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doi: 10.3934/dcdsb.2022049
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Mortality can produce limit cycles in density-dependent models with a predator-prey relationship

Tahani Mtar a, , Radhouane Fekih-Salem a,c,, and  Tewfik Sari b,

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

* Corresponding author: Radhouane Fekih-Salem

Received  July 2021 Revised  January 2022 Early access March 2022

Fund Project: Supported by the UNESCO ICIREWARD project ANUMAB and the Euro-Mediterranean research network TREASURE (http://www.inrae.fr/treasure).

We study an interspecific, density-dependent model of two species competing for a single nutrient in a chemostat, allowing for a predator-prey relationship between them. We have previously examined the system in the absence of species mortality, showing that multiple positive steady states can appear and disappear through a saddle-node or transcritical bifurcation. In this paper we include mortality. We give a complete analysis for the existence and local stability of all steady states of the three-dimensional system that cannot be reduced to two dimensional ones. Specializing the forms of the rate functions, we show how mortality destabilizes the positive steady state and that stable limit cycles emerge through supercritical Hopf bifurcations. To describe how the process behaves with respect to the choice of dilution rate and input concentration as control parameters, we determine the operating diagram theoretically and also numerically by using the software package MATCONT. The bifurcation diagram based on the input concentration shows various types of bifurcations of steady states, and coexistence either at a positive steady state or via sustained oscillations.

Keywords: Density-dependence, Hopf Bifurcation, limit cycle, mortality, operating diagram, predator-prey relationship.
Mathematics Subject Classification: Primary: 34A34, 34D20; Secondary: 92B05, 92D25.
Citation: Tahani Mtar, Radhouane Fekih-Salem, Tewfik Sari. Mortality can produce limit cycles in density-dependent models with a predator-prey relationship. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022049
References:
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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 Contin. Dyn. Syst. Ser. 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 Contin. Dyn. Syst. Ser. B, 25 (2020), 2093-2120.  doi: 10.3934/dcdsb.2019203.

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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.

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M. Dali-Youcef, A. Rapaport and T. Sari, Study of performance criteria of serial configuration of two chemostats, Math. Biosci. Eng., 17 (2020), 6278-6309.  doi: 10.3934/mbe.2020332.

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R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, J. Math. Anal. Appl., 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055.

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R. Fekih-Salem, C. Lobry and T. Sari, A density-dependent model of competition for one resource in the chemostat, Math. Biosc., 268 (2017), 104-122.  doi: 10.1016/j.mbs.2017.02.007.

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R. Fekih-Salem, A. Rapaport and T. Sari, Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses, Appl. Math. Modell., 40 (2016), 7656-7677.  doi: 10.1016/j.apm.2016.03.028.

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R. Fekih-Salem and T. Sari, Properties of the chemostat model with aggregated biomass and distinct removal rates, SIAM J. Appl. Dyn. Syst. (SIADS), 18 (2019), 481-509.  doi: 10.1137/18M1171801.

[20]

R. Fekih-Salem and T. Sari, Operating diagram of a flocculation model in the chemostat, ARIMA J., 31 (2020), 45-58.  doi: 10.46298/arima.5593.

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B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2 (2008), 1-13.  doi: 10.1080/17513750801942537.

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M. Hanaki, J. Harmand, Z. Mghazli, A. Rapaport, T. Sari and P. Ugalde, Mathematical study of a two-stage anaerobic model when the hydrolysis is the limiting step, Processes, 9 (2021). doi: 10.3390/pr9112050.

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J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Chemostat and bioprocesses set., Vol. 1. ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2017. doi: 10.1002/9781119437215.

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S.-B. Hsu, C. A. Klausmeier and C.-J. Lin, Analysis of a model of two parallel food chains, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 337-359.  doi: 10.3934/dcdsb.2009.12.337.

[27]

Z. Khedim, B. Benyahia, B. Cherki, T. Sari and J. Harmand, Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates, Appl. Math. Model., 61 (2018), 351-376.  doi: 10.1016/j.apm.2018.04.020.

[28]

B. W. Kooi and M. P. Boer, Chaotic behaviour of a predator-prey system in the chemostat, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 10 (2003), 259-272. 

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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.

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C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of n species in the presence of a single resource, C. R. Biol., 329 (2006), 40-46.  doi: 10.1016/j.crvi.2005.10.004.

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C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models, Electron. J. Diff. Equ., 125 (2007), 1-10. 

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C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 199-204.  doi: 10.1016/j.crma.2004.12.021.

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C. Lobry, A. Rapaport and F. Mazenc, Sur un modèle densité-dépendant de compétition pour une ressource, C. R. Biol., 329 (2006), 63-70.  doi: 10.1016/j.crvi.2005.11.004.

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T. Mtar, R. Fekih-Salem and T. Sari, Effect of the mortality on a density-dependent model with a predator-prey relationship, CARI'2020, Proceedings of the 15th African Conference on Research in Computer Science and Applied Mathematics, (2020).

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S. Nouaoura, N. Abdellatif, R. Fekih-Salem and T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, SIAM J. Appl. Math., 81 (2021), 1264-1286.  doi: 10.1137/20M1353897.

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S. Nouaoura, R. Fekih-Salem, N. Abdellatif and T. Sari, Mathematical analysis of a three-tiered food-web in the chemostat, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5601-5625.  doi: 10.3934/dcdsb.2020369.

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S. Nouaoura, R. Fekih-Salem, N. Abdellatif and T. Sari, Operating diagrams for a three-tiered microbial food web in the chemostat, Preprint HAL, (2021).

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A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.

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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 Contin. Dyn. Syst. Ser. 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 Contin. Dyn. Syst. Ser. B, 25 (2020), 2093-2120.  doi: 10.3934/dcdsb.2019203.

[4]

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.

[5]

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.

[6]

M. P. Boer, B. W. Kooi and S. A. L. M. Kooijman, Food chain dynamics in the chemostat, Math. Biosci., 150 (1998), 43-62.  doi: 10.1016/S0025-5564(98)00010-8.

[7]

F. Borsali and K. Yadi, Contribution to the study of the effect of the interspecificity on a two nutrients competition model, Int. J. Biomath., 8 (2015), 1550008, 17 pp. doi: 10.1142/S1793524515500084.

[8]

M. Dali-Youcef, A. Rapaport and T. Sari, Study of performance criteria of serial configuration of two chemostats, Math. Biosci. Eng., 17 (2020), 6278-6309.  doi: 10.3934/mbe.2020332.

[9]

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), Paper No. 31, 21 pp. doi: 10.1051/mmnp/2018037.

[10]

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.

[11]

M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1129-1148.  doi: 10.3934/dcdsb.2020156.

[12]

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.

[13]

A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.  doi: 10.1080/13873950701742754.

[14]

M. El-Hajji, How can inter-specific interferences explain coexistence or confirm the competitive exclusion principle in a chemostat?, Int. J. Biomath., 11 (2018), 1850111, 20 pp. doi: 10.1142/S1793524518501115.

[15]

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.

[16]

R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, J. Math. Anal. Appl., 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055.

[17]

R. Fekih-Salem, C. Lobry and T. Sari, A density-dependent model of competition for one resource in the chemostat, Math. Biosc., 268 (2017), 104-122.  doi: 10.1016/j.mbs.2017.02.007.

[18]

R. Fekih-Salem, A. Rapaport and T. Sari, Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses, Appl. Math. Modell., 40 (2016), 7656-7677.  doi: 10.1016/j.apm.2016.03.028.

[19]

R. Fekih-Salem and T. Sari, Properties of the chemostat model with aggregated biomass and distinct removal rates, SIAM J. Appl. Dyn. Syst. (SIADS), 18 (2019), 481-509.  doi: 10.1137/18M1171801.

[20]

R. Fekih-Salem and T. Sari, Operating diagram of a flocculation model in the chemostat, ARIMA J., 31 (2020), 45-58.  doi: 10.46298/arima.5593.

[21]

B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2 (2008), 1-13.  doi: 10.1080/17513750801942537.

[22]

M. Hanaki, J. Harmand, Z. Mghazli, A. Rapaport, T. Sari and P. Ugalde, Mathematical study of a two-stage anaerobic model when the hydrolysis is the limiting step, Processes, 9 (2021). doi: 10.3390/pr9112050.

[23]

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.

[24]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Chemostat and bioprocesses set., Vol. 1. ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2017. doi: 10.1002/9781119437215.

[25]

J. Harmand, A. Rapaport, D. Dochain and C. Lobry, Microbial ecology and bioprocess control: Opportunities and challenges, Journal of Process Control, 18 (2008), 865-875.  doi: 10.1016/j.jprocont.2008.06.017.

[26]

S.-B. Hsu, C. A. Klausmeier and C.-J. Lin, Analysis of a model of two parallel food chains, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 337-359.  doi: 10.3934/dcdsb.2009.12.337.

[27]

Z. Khedim, B. Benyahia, B. Cherki, T. Sari and J. Harmand, Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates, Appl. Math. Model., 61 (2018), 351-376.  doi: 10.1016/j.apm.2018.04.020.

[28]

B. W. Kooi and M. P. Boer, Chaotic behaviour of a predator-prey system in the chemostat, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 10 (2003), 259-272. 

[29]

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.

[30]

C. Lobry and J. Harmand, A new hypothesis to explain the coexistence of n species in the presence of a single resource, C. R. Biol., 329 (2006), 40-46.  doi: 10.1016/j.crvi.2005.10.004.

[31]

C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models, Electron. J. Diff. Equ., 125 (2007), 1-10. 

[32]

C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 199-204.  doi: 10.1016/j.crma.2004.12.021.

[33]

C. Lobry, A. Rapaport and F. Mazenc, Sur un modèle densité-dépendant de compétition pour une ressource, C. R. Biol., 329 (2006), 63-70.  doi: 10.1016/j.crvi.2005.11.004.

[34]

MAPLE [Software], Version 13.0, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario, (2009).

[35]

MATCONT [Software], Matcontsoft, (2021).

[36]

T. Mtar, R. Fekih-Salem and T. Sari, Interspecific density-dependent model of predator-prey relationship in the chemostat, Int. J. Biomath., 14 (2021), 2050086, 22 pp. doi: 10.1142/S1793524520500862.

[37]

T. Mtar, R. Fekih-Salem and T. Sari, Effect of the mortality on a density-dependent model with a predator-prey relationship, CARI'2020, Proceedings of the 15th African Conference on Research in Computer Science and Applied Mathematics, (2020).

[38]

S. Nouaoura, N. Abdellatif, R. Fekih-Salem and T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, SIAM J. Appl. Math., 81 (2021), 1264-1286.  doi: 10.1137/20M1353897.

[39]

S. Nouaoura, R. Fekih-Salem, N. Abdellatif and T. Sari, Mathematical analysis of a three-tiered food-web in the chemostat, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5601-5625.  doi: 10.3934/dcdsb.2020369.

[40]

S. Nouaoura, R. Fekih-Salem, N. Abdellatif and T. Sari, Operating diagrams for a three-tiered microbial food web in the chemostat, Preprint HAL, (2021).

[41]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.

[42]

T. Sari and B. Benyahia, The operating diagram for a two-step anaerobic digestion model, Nonlinear Dynam., 105 (2021), 2711-2737.  doi: 10.1007/s11071-021-06722-7.

[43]

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Figure 1.  Existence of solutions of equation $ \widetilde{f}_2(x_1,0) = 0 $. The color in this figure and all other Figures is available online only
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Figure 2.  Number of positive steady states: (a) $ \mbox{Case 1} $: no positive steady state when $ (S_{in}, D) = (0.26,0.1) $, (b) $ \mbox{Case 2} $: an odd number when $ (S_{in}, D) = (0.35,0.1) $, $ {} (c) $ $ \mbox{Case 3} $: an even number when $ (S_{in}, D) = (4.5,0.8) $
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Figure 3.  Curves of the function $ C_3(S) $ for different values of $ D $ when $ S>\max(\lambda_1(D),\bar{\lambda}_2(D)) $
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Figure 4.  (a) Steady-state characteristics describing the local asymptotic behavior of the positive steady state $ \mathcal{E}^* $ when $ D = 0.25 $. Magnification for (b) $ S \leqslant 2 $ and (c) $ 0.32 \leqslant S \leqslant 0.36 $
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Figure 5.  Case $ D = 0.25 \leqslant D^* \simeq 0.2648 $: (a) Change of sign of $ C_4 $ when $ S_1\simeq 0.3299 $ (or equivalently $ \sigma_3 \simeq 0.5255 $), $ S_2\simeq 0.3423 $ (or equivalently $ \sigma_4 \simeq 0.7159 $) and $ S_3\simeq 1.4365 $ (or equivalently $ \sigma_5 \simeq 12.4809 $). (b) Magnification for $ S \in [0.316,1.7] $ and (c) magnification for $ S \in [0.316,0.361] $
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Figure 6.  Operating diagram of (1) in MAPLE. (b)-(e) Magnification when $ (S_{in},D)\in [0,2.6]\times[0,0.3] $. (c)-(f) Magnification when $ (S_{in},D)\in [0,0.6]\times[0,0.3] $
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Figure 7.  Operating diagram of (1) in MATCONT. (b) Magnification when $ (S_{in},D)\in [0,2.6]\times[0,0.3] $. (c) Magnification when $ (S_{in},D)\in [0,0.6]\times[0,0.3] $
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Figure 8.  Operating diagram of (1) showing the disappearance of the region $ \mathcal{J}_4 $ when $ a_1 $ and $ a_2 $ diminish
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Figure 9.  Scilab simulation showing projections of the $ \omega $-limit set in variable $ S $ when $ D = 0.25 $: (a) emergence and the disappearance of limit cycle at $ \sigma_3 $ and $ \sigma_4 $ for $ S_{in}\in[0.3,0.8] $; (b) emergence of limit cycle at $ \sigma_5 $ for $ S_{in}\in[0.8,30] $
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Figure 10.  (a) Variation of the pair of complex-conjugate eigenvalues (33) as $ S_{in} $ increases from 0 to 40 when $ D = 0.25 $. (b) Magnification on $ \lambda_{\pm} $ for $ S_{in}\in [0.4,0.8] $
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Figure 11.  Case $ S_{in} = 0.5 \in (\sigma_2,\sigma_3) $ and $ D = 0.25 $: convergence to $ \mathcal{E}^* $
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Figure 12.  Case $ S_{in} = 0.6 \in (\sigma_3,\sigma_4) $ and $ D = 0.25 $: convergence towards a stable limit cycle (in red)
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Figure 13.  Case $ S_{in} = 15 >\sigma_5 $ and $ D = 0.25 $: convergence to a stable limit cycle showing the sustained oscillations
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Table 1.  Necessary and sufficient conditions of existence and stability of steady states of system (1) where $ c_4 $ is defined by (20)
Existence Local stability
$ \mathcal{E}_0 $ always exists $ S_{in}<\lambda_1(D) $
$ \mathcal{E}_1 $ $ S_{in}>\lambda_1(D) $ $ S_{in}< \varphi(D) $
$ \mathcal{E}^* $ (16) has a solution $ F_2^{'}\left(x_1^*\right)>F_1^{'}\left(x_1^*\right) $ and $ c_4(S_{in},D)>0 $
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Existence Local stability
$ \mathcal{E}_0 $ always exists $ S_{in}<\lambda_1(D) $
$ \mathcal{E}_1 $ $ S_{in}>\lambda_1(D) $ $ S_{in}< \varphi(D) $
$ \mathcal{E}^* $ (16) has a solution $ F_2^{'}\left(x_1^*\right)>F_1^{'}\left(x_1^*\right) $ and $ c_4(S_{in},D)>0 $
Table 2.  The set $ \Upsilon $ and the corresponding colors in Figs. 6 and 7 where $ \varphi(D) $ and $ c_4 $ are defined by (27) and (20), resp
$ \Upsilon $ Color
$ \Upsilon_1=\left\{ (S_{in},D): S_{in} = \lambda_1(D) \right\} $ Black
$ \Upsilon_2=\left\{ (S_{in},D): S_{in} = \varphi(D) \right\} $ Blue
$ \Upsilon_3:=\left\{(S_{in},D): S_{in}=S_{in}^{\rm SN}(D) \right\} $
$ \Upsilon_4=\left\{ (S_{in}, D): c_4(S_{in}, D)=0 \right\} $ Green
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$ \Upsilon $ Color
$ \Upsilon_1=\left\{ (S_{in},D): S_{in} = \lambda_1(D) \right\} $ Black
$ \Upsilon_2=\left\{ (S_{in},D): S_{in} = \varphi(D) \right\} $ Blue
$ \Upsilon_3:=\left\{(S_{in},D): S_{in}=S_{in}^{\rm SN}(D) \right\} $
$ \Upsilon_4=\left\{ (S_{in}, D): c_4(S_{in}, D)=0 \right\} $ Green
Table 3.  Existence and stability of steady states according to regions in the operating diagrams of Figs. 6 and 7. The letter S (resp. U) means stable (resp. unstable) steady state. Absence of letter means that the corresponding steady state does not exist
Condition Region Color $ \mathcal{E}_0 $ $ \mathcal{E}_1 $ $ \mathcal{E}^* $
$ S_{in}<\lambda_1(D) $ $ \mathcal{J}_1 $ Cyan S
$ \lambda_1(D)<S_{in}<\varphi(D) $ $ \mathcal{J}_2 $ Pink U S
$ \varphi(D)<S_{in} $ and $ c_4(S_{in}, D)>0 $ $ \mathcal{J}_3 $ Grey U U S
$ \varphi(D)<S_{in} $ and $ c_4(S_{in}, D)<0 $ $ \mathcal{J}_4 $ Yellow U U U
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Condition Region Color $ \mathcal{E}_0 $ $ \mathcal{E}_1 $ $ \mathcal{E}^* $
$ S_{in}<\lambda_1(D) $ $ \mathcal{J}_1 $ Cyan S
$ \lambda_1(D)<S_{in}<\varphi(D) $ $ \mathcal{J}_2 $ Pink U S
$ \varphi(D)<S_{in} $ and $ c_4(S_{in}, D)>0 $ $ \mathcal{J}_3 $ Grey U U S
$ \varphi(D)<S_{in} $ and $ c_4(S_{in}, D)<0 $ $ \mathcal{J}_4 $ Yellow U U U
Table 4.  Nature of bifurcations of the steady states of (1) by crossing to the surfaces of $ \Upsilon $. The letter TB (resp. SHB) means a transcritical bifurcation (resp. Supercritical Hopf bifurcation)
Subset Transition Bifurcation
$ \Upsilon_1 $ $ \mathcal{J}_1 $ to $ \mathcal{J}_2 $ TB: $ \mathcal{E}_0=\mathcal{E}_1 $
$ \Upsilon_2 $ $ \mathcal{J}_2 $ to $ \mathcal{J}_3 $ TB: $ \mathcal{E}_1=\mathcal{E}^* $
$ \Upsilon_4 $ $ \mathcal{J}_3 $ to $ \mathcal{J}_4 $ SHB: $ \mathcal{E}^* $
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Subset Transition Bifurcation
$ \Upsilon_1 $ $ \mathcal{J}_1 $ to $ \mathcal{J}_2 $ TB: $ \mathcal{E}_0=\mathcal{E}_1 $
$ \Upsilon_2 $ $ \mathcal{J}_2 $ to $ \mathcal{J}_3 $ TB: $ \mathcal{E}_1=\mathcal{E}^* $
$ \Upsilon_4 $ $ \mathcal{J}_3 $ to $ \mathcal{J}_4 $ SHB: $ \mathcal{E}^* $
Table 5.  Definitions of the critical values $ \sigma_i $, $ i = 1,\ldots,5 $ of $ S_{in} $ and the corresponding nature of bifurcations when $ D = 0.25 $
Definition Value Bifurcation
$ \sigma_1=\lambda_1(D) $ 0.31884 TB
$ \sigma_2=\varphi(D) $ 0.35394 TB
$ \sigma_3 $ is the first solution of equation $ c_4(S_{in})=0 $ 0.52555 SHB
$ \sigma_4 $ is the second solution of equation $ c_4(S_{in})=0 $ 0.71593 SHB
$ \sigma_5 $ is the third solution of equation $ c_4(S_{in})=0 $ 12.4809 SHB
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Definition Value Bifurcation
$ \sigma_1=\lambda_1(D) $ 0.31884 TB
$ \sigma_2=\varphi(D) $ 0.35394 TB
$ \sigma_3 $ is the first solution of equation $ c_4(S_{in})=0 $ 0.52555 SHB
$ \sigma_4 $ is the second solution of equation $ c_4(S_{in})=0 $ 0.71593 SHB
$ \sigma_5 $ is the third solution of equation $ c_4(S_{in})=0 $ 12.4809 SHB
Table 6.  Existence and stability of steady states according to $ S_{in} $
Interval of $ S_{in} $ $ \mathcal{E}_0 $ $ \mathcal{E}_1 $ $ \mathcal{E}^* $
$ (0,\sigma_1) $ S
$ (\sigma_1,\sigma_2) $ U S
$ (\sigma_2,\sigma_3) $ U U S
$ (\sigma_3,\sigma_4) $ U U U
$ (\sigma_4,\sigma_5) $ U U S
$ (\sigma_5,+\infty) $ U U U
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Interval of $ S_{in} $ $ \mathcal{E}_0 $ $ \mathcal{E}_1 $ $ \mathcal{E}^* $
$ (0,\sigma_1) $ S
$ (\sigma_1,\sigma_2) $ U S
$ (\sigma_2,\sigma_3) $ U U S
$ (\sigma_3,\sigma_4) $ U U U
$ (\sigma_4,\sigma_5) $ U U S
$ (\sigma_5,+\infty) $ U U U
Table 7.  Parameter values used for model (1) when the growth rates $ f_1 $ and $ f_2 $ are given by (31).
Parameter $ m_1 $ $ K_1 $ $ L_1 $ $ m_2 $ $ K_2 $ $ L_2 $ $ \alpha_1 $ $ \alpha_2 $ $ a_1 $ $ a_2 $
Fig. 2(c) 2.75 2 1.2 2.95 1.8 1.5 $ 10^{-3} $ 0.1 0.95 0.7
Figs. 1, 2(a, b), 6, 7, 9–13 0.3 0.2
Fig. 8(a) 4 2 3 8 0.1 0.2 1 1 0.3 0.05
Fig. 8(b) 0.1 0.05
Fig. 8(c) 0 0
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Parameter $ m_1 $ $ K_1 $ $ L_1 $ $ m_2 $ $ K_2 $ $ L_2 $ $ \alpha_1 $ $ \alpha_2 $ $ a_1 $ $ a_2 $
Fig. 2(c) 2.75 2 1.2 2.95 1.8 1.5 $ 10^{-3} $ 0.1 0.95 0.7
Figs. 1, 2(a, b), 6, 7, 9–13 0.3 0.2
Fig. 8(a) 4 2 3 8 0.1 0.2 1 1 0.3 0.05
Fig. 8(b) 0.1 0.05
Fig. 8(c) 0 0
Table 8.  Break-even concentration, notations and auxiliary functions
λ1(D) S = λ1(D) is the solution of equation f1(S, 0)=α1D+a1
It is defined for D < , (f1(+∞, 0) − a1)=α1, see(4)
$\tilde{x}_{1} $ $\tilde{x}_{1}=\frac{D}{D_{1}}\left(S_{i n}-\lambda_{1}(D)\right), \text { see }(5)$
$\widetilde{f}_{1}\left(x_{1}, x_{2}\right)$ $\widetilde{f}_{1}\left(x_{1}, x_{2}\right)=f_{1}\left(S_{i n}-\frac{D_{1}}{D} x_{1}-\frac{D_{2}}{D} x_{2}, x_{2}\right)-D_{1}, \text { see }(11) $
$\widetilde{f}_{2}\left(x_{1}, x_{2}\right)$ $\widetilde{f}_{2}\left(x_{1}, x_{2}\right)=f_{2}\left(S_{i n}-\frac{D_{1}}{D} x_{1}-\frac{D_{2}}{D} x_{2}, x_{1}\right)-D_{2}, \text { see }(11)$
F1(x1) x2 = F1(x1) is the unique solution of equation $\widetilde{f}_{1}$(x1, x2) = 0
It is defined for 0 ≤ x1 ≤ $\tilde{x}_{1} $, see Lemma 1
F2(x1) x2 = F2(x1) is the unique solution of equation $\widetilde{f}_{2}$ (x1, x2) = 0
It is defined for $x_{1}^{1}$ ≤ x1 ≤ $x_{1}^{2}$, where $x_{1}^{1}$ and $x_{1}^{2}$ are
the solutions of equation $\widetilde{f}_{2}$(x1, 0) = 0, see Lemma 3
$\left(x_{1}^{*}, x_{2}^{*}\right)$ $\left(x_{1}^{*}, x_{2}^{*}\right)$ is a solution of x2 = F1(x1) = F2(x1), see Prop. 3
X1(S, D) x1 = X1(S, D) is the solution of equation f2(S, x1) = α2D + a2
It is defined for S > λ2(D), where λ2(D) is the unique solution,
if it exists, of equation f2(S, +∞) = α2D + a2, see (26)
X2(S, D) x2 = X2(S, D) is the solution of equation f1(S, x2) = α1D + a1
It is defined for λ1(D) ≤ S < λ1(D), where λ1(D) is the unique solution,
if it exists, of equation f1(S, +∞) = α1D + a1, see (25)
$\varphi$(D) $\varphi(D)=\lambda_{1}(D)+\frac{D_{1}}{D} X_{1}\left(\lambda_{1}(D), D\right), \text { see }(27)$
K(S, D) K(S, D) = D1X1(S, D) + D2X2(S, D), see (30)
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λ1(D) S = λ1(D) is the solution of equation f1(S, 0)=α1D+a1
It is defined for D < , (f1(+∞, 0) − a1)=α1, see(4)
$\tilde{x}_{1} $ $\tilde{x}_{1}=\frac{D}{D_{1}}\left(S_{i n}-\lambda_{1}(D)\right), \text { see }(5)$
$\widetilde{f}_{1}\left(x_{1}, x_{2}\right)$ $\widetilde{f}_{1}\left(x_{1}, x_{2}\right)=f_{1}\left(S_{i n}-\frac{D_{1}}{D} x_{1}-\frac{D_{2}}{D} x_{2}, x_{2}\right)-D_{1}, \text { see }(11) $
$\widetilde{f}_{2}\left(x_{1}, x_{2}\right)$ $\widetilde{f}_{2}\left(x_{1}, x_{2}\right)=f_{2}\left(S_{i n}-\frac{D_{1}}{D} x_{1}-\frac{D_{2}}{D} x_{2}, x_{1}\right)-D_{2}, \text { see }(11)$
F1(x1) x2 = F1(x1) is the unique solution of equation $\widetilde{f}_{1}$(x1, x2) = 0
It is defined for 0 ≤ x1 ≤ $\tilde{x}_{1} $, see Lemma 1
F2(x1) x2 = F2(x1) is the unique solution of equation $\widetilde{f}_{2}$ (x1, x2) = 0
It is defined for $x_{1}^{1}$ ≤ x1 ≤ $x_{1}^{2}$, where $x_{1}^{1}$ and $x_{1}^{2}$ are
the solutions of equation $\widetilde{f}_{2}$(x1, 0) = 0, see Lemma 3
$\left(x_{1}^{*}, x_{2}^{*}\right)$ $\left(x_{1}^{*}, x_{2}^{*}\right)$ is a solution of x2 = F1(x1) = F2(x1), see Prop. 3
X1(S, D) x1 = X1(S, D) is the solution of equation f2(S, x1) = α2D + a2
It is defined for S > λ2(D), where λ2(D) is the unique solution,
if it exists, of equation f2(S, +∞) = α2D + a2, see (26)
X2(S, D) x2 = X2(S, D) is the solution of equation f1(S, x2) = α1D + a1
It is defined for λ1(D) ≤ S < λ1(D), where λ1(D) is the unique solution,
if it exists, of equation f1(S, +∞) = α1D + a1, see (25)
$\varphi$(D) $\varphi(D)=\lambda_{1}(D)+\frac{D_{1}}{D} X_{1}\left(\lambda_{1}(D), D\right), \text { see }(27)$
K(S, D) K(S, D) = D1X1(S, D) + D2X2(S, D), see (30)
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