Journal of Guangxi Normal University(Natural Science Edition) ›› 2023, Vol. 41 ›› Issue (1): 17-23.doi: 10.16088/j.issn.1001-6600.2022061703

Previous Articles     Next Articles

Logistic Diffusion Problem and Its Analysis on Three Types of Domains

ZHANG Mengyun1, GE Jing2, LIN Zhigui3*   

  1. 1. School of Applied Mathematics, Nanjing University of Finance Economics, Nanjing Jiangsu 210003, China;
    2. School of Mathematics and Statistics, Huaiyin Normal University, Huaian Jiangsu 223300, China;
    3. School of Mathematical Science, Yangzhou University, Yangzhou Jiangsu 225002, China
  • Received:2022-06-17 Revised:2022-07-18 Online:2023-01-25 Published:2023-03-07

PDF (PC)

323

Abstract: Reaction-diffusion problems in several different types of domains are investigated in this paper, which describes spatial expansion of population. Compared with the classical Cauchy problem, diffusion problems are presented in fixed domains, evolving domains and free boundaries. According to the theoretical results of the corresponding problems, the differences and connections of diffusion in different domains are analyzed.

Key words: reaction-diffusion, free boundary, evolving domain, expansion and extinction

CLC Number: 

  • O29
[1] 陈兰荪, 宋新宇, 陆征一. 数学生态学模型与研究方法[M]. 成都:四川科学技术出版社, 2003.
[2]MALTHUS T R. An essay on the principle of population[M]. London: J. Johnson, 1798.
[3]VERHULST P F. Notice sur la loi que la population suit dans son accroissement[J]. Correspondence Mathematique et Physique (Ghent), 1838,10: 113-121.
[4]CANTRELL R S, COSNER C. Spatial ecology via reaction-diffusion equations[M]. Chichester, UK: John Wiley and Sons Ltd., 2003.
[5]叶其孝, 李正元, 王明新, 等. 反应扩散方程引论[M]. 2版.北京: 科学出版社, 2011.
[6]林支桂. 数学生态学导引[M].北京: 科学出版社, 2013.
[7]FISHER R A. The wave of advance of advantageous genes[J]. Annals of Eugenics, 1937, 7: 335-369.
[8]KOLMOGOROV A N, PETROVSKI I G, PISKUNOV N S. A study of the equation of diffusion with increase in the quantity of matter, and its application on a biological problem[J]. Moscow University Bulletin of Mathematics, 1937, 1: 1-25.
[9]ARONSON D G, WEINBERGER H F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propgation[J]. Lecture Notes in Mathematics, 1975, 446: 5-49.
[10]ARONSON D G, WEINBERGER H F. Multidimensional nonlinear diffusion arising in population genetics[J]. Advances in Mathematics, 1978, 30(1):33-76.
[11]KOT M. Elements of mathematical ecology[M]. Cambridge: Cambridge University Press, 2001:282.
[12]HUTSON V, LPEZ-GMEZ J, MISCHAIKOW K, et al. Limit behaviour for a competing species problem with diffusion[M]// AGARWAL R P. Dynamical systems and applications. Singapore: World Scientific, 1995: 343-358.
[13]DOCKERY J, HUTSON V, MISCHAIKOW K, et al. The evolution of slow dispersal rates: a reaction diffusion model[J]. Journal of Mathematical Biology, 1998, 37(1): 61-83.
[14]GAUSE G F. The struggle for existence[M]. Baltimore: Williams and Wilkins, 1934.
[15]HARDIN G. The competitive exclusion principle[J]. Science, 1960, 131(3409): 1292-1297.
[16]楼元. 空间生态学中的一些反应扩散方程模型[J]. 中国科学: 数学, 2015, 45(10): 1619-1634.
[17]闫宝强, 程红梅, 房钦贺, 等. 反应扩散方程及其应用研究进展[J]. 山东师范大学学报(自然科学版). 2021, 36(3): 253-266.
[18]ALLEN L J S, BOLKER B M, LOU Y, et al. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model[J]. Discrete and Continuous Dynamical Systems, 2008, 21(1):1-20.
[19]DU Y H, LIN Z G. Spreading-vanishing dichotomy in the diffusion logistic model with a free boundary[J]. SIAM Journal on Mathematical Analysis, 2010, 42(1): 377-405.
[20]GU H, LOU B D, ZHOU M L. Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries[J]. Journal of Functional Analysis, 2015, 269(6): 1714-1768.
[21]KANEKO Y. Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations[J]. Nonlinear Analysis: Real World Applications, 2014,18: 121-140.
[22]DU Y H, LIN Z G. The diffusive competition model with a free boundary: invasion of a superior or inferior competitor[J]. Discrete and Continuous Dynamical Systems-B, 2014, 19(10): 3105-3132.
[23]WU C. H. The minimal habitat size for spreading in a weak competition system with two free boundaries[J]. Journal of Differential Equations, 2015, 259(3): 873-897.
[24]WANG M X, SHENG W J, ZHANG Y. Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary[J]. Journal of Mathematical Analysis and Applications, 2016, 441(1): 309-329.
[25]WANG M X, ZHAO J F. A free boundary problem for the prey-predator model with double free boundaries[J]. Journal of Dynamics and Differential Equations, 2017, 29(3): 957-979.
[26]CHEN X F, FRIEDMAN A. A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth[J]. SIAM Journal on Mathematical Analysis, 2003, 35(4): 974-986.
[27]SUN W L, CARABALLO T, HAN X Y, et al. A free boundary tumor model with time dependent nutritional supply[J]. Nonlinear Analysis: Real World Applications, 2020, 53: 103063.
[28]CHEN X F, FRIEDMAN A. A free boundary problem arising in a model of wound healing[J]. SIAM Journal on Mathematical Analysis, 2000, 32(4): 778-800.
[29]MERZ W, RYBKA P. A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon[J]. Journal of Mathematical Analysis and Applications, 2004, 292(2): 571-588.
[30]RAO R R, BAER T A, NOBLE D R. Computationaluid mechanics for free and moving boundary problems[J]. International Journal for Numerical Methods in Fluids, 2012, 68(11): 1341-1342.
[31]LIANG J, HU B, JIANG L S. Optimal convergence rate of the binomial tree scheme for American options with jump diffusion and their free boundaries[J]. SIAM Journal on Financial Mathematics, 2010, 1(1): 30-65.
[32]DU Y H, GUO Z M. Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II[J]. Journal of Differential Equations, 2011, 250(12): 4336-4366.
[33]DU Y H, GUO Z M. The stefan problem for the Fisher-KPP equation[J]. Journal of Differential Equations, 2012, 253(3): 996-1035.
[34]WANG M X. The diffusive logistic equation with a free boundary and sign-changing coefficient[J]. Journal of Differential Equations, 2015, 258(4): 1252-1266.
[35]ZHOU P, XIAO D M. The diffusive logistic model with a free boundary in heterogeneous environment[J]. Journal of Differential Equations, 2014, 256(6): 1927-1954.
[36]DU Y H, LOU B D. Spreading and vanishing in nonlinear diffusion problems with free boundaries[J]. Journal of the European Mathematical Society, 2015, 17(10): 2673-2724.
[37]DU Y H, MATSUAZAWA H, ZHOU M L. Sharp estimate to the spreading speed determined by nonlinear free boundary problems[J]. SIAM Journal on Mathematical Analysis, 2014, 46(1): 375-396.
[38]BAO W D, DU Y H, LIN Z G, et al. Free boundary models for mosquito range movement driven by climate warming[J]. Journal of Mathematical Biology, 2018, 76(4): 841-875.
[39]KONDO S, ASAI R. A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus[J]. Nature, 1995, 376(6543): 765-768.
[40]CRAMPIN E J, GAFNEY E A, MAINI P K. Reaction and diffusion on growing domains: scenarios for robust pattern formation[J]. Bulletin of Mathematical Biology, 1999, 61(6): 1093-1120.
[41]VAREA C, ARAGN J L, BARRIO R. Turing patterns on a sphere[J]. Physical Review E, 1999, 60(4): 4588-4592.
[42]PLAZA R G, SNCHEZ-GARDUO F, PADILLA P, et al. The effect of growth and curvature on pattern formation[J]. Journal of Dynamics and Differential Equations, 2004, 16(4): 1093-1121.
[43]MADZVAMUSE A. Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains[J]. Journal of Computational Physics, 2006, 214(1): 239-263.
[44]MADZVAMUSE A, GAFNEY E A, MAINI P K. Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains[J]. Journal of Mathematical Biology, 2010, 61(1): 133-164.
[45]TANG Q L, ZHANG L, LIN Z G. Asymptotic profile of species migrating on a growing habitat[J]. Acta Applicandae Mathematicae, 2011, 116(2): 227.
[46]JIANG D H, WANG Z C. The diffusive logistic equation on periodically evolving domains[J]. Journal of Mathematical Analysis and Applications, 2018, 458(1): 93-111.
[1] ZHANG Jie, LI Xiaojun. Existence of Uniform Random Attractor for NonautonomousStochastic Reaction-diffusion Equations on Unbounded Domains [J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(2): 134-143.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] LIU Guolun, SONG Shuxiang, CEN Mingcan, LI Guiqin, XIE Lina. Design of Bandwidth Tunable Band-Stop Filter[J]. Journal of Guangxi Normal University(Natural Science Edition), 2018, 36(3): 1 -8 .
[2] HU Wenjun. Delay Consensus of Leader-following Multi-agent Systems via the Adaptive Distributed Control[J]. Journal of Guangxi Normal University(Natural Science Edition), 2018, 36(1): 70 -75 .
[3] LI Yi-chun, DONG De-xin, WANG Yi-bing. Transport Time Scale in the Beilun River Estuary and Its Adjacent Area[J]. Journal of Guangxi Normal University(Natural Science Edition), 2015, 33(2): 56 -63 .
[4] XU Jiu-cheng, LI Xiao-yan, LI Shuang-qun, ZHANG Ling-jun. Feature Images Retrieval Method of Tolerance Granular-basedMulti-level Texture[J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(1): 186 -187 .
[5] ZHAO Ming, NIU Ya-lan, ZHONG Jin-xiu, LIANG Feng-jun, LUO Jiang-sheng, ZHENG Mu-hua. Effects of Average Degree on Network Dynamics[J]. Journal of Guangxi Normal University(Natural Science Edition), 2012, 30(3): 88 -93 .
[6] ZHOU Shan-yi. Progress of Taxonomic Study on Formicidae (Hymenoptera) in China[J]. Journal of Guangxi Normal University(Natural Science Edition), 2012, 30(3): 244 -251 .
[7] XU Lunhui,HUANG Baoshan,ZHONG Haixing. Time Window Model and Algorithm with AGV System Path Planning[J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(3): 1 -8 .
[8] ZENG Wei, LONG Yongcai, GAO Ge, LI Wande, HU Yuanfang,LI Xuezhu, WANG Yufeng, SHAO Shimiao, LUO Xu. Bird Diversity of Gulinqing Nature Reserve, Yunnan, China[J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(4): 113 -123 .
[9] ZHENG Tao, ZHOU Xinran, ZHANG Long. Global Asymptotic Stability of Predator-Competition-Cooperative Hybrid Population Models of Three Species[J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(5): 64 -70 .
[10] YANG Zhou, FAN Yixing, ZHU Xiaofei, GUO Jiafeng, WANG Yue. Survey on Modeling Factors of Neural Information Retrieval Model[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(2): 1 -12 .
Copyright © Journal of Guangxi Normal University(Natural Science Edition), All Rights Reserved.
Tel: 0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
Powered by Beijing Magtech Co. Ltd