Vol. 28 ( 2018 ) No. 1 J. of Math. (PRC) Prey-Taxis Holling-Tanner, (, 730070) : prey-taxis Holling-Tanner.,,.. : Holling-Tanner ; prey-taxis; ; MR(2010) : 35B32; 35B36 : O175.26 : A : 0255-7797(2018)01-0140-07 1 prey-taxis Holling-Tanner u t = d 1 u + au u 2 uv m + u, v t = d 2 v χ (v u) + bv v2 ru, u ν = v ν = 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x Ω, (x, t) Ω (0, ), (x, t) Ω (0, ), x Ω (0, ), (1.1) u, v, a, b, ru, u/(m + u) Holling II,. a, m, b, r. d 1, d 2, χ (v u) prey-taxis. Neumann, Ω. (1.1) ( (1.1) d 1 = d 2 = χ = 0) Holling- Tanner [1], Robert May Holling [2],,, [3 5]., [6, 7] Holling-Tanner ( (1.1) χ = 0),,.,,., [8, 9]. : 2016-04-08 : 2016-09-01 : (11061031). : (1992 ),,,, :.
No. 1 : Prey-Taxis Holling-Tanner 141,. (1.1) prey-taxis χ (v u), χ > 0,,. χ < 0,,,, [10]. prey-taxis.,,,., (1.1), (u, v ). Crandall-Rabinowitz [11], χ, (1.1),. 2, (1.1) (u, v ),, u = 1 2 (a m br + 4am + (m + br a) 2 ), v = bru. f(u, v) = au u 2 uv v2, g(u, v) = bv m + u ru, f 1 = f u (u, v ) = a 2u mv (m + u ), 2 = f v (u, v ) = u m + u < 0, g 1 = g u (u, v ) = b 2 r > 0, g 2 = g v (u, v ) = b < 0., f 1 + g 2 < 0, f 1 g 2 g 1 > 0 (H0). (u, v ) (1.1). 0 = µ 0 < µ 1 µ 2 µ 3 Ω Neumann, µ i m i 1, φ ij, 1 j m i µ i, {φ ij i 0, 1 j m i } L 2 (Ω), X ij = {cφ ij : c R 2 }, X i = m i j=1 X ij X = + i=1 X i. (1.1) (u, v ). ( ) d 1 + f 1 L 0 =. (2.1) χv + g 1 d 2 + g 2 ( i 0, X i L 0, ) η L 0 X i η µ i d 1 + f 1. η 2 P µ i χv i (µ i, χ)η + Q i (µ i, χ) = 0, + g 1 µ i d 2 + g 2 P i (µ i, χ) = (d 1 + d 2 )µ i + f 1 + g 2, (2.2) Q i (µ i, χ) = d 1 d 2 µ 2 i (χv + d 2 f 1 + d 1 g 2 )µ i + f 1 g 2 g 1. (2.3)
142 Vol. 38, P i (µ i, χ) = P i (µ i, 0), Q i (µ i, χ) = Q i (µ i, 0) χv µ i. χ i = Q i(µ i, 0) v µ i = d 1d 2 µ 2 i (d 2 f 1 + d 1 g 2 )µ i + f 1 g 2 g 1 v µ i, i 1. (2.4) < 0 (H0),. 2.1 (i) P i (µ i, 0) < 0, Q i (µ i, 0) > 0 i 1. χ > 0, (1.1) (u, v ). (ii) i 0, P i (µ i, 0) < 0, i 1 Q i (µ i, 0) < 0. Λ 1 = {i i 1, Q i (µ i, 0) < 0}. χ > max χ i, (1.1) i Λ 1 (u, v ). (iii) P i (µ i, 0) < 0, Q i (µ i, 0) > 0 i 1. χ < max χ i, 1 i + (1.1) (u, v ). 2.1(i), (χ = 0) (χ > 0). 2.1(ii),,,.,,. 2.1(iii),,.,. 3 f 1 < 0, (2.2) (2.3) P i (µ i, 0) < 0, Q i (µ i, 0) > 0 i 1. 2.1 (i) (iii). f 1 < 0, χ < 0,,., Ω = (0, l), l > 0.. d 1 u + au u 2 uv m + u = 0, d 2 v χ (v u) + bv v2 ru = 0, x (0, l), x (0, l), u (x) = v (x) = 0, x = 0, l., µ i = (πi/l) 2, φ 0 (x) = 1/ l, φ i (x) = 2/l cos iπx/l, i 1. (3.1) Y = L 2 (0, l) L 2 (0, l) Hilbert, (U 1, U 2 ) Y = (u 1, u 2 ) L2 (0,l) + (v 1, v 2 ) L2 (0,l), X = {(u, v) : u, v L 2 (0, l), u = v = 0, x = 0, x = l}. F : R X Y, ( ) d 1 u + f(u, v) F (χ; u, v) =. d 2 v χ (v u) + g(u, v), Γ = {χ; (u, v )} R X.
No. 1 : Prey-Taxis Holling-Tanner 143, (χ; u, v ), F (χ; u, v) (u, v) F (u,v) (χ; u, v) (χ; u, v ). 2, (2.1) L 0 χ = χ i, L 0 (χ i ) = F (u,v) (χ i ; u, v ). (2.3) χ = χ i Q i (µ i, χ i ) = 0, F (u,v) (χ i ; u, v ). i 1, (χ i ; u, v ). 3.1 f 1 < 0. i, i j, χ i χ j, δ (3.1) (χ i ; u, v ) χ(s) = χ i + s 2 χ(s), u(s) = u + sφ i + s 2 ũ(s), v(s) = v + sb i φ i + s 2 ṽ(s), s ( δ, δ), (3.2) b i = d1µi f1 < 0, χ(s), ũ(s), ṽ(s) s, 0. l 0 (ũ(s) + b i ṽ(s))φ i dx = KerF (u,v) (χ i ; u, v ) = span {U 0 }, U 0 = (φ i, b i φ i ), b i = d 1µ i f 1 < 0. F (u,v) (χ i ; u, v ) F (u,v) (χ i; u, v ). KerF (u,v) (χ i; u, v ) = span {U 0 }, U 0 = (φ i, b i φ i ), b i = Frédholm, RangeF (u,v) (χ i ; u, v ) = [KerF (u,v) (χ i; u, v )]. codim RangeF (u,v) (χ i ; u, v ) = 1. l Fχ,(u,v) (χ i ; u, v )U 0, U0 = µ i v b i φ 2 i dx > 0, 0 d 2 µ i g 2 < 0. F χ,(u,v) (χ i ; u, v )U 0 / RangeF (u,v) (χ i ; u, v ). [11], (3.1) (χ i ; u, v ) (χ(s); u(s), v(s)), u(s) = u + sφ i + s 2 ũ(s), v(s) = v + sb i φ i + s 2 ṽ(s), χ(s) = χ i + sβ(s), s ( δ, δ), χ(s), ũ(s), ṽ(s) s, l 0 (ũ(s) + b i ṽ(s))φ i dx = 0. (3.2), β(0) = 0., f 11 = f uu (u, v ), f 12 = f uv (u, v ), g 11 = g uu (u, v ), g 12 = g uv (u, v ). (χ(s); u(s), v(s)) (3.1), s s = 0 2d 1 ũ(0) + 2f 1 ũ(0) + 2 ṽ(0) + f 11 φ 2 i + 2f 12 b i φ 2 i = 0, (3.3) (3.3) φ i L 2 -, Green s φ 2 i, φ i = 0, ũ(0), φ i = λ i ũ(0), φ i, ṽ(0), φ i = λ i ṽ(0), φ i, (3.4) ( λ i d 1 + f 1 ) ũ(0), φ i + ṽ(0), φ i = 0. (3.5)
144 Vol. 38, (χ(s); u(s), v(s)) (3.1), s s = 0 2d 2 ṽ(0) + 2χ (0)v µ i φ i 2χ i b i {(φ i) 2 µ i φ 2 i } 2χ i v ũ(0) + 2g 1 ũ(0) + 2g 2 ṽ(0) + g 11 φ 2 i + 2g 12 b i φ 2 i + g 22 b 2 i φ 2 i = 0. (3.6) (3.6) φ i L 2 -, (3.4), (φ i) 2, φ i = 0 χ (0)v λ i φ i, φ i + (g 2 d 2 µ i ) ṽ(0), φ i + (g 1 + χ i v µ i ) ũ(0), φ i = 0. (3.7) χ i (2.4), g 1 + χ i v µ i = (f1 d1µi)(g2 d2µi), (3.7) χ (0)v λ i φ i, φ i + g 2 d 2 µ i [(f 1 d 1 λ i ) ũ(0), φ i + ṽ(0), φ i ] = 0, (3.8) (3.5) (3.8), β(0) = χ (0) = 0.. 4, 3.1. χ(0), (3.2) (χ i ; u, v ) Γ. χ(0) > 0, ; χ(0) < 0,. χ(0) L 2 - (A, B, C, D) := ( ũ(0), φ 2 i, ũ(0), (φ i) 2, ṽ(0), φ 2 i, ṽ(0), (φ i) 2 ) (4.1). 4.1 (3.2) χ(s) χ(0) = 1 µ iv [( f11+f12 + g b 11 + g 12 b i + χ i µ i b i )A χ i b i B + ( f12 + g i b 12 + g 22 b i )C i +χ i D + 1 ( f111+3f112bi + g 4l b 111 + 3g 112 b i + 3g 122 b 2 i )], i (4.2) f 111 := f uuu (u, v ), f 112 := f uuv (u, v ), g 111 := g uuu (u, v ), g 112 := g uuv (u, v ), g 122 := g uvv (u, v ). (A, B, C, D),. 4.2 (A, B, C, D) 2f 1 4d 1 µ i 4d 1 2 0 A 4d 1 µ 2 i 2f 1 4d 1 µ i 0 2 2g 1 + 4λ i χ i v 4χ i v 2g 2 4d 2 µ i 4d 2 B C 4λ 2 i χ i v 2g 1 + 4λ i χ i v 4d 2 µ 2 i 2g 2 4d 2 µ i D 3 (f 2l 11 + 2f 12 b i ) (πi)2 (f 2l = 3 11 + 2f 12 b i ) 3(χ l iµ i b i + 1g 2 11 + g 12 b i + 1g 2 22b 2 i ) + (πi)2 χ l 3 i b i. (4.3) (πi)2 (χ l 3 i µ i b i + 1g 2 11 + g 12 b i + 1g 2 22b 2 i ) + 3(πi)4 χ l 5 i b i
No. 1 : Prey-Taxis Holling-Tanner 145 (3.3) (3.6) φ 2 i (φ i) 2 L 2 -, φ 2 i, φ 2 i = 3 2l, φ2 i, (φ i) 2 = (πi)2 2l, 3 (φ i) 2, (φ i) 2 = 3(πi)4, 2l 5 (4.3).. 4.1 (χ(s); u(s), v(s)) (3.1), s s = 0 6d 1 ũ (0) + 6f 1 ũ (0) + 6 ṽ (0) + 6(f 11 + f 12 b i )ũ(0)φ i + 6f 12 ṽ(0)φ i +(f 111 + 3f 112 b i )φ 3 i = 0, (4.4) 6d 2 ṽ (0) 6 χ(0)v φ i 6χ i {ṽ(0)φ i + b i φ i ũ(0) + v ũ (0)} + 6g 1 ũ (0) + 6g 2 ṽ (0) +6(g 11 + g 12 b i )ũ(0)φ i + 6(g 12 + g 22 b i )ṽ(0)φ i + (g 111 + 3g 112 b i + 3g 122 b 2 i )φ 3 i = 0. (4.5) (4.4), (4.5) φ i L 2 -, 6(f 1 d 1 λ i ) ũ (0), φ i + 6 ṽ (0), φ i + 6(f 11 + f 12 b i ) ũ(0), φ 2 i +6f 12 ṽ(0), φ 2 i + (f 111 + 3f 112 b i ) φ 3 i, φ i = 0, (4.6) 6d 2 µ i ṽ (0), φ i + 6 χ(0)v µ i φ i, φ i + 6χ i { ṽ(0), (φ i) 2 b i ũ(0), (φ i) 2 µ i φ 2 i +v µ i ũ (0), φ i } + 6g 1 ũ (0), φ i + 6g 2 ṽ (0), φ i + 6(g 11 + g 12 b i ) ũ(0), φ 2 i +6(g 12 + g 22 b i ) ṽ(0), φ 2 i + (g 111 + 3g 112 b i + 3g 122 b 2 i ) φ 3 i, φ i = 0. (4.7) φ i, φ i = 1, (4.6) (4.7) ũ (0), φ i ṽ (0), φ i, (4.7) (4.6) (4.2). 4.1. [1] Sáez E, González-Olivares E. Dynamies of a predator-prey model [J]. SIAM J. Appl. Math., 1999, 59(3): 1867 1878. [2] Holling C S. The functional response of invertebrate predators to prey density [J]. Mem. Ent. Soc. Can., 1965, 45(4): 3 60. [3] Braza P A. The bifurcation structure of the Holling-Tanner model for predator-prey interations using two-timings [J]. SIAM J. Appl. Math., 2003, 63(2): 889 904. [4] Collings J B. Bifurcation and stability analysis of a tempreture-dependent of predator-prey interation model incorporating a prey-refuge [J]. Bull. Math. Biol., 1995, 57(3): 63 76. [5] May R M. Limit cycles in predator-prey communities [J]. Sci., 1972, 77(2) 900 902. [6] Peng Rui, Wang Mingxin. Positive steady-states of the Holling-Tanner prey-predator model with diffusion [J]. Proc. Roy. Soc. Edinburgh. Sect A., 2005, 135(4): 149 164. [7] Peng Rui, Wang Mingxin. Global stability of the equilibrium of a diffusive Holling-Tanner preypredator model [J]. Appl. Math. Lett., 2007, 20(3): 664 670. [8]. [J]., 2012, 32(6): 1129 1135. [9] Chen Xueyong, Zhang Jincai. The behavior of the solutions to a multi-group chemotaxis model with reproduction term [J]. J. Math., 2011, 31(5): 853 860.
146 Vol. 38 [10] Wang Xiaoli, Wang Wendi, Zhang Guohong. Global bifurcation of solutions for a predator-prey model with prey-taxis [J]. Math. Meth. Appl. Sci., 2015, 38(2): 431 443. [11] Crandall M G, Rabinowitz P H. Bifurcation from simple eigenvalues [J]. J. Funct. Anal., 1971, 8(4):321 340. BIFURCATION STRUCTURES FOR A HOLLING-TANNER PREDATOR-PREY MODEL WITH PREY-TAXIS ZHANG Xiao-jie, ZHANG Li-na (College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070,China) Abstract: In this paper, we study pattern formations in a Holling-Tanner predator-prey model with prey-taxis. By using the linear analysis method and the classical bifurcation theory, it is proved that the branches of nonconstant solutions can bifurcate from the positive equilibrium only when the retreating behavior of predators occurs. Furthermore, the directions of the branches near the bifurcation points are obtained. Keywords: Holling-Tanner predator-prey model; prey-taxis; bifurcation; non-constant positive steady-states 2010 MR Subject Classification: 35B32; 35B36