Prey-Taxis Holling-Tanner

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Prey-Taxis Holling-Tanner"

Τηθύς Ελευθερόπουλος
5 χρόνια πριν
Προβολές:

1 Vol. 28 ( 2018 ) No. 1 J. of Math. (PRC) Prey-Taxis Holling-Tanner, (, ) : prey-taxis Holling-Tanner.,,.. : Holling-Tanner ; prey-taxis; ; MR(2010) : 35B32; 35B36 : O : A : (2018) 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]. : : : ( ). : (1992 ),,,, :.

2 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)

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 (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.

4 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)

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 g 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 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 f 12 b i ) (πi)2 (f 2l = f 12 b i ) 3(χ l iµ i b i + 1g 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 g 12 b i + 1g 2 22b 2 i ) + 3(πi)4 χ l 5 i b i

6 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) (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 f 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 g 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 f 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 g 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) [1] Sáez E, González-Olivares E. Dynamies of a predator-prey model [J]. SIAM J. Appl. Math., 1999, 59(3): [2] Holling C S. The functional response of invertebrate predators to prey density [J]. Mem. Ent. Soc. Can., 1965, 45(4): [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): [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): [5] May R M. Limit cycles in predator-prey communities [J]. Sci., 1972, 77(2) [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): [7] Peng Rui, Wang Mingxin. Global stability of the equilibrium of a diffusive Holling-Tanner preypredator model [J]. Appl. Math. Lett., 2007, 20(3): [8]. [J]., 2012, 32(6): [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):

7 146 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): [11] Crandall M G, Rabinowitz P H. Bifurcation from simple eigenvalues [J]. J. Funct. Anal., 1971, 8(4): 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 ,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

Σχετικά έγγραφα

J. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n

J. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n Vol. 35 ( 215 ) No. 5 J. of Math. (PRC) a, b, a ( a. ; b., 4515) :., [3]. : ; ; MR(21) : 35Q4 : O175. : A : 255-7797(215)5-15-7 1 [1] : [ ( ) ] ε 2 n n t + div 6 n (nt ) + n V =, (1.1) n div(n T ) = n