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A curve of coexistence equilibria
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Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator
November
2020, 13(11): 2975-3004.
doi: 10.3934/dcdss.2020192
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 and Continuous Dynamical Systems - S,
2020, 13
(11)
: 2975-3004.
doi: 10.3934/dcdss.2020192
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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. |
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)
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