捕食-食饵模型一直是生态系统中很重要的一个研究课题, 它描述了自然界最主要的一部分种群间的关系, 帮助我们更清晰地了解到物种之间的动态关系. 近年来, 生物学家们发现恐惧效应会对捕食-食饵系统的动力学行为产生明显影响. 关于恐惧效应对捕食-食饵模型的影响, 目前已经有很多相关的研究. 但是恐惧效应对带有空间异质模型以及交叉扩散型模型影响的研究还尚有欠缺. 因此, 本文主要分为两部分来讨论恐惧效应的影响.

本文第一部分研究了齐次 Neumann 边值条件下具有恐惧效应和空间异质的Holling III 型捕食-食饵模型. 首先运用线性化算子的谱理论, 得到了平凡解和半平凡解的局部渐近稳定性. 其次, 利用抛物型偏微分方程的比较原理, 得到了平凡解和半平凡解的全局吸引性. 然后利用不动点定理, 得到了正稳态解的存在性. 最后, 运用数值模拟直观地验证了文章的理论结果.

本文第二部分研究了齐次 Dirichlet 边值条件下具有恐惧效应和交叉扩散的捕食-食饵模型稳态解的性质, 包括模型解的稳定性、先验估计以及正解的存在性. 首先,研究了模型稳态解的稳定性, 并且利用比较方法得到了先验估计; 其次, 运用正锥中的不动点定理讨论了正稳态解的存在性. 最后, 利用数值模拟研究了交叉扩散系数对正稳态解的影响, 得出了一些新的结论.
 

The predator-prey model has always been an important research topic in ecosystems. It describes the relationships between the most important parts of the population in nature, helping us to have a clearer understanding of the dynamic relationships between species. In recent years, biologists have found that the fear effect has a significant impact on the dynamic behavior of predator-prey systems. There have been many related studies on the impact of fear effects on predator-prey models. However, the research on the influence of fear effect on models with spatial heterogeneity and cross diffusion models is still lacking. Therefore, this article is mainly divided into two parts to discuss the impact of fear effects.

The first part of this article investigates the Holling III predator-prey model with fear effect and spatial heterogeneity under homogeneous Neumann boundary conditions. Firstly, using
the spectral theory of linearization operators, the local asymptotic stability of trivial and semi trivial solutions was obtained. Secondly, using the comparison principle of parabolic partial differential equations, the global attractiveness of trivial and semi trivial solutions is obtained. Then, using the fixed point theorem, the existence of positive steady-state solutions was obtained. Finally, numerical simulations were used to visually validate the theoretical
results of the article.

The second part of this article investigates the properties of steady-state solutions for a predator-prey model with fear effects and cross-diffusion under homogeneous Dirichlet boundary value conditions, including the stability of the model solution, prior estimation, and the existence of positive solutions. Firstly, the stability of the steady-state solution of the model is studied, and a priori estimation is obtained by comparative method; Secondly,
the fixed point theorem in a cone is used to establish the existence of positive steady-state solutions. Finally, the influence of cross-diffusion coefficient on the positive steady state solution is studied by numerical simulation, and some new conclusions are drawn.
 

[1] 占萌颖, 汪艳梅, 万易等. 恐惧因素对食蚜蝇-蚜虫种群稳定性的影响[J]. 数学的实践与认识,2021, 51(13): 7.

[2] LOU Y, NI W M. Diffusion vs cross-diffusion: an elliptic approach[J]. Journal of Differential Equations, 1999, 154(1): 157-190.

[3] WU Y, Ni W M, YUAN L. On the global existence of a cross-diffusion system[J]. Discrete and Continuous Dynamical Systems-Series, 2012, 4(2): 193-203.

[4] LOTKA A J. Elements of physical biology[J]. Nature, 1925.

[5] VOLTERRA V. Fluctuations in the abundance of aspecies considered mathematically[J]. Nature, 1926, 118(1): 558-560.

[6] 何梦昕. 具脉冲扰动Lotka-Volterra模型的稳定性研究[D]. 福建师范大学, 2018.

[7] CRESSWELL W. Predation in bird populations[J]. Journal of Ornithology, 2011, 152(1): 251-263.

[8] ANTWI-FORDJOUR K, PARSHAD R D, THOMPSON H E, et al. Fear-driven extinction and (de)stabilization in a predator-prey model incorporating prey herd behavior and mutual interference[J]. AIMS Mathematics, 2023, 8(2): 3353-3377.

[9] CLINCHY M, SHERIFF M J, ZANETTE L Y. Predator-induced stress and the ecology of fear[J]. Functional Ecology, 2013, 27(1): 56-65.

[10] ZANETTE L Y, WHITE A F, ALLEN M C, et al. Perceived predation risk reduces the number of offspring songbirds produce per year[J]. Science, 2011, 334(6061): 1398-1401.

[11] WANG X Y, ZANETTE L Y, ZOU X F. Modelling the fear effect in predator-prey interactions[J]. Journal of Mathematical Biology, 2016, 73(5): 1179-1204.

[12] CHEN S S, LIU Z H, SHI J P. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model with fear effect[J]. Journal of Nonlinear Modeling and Analysis, 2019, 1: 47-56.

[13] WANG J, CAI Y L, FU S M, et al. The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge[J]. Chaos, 2019, 29(8): 083109.

[14] LIU Q, JIANG D Q. Influence of the fear factor on the dynamics of a stochastic predator-prey model[J]. Applied Mathematics Letters, 2021, 112: 106756.

[15] HUANG Y, CHEN F, LI Z. Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge[J]. Applied Mathematics and Computation, 2006, 182(1): 672-683.

[16] YAN X, LI W. Stability and hopf bifurcation for a delayed prey-predator system with diffusion effects[J]. Applied Mathematics and Computation, 2008, 192(2): 552-566.

[17] KUTO K. Stability of steady-state solutions to a prey-predator system with cross-diffusion[J]. Journal of Differential Equations,2004, 197(2): 293-314.

[18] 李景荣, 李艳玲, 闫焱. 一类带交叉扩散项的捕食模型正解的存在性[J]. 陕西师范大学学报(自然科学版), 2009, 37(01): 20-24.

[19] CRAMPIN E J, HACKBORN W W, MAINI P K. Pattern formation in reaction-diffusion models with nonuniform domain growth[J]. Bulletin of Mathematical Biology, 2002, 64(4): 747-769.

[20] HOLLING C. Some characteristics of simple types of predation and parasitism[J]. The Canadian Entomologist, 1959, 91(7): 385-398.

[21] KEMPF A, FLOETER J, TEMMING A. Predator-prey overlap induced Holling type III functional response in the north sea fish assemblage[J]. Marine Ecology Progress Series, 2008, 367: 295-308.

[22] LIU Z, ZHONG S. An impulsive periodic predator-prey system with Holling type III functional response and diffusion[J]. Applied Mathematical Modelling, 2012, 36(12): 5976-5990.

[23] MA H. A qualitative analysis of system with the Holling III functional response of predator to prey[J]. Journal of Biomathematics, 1999, 14(1): 33-36.

[24] 何德材, 黄文韬. 一类被开发的Holling Ⅲ类功能反应模型的定性分析[J]. 生物数学学报, 2009,24(1): 7.

[25] 叶其孝, 李正元, 王明新等. 反应扩散方程引论[M]. 科学出版社, 2013: 78-102.

[26] ALLEN L J S, BOLKER B M, LOU Y. Asymptotic profiles of the steady states for an SIS epidemic reaction diffusion model[J]. Discrete and Continuous Dynamical Systems, 2008, 21(1): 1-20.

[27] 黄振友. 泛函分析[M]. 南京东南大学出版社, 2019, 08.

[28] 张建梅. 谈拓扑度(紧)同伦不变性的应用规律[J]. 黑龙江农垦师专学报, 2002, (03): 71-73.

[29] SAMADDAR S, DHAR M, BHATTACHARYA P. Effect of fear on prey-predator dynamics: Exploring the role of prey refuge and additional food[J]. Chaos, 2020, 30(6): 063129.

[30] HECK K L, THOMAN T A. Experiments on predator-prey interactions in vegetated aquatic habitats[J]. Journal of Experimental Marine Biology and Ecology, 1981, 53(2-3): 125-134.

[31] PECKARSKY B L. Predator-prey interactions between stoneflies and mayflies: behavioral observations[J]. Ecology, 1980, 61(4): 932.

[32] LI S B, XIAO Y N, DONG Y Y. Diffusive predator-prey models with fear effect in spatially heterogeneous environment[J]. Electronic Journal of Differential Equations, 2021, 2021: 1-31.

[33] RUAN W H, FENG W. On the fixed point index and multiple steady-state solutions of reactiondiffusion systems[J]. Differential and Integral Equations, 1995, 8(2): 371-391.

[34] DUBEY B, DAS B, HUSSAIN J A. predator-prey interaction model with self and cross-diffusion[J]. Ecological Modelling, 2001, 141(1-3), 67-76.

[35] CHEN X, HAMBROCK R, YUAN L. Evolution of conditional dispersal: a reaction-diffusionadvection model[J]. Journal of Mathematical Biology, 2008, 57(3): 361-386.

[36] CREEL S, CHRISTIANSON D. Relationships between direct predation and risk effects[J]. Trends in Ecology and Evolution, 2008, 23(4): 194-201.

[37] STEVEN L L. Nonlethal effects in the ecology of predator-prey interactions[J]. BioScience, 1998,48: 25-34.

[38] LI L. Coexistence theorems of steady states for predator-prey interacting systems[J]. Transactions of the American Mathematical Society, 1988, 305(1): 143-166.

[39] AMANN H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces[J]. Siam Review, 1976, 62-709.

[40] FRAILE J M, KOCH M P, LOPEZ-GOMEZ J, et al. Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation[J], Journal of Differential Equations, 1996, 127: 295-319.

[41] JIN H, WANG Z. Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion[J], European Journal of Applied Mathematics, 2021, 32(4): 652-682.

[42] KUTO K A. strongly coupled diffusion effect on the stationary solution set of a prey-predator model[J]. Advances in Differential Equations, 2007, 12(2): 145-172.

[43] KUTO K, YAMADA Y. Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type[J]. Differential and Integral Equations, 2009, 22(7).

[44] KADOTA T, KUTO K. Positive steady states for a prey-predator model with some nonlinear diffusion terms[J]. Journal of Mathematical Analysis and Applications, 2006, 323(2): 1387-1401.

[45] LI S B, MA R Y. Positive steady-state solutions for predator-prey systems with prey-taxis and Dirichlet conditions[J]. Nonlinear Analysis: Real World Applications, 2022, 68: 1468-1218.

[46] GUI C F, LOU Y. Uniqueness and nonuniqueness of coexistence states in the lotka-volterra competition model[J]. Communications on Pure and Applied Mathematics, 1994, 47: 1571-1594.

[47] DANCER E N. On the indices of fixed points of mappings in cones and applications[J]. Journal of Mathematical Analysis and Applications, 1983, 91(1): 131-151.

[48] NAKASHIMA K, YAMADA Y. Positive steady states for prey-predator models with crossdiffusion[J]. Advances in Differential Equations, 1996, 1(6): 1099-1122.

[49] WANG J, SHI J, WEI J. Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey[J]. Journal of Differential Equations, 2011, 251(4-5): 1276-1304.

[50] DANCER E N. On positive solutions of some pairs of differential equations[J]. Journal of Differential Equations, 1985, 60(2): 236–258.

[51] GILBARG D, TRUDINGER N S. Elliptic partial differential equations of second order[M]. Springer-Verlag, 1983.