
-
Previous Article
Stability analysis of an equation with two delays and application to the production of platelets
- DCDS-S Home
- This Issue
-
Next Article
Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator
A nutrient-prey-predator model: Stability and bifurcations
1. | Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA |
2. | Department of Mathematics, Faculty of Science, Al-Hussein Bin Talal University, Ma'an, P.O.Box (20), Jordan |
We model a nutrient-prey-predator system in a chemostat with general functional responses, using the input concentration of nutrient as the bifurcation parameter. We study changes in the existence and the stability of isolated equilibria, as well as changes in the global dynamics, as the nutrient concentration varies. The bifurcations of the system are analytically verified and we identify conditions under which an equilibrium undergoes a Hopf bifurcation and a limit cycle appears. Numerical simulations for specific functional responses illustrate the general results.
References:
[1] |
M. Ballyk, I. Jawarneh and R. Staffeldt, A nutrient-prey-predator model: Stability and bifurcations, preprint, arXiv: 1812.09964. |
[2] |
G. Butler, H. I. Freedman and P. Waltman,
Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430.
doi: 10.1090/S0002-9939-1986-0822433-4. |
[3] |
H. I. Freedman and P. Waltman,
Persistence in models of three interacting predator-prey populations, Math. Biosci., 68 (1984), 213-231.
doi: 10.1016/0025-5564(84)90032-4. |
[4] |
T. C. Gard, Mathematical analysis of some resource-prey-predator models: Application to an NPZ microcosm model, in Population Biology, Lecture Notes in Biomath., 52, Springer, Berlin, 1983,275–282.
doi: 10.1007/978-3-642-87893-0_34. |
[5] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981. |
[6] |
S. B. Hsu,
Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[7] |
S. Lang, Analysis II, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1969. |
[8] |
J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, With Applications, Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961. |
[9] |
B. Li and Y. Kuang,
Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.
doi: 10.1006/jmaa.1999.6655. |
[10] |
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976.
doi: 10.1007/978-1-4612-6374-6. |
[11] |
G. S. K. Wolkowicz,
Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.
doi: 10.1016/0025-5564(89)90025-4. |
[12] |
T. Zhang and W. Wang,
Hopf bifurcation and bistability of a nutrient-phytoplankton-zooplankton model, Appl. Math. Model., 36 (2012), 6225-6235.
doi: 10.1016/j.apm.2012.02.012. |
show all references
References:
[1] |
M. Ballyk, I. Jawarneh and R. Staffeldt, A nutrient-prey-predator model: Stability and bifurcations, preprint, arXiv: 1812.09964. |
[2] |
G. Butler, H. I. Freedman and P. Waltman,
Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430.
doi: 10.1090/S0002-9939-1986-0822433-4. |
[3] |
H. I. Freedman and P. Waltman,
Persistence in models of three interacting predator-prey populations, Math. Biosci., 68 (1984), 213-231.
doi: 10.1016/0025-5564(84)90032-4. |
[4] |
T. C. Gard, Mathematical analysis of some resource-prey-predator models: Application to an NPZ microcosm model, in Population Biology, Lecture Notes in Biomath., 52, Springer, Berlin, 1983,275–282.
doi: 10.1007/978-3-642-87893-0_34. |
[5] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981. |
[6] |
S. B. Hsu,
Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[7] |
S. Lang, Analysis II, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1969. |
[8] |
J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, With Applications, Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961. |
[9] |
B. Li and Y. Kuang,
Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.
doi: 10.1006/jmaa.1999.6655. |
[10] |
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976.
doi: 10.1007/978-1-4612-6374-6. |
[11] |
G. S. K. Wolkowicz,
Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.
doi: 10.1016/0025-5564(89)90025-4. |
[12] |
T. Zhang and W. Wang,
Hopf bifurcation and bistability of a nutrient-phytoplankton-zooplankton model, Appl. Math. Model., 36 (2012), 6225-6235.
doi: 10.1016/j.apm.2012.02.012. |






[1] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[2] |
Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 |
[3] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013 |
[4] |
Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012 |
[5] |
Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321 |
[6] |
Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 |
[7] |
Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 |
[8] |
John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 |
[9] |
Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 |
[10] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[11] |
Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 |
[12] |
Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 |
[13] |
R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure and Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 |
[14] |
Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 |
[15] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[16] |
Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 |
[17] |
Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022069 |
[18] |
Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations and Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143 |
[19] |
Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 |
[20] |
Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]