Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3

Tia Kessé Thiban; Nindjin Aka Fulgence; Okou Hypolithe; N’Guessan Tetchi Albin

doi:doi:10.11648/j.ajam.20180606.11

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Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3

Tia Kessé Thiban, Nindjin Aka Fulgence, Okou Hypolithe, N’Guessan Tetchi Albin

Published in American Journal of Applied Mathematics (Volume 6, Issue 6)

Received: 4 December 2018 Accepted: 10 January 2019 Published: 14 February 2019

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Abstract

This current paper investigates a predator-prey model from Holling-II type and Leslie Gower modified with diffusion and two time delays in dimension three. Firstly, we demonstrate that its solutions are positive and globally bounded. Secondly, we study the local stability of six equilibria points of from one is located within the relevant domain. Under certain conditions, it reveals that among the equilibria points, some are locally stable. Finally, we focus on the global stability of the positive interior equilibrium point. We show that the global stability set out due to the lack of time delays is kept until a certain threshold value of time delays above which a change in the stability is observed. Thus, the global convergence analysis towards the positive interior equilibrium point demonstrate the importance and impacts of the time delay in the stability of our model.

Published in	American Journal of Applied Mathematics (Volume 6, Issue 6)
DOI	10.11648/j.ajam.20180606.11
Page(s)	167-185
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), 2019. Published by Science Publishing Group

Keywords

Holling-2, Leslie-Gower, Boundedness, Lyapunov’s Functional, Equilibrium Point, Local Stability, Global Stability, Time Delay

References

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[11]	Z. Zhang, H. Yang, and J. Liu, Stability and Hopf Bifurcation in a Modified Holling-Tanner Predator-Prey System with Multiple Delays. Hindawi Publishing Corporation, Abstract and Applied Analysis, vol. 2012, pp. 1-19, (2012).
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[14]	Z. Ma, H. Tang, S. Wang, and T. Wang, Bifurcation of a predator-prey system with generation delay and habitat complexity. J. Korean Math. Soc, Vol. 55 N°1, pp. 43-58 DOI: 10.4134/JKMS.j160717, (2018).
[15]	R. Sivasamy, K. Sathiyanathan, Dynamics of a diffusive Leslie-Gower predator-prey model with nonlinear prey harvesting. Annual Review of Chaos Theory, Bifurcations and Dynamical Systems, Vol. 8, n°3 pp. 1-20, (2018).
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APA Style

Tia Kessé Thiban, Nindjin Aka Fulgence, Okou Hypolithe, N’Guessan Tetchi Albin. (2019). Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3. American Journal of Applied Mathematics, 6(6), 167-185. https://doi.org/10.11648/j.ajam.20180606.11

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ACS Style

Tia Kessé Thiban; Nindjin Aka Fulgence; Okou Hypolithe; N’Guessan Tetchi Albin. Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3. Am. J. Appl. Math. 2019, 6(6), 167-185. doi: 10.11648/j.ajam.20180606.11

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AMA Style

Tia Kessé Thiban, Nindjin Aka Fulgence, Okou Hypolithe, N’Guessan Tetchi Albin. Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3. Am J Appl Math. 2019;6(6):167-185. doi: 10.11648/j.ajam.20180606.11

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@article{10.11648/j.ajam.20180606.11,
  author = {Tia Kessé Thiban and Nindjin Aka Fulgence and Okou Hypolithe and N’Guessan Tetchi Albin},
  title = {Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {6},
  pages = {167-185},
  doi = {10.11648/j.ajam.20180606.11},
  url = {https://doi.org/10.11648/j.ajam.20180606.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180606.11},
  abstract = {This current paper investigates a predator-prey model from Holling-II type and Leslie Gower modified with diffusion and two time delays in dimension three. Firstly, we demonstrate that its solutions are positive and globally bounded. Secondly, we study the local stability of six equilibria points of from one is located within the relevant domain. Under certain conditions, it reveals that among the equilibria points, some are locally stable. Finally, we focus on the global stability of the positive interior equilibrium point. We show that the global stability set out due to the lack of time delays is kept until a certain threshold value of time delays above which a change in the stability is observed. Thus, the global convergence analysis towards the positive interior equilibrium point demonstrate the importance and impacts of the time delay in the stability of our model.},
 year = {2019}
}

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TY  - JOUR
T1  - Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3
AU  - Tia Kessé Thiban
AU  - Nindjin Aka Fulgence
AU  - Okou Hypolithe
AU  - N’Guessan Tetchi Albin
Y1  - 2019/02/14
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20180606.11
DO  - 10.11648/j.ajam.20180606.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 167
EP  - 185
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180606.11
AB  - This current paper investigates a predator-prey model from Holling-II type and Leslie Gower modified with diffusion and two time delays in dimension three. Firstly, we demonstrate that its solutions are positive and globally bounded. Secondly, we study the local stability of six equilibria points of from one is located within the relevant domain. Under certain conditions, it reveals that among the equilibria points, some are locally stable. Finally, we focus on the global stability of the positive interior equilibrium point. We show that the global stability set out due to the lack of time delays is kept until a certain threshold value of time delays above which a change in the stability is observed. Thus, the global convergence analysis towards the positive interior equilibrium point demonstrate the importance and impacts of the time delay in the stability of our model.
VL  - 6
IS  - 6
ER  -

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Author Information

Tia Kessé Thiban

Research and Formation Unit of Mathematics and Computer Science, Félix Houphou?t Boigny University of Abidjan Cocody, Abidjan, Ivory Coast
Nindjin Aka Fulgence

Research and Formation Unit of Mathematics and Computer Science, Félix Houphou?t Boigny University of Abidjan Cocody, Abidjan, Ivory Coast
Okou Hypolithe

Research and Formation Unit of Mathematics and Computer Science, Félix Houphou?t Boigny University of Abidjan Cocody, Abidjan, Ivory Coast
N’Guessan Tetchi Albin

Research and Formation Unit of Mathematics and Computer Science, Félix Houphou?t Boigny University of Abidjan Cocody, Abidjan, Ivory Coast

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Plain Text BibTeX RIS

APA Style

Tia Kessé Thiban, Nindjin Aka Fulgence, Okou Hypolithe, N’Guessan Tetchi Albin. (2019). Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3. American Journal of Applied Mathematics, 6(6), 167-185. https://doi.org/10.11648/j.ajam.20180606.11

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ACS Style

Tia Kessé Thiban; Nindjin Aka Fulgence; Okou Hypolithe; N’Guessan Tetchi Albin. Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3. Am. J. Appl. Math. 2019, 6(6), 167-185. doi: 10.11648/j.ajam.20180606.11

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AMA Style

Tia Kessé Thiban, Nindjin Aka Fulgence, Okou Hypolithe, N’Guessan Tetchi Albin. Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3. Am J Appl Math. 2019;6(6):167-185. doi: 10.11648/j.ajam.20180606.11

Copy | Download

@article{10.11648/j.ajam.20180606.11,
  author = {Tia Kessé Thiban and Nindjin Aka Fulgence and Okou Hypolithe and N’Guessan Tetchi Albin},
  title = {Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {6},
  pages = {167-185},
  doi = {10.11648/j.ajam.20180606.11},
  url = {https://doi.org/10.11648/j.ajam.20180606.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180606.11},
  abstract = {This current paper investigates a predator-prey model from Holling-II type and Leslie Gower modified with diffusion and two time delays in dimension three. Firstly, we demonstrate that its solutions are positive and globally bounded. Secondly, we study the local stability of six equilibria points of from one is located within the relevant domain. Under certain conditions, it reveals that among the equilibria points, some are locally stable. Finally, we focus on the global stability of the positive interior equilibrium point. We show that the global stability set out due to the lack of time delays is kept until a certain threshold value of time delays above which a change in the stability is observed. Thus, the global convergence analysis towards the positive interior equilibrium point demonstrate the importance and impacts of the time delay in the stability of our model.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Stability Study of a Holling-II Type Model and Leslie-Gower Modified with Diffusion and Time Delays in Dimension 3
AU  - Tia Kessé Thiban
AU  - Nindjin Aka Fulgence
AU  - Okou Hypolithe
AU  - N’Guessan Tetchi Albin
Y1  - 2019/02/14
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20180606.11
DO  - 10.11648/j.ajam.20180606.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 167
EP  - 185
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180606.11
AB  - This current paper investigates a predator-prey model from Holling-II type and Leslie Gower modified with diffusion and two time delays in dimension three. Firstly, we demonstrate that its solutions are positive and globally bounded. Secondly, we study the local stability of six equilibria points of from one is located within the relevant domain. Under certain conditions, it reveals that among the equilibria points, some are locally stable. Finally, we focus on the global stability of the positive interior equilibrium point. We show that the global stability set out due to the lack of time delays is kept until a certain threshold value of time delays above which a change in the stability is observed. Thus, the global convergence analysis towards the positive interior equilibrium point demonstrate the importance and impacts of the time delay in the stability of our model.
VL  - 6
IS  - 6
ER  -

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