (3.2) See Laetsch [8] for the derivation of the time map formula. So positive solutions uλ of (1.1) correspond to ‖uλ‖∞ = 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 θ(u) = 2F (u)− uf(u) = r 3q u3 + u2 1 +mu+ u2 − 2 ∫ u 0 t 1 +mt+ t2 dt. (3.3) Lemma 3.2. Consider f(u) = ru(1− u q )− u 1 +mu+ u2 EJDE-2021/10 BIFURCATION CURVES FOR A DIFFUSIVE LOGISTIC PROBLEM 9 with m ≥ 1, q, r > 0. If (q, r) ∈ R1 ∪ R̂2, then there exists a positive number B ∈ (0, β) such that θ′′(u) = −uf ′′(u)  = 0 for u = B, < 0 on (0, B), > 0 on (B, β). Proof. For m ≥ 1 and q, r > 0, by (3.3), we have θ′(u) = f(u)− uf ′(u) = u2 [r q − m+ 2u (1 +mu+ u2)2 ] , θ′′(u) = −uf ′′(u) = 2u [r q − m+ 3u− u3 (1 +mu+ u2)3 ] = 2u [r q − I(u) ] , where I(u) ≡ m+ 3u− u3 (1 +mu+ u2)3 . We compute that I ′(u) = 3u4 − 18u2 − 12mu+ 3(1−m2) (1 +mu+ u2)4 . Now we define p(u) = 3u4 − 18u2 − 12mu+ 3(1−m2) and obtain p′(u) = 12(u3 − 3u−m) , p′′(u) = 36(u2 − 1). Therefore, p′′(1) = 0, p′′(u) < 0 on [0, 1), p′′(u) > 0 on (1,∞). (3.4) Since p(0) = 3(1 −m2) ≤ 0, p′(0) = −12m < 0, limu→∞ p(u) = ∞, and by (3.4), there exists C > 0 such that p(C) = 0, p(u) < 0 on (0, C), p(u) > 0 on (C,∞). Then I ′(C) = 0, I ′(u) < 0 on (0, C), I ′(u) > 0 on (C,∞). (3.5) By (3.5), I(0) = m ≥ 1 and limu→∞ I(u) = 0, there exists a D with 0 < D < C such that I(D) = 0, I(u) > 0 on (0, D), I(u) < 0 on (D,∞). 10 J.-Y. CIOU, T.-S. YEH EJDE-2021/10 It follows that maxu∈[0,∞) I(u) = I(0) = m and I(u) is strictly decreasing on [0, D]. So we obtain that for 0 < r/q ≤ m, there exists B > 0 such that I(B) = r/q and θ′′(B) = 0, θ′′(u) < 0 on (0, B), θ′′(u) > 0 on (B,∞). (3.6) In addition, if 0 < r/q ≤ m, we have θ′(u) = u2 [r q − m+ 2u (1 +mu+ u2)2 ] > 0 (3.7) for u large enough. By (3.6), (3.7) and θ(0) = θ′(0) = 0, there exists E > B such that θ(0) = θ(E) = 0, θ(u) < 0 on (0, E), θ(u) > 0 on (E,∞). (3.8) (i) For (q, r) ∈ R1, we know that 0 < r/q < m and ∫ β 0 f(u)du > 0 by f(u) > 0 on (0, β). Thus, θ(β) = 2F (β)− βf(β) = 2 ∫ β 0 f(u)du > 0 by f(β) = 0. It follows that β > E > B by (3.8). So we obtain B ∈ (0, β) and θ′′(u) = −uf ′′(u)  = 0 for u = B, < 0 on (0, B), > 0 on (B, β) by (3.6). (ii) For (q, r) ∈ R̂2, we know that 0 < r/q < m and ∫ β 0 f(u)du > 0. Thus, θ(β) = 2F (β)− βf(β) = 2 ∫ β 0 f(u)du > 0 by f(β) = 0. It follows that β > E > B by (3.8). So we obtain B ∈ (0, β) and θ′′(u) = −uf ′′(u)  = 0 for u = B, < 0 on (0, B), > 0 on (B, β) by (3.6). The proof of Lemma 3.2 is complete. � By Lemmas 3.1 (i), 3.2, and a slight modification of the proof of [22, Theorem], we obtain the following Lemma. Lemma 3.3. Consider (1.1) with m ≥ 1, q, r > 0. If (q, r) ∈ R1, then lim α→0+ T (α) = { π 2 √ f ′(0) ∈ (0,∞), if f ′(0) > 0, ∞, if f ′(0) = 0, lim α→β− T (α) =∞ . In addition, T (α) has exactly one critical point, a minimum, on (0, β). EJDE-2021/10 BIFURCATION CURVES FOR A DIFFUSIVE LOGISTIC PROBLEM 11 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 curve S̄ of (1.1) is a C-shaped curve on the (λ, ‖u‖∞)- plane. More precisely, S̄ consists of a continuous curve with exactly one turning point, (λ∗, ‖uλ∗‖∞) such that 0 < λ∗ < λ̂ ≤ ∞ and 0 < ‖uλ∗‖∞ < 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. � By Lemmas 3.1 (ii), 3.2, and [23, Theorem 1, Notes 1, 2 and Remark 2], we obtain the following Lemma. Lemma 3.4. Consider (1.1) with m ≥ 1, q, r > 0. If (q, r) ∈ R̂2, then lim α→γ+ T (α) = lim α→β− T (α) =∞. In addition, T (α) has exactly one critical point, a minimum, on (γ, β). Proof of Theorem 2.2. By (3.2) and Lemma 3.4, the results in (2.4) hold, and the bifurcation curve S̄ of (1.1) is a C-shaped curve on the (λ, ‖u‖∞)-plane. More pre- cisely, S̄ consists of a continuous curve with exactly one turning point, (λ∗, ‖uλ∗‖∞) such that 0 < λ∗ < ∞ and γ < ‖uλ∗‖∞ < β, where the curve turns to the right. 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Yeh; S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J. Differential Equations, 255 (2013), 812-839. [25] T.-S. Yeh; Classification of bifurcation diagrams for a multiparameter diffusive logistic prob- lem with Holling type-IV functional response, J. Math. Anal. Appl., 418 (2014), 283-304. [26] T.-S. Yeh; S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-III functional response, Comm. Pure Appl. Anal., 16 (2017), 645-670. Jyun-Yuan Ciou Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan ROC Email address: b7626529@gmail.com Tzung-Shin Yeh Department of Applied Mathematics, National University of Tainan, Tainan 700, Tai- wan, ROC Email address: tsyeh@mail.nutn.edu.tw 1. Introduction 2. Main results 3. Proofs of main results Acknowledgments References