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The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure

In this paper, fear effect and stage structure are introduced in a free boundary problem of a prey-predator model. This system simulates the spread of an invasive or newly introduced predator species, taking into account the presence of both immature and mature stages of prey that are affected by fear of the predator. The predator's predation behavior on adult prey induces fear in the prey, which in turn causes the prey to seek out safer habitats. While this short-term survival strategy may be effective, it ultimately leads to a decrease in the prey's long-term survival fitness, including reduced reproductive ability. Consequently, the overall population of prey is expected to decline over the long term. The existence and uniqueness of the solution is given, and the comparison principle is used to discuss the long-term behavior of the solution by constructing a sequence of upper and lower solutions. We obtain a spreading–vanishing dichotomy for this model, in other words, when the predator can only spread in a limited area, the predator will eventually become extinct, the population density of the two stages of prays will tend to two positive constants, and when the predator can spread to infinity, the predator ultimately survives, and their population density, defined as (u, v, w) will eventually tend to (u*, v*, w*) which we defined blow.

Fear Effect, Stage Structure, Free Boundary Problem, Asymptotic Property

APA Style

Chao Shao, Jingfu Zhao. (2023). The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure. Mathematics and Computer Science, 8(2), 62-67. https://doi.org/10.11648/j.mcs.20230802.15

ACS Style

Chao Shao; Jingfu Zhao. The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure. Math. Comput. Sci. 2023, 8(2), 62-67. doi: 10.11648/j.mcs.20230802.15

AMA Style

Chao Shao, Jingfu Zhao. The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure. Math Comput Sci. 2023;8(2):62-67. doi: 10.11648/j.mcs.20230802.15

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Wang, Mingxin. On some Free Boundary Problems of the Prey-predator Model [J]. Journal of Differential Equations, 2014, 256 (10): 3365-3394.
2. Zhao J, Wang M. A free boundary problem of a predator–prey model with higher dimension and heterogeneous environment [J]. Nonlinear Analysis Real World Applications, 2014, 16: 250-263.
3. Zhao M, Li W, Cao J. A PREY-PREDATOR MODEL WITH A FREE BOUNDARY AND SIGN-CHANGING COEFFICIENT IN TIME-PERIODIC ENVIRONMENT [J]. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 2017, 22 (4): 3295-3316.
4. Zhu Z, Wu R, Lai L, et al. The influence of fear effect to the Lotka-Volterra predator–prey system with predator has other food resource [J]. Advances in Difference Equations, 2020 (1): 1-13.
5. Mengxin He, Zhong Li. Stability of a fear effect predator–prey model with mutual interference or group defense [J]. Journal of Biological Dynamics, 2022, 16 (1): 480-498.
6. Nazmul Sk, Pankaj Kumar Tiwari, Samares Pal, et al. A delay non-autonomous model for the combined effects of fear, prey refuge and additional food for predator [J]. Journal of Biological Dynamics, 2021, 15 (1): 580-622.
7. Renxiu Xue, Yuanfu Shao, Minjuan Cui. Analysis of a stochastic predator–prey system with fear effect and Lévy noise [J]. Advances in Continuous and Discrete Models 2022, 72.
8. Lazarus K, Agus S, Isnani D, et al. Hopf bifurcation and stability analysis of the Rosenzweig-MacArthur predator-prey model with stage-structure in prey [J]. Mathematical Biosciences and Engineering, 2020, 17 (4): 4080-4097.
9. Shi W, Huang Y, Wei C, et al. A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey [J]. Advances in Mathematical Physics, 2021. DOI: 10.1155/2021/9479012.
10. Wang, J, Wang, M. A predator–prey model with taxis mechanisms and stage structure for the predator [J]. Nonlinearity, 2020, 33 (7), 3134–3172.
11. Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments [J]. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2): 415-421.
12. Mingxin Wang, Jingfu Zhao. A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries [J], J Dyn Diff Equat (2017) 29: 957–979.
13. Wang M, Zhang Q. Dynamics for the diffusive Leslie-Gower model with double free boundaries [J]. Discrete and continuous dynamical systems, 2018, 38: 2591-2607.
14. Jingfu Zhao et al. A diffusive stage-structured model with a free boundary. Boundary Value Problems (2018): 138.
15. Xu, R., Chaplain, M. A. J., Davidson, F. A.: Modelling and analysis of a competitive model with stage structure. Math. Comput. Model. 41, 150–175 (2005).