一类带有交叉扩散的捕食-食饵模型的正解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 545-559.

一类带有交叉扩散的捕食-食饵模型的正解

袁海龙^1,^2,*(),王玉萍¹,李艳玲³

¹ 陕西科技大学文理学院西安 710021
² 西安交通大学数学与统计学院西安 710049
³ 陕西师范大学数学与信息科学学院西安 710119

收稿日期:2018-04-20 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 袁海龙 E-mail:yuanhailong@sust.edu.cn
基金资助:
国家自然科学基金(11271236);国家自然科学基金(61672021);国家自然科学基金(61872227);陕西科技大学博士科研启动基金(2017BJ-44)

Positive Solutions of a Predator-Prey Model with Cross Diffusion

Hailong Yuan^1,^2,*(),Yuping Wang¹,Yanling Li³

¹ School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021
² School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
³ School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119

Received:2018-04-20 Online:2019-06-26 Published:2019-06-27
Contact: Hailong Yuan E-mail:yuanhailong@sust.edu.cn
Supported by:
the NSFC(11271236);the NSFC(61672021);the NSFC(61872227);the Natural Science Foundation of Shaanxi University of Science and Technology(2017BJ-44)

摘要/Abstract

摘要：

该文研究了一类在齐次Dirichlet边界条件下的带有交叉扩散的捕食-食饵模型.首先，根据Leray-Schauder度理论，建立了系统的正解的存在性；其次，当参数m=β且充分大时，分别研究了正则扰动方程和奇异扰动方程的正解的存在性，和借助分歧理论说明奇异系统的正解在a^*处爆破；最后，建立了系统正解的多解性.

关键词: 交叉扩散, 分歧, 正解

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

中图分类号:

O175.26

袁海龙,王玉萍,李艳玲. 一类带有交叉扩散的捕食-食饵模型的正解[J]. 数学物理学报, 2019, 39(3): 545-559.

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.

参考文献 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

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