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

Chao Shao; Jingfu Zhao

doi:doi:10.11648/j.mcs.20230802.15

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

Chao Shao, Jingfu Zhao

Published in Mathematics and Computer Science (Volume 8, Issue 2)

Received: 25 March 2023 Accepted: 15 April 2023 Published: 23 April 2023

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Abstract

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.

Published in	Mathematics and Computer Science (Volume 8, Issue 2)
DOI	10.11648/j.mcs.20230802.15
Page(s)	62-67
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

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

References

[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).

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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

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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

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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

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@article{10.11648/j.mcs.20230802.15,
  author = {Chao Shao and Jingfu Zhao},
  title = {The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure},
  journal = {Mathematics and Computer Science},
  volume = {8},
  number = {2},
  pages = {62-67},
  doi = {10.11648/j.mcs.20230802.15},
  url = {https://doi.org/10.11648/j.mcs.20230802.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.15},
  abstract = {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.},
 year = {2023}
}

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TY  - JOUR
T1  - The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure
AU  - Chao Shao
AU  - Jingfu Zhao
Y1  - 2023/04/23
PY  - 2023
N1  - https://doi.org/10.11648/j.mcs.20230802.15
DO  - 10.11648/j.mcs.20230802.15
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 62
EP  - 67
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20230802.15
AB  - 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.
VL  - 8
IS  - 2
ER  -

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Author Information

Chao Shao

College of Science, Zhongyuan University of Technology, Zhengzhou, China
Jingfu Zhao

College of Science, Zhongyuan University of Technology, Zhengzhou, China

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Plain Text BibTeX RIS

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

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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

Copy | Download

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

Copy | Download

@article{10.11648/j.mcs.20230802.15,
  author = {Chao Shao and Jingfu Zhao},
  title = {The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure},
  journal = {Mathematics and Computer Science},
  volume = {8},
  number = {2},
  pages = {62-67},
  doi = {10.11648/j.mcs.20230802.15},
  url = {https://doi.org/10.11648/j.mcs.20230802.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.15},
  abstract = {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.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - The Free Boundary Problem of a Predator-Prey Model with Fear Effect and Stage Structure
AU  - Chao Shao
AU  - Jingfu Zhao
Y1  - 2023/04/23
PY  - 2023
N1  - https://doi.org/10.11648/j.mcs.20230802.15
DO  - 10.11648/j.mcs.20230802.15
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 62
EP  - 67
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20230802.15
AB  - 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.
VL  - 8
IS  - 2
ER  -

Copy | Download