Figure 1.
A curve of coexistence equilibria
-
Previous Article
PID parameter optimization algorithm for pressure control of heating system of ground source heat pump
- DCDS-S Home
- This Issue
-
Next Article
Combined effects for non-autonomous singular biharmonic problems
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.
Mathematics Subject Classification:
Primary: 37G10; Secondary: 34C23, 92D25, 34A34.
Citation:
Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh.
A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020192
![]()
References:
| [1] |
M. Ballyk, I. Jawarneh and R. Staffeldt, A nutrient-prey-predator model: Stability and bifurcations, preprint, arXiv: 1812.09964. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. |
Figure 2.
Comparison of real parts. For the solid curve, $ D{ = }D_1{ = }D_2 = 1 $ ; for the curve of symbols, $ D{ = }1 $ , $ D_1{ = }1.2 $ and $ D_2{ = }1.3 $
Figure 3.
Before and after a Hopf bifurcation: $ D{ = }D_1{ = }D_2{ = }1 $ and Holling type Ⅱ rate functions
Figure 4.
Before and after Hopf bifurcation: $ D{ = }1 $ , $ D_1{ = }1.2 $ and $ D_2{ = }1.3 $ and Holling type Ⅱ rate functions
Figure 5.
Real part using rate functions (23)
Figure 6.
Before and after Hopf bifurcation: $ D{ = }1 $ , $ D_1{ = }1.2 $ , and $ D_2{ = }1.1 $ and using rate functions (23)
| [1] |
Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 |
| [2] |
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 |
| [3] |
Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321 |
| [4] |
Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 |
| [5] |
Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 |
| [6] |
John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 |
| [7] |
Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 |
| [8] |
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 |
| [9] |
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 |
| [10] |
R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 |
| [11] |
Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 |
| [12] |
Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 |
| [13] |
Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations & Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143 |
| [14] |
Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 |
| [15] |
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 |
| [16] |
Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 |
| [17] |
Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 |
| [18] |
Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 |
| [19] |
Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529 |
| [20] |
Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 |
2018 Impact Factor: 0.545
Tools
Metrics
Other articles
by authors
[Back to Top]