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Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve
Mathematical analysis of a three-tiered food-web in the chemostat
a. | University of Tunis El Manar, National Engineering School of Tunis, LAMSIN, 1002, Tunis, Tunisia |
b. | ITAP, Univ Montpellier, INRAE, Institut Agro, Montpellier, France |
c. | University of Monastir, Higher Institute of Computer Science of Mahdia, 5111, Mahdia, Tunisia |
d. | University of Manouba, National School of Computer Science, 2010, Manouba, Tunisia |
A mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web is investigated. The model is a six-dimensional system of ordinary differential equations. In our study, the phenol and the hydrogen inflowing concentrations are taken into account as well as the maintenance terms. The case of a large class of growth kinetics is considered, instead of specific kinetics. We show that the system can have up to eight types of steady states and we analytically determine the necessary and sufficient conditions for their existence according to the operating parameters. In the particular case without maintenance, the local stability conditions of all steady states are determined. The bifurcation diagram shows the behavior of the process by varying the concentration of influent chlorophenol as the bifurcating parameter. It shows that the system exhibits a bi-stability where the positive steady state can lose stability undergoing a supercritical Hopf bifurcation with the emergence of a stable limit cycle.
References:
[1] |
N. Abdellatif, R. Fekih-Salem and T. Sari,
Competition for a single resource and coexistence of several species in the chemostat, Math. Biosci. Eng., 13 (2016), 631-652.
doi: 10.3934/mbe.2016012. |
[2] |
M. Ballyk, R. Staffeldt and I. Jawarneh,
A nutrient-prey-predator model: Stability and bifurcations, Discrete & Continuous Dyn. Syst. - S, 13 (2020), 2975-3004.
doi: 10.3934/dcdss.2020192. |
[3] |
B. Bar and T. Sari,
The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete & Continuous Dyn. Syst. - B, 25 (2020), 2093-2120.
doi: 10.3934/dcdsb.2019203. |
[4] |
D. J. Batstone, J. Keller, I. Angelidaki, S. V. Kalyhuzhnyi, S. G. Pavlosthathis, A. Rozzi, W. T. M. Sanders, H. Siegrist and V. A. Vavilin,
The IWA anaerobic digestion model no 1 (ADM1), Water Sci Technol., 45 (2002), 66-73.
doi: 10.2166/wst.2002.0292. |
[5] |
B. Benyahia, T. Sari, B. Cherki and J. Harmand,
Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, J. Proc. Control, 22 (2012), 1008-1019.
doi: 10.1016/j.jprocont.2012.04.012. |
[6] |
O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J-P. Steyer,
Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnol. Bioeng., 75 (2001), 424-438.
doi: 10.1002/bit.10036. |
[7] |
Y. Daoud, N. Abdellatif, T. Sari and J. Harmand,
Steady state analysis of a syntrophic model: The effect of a new input substrate concentration, Math. Model. Nat. Phenom., 13 (2018), 1-22.
doi: 10.1051/mmnp/2018037. |
[8] |
P. De Leenheer, D. Angeli and E. D. Sontag,
Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.
doi: 10.1016/j.jmaa.2006.02.036. |
[9] |
M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete & Continuous Dyn. Syst. - B, (2020).
doi: 10.3934/dcdsb.2020156. |
[10] |
M. Dellal, M. Lakrib and T. Sari,
The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Math. Biosci., 302 (2018), 27-45.
doi: 10.1016/j.mbs.2018.05.004. |
[11] |
N. Dimitrova and M. Krastanov,
Nonlinear stabilizing control of an uncertain bioprocess model, Int. J. Appl. Math. Comput. Sci., 19 (2009), 441-454.
doi: 10.2478/v10006-009-0036-0. |
[12] |
M. El Hajji, F. Mazenc and J. Harmand,
A mathematical study of a syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656.
doi: 10.3934/mbe.2010.7.641. |
[13] |
M. El Hajji, N. Chorfi and M. Jleli, Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat, Electron. J. Differ. Eq., 2017, Paper No. 255, 13 pp. https://ejde.math.txstate.edu/Volumes/2017/255/elhajji.pdf. |
[14] |
R. Fekih-Salem, N. Abdellatif, T. Sari and J. Harmand, Analyse mathématique d'un modèle de digestion anaérobie à trois étapes, {ARIMA Rev.}, 17 (2014), 53–71. http://arima.inria.fr/017/017003.html. |
[15] |
R. Fekih-Salem, C. Lobry and T. Sari,
A density-dependent model of competition for one resource in the chemostat, Math. Biosci., 286 (2017), 104-122.
doi: 10.1016/j.mbs.2017.02.007. |
[16] |
S. R. Hansen and S. P. Hubbell,
Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.
doi: 10.1126/science.6767274. |
[17] |
J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2017. |
[18] |
P. A. Hoskisson and G. Hobbs,
Continuous culture–making a comeback?, Microbiology, 151 (2005), 3153-3159.
doi: 10.1099/mic.0.27924-0. |
[19] |
S.-B. Hsu, C. A. Klausmeier and C.-J. Lin,
Analysis of a model of two parallel food chains, Discrete & Continuous Dyn. Syst. - B, 12 (2009), 337-359.
doi: 10.3934/dcdsb.2009.12.337. |
[20] |
J. Monod, La technique de culture continue: Théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184–204
doi: 10.1016/B978-0-12-460482-7.50023-3. |
[21] |
S. Nouaoura, N. Abdellatif, R. Fekih-Salem and T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, preprint (2020), hal-02540350. https://hal.archives-ouvertes.fr/hal-02540350. |
[22] |
A. Rapaport and M. Veruete,
A new proof of the competitive exclusion principle in the chemostat, Discrete & Continuous Dyn. Syst. - B, 24 (2019), 3755-3764.
doi: 10.3934/dcdsb.2018314. |
[23] |
T. Sari, M. El Hajji and J. Harmand,
The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat, Math. Biosci. Eng., 9 (2012), 627-645.
doi: 10.3934/mbe.2012.9.627. |
[24] |
T. Sari and J. Harmand,
A model of a syntrophic relationship between two microbial species in a chemostat including maintenance, Math. Biosci., 275 (2016), 1-9.
doi: 10.1016/j.mbs.2016.02.008. |
[25] |
T. Sari and M. J. Wade,
Generalised approach to modelling a three-tiered microbial food-web, Math. Biosci., 291 (2017), 21-37.
doi: 10.1016/j.mbs.2017.07.005. |
[26] |
M. Sbarciog, M. Loccufier and E. Noldus,
Determination of appropriate operating strategies for anaerobic digestion systems, Biochem. Eng. J., 51 (2010), 180-188.
doi: 10.1016/j.bej.2010.06.016. |
[27] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() ![]() |
[28] |
S. Sobieszek, M. J. Wade and G. S. K. Wolkowicz,
Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Math. Biosci. Eng., 17 (2020), 7045-7073.
doi: 10.3934/mbe.2020363. |
[29] |
M. J. Wade, Not Just Numbers: Mathematical Modelling and Its Contribution to Anaerobic Digestion Processes, Processes, 8 (2020), 888.
doi: 10.3390/pr8080888. |
[30] |
M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J.-J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry,
Perspectives in mathematical modelling for microbial ecology, Ecol. Modell., 321 (2016), 64-74.
doi: 10.1016/j.ecolmodel.2015.11.002. |
[31] |
M. J. Wade, R. W. Pattinson, N. G. Parker and J. Dolfing,
Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web', J. Theor. Biol., 389 (2016), 171-186.
doi: 10.1016/j.jtbi.2015.10.032. |
[32] |
M. Weedermann,
Analysis of a model for the effects of an external toxin on anaerobic digestion, Math. Biosci. Eng., 9 (2012), 445-459.
doi: 10.3934/mbe.2012.9.445. |
[33] |
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. |
[34] |
A. Xu, J. Dolfing, T. P. Curtis, G. Montague and E. Martin,
Maintenance affects the stability of a two-tiered microbial 'food chain'?, J. Theor. Biol., 276 (2011), 35-41.
doi: 10.1016/j.jtbi.2011.01.026. |
show all references
References:
[1] |
N. Abdellatif, R. Fekih-Salem and T. Sari,
Competition for a single resource and coexistence of several species in the chemostat, Math. Biosci. Eng., 13 (2016), 631-652.
doi: 10.3934/mbe.2016012. |
[2] |
M. Ballyk, R. Staffeldt and I. Jawarneh,
A nutrient-prey-predator model: Stability and bifurcations, Discrete & Continuous Dyn. Syst. - S, 13 (2020), 2975-3004.
doi: 10.3934/dcdss.2020192. |
[3] |
B. Bar and T. Sari,
The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete & Continuous Dyn. Syst. - B, 25 (2020), 2093-2120.
doi: 10.3934/dcdsb.2019203. |
[4] |
D. J. Batstone, J. Keller, I. Angelidaki, S. V. Kalyhuzhnyi, S. G. Pavlosthathis, A. Rozzi, W. T. M. Sanders, H. Siegrist and V. A. Vavilin,
The IWA anaerobic digestion model no 1 (ADM1), Water Sci Technol., 45 (2002), 66-73.
doi: 10.2166/wst.2002.0292. |
[5] |
B. Benyahia, T. Sari, B. Cherki and J. Harmand,
Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, J. Proc. Control, 22 (2012), 1008-1019.
doi: 10.1016/j.jprocont.2012.04.012. |
[6] |
O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J-P. Steyer,
Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnol. Bioeng., 75 (2001), 424-438.
doi: 10.1002/bit.10036. |
[7] |
Y. Daoud, N. Abdellatif, T. Sari and J. Harmand,
Steady state analysis of a syntrophic model: The effect of a new input substrate concentration, Math. Model. Nat. Phenom., 13 (2018), 1-22.
doi: 10.1051/mmnp/2018037. |
[8] |
P. De Leenheer, D. Angeli and E. D. Sontag,
Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.
doi: 10.1016/j.jmaa.2006.02.036. |
[9] |
M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete & Continuous Dyn. Syst. - B, (2020).
doi: 10.3934/dcdsb.2020156. |
[10] |
M. Dellal, M. Lakrib and T. Sari,
The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Math. Biosci., 302 (2018), 27-45.
doi: 10.1016/j.mbs.2018.05.004. |
[11] |
N. Dimitrova and M. Krastanov,
Nonlinear stabilizing control of an uncertain bioprocess model, Int. J. Appl. Math. Comput. Sci., 19 (2009), 441-454.
doi: 10.2478/v10006-009-0036-0. |
[12] |
M. El Hajji, F. Mazenc and J. Harmand,
A mathematical study of a syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656.
doi: 10.3934/mbe.2010.7.641. |
[13] |
M. El Hajji, N. Chorfi and M. Jleli, Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat, Electron. J. Differ. Eq., 2017, Paper No. 255, 13 pp. https://ejde.math.txstate.edu/Volumes/2017/255/elhajji.pdf. |
[14] |
R. Fekih-Salem, N. Abdellatif, T. Sari and J. Harmand, Analyse mathématique d'un modèle de digestion anaérobie à trois étapes, {ARIMA Rev.}, 17 (2014), 53–71. http://arima.inria.fr/017/017003.html. |
[15] |
R. Fekih-Salem, C. Lobry and T. Sari,
A density-dependent model of competition for one resource in the chemostat, Math. Biosci., 286 (2017), 104-122.
doi: 10.1016/j.mbs.2017.02.007. |
[16] |
S. R. Hansen and S. P. Hubbell,
Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.
doi: 10.1126/science.6767274. |
[17] |
J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2017. |
[18] |
P. A. Hoskisson and G. Hobbs,
Continuous culture–making a comeback?, Microbiology, 151 (2005), 3153-3159.
doi: 10.1099/mic.0.27924-0. |
[19] |
S.-B. Hsu, C. A. Klausmeier and C.-J. Lin,
Analysis of a model of two parallel food chains, Discrete & Continuous Dyn. Syst. - B, 12 (2009), 337-359.
doi: 10.3934/dcdsb.2009.12.337. |
[20] |
J. Monod, La technique de culture continue: Théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184–204
doi: 10.1016/B978-0-12-460482-7.50023-3. |
[21] |
S. Nouaoura, N. Abdellatif, R. Fekih-Salem and T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, preprint (2020), hal-02540350. https://hal.archives-ouvertes.fr/hal-02540350. |
[22] |
A. Rapaport and M. Veruete,
A new proof of the competitive exclusion principle in the chemostat, Discrete & Continuous Dyn. Syst. - B, 24 (2019), 3755-3764.
doi: 10.3934/dcdsb.2018314. |
[23] |
T. Sari, M. El Hajji and J. Harmand,
The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat, Math. Biosci. Eng., 9 (2012), 627-645.
doi: 10.3934/mbe.2012.9.627. |
[24] |
T. Sari and J. Harmand,
A model of a syntrophic relationship between two microbial species in a chemostat including maintenance, Math. Biosci., 275 (2016), 1-9.
doi: 10.1016/j.mbs.2016.02.008. |
[25] |
T. Sari and M. J. Wade,
Generalised approach to modelling a three-tiered microbial food-web, Math. Biosci., 291 (2017), 21-37.
doi: 10.1016/j.mbs.2017.07.005. |
[26] |
M. Sbarciog, M. Loccufier and E. Noldus,
Determination of appropriate operating strategies for anaerobic digestion systems, Biochem. Eng. J., 51 (2010), 180-188.
doi: 10.1016/j.bej.2010.06.016. |
[27] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() ![]() |
[28] |
S. Sobieszek, M. J. Wade and G. S. K. Wolkowicz,
Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Math. Biosci. Eng., 17 (2020), 7045-7073.
doi: 10.3934/mbe.2020363. |
[29] |
M. J. Wade, Not Just Numbers: Mathematical Modelling and Its Contribution to Anaerobic Digestion Processes, Processes, 8 (2020), 888.
doi: 10.3390/pr8080888. |
[30] |
M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J.-J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry,
Perspectives in mathematical modelling for microbial ecology, Ecol. Modell., 321 (2016), 64-74.
doi: 10.1016/j.ecolmodel.2015.11.002. |
[31] |
M. J. Wade, R. W. Pattinson, N. G. Parker and J. Dolfing,
Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web', J. Theor. Biol., 389 (2016), 171-186.
doi: 10.1016/j.jtbi.2015.10.032. |
[32] |
M. Weedermann,
Analysis of a model for the effects of an external toxin on anaerobic digestion, Math. Biosci. Eng., 9 (2012), 445-459.
doi: 10.3934/mbe.2012.9.445. |
[33] |
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. |
[34] |
A. Xu, J. Dolfing, T. P. Curtis, G. Montague and E. Martin,
Maintenance affects the stability of a two-tiered microbial 'food chain'?, J. Theor. Biol., 276 (2011), 35-41.
doi: 10.1016/j.jtbi.2011.01.026. |








Definition | |
Let | |
Let | |
Definition | |
Let | |
Let | |
SS1 | |
SS2 | |
SS3 | |
SS4 | |
SS5 | |
SS6 | |
SS7 | |
SS8 |
SS1 | |
SS2 | |
SS3 | |
SS4 | |
SS5 | |
SS6 | |
SS7 | |
SS8 |
Existence conditions | |
SS1 | Always exists |
SS2 | |
SS3 | |
SS4 | |
SS5 | |
SS6 | |
SS7 | |
SS8 |
Existence conditions | |
SS1 | Always exists |
SS2 | |
SS3 | |
SS4 | |
SS5 | |
SS6 | |
SS7 | |
SS8 |
Stability conditions | |
SS1 | |
SS2 | |
SS3 | |
SS4 | |
SS5 | |
SS6 | |
SS7 | |
SS8 |
Stability conditions | |
SS1 | |
SS2 | |
SS3 | |
SS4 | |
SS5 | |
SS6 | |
SS7 | |
SS8 |
Parameter | Wade et al. [31] | Unit |
29 26 35 | | |
0.053 0.302 3.5 2.5 | | |
| 0.019 0.04 0.06 |
Parameter | Wade et al. [31] | Unit |
29 26 35 | | |
0.053 0.302 3.5 2.5 | | |
| 0.019 0.04 0.06 |
Auxiliary function | Definition domain |
|
|
|
|
|
|
|
|
|
|
Auxiliary function | Definition domain |
|
|
|
|
|
|
|
|
|
|
Definition | Value |
Definition | Value |
Interval of |
SS1 | SS2 | SS3 | SS5 | SS6 | ||
U | S | ||||||
U | S | U | |||||
U | S | U | U | U | |||
U | U | U | U | U | S | ||
U | U | S | U | U | |||
U | U | S | U | U | U | ||
U | U | S | U | U | S |
Interval of |
SS1 | SS2 | SS3 | SS5 | SS6 | ||
U | S | ||||||
U | S | U | |||||
U | S | U | U | U | |||
U | U | U | U | U | S | ||
U | U | S | U | U | |||
U | U | S | U | U | U | ||
U | U | S | U | U | S |
Type of the bifurcation | |
Transcritical bifurcation of SS1 and SS3 | |
Saddle-node bifurcation of |
|
Transcritical bifurcation of SS2 and SS5 | |
Transcritical bifurcation of SS3 and SS5 | |
Transcritical bifurcation of |
|
Disappearance of the stable limit cycle | |
Supercritical Hopf bifurcation |
Type of the bifurcation | |
Transcritical bifurcation of SS1 and SS3 | |
Saddle-node bifurcation of |
|
Transcritical bifurcation of SS2 and SS5 | |
Transcritical bifurcation of SS3 and SS5 | |
Transcritical bifurcation of |
|
Disappearance of the stable limit cycle | |
Supercritical Hopf bifurcation |
SS1 | SS2 | SS3 | SS5 | SS6 | ||
Red | Blue | Purple | Dark Green | Magenta | Green | Cyan |
SS1 | SS2 | SS3 | SS5 | SS6 | ||
Red | Blue | Purple | Dark Green | Magenta | Green | Cyan |
Existence conditions | Stability conditions | |
SS1 | Always exists | |
SS2 | ||
SS3 | with | |
SS4 | ||
SS5 | ||
SS6 |
Existence conditions | Stability conditions | |
SS1 | Always exists | |
SS2 | ||
SS3 | with | |
SS4 | ||
SS5 | ||
SS6 |
[1] |
Marion Weedermann. Analysis of a model for the effects of an external toxin on anaerobic digestion. Mathematical Biosciences & Engineering, 2012, 9 (2) : 445-459. doi: 10.3934/mbe.2012.9.445 |
[2] |
S. Ouchtout, Z. Mghazli, J. Harmand, A. Rapaport, Z. Belhachmi. Analysis of an anaerobic digestion model in landfill with mortality term. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2333-2346. doi: 10.3934/cpaa.2020101 |
[3] |
Miled El Hajji, Frédéric Mazenc, Jérôme Harmand. A mathematical study of a syntrophic relationship of a model of anaerobic digestion process. Mathematical Biosciences & Engineering, 2010, 7 (3) : 641-656. doi: 10.3934/mbe.2010.7.641 |
[4] |
Donald L. DeAngelis, Joel C. Trexler, Douglas D. Donalson. Food web dynamics in a seasonally varying wetland. Mathematical Biosciences & Engineering, 2008, 5 (4) : 877-887. doi: 10.3934/mbe.2008.5.877 |
[5] |
Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244 |
[6] |
Bachir Bar, Tewfik Sari. The operating diagram for a model of competition in a chemostat with an external lethal inhibitor. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2093-2120. doi: 10.3934/dcdsb.2019203 |
[7] |
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 |
[8] |
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 |
[9] |
Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665 |
[10] |
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 |
[11] |
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 |
[12] |
Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319 |
[13] |
Rafael Labarca, Solange Aranzubia. A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1745-1776. doi: 10.3934/dcds.2018072 |
[14] |
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 |
[15] |
Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823 |
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