一类带有交叉扩散的捕食-食饵模型的正解

Abstract

Abstract:

A predator-prey model with cross diffusion under homogeneous Dirichlet boundary conditions is investigated. Firstly, the existence of positive solutions can be established by the Leray-Schauder degree theory. Secondly, we consider that the existence of positive solutions of the regular perturbation system and the singular perturbation system when m=β is sufficiently large, respectively, and moreover, we show that the positive solutions of the singular perturbation system will blow up along the continuum at a^* by the bifurcation theory. Finally, the multiplicity results of positive solutions of system is also considered.

Key words: Cross diffusion, Bifurcation, Positive solutions

CLC Number:

O175.26

Hailong Yuan,Yuping Wang,Yanling Li. Positive Solutions of a Predator-Prey Model with Cross Diffusion[J].Acta mathematica scientia,Series A, 2019, 39(3): 545-559.

Trendmd

References 25

1	Crandall M G , Rabinowitz P H . Bifurcation from simple eigenvalues. J Funct Anal, 1971, 8: 321- 340 doi: 10.1016/0022-1236(71)90015-2
2	Dancer E N . On the indices of fixed points of mapping in cones and applications. J Math Anal Appl, 1983, 91: 131- 151 doi: 10.1016/0022-247X(83)90098-7
3	Dancer E N . Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. Bull London Math Soc, 2002, 34: 533- 538 doi: 10.1112/S002460930200108X
4	Du Y H , Lou Y . Some uniqueness and exact multiplicity results for a predator-prey model. Trans Amer Math Soc, 1997, 6: 2443- 2475
5	Du Y H , Lou Y . S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predatorprey model. J Differential Equations, 1998, 144: 390- 440 doi: 10.1006/jdeq.1997.3394
6	Du Y H , Shi J P . A diffusive predator-prey model with a protection zone. J Differential Equations, 2006, 229: 63- 91 doi: 10.1016/j.jde.2006.01.013
7	Du Y H , Peng R , Wang M X . Effect of a protection zone in the diffusive Leslie predator-prey model. J Differential Equations, 2009, 246: 3932- 3956 doi: 10.1016/j.jde.2008.11.007
8	李海侠. 一类食物链模型正解的稳定性和唯一性. 数学物理学报, 2017, 37A (6): 1094- 1104 doi: 10.3969/j.issn.1003-3998.2017.06.009
	Li H X . Stability and uniqueness of positive solutions for a food-chain model. Acta Math Sci, 2017, 37A (6): 1094- 1104 doi: 10.3969/j.issn.1003-3998.2017.06.009
9	Li L . Coexistence theorems of steady states for predator-prey interacting systems. Trans Amer Math Soc, 1988, 305: 143- 166 doi: 10.1090/S0002-9947-1988-0920151-1
10	Li L . On positive solutions of a nonlinear equilibrium boundary values problem. J Math Anal Appl, 1989, 138: 537- 549 doi: 10.1016/0022-247X(89)90308-9
11	López-Gómez J , Pardo R . Existence and uniqueness of coexistence states for the predator-prey LotkaVolterra model with diffusion on intervals. Differential Integral Equations, 1993, 6: 1025- 1031
12	López-Gómez J . Spectral Theorey and Nonlinear Functional Analysis. Boca Raton, FL: Chapman and Hall/CRC, 2001
13	Kadota T , Kuto K . Positive steady states for a prey-predator model with some nonlinear diffusion terms. J Math Anal Appl, 2006, 323: 1387- 1401 doi: 10.1016/j.jmaa.2005.11.065
14	Kuto K , Yamada Y . Multiple coexistence states for a prey-predator system with cross-diffusion. J Differential Equations, 2004, 197: 315- 348 doi: 10.1016/j.jde.2003.08.003
15	Kuto K . A strongly coupled diffusion effect on the stationary solution set of a prey-predator model. Adv Diff Eqns, 2007, 12: 145- 172
16	Kuto K , Yamada Y . Coexistence problem for a prey-predator model with density-dependent diffusion. Nonlinear Anal, 2009, 71: 2223- 2232 doi: 10.1016/j.na.2009.05.014
17	Kuto K , Yamada Y . Positive solutions for Lotka-Volterra competition systems with cross-diffsuion. Appl Anal, 2010, 89: 1037- 1066 doi: 10.1080/00036811003627534
18	Lou Y , Ni W M . Diffusion, self-diffusion and cross-diffusion. J Differential Equations, 1996, 131: 79- 131 doi: 10.1006/jdeq.1996.0157
19	Lou Y , Ni W M . Diffusion vs cross-diffusion:An elliptic approach. J Differential Equations, 1999, 154: 157- 190 doi: 10.1006/jdeq.1998.3559
20	Lou Y , Ni W M , Wu Y P . On the global existence of a cross-diffusion system. Discrete Contin Dyn Syst, 1998, 4: 193- 203 doi: 10.3934/dcdsa
21	Lou Y , Ni W M , Yotsutani S . On a limiting system in the Lotka-Volterra competition with cross-diffsuion. Discrete Contin Dyn Syst, 2004, 10: 435- 458
22	Nakashima K , Yamada Y . Positive steady states for prey-predator models with cross-diffusion. Adv Diff Eqns, 1996, 6: 1099- 1122
23	Rabinowitz P H . Some global results for nonlinear eigenvalue problems. J Funct Anal, 1971, 7: 487- 513 doi: 10.1016/0022-1236(71)90030-9
24	Wang Y X , Li W T . Stationary problem of a predator-prey system with nonlinear diffusion effects. Comput Math Appl, 2015, 70: 2102- 2124 doi: 10.1016/j.camwa.2015.08.033
25	袁海龙, 李艳玲. 一类具有Lotka-Volterra竞争模型共存解的存在性与稳定性. 数学物理学报, 2017, 36A (1): 173- 184 doi: 10.3969/j.issn.1003-3998.2017.01.016
	Yuan H L , Li Y L . Existence and stability of coexistence states for a Lotka-Volterra competition model. Acta Math Sci, 2017, 36A (1): 173- 184 doi: 10.3969/j.issn.1003-3998.2017.01.016

[1]	Xue Zhang,Yuhuai Sun. Dynamical Analysis and Traveling Wave Solutions for Generalized (3+1)-Dimensional Kadomtsev-Petviashvili Equation [J]. Acta mathematica scientia,Series A, 2019, 39(3): 501-509.
[2]	Haiyin Li. Hopf Bifurcation of Delayed Density-Dependent Predator-Prey Model [J]. Acta mathematica scientia,Series A, 2019, 39(2): 358-371.
[3]	Shen Wenguo. Interval Bifurcation for the p-Laplacian Equation and Its Applications [J]. Acta mathematica scientia,Series A, 2018, 38(4): 770-778.
[4]	Liao Jiafeng, Li Hongying. Existence and Multiplicity of Positive Solutions for the Superlinear Kirchhoff-Type Equations with Critical Sobolev Exponent [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1119-1124.
[5]	Shen Wenguo. Unilateral Global Interval Bifurcation and One-Sign Solutions for the Half-Linear Periodic Problems [J]. Acta mathematica scientia,Series A, 2017, 37(5): 902-916.
[6]	Yang Jihua, Zhang Erli, Liu Mei. Dynamic Analysis and Chaos Control of a Finance System with Delayed Feedbacks [J]. Acta mathematica scientia,Series A, 2017, 37(4): 767-782.
[7]	Wang Heyuan, Cui Jin. The Analysis of Global Stability and Numerical Simulation of Chaos Behaviors of Rotating Flow [J]. Acta mathematica scientia,Series A, 2017, 37(4): 783-792.
[8]	Zhou Jun. Turing Instability and Hopf Bifurcation of a Bimolecular Model with Autocatalysis and Saturation Law [J]. Acta mathematica scientia,Series A, 2017, 37(2): 366-373.
[9]	Wang Heyuan. The Dynamical Behaviors and the Numerical Simulation of a Five-Mode Lorenz-Like System of the MHD Equations for a Two-Dimensional Incompressible Fluid on a Torus [J]. Acta mathematica scientia,Series A, 2017, 37(1): 199-216.
[10]	Wan Aying, Yi Fengqi, Zheng Lifei. Hopf Bifurcation Analysis and Oscillatory Patterns of A Diffusive Gierer-Meinhardt Model [J]. Acta mathematica scientia,Series A, 2015, 35(2): 381-394.
[11]	LING Lin, LIU Su-Yu, JIANG Gui-Rong. Bifurcation Analysis of a SIRS Epidemic Model with Saturating Contact Rate and Vertical Transmission [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1415-1425.
[12]	LI Bo, LIU Han. Positive Solutions of the Predator-Prey Model with Diffusion and Cross Diffusion [J]. Acta mathematica scientia,Series A, 2014, 34(6): 1578-1586.
[13]	ZHANG Wen-Li, ZHONG Li-Nan. Multiplicity of Positive Solutions for a (p, q-Laplacian Elliptic Systems in R^N [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1264-1274.
[14]	WANG Xue-Chen, WEI Jun-Jie. The Effect of Delay on a Diffusive Predator-Prey System with Ivlev-Type Functional Response [J]. Acta mathematica scientia,Series A, 2014, 34(2): 234-250.
[15]	CAI Dong-Han, YE Hui. Bifurcation and Multiple Growth Paths in an Economic Growth Model with Demographic Transition [J]. Acta mathematica scientia,Series A, 2014, 34(1): 9-15.

Positive Solutions of a Predator-Prey Model with Cross Diffusion

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 25

Related Articles 15

Metrics

Comments

Recommended 0