|
|
- 2018
一类空间退化的捕食-食饵模型的全局分歧结构
|
Abstract:
摘要： 研究了一类空间退化异质环境中带有Holling II型反应函数的捕食-食饵模型。当食饵的生长率较弱时, 通过比较原理给出了任意正稳态解的先验估计, 再利用全局分歧理论证明了正稳态解集合形成一条有界的全局分歧曲线;当食饵的生长率较强时, 通过反证法得到了任意正稳态解的先验估计, 并利用全局分歧理论证明了正稳态解集合形成一条无界的全局分歧曲线。
Abstract: A predator-prey model with Holling type II response function in a spatially degenerate heterogeneous environment is studied. When the prey growth rate is weak, a priori estimates of any positive steady-state solution is first obtained by the comparison principle, and then we show that the set of positive steady-state solutions forms a bounded global bifurcation curve by the global bifurcation theory. When the prey growth rate is strong, a priori estimates of any positive steady-state solution is first obtained by using reduction to absurdity, and then we show that the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory
| [1] | DONG Yaying, ZHANG Shunli, LI Shanbing, et al. Qualitative analysis of a predator-prey model with Crowley-Martin functional response[J]. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2015, 25(9):1550110(19 pages). |
| [2] | DU Yihong, LOU Yuan. Some uniqueness and exact multiplicity results for a predator-prey model[J]. Transactions of the American Mathematical Society, 1997, 349(6):2443-2475. |
| [3] | HUFFAKER C B. Exerimental studies on predation: despersion factors and predator-prey oscilasions[J]. Hilgardia, 1958, 27:343-383. |
| [4] | DANCER E N, DU Yihong. On a free boundary problem arising from population biology[J]. Indiana University Mathematics Journal, 2003, 52(1)51-67. |
| [5] | DU Yihong, HUANG Qingguang. Blow-up solutions for a class of semilinear elliptic and parabolic equations[J]. SIAM Journal on Mathematical Analysis, 1999, 31(1)1-18. |
| [6] | LóPEZ-GóMEZ J, SABINA de Lis J C. First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs[J]. Journal of Differential Equations, 1998, 148(1):47-64. |
| [7] | LóPEZ-GóMEZ J. Spectral theory and nonlinear functional analysis[M]. Florida: Chapman and Hall/CRC Research Notes in Mathematics, 2001. |
| [8] | DANCER E N, DU Yihong. Effects of certain degeneracies in the predator-prey model[J]. SIAM Journal on Mathematical Analysis, 2002, 34(2)292-314. |
| [9] | LI Lige. Coexistence theorems of steady-states for predator-prey interacting systems[J]. Transactions of the American Mathematical Society, 1988, 305(1):143-166. |
| [10] | LI Shanbing, WU Jianhua. Qualitative analysis of a predator-prey model with predator saturation and competition[J]. Acta Applicandae Mathematicae, 2016, 141:165-185. |
| [11] | 叶其孝,李正元.反应扩散引论[M]. 北京:科学出版社, 1990. YE Qixiao, LI Zhengyuan. Introduction to reaction-diffusion equations[M]. Beijing: Science Press, 1990. |
| [12] | BLAT J, BROWN K J. Global bifurcation of positive solutions in some systems of elliptic equations[J]. SIAM Journal on Mathematical Analysis, 1986, 17(6):1339-1353. |
| [13] | CASAL A, EILBECK J C, LóPEZ-GóMEZ J. Existence and uniqueness of coexistence states for a predator-prey model with diffusion[J]. Differential Integral Equations, 1994, 7(2):411-439. |
| [14] | DU Yihong, LOU Yuan. S-shaped global solution curve and Hopf bifurcation of positive solutions to a predator-prey model[J]. Journal of Differential Equations, 1998, 144(2):390-440. |
| [15] | CRANDALL M G, RABINOWITZ P H. Bifurcation from simple eigenvalues[J]. Journal of Functional Analysis, 1971, 8:321-340. |
| [16] | LI Shanbing, WU Jianhua, LIU Sanyang. Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone[J]. Calculus of Variations and Partial Differential Equations, 2017, 56(3):82(35 pages). |
| [17] | DONG Yaying, LI Shanbing, LI Yanling. Multiplicity and uniqueness of positive solutions for a predator-prey model with C-M functional response[J]. Acta Applicandae Mathematicae, 2015, 139:187-206. |
