American Institute of Mathematical Sciences

Advanced
November  2020, 13(11): 2975-3004. doi: 10.3934/dcdss.2020192

A nutrient-prey-predator model: Stability and bifurcations

Mary Ballyk 1, , Ross Staffeldt 1,, and  Ibrahim Jawarneh 2,

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

* Corresponding author: Ross Staffeldt

Received  April 2019 Revised  October 2019 Published  December 2019

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.

Keywords: Chemostat, Hopf bifurcation, coexistence equilibrium.
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, 2020, 13 (11) : 2975-3004. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A curve of coexistence equilibria
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Before and after a Hopf bifurcation: $ D{ = }D_1{ = }D_2{ = }1 $ and Holling type Ⅱ rate functions
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Real part using rate functions (23)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Before and after Hopf bifurcation: $ D{ = }1 $, $ D_1{ = }1.2 $, and $ D_2{ = }1.1 $ and using rate functions (23)
Figure Options

Download full-size image

Download as PowerPoint slide

[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]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013
[3]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[4]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342
[5]

Manuel Friedrich, Martin Kružík, Ulisse Stefanelli. Equilibrium of immersed hyperelastic solids. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021003
[6]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
[7]

Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140
[8]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
[9]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034
[10]

Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298
[11]

Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020369
[12]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[13]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[14]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[15]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053
[16]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293
[17]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[18]

Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164
[19]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[20]

Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (132)
  • HTML views (316)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences