交错扩散对具有保护区域的Beddington-DeAngelis型捕食模型共存解的影响
The Effect of Cross-Diff on Beddington-DeAngelis Type Predator-Prey Model with Protection Zone
DOI: 10.12677/PM.2021.115097, PDF, HTML, 下载: 368  浏览: 509  国家自然科学基金支持
作者: 闫凯, 张丽娜*, 蔺娜娜：西北师范大学数学与统计学院, 甘肃 兰州
关键词: 捕食模型；Beddington-DeAngelis 型功能反应函数；保护区域；交错扩散；共存解；Predator-Prey Model； Beddington-DeAngelis Functional Response； Protection Zone； Cross-Diffusion； Coexistence Solutions
摘要: 本文研究齐次 Neumann边界条件下交错扩散对具有保护区域的 Beddington-DeAngelis 型捕食模型共存解的影响, 其中交错扩散表示食饵躲避猎物. 首先应用线性化方法分析非负常数平衡解的稳定性. 其次应用最大值原理给出正解的先验估计. 最后应用分歧理论讨论共存解的存在性. 结果表明, 交错扩散有助于物种的共存。
Abstract: The effect of cross-diffusion on Beddington-DeAngelis type predator-prey model with protection zone and the homogeneous Neumann boundary conditions is considered, where the cross-diffusion represents the tendency of prey to keep away from its predator. Firstly, the stability of the nonnegative constant steady state solutions is analyzed by the linearization method. Secondly, a priori estimates of positive steady state solutions is given by applying maximum principle. Finally, the existence of coexistence solutions is discussed by using bifurcation theory. As a result, it is shown that the cross-diffusion is beneficial for species coexistence.
文章引用：闫凯, 张丽娜, 蔺娜娜. 交错扩散对具有保护区域的Beddington-DeAngelis型捕食模型共存解的影响[J]. 理论数学, 2021, 11(5): 841-850. https://doi.org/10.12677/PM.2021.115097

参考文献

[1] Beddington, J.R. (1975) Mutual Interference between Parasites or Predators and Its Effect on Searching Efficiency. Journal of Animal Ecology, 44, 331-340.
https://doi.org/10.2307/3866
[2] Deangelis, D.L., Goldstein, R.A. and O’Neill, R.V. (1975) A Model for Tropic Interaction. Ecology, 56, 881-892.
https://doi.org/10.2307/1936298
[3] Du, Y.H. and Shi, J.P. (2006) A Diffusive Predator-Prey Models with a Protection Zone. Journal of Differential Equations, 229, 63-91.
https://doi.org/10.1016/j.jde.2006.01.013
[4] Shigesada, N., Kawasaki, K. and Teramoto, E. (1979) Spatial Segregation of Interacting Species. Journal of Theoretical Biology, 79, 83-99.
https://doi.org/10.1016/0022-5193(79)90258-3
[5] Wang, L., Wu, Y.P. and Xu, Q. (2017) Instability of Spiky Steady States for S-K-T Biological Competing Model with Cross-Diffusion. Nonlinear Analysis, 159, 424-457.
https://doi.org/10.1016/j.na.2017.02.026
[6] Li, S., Liu, S., Wu, J. and Dong, Y. (2017) Positive Solutions for Lotka-Volterra Competition System with Large Cross-Diffusion in a Spatially Heterogeneous Environment. Nonlinear Analysis: Real World Applications, 36, 1-19.
https://doi.org/10.1016/j.nonrwa.2016.12.004
[7] Peng, R., Wang, M.X. and Yang, G.Y. (2008) Stationary Patterns of the Holling-Tanner Prey- Predator Model with Diffusion and Cross-Diffusion. Applied Mathematics and Computation, 196, 570-577.
https://doi.org/10.1016/j.amc.2007.06.019
[8] Ling, Z., Zhang, L. and Lin, Z.G. (2014) Turing Pattern Formation in a Predator-Prey System with Cross Diffusion. Applied Mathematical Modelling, 38, 5022-5032.
https://doi.org/10.1016/j.apm.2014.04.015
[9] He, X., Zheng, S.N. (2017) Protection Zone in a Diffusive Predator-Prey Model with Beddington-DeAngelis Functional Response. Journal of Mathematical Biology, 75, 239-257.
https://doi.org/10.1007/s00285-016-1082-5
[10] Zeng, X.Z., Zeng, W.T. and Liu, L.Y. (2018) Effect of the Protection Zone on Coexistence of the Species for a Ratio-Dependent Predator-Prey Model. Journal of Mathematical Analysis and Applications, 462, 1605-1626.
https://doi.org/10.1016/j.jmaa.2018.02.060
[11] Crandall, M.G. and Rabinowitz, P.H. (1971) Bifurcation from Simple Eigenvalues. Journal of Functional Analysis, 8, 321-340.
https://doi.org/10.1016/0022-1236(71)90015-2
[12] Ye, Q.X., Li, Z.Y., Wang, M.X. and Wu, Y.P. (2011) Introduction to Reaction-Diffusion Equations. 2nd Edition, Science Press, Beijing. (In Chinese)
[13] Oeda, K. (2011) Effect of Cross-Diffusion on the Stationary Problem of a Prey-Predator Model with a Protection Zone. Journal of Differential Equations, 250, 3988-4009.
https://doi.org/10.1016/j.jde.2011.01.026

为你推荐