
-
Previous Article
On the isoperimetric problem with perimeter density $r^p$
- CPAA Home
- This Issue
-
Next Article
Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation
On a predator prey model with nonlinear harvesting and distributed delay
1. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain |
2. | Department of Engineering, Niccolò Cusano University, via Don Carlo Gnocchi 3, 00166 Roma, Italy |
3. | Department of Management, Università Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy |
A predator prey model with nonlinear harvesting (Holling type-Ⅱ) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some numerical simulations.
References:
[1] |
L. Chang, G. Q. Sun, Z. Jin and Z. Wang,
Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput, 256(C) (2015), 540-550.
|
[2] |
T. Das, R. N. Mukherjee and K. S. Chaudhari,
Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462.
|
[3] |
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. |
[4] |
R. P. Gupta and P. Chandra,
Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
|
[5] |
R. P. Gupta, P. Chandra and M. Banerjee,
Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete and continuous dynamical systems series B, 20 (2015), 423-443.
|
[6] |
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. |
[7] |
S. V. Krishna, P. D. N. Srinivasu and B. Prasad Kaymakcalan,
Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.
|
[8] |
J. Liu and L. Zhang,
Bifurcation analysis in a prey-predator model with nonlinear predator harvesting, Journal of the Franklin Institute, 353 (2016), 4701-4714.
|
[9] |
N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. |
[10] |
T. Pradhan and K. S. Chaudhuri,
Bioeconomic harvesting of a schooling fish species: A dynamic reaction model, Korean J. Comput. Appl. Math., 6 (1999), 127-141.
|
[11] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Graduate Studies in Mathematics Volume: 118 (2011), 405 pp. |
[12] |
H. L. Smith and X. Q. Zhao,
Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
|
[13] |
Y. Song, Y. Peng and M. Han,
Travelling wave fronts in the diffusive single species model with Allee effect and distributed delay, Appl. Math. Comput., 152 (2004), 483-497.
|
[14] |
P. D. N. Srinivasu,
Bioeconomics of a renewable resource in presence of a predator, Nonlin. Anal. Real World Appl., 2 (2001), 497-506.
|
[15] |
J. Wu and X. Zou,
Traveling wave fronts of reaction-diffusion systems with delay, J.Dyn.Diff.Equ., 13 (2001), 651-687.
|
[16] |
R. Xu and Z. Ma,
An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.
|
show all references
References:
[1] |
L. Chang, G. Q. Sun, Z. Jin and Z. Wang,
Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput, 256(C) (2015), 540-550.
|
[2] |
T. Das, R. N. Mukherjee and K. S. Chaudhari,
Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462.
|
[3] |
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. |
[4] |
R. P. Gupta and P. Chandra,
Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
|
[5] |
R. P. Gupta, P. Chandra and M. Banerjee,
Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete and continuous dynamical systems series B, 20 (2015), 423-443.
|
[6] |
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. |
[7] |
S. V. Krishna, P. D. N. Srinivasu and B. Prasad Kaymakcalan,
Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.
|
[8] |
J. Liu and L. Zhang,
Bifurcation analysis in a prey-predator model with nonlinear predator harvesting, Journal of the Franklin Institute, 353 (2016), 4701-4714.
|
[9] |
N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. |
[10] |
T. Pradhan and K. S. Chaudhuri,
Bioeconomic harvesting of a schooling fish species: A dynamic reaction model, Korean J. Comput. Appl. Math., 6 (1999), 127-141.
|
[11] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Graduate Studies in Mathematics Volume: 118 (2011), 405 pp. |
[12] |
H. L. Smith and X. Q. Zhao,
Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
|
[13] |
Y. Song, Y. Peng and M. Han,
Travelling wave fronts in the diffusive single species model with Allee effect and distributed delay, Appl. Math. Comput., 152 (2004), 483-497.
|
[14] |
P. D. N. Srinivasu,
Bioeconomics of a renewable resource in presence of a predator, Nonlin. Anal. Real World Appl., 2 (2001), 497-506.
|
[15] |
J. Wu and X. Zou,
Traveling wave fronts of reaction-diffusion systems with delay, J.Dyn.Diff.Equ., 13 (2001), 651-687.
|
[16] |
R. Xu and Z. Ma,
An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.
|










[1] |
Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633 |
[2] |
Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395 |
[3] |
Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243 |
[4] |
Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 |
[5] |
MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 |
[6] |
Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233 |
[7] |
Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051 |
[8] |
Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071 |
[9] |
Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay and nonlinear birth rate. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 735-752. doi: 10.3934/dcdsb.2005.5.735 |
[10] |
Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451 |
[11] |
Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra, Maria dM Vivanco. A model for stem cell population dynamics with regulated maturation delay. Conference Publications, 2011, 2011 (Special) : 32-43. doi: 10.3934/proc.2011.2011.32 |
[12] |
Meng Liu, Ke Wang. Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2495-2522. doi: 10.3934/dcds.2013.33.2495 |
[13] |
Meng Liu, Chuanzhi Bai, Yi Jin. Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108 |
[14] |
Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515 |
[15] |
Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51 |
[16] |
Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 |
[17] |
Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122 |
[18] |
Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745 |
[19] |
Silviu-Iulian Niculescu, Peter S. Kim, Keqin Gu, Peter P. Lee, Doron Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 129-156. doi: 10.3934/dcdsb.2010.13.129 |
[20] |
Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]