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doi: 10.3934/dcdsb.2021120

A competition model in the chemostat with allelopathy and substrate inhibition

Mohamed Dellal a,c,, , Bachir Bar b,c, and  Mustapha Lakrib c,

a. 

Ibn Khaldoun University, 14000 Tiaret, Algeria
b. 

Ecole Normale Supérieure, 27000 Mostaganem, Algeria
c. 

LDM, Djillali Liabès University, 22000 Sidi Bel Abbès, Algeria

* Corresponding author: Mohamed Dellal

Received  August 2020 Revised  February 2021 Published  April 2021

A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a super-critical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. This diagram is very useful to understand the model from both the mathematical and biological points of view.

Keywords: Chemostat, inhibitor, stability, Hopf bifurcation, operating diagram.
Mathematics Subject Classification: Primary: 34C23, 34D20; Secondary: 92B05, 92D25.
Citation: Mohamed Dellal, Bachir Bar, Mustapha Lakrib. A competition model in the chemostat with allelopathy and substrate inhibition. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021120
References:
[1]

N. Abdellatif, R. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Mathematical Biosciences and Engineering, 13 (2016), 631-652.  doi: 10.3934/mbe.2016012.  Google Scholar

[2]

J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.  doi: 10.1002/bit.260100602.  Google Scholar

[3]

B. Bar and T. Sari, The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete and Continuous Dynamical Systems–B, 25 (2020), 2093-2120.  doi: 10.3934/dcdsb.2019203.  Google Scholar

[4]

G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.  doi: 10.1137/0145006.  Google Scholar

[5]

M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete and Continuous Dynamical Systems–B, 26 (2021), 1129-1148.  doi: 10.3934/dcdsb.2020156.  Google Scholar

[6]

M. Dellal, M. Lakrib and T. Sari, The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Mathematical Biosciences, 302 (2018), 27-45.  doi: 10.1016/j.mbs.2018.05.004.  Google Scholar

[7]

R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055.  Google Scholar

[8]

P. Fergola, M. Cerasuolo, A. Pollio, G. Pinto and M. Della Grecac., Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecological Modelling, 208 (2007), 205-214.  doi: 10.1016/j.ecolmodel.2007.05.024.  Google Scholar

[9]

P. Fergola, J. Li and Z. Ma, On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ricerche di Matematica, 60 (2011), 313-332.  doi: 10.1007/s11587-011-0108-y.  Google Scholar

[10]

H. Fgaier, M. Kalmokoff, T. Ells and H. J. Eberl, An allelopathy based model for the Listeria overgrowth phenomenon, Mathematical Biosciences, 247 (2014), 13-26.  doi: 10.1016/j.mbs.2013.10.008.  Google Scholar

[11]

G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, (1934). Google Scholar

[12]

B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, 2 (2008), 1-13.  doi: 10.1080/17513750801942537.  Google Scholar

[13]

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.  Google Scholar

[14]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[15]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Vol. 1, ISTE, London, John Wiley and Sons, Inc. Hoboken, NJ, 2017.  Google Scholar

[16]

J. Heßeler, J. K. Schmidt, U. Reichl and D. Flockerzi, Coexistence in the chemostat as a result of metabolic by-products, Journal of Mathematical Biology, 53 (2006), 556-584.  doi: 10.1007/s00285-006-0012-3.  Google Scholar

[17]

S. B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[18]

S. B. Hsu, T. K. Luo and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor, Journal of Mathematical Biology, 34 (1995), 225-238.  doi: 10.1007/BF00178774.  Google Scholar

[19]

S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan Journal of Industrial and Applied Mathematics, 15 (1998), 471-490.  doi: 10.1007/BF03167323.  Google Scholar

[20]

Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 3$^rd$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[21]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

[22]

R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics, Journal of Theoretical Biology, 122 (1986), 83-93.  doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar

[23]

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

[24]

I. P. Martines, H. V. Kojouharov and J. P. Grover, A chemostat model of resource competition and allelopathy, Applied Mathematics and Computation, 215 (2009), 573-582.  doi: 10.1016/j.amc.2009.05.033.  Google Scholar

[25]

S. Pavlou, Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.  doi: 10.1016/S0168-1656(99)00011-5.  Google Scholar

[26]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827.  Google Scholar

[27]

M. Scheffer, S. Rinaldi, J. Huisman and F. J. Weissing, Why plankton communities have no equilibrium: Solutions to the paradox, Hydrobiologia, 491 (2003), 9-18.  doi: 10.1023/A:1024404804748.  Google Scholar

[28]

H. L. Smith and B. Tang, Competition in the gradostat: The role of the communication rate, Journal of Mathematical Biology, 27 (1989), 139-165.  doi: 10.1007/BF00276100.  Google Scholar

[29] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[30]

S. Sobieszek, G. S. K. Wolkowicz and M. J. Wade, Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Mathematical Biosciences and Engineering, 17 (2020), 7045-7073.  doi: 10.3934/mbe.2020363.  Google Scholar

[31]

M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecological Modelling, 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002.  Google Scholar

[32]

M. Weedermann, G. Seo and G. Wolkowicz, Mathematical model of anaerobic digestion in a chemostat: Effects of syntrophy and inhibition, Journal of Biological Dynamics, 7 (2013), 59-85.  doi: 10.1080/17513758.2012.755573.  Google Scholar

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, Mathematical Biosciences and Engineering, 13 (2016), 631-652.  doi: 10.3934/mbe.2016012.  Google Scholar

[2]

J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.  doi: 10.1002/bit.260100602.  Google Scholar

[3]

B. Bar and T. Sari, The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete and Continuous Dynamical Systems–B, 25 (2020), 2093-2120.  doi: 10.3934/dcdsb.2019203.  Google Scholar

[4]

G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.  doi: 10.1137/0145006.  Google Scholar

[5]

M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete and Continuous Dynamical Systems–B, 26 (2021), 1129-1148.  doi: 10.3934/dcdsb.2020156.  Google Scholar

[6]

M. Dellal, M. Lakrib and T. Sari, The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Mathematical Biosciences, 302 (2018), 27-45.  doi: 10.1016/j.mbs.2018.05.004.  Google Scholar

[7]

R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari, Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.  doi: 10.1016/j.jmaa.2012.07.055.  Google Scholar

[8]

P. Fergola, M. Cerasuolo, A. Pollio, G. Pinto and M. Della Grecac., Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecological Modelling, 208 (2007), 205-214.  doi: 10.1016/j.ecolmodel.2007.05.024.  Google Scholar

[9]

P. Fergola, J. Li and Z. Ma, On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ricerche di Matematica, 60 (2011), 313-332.  doi: 10.1007/s11587-011-0108-y.  Google Scholar

[10]

H. Fgaier, M. Kalmokoff, T. Ells and H. J. Eberl, An allelopathy based model for the Listeria overgrowth phenomenon, Mathematical Biosciences, 247 (2014), 13-26.  doi: 10.1016/j.mbs.2013.10.008.  Google Scholar

[11]

G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, (1934). Google Scholar

[12]

B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, 2 (2008), 1-13.  doi: 10.1080/17513750801942537.  Google Scholar

[13]

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.  Google Scholar

[14]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[15]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Vol. 1, ISTE, London, John Wiley and Sons, Inc. Hoboken, NJ, 2017.  Google Scholar

[16]

J. Heßeler, J. K. Schmidt, U. Reichl and D. Flockerzi, Coexistence in the chemostat as a result of metabolic by-products, Journal of Mathematical Biology, 53 (2006), 556-584.  doi: 10.1007/s00285-006-0012-3.  Google Scholar

[17]

S. B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[18]

S. B. Hsu, T. K. Luo and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor, Journal of Mathematical Biology, 34 (1995), 225-238.  doi: 10.1007/BF00178774.  Google Scholar

[19]

S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan Journal of Industrial and Applied Mathematics, 15 (1998), 471-490.  doi: 10.1007/BF03167323.  Google Scholar

[20]

Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 3$^rd$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[21]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

[22]

R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics, Journal of Theoretical Biology, 122 (1986), 83-93.  doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar

[23]

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

[24]

I. P. Martines, H. V. Kojouharov and J. P. Grover, A chemostat model of resource competition and allelopathy, Applied Mathematics and Computation, 215 (2009), 573-582.  doi: 10.1016/j.amc.2009.05.033.  Google Scholar

[25]

S. Pavlou, Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.  doi: 10.1016/S0168-1656(99)00011-5.  Google Scholar

[26]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827.  Google Scholar

[27]

M. Scheffer, S. Rinaldi, J. Huisman and F. J. Weissing, Why plankton communities have no equilibrium: Solutions to the paradox, Hydrobiologia, 491 (2003), 9-18.  doi: 10.1023/A:1024404804748.  Google Scholar

[28]

H. L. Smith and B. Tang, Competition in the gradostat: The role of the communication rate, Journal of Mathematical Biology, 27 (1989), 139-165.  doi: 10.1007/BF00276100.  Google Scholar

[29] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[30]

S. Sobieszek, G. S. K. Wolkowicz and M. J. Wade, Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Mathematical Biosciences and Engineering, 17 (2020), 7045-7073.  doi: 10.3934/mbe.2020363.  Google Scholar

[31]

M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecological Modelling, 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002.  Google Scholar

[32]

M. Weedermann, G. Seo and G. Wolkowicz, Mathematical model of anaerobic digestion in a chemostat: Effects of syntrophy and inhibition, Journal of Biological Dynamics, 7 (2013), 59-85.  doi: 10.1080/17513758.2012.755573.  Google Scholar

Figure 1.  Growth function and definitions of break-even concentrations (a): $ f_1 $ of Monod type; (b): $ f_2 $ of Haldane type
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Figure 2.  Graphs of $ f_1 $ (in red) and $ (1\!-\!k)f_2 $ (in blue) when equation $ f_1(S)\! = \!(1\!-\!k)f_2(S) $ has a positive solution $ S\! = \!\overline{S} $ and graphical depiction of $ I_{c_1} $ and $ I_{c_2} $. $ \rm(a) $: $ I_{c_1} = \left(\overline{D},(1\!-\!k)f_2(S_2^m)\right] $ and $ I_{c_2} = \left(0,(1\!-\!k)f_2(S_2^m)\right] $. $ \rm(b) $: $ I_{c_1} = \emptyset $ and $ I_{c_2} = \left(0,\overline{D}\right) $ where $ \overline{D} = f_1\left(\overline{S}\right) = f_2\left(\overline{S}\right) $. Intervals $ I_{c_1} $ and $ I_{c_2} $ are defined by (14)
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Figure 3.  Graphs of $ F_1 $ (in green) and $ F_2 $ (in magenta) when equation $ f_1(S)\! = \!(1\!-\!k)f_2(S) $ has a positive solution $ S\! = \!\overline{S} $
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Figure 4.  The graphs of $ F_3(x_{c_2}) $ and $ A(x_{c_2}) $, showing the relative positions of the roots $ x_i = x_i(D) $, $ i = 0,2 $, of $ F_3(x_{c_2}) $ with respect to the root $ x_0 = x_0(D) $ of $ A(x_{c_2}) $, when $ D\in I_3 $
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Figure 2. The curves $ \Gamma_i $, $ i = 0\cdots 9 $, defined in Table 3, separate the operating plane $ (D,S^0) $ into fifteen regions labeled $ \mathcal J_k $, $ k = 0...14 $. The existence and stability of equilibria $ E_0 $, $ E_1 $, $ E_2^j $ and $ E_c^j $ in the regions $ \mathcal J_0 $, $ \mathcal J_1 $, ...., $ \mathcal J_{14} $ of these diagrams are shown by Table 5">Figure 5.  Illustrative operating diagrams corresponding to cases (a) and (b) in Figure 2. The curves $ \Gamma_i $, $ i = 0\cdots 9 $, defined in Table 3, separate the operating plane $ (D,S^0) $ into fifteen regions labeled $ \mathcal J_k $, $ k = 0...14 $. The existence and stability of equilibria $ E_0 $, $ E_1 $, $ E_2^j $ and $ E_c^j $ in the regions $ \mathcal J_0 $, $ \mathcal J_1 $, ...., $ \mathcal J_{14} $ of these diagrams are shown by Table 5
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Table 6, Case 1">Figure 6.  Operating diagram with biological parmeters given in Table 6, Case 1
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Figure 7) creating a stable limit cycle. We use the color codes; Green: initial conditions, Red: local attractors and Blue: unstable equilibria">Figure 8.  (a): $ (S^0,D) = (15,0.53)\in \mathcal J_{12} $. In this case we have bi-stability of $ E_c^2 $ and $ E_2^1 $. (b): $ (S^0,D) = (15,0.51) \in\mathcal J_{11} $. In this case $ E_c^2 $ loses its stability through a super-critical Hopf bifurcation (see Figure 7) creating a stable limit cycle. We use the color codes; Green: initial conditions, Red: local attractors and Blue: unstable equilibria
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Table 6, Case 1, and $ S^0 = 15 $. (a): Variation of a pair of complex-conjugate eigenvalues. (b): The real part of the eigenvalues showing that its change of stability at $ D = D_{crit}\approx 0.521403 $ indicating a Hopf bifurcation">Figure 7.  Hopf bifurcation. Biological values are in Table 6, Case 1, and $ S^0 = 15 $. (a): Variation of a pair of complex-conjugate eigenvalues. (b): The real part of the eigenvalues showing that its change of stability at $ D = D_{crit}\approx 0.521403 $ indicating a Hopf bifurcation
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Figure 9.  Homoclinic bifurcation (a): $ (S^0,D) = (15,0.508105) $. After the Hopf bifurcation, the limit cycle gets larger. (b): $ (S^0,D) = (15,0.50) $. The limit cycle loses its stability (through homoclinic bifurcation) and the only attractor remaining is $ E_2^1 $. We use the color codes, Green: initial conditions, Red: local attractors and Blue: unstable equilibria
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Figure 10.  One parameter bifurcation diagram for the homoclinic bifurcation. We plot the projections of the $ \omega $-limit set in variables $ \{S,y\} $ for $ D\in[0.5,0.53] $, which reveals the emergence of limit cycle through a Hopf bifurcation and its disappearance through a homoclinic bifurcation. Solid line is for stable fixed point (dashed when unstable). H: Hopf bifurcation
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Table 6, Case 2. (b): A zoom of the operating diagram near regions $ \mathcal J_{13} $ and $ \mathcal J_{14} $">Figure 11.  (a): Operating diagram corresponding to Table 6, Case 2. (b): A zoom of the operating diagram near regions $ \mathcal J_{13} $ and $ \mathcal J_{14} $
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Figure 11(b)). In this case there is tri-stability of equilibria $ E_c^2 $, $ E_2^1 $ and $ E_1 $. (b): $ (D,S^0) = (0.25785,3.5)\in\mathcal J_{13} $ (see Figure 11(b)). Tri-stability of equilibria $ E_2^1 $, $ E_1 $ and a stable limit cycle">Figure 12.  Tri-stability (a): $ (D,S^0) = (0.26,3.5)\in\mathcal J_{14} $ (see Figure 11(b)). In this case there is tri-stability of equilibria $ E_c^2 $, $ E_2^1 $ and $ E_1 $. (b): $ (D,S^0) = (0.25785,3.5)\in\mathcal J_{13} $ (see Figure 11(b)). Tri-stability of equilibria $ E_2^1 $, $ E_1 $ and a stable limit cycle
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Table. 6 case 2, and $ S^0 = 5 $. (a): Saddle node bifurcation of $ E_2^1 $ and $ E_2^2 $. (b): Saddle node bifurcation of $ E_c^1 $ and $ E_c^2 $. Solid line is for stable fixed point; dashed when unstable. H: Hopf bifurcation. LP: Limit Point (Saddle-node). PD: Period Doubling">Figure 13.  One parameter bifurcation diagram. Biological values are in Table. 6 case 2, and $ S^0 = 5 $. (a): Saddle node bifurcation of $ E_2^1 $ and $ E_2^2 $. (b): Saddle node bifurcation of $ E_c^1 $ and $ E_c^2 $. Solid line is for stable fixed point; dashed when unstable. H: Hopf bifurcation. LP: Limit Point (Saddle-node). PD: Period Doubling
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Figure 14.  Bifurcation diagrams of the limit cycle. (a) Continuation of the limit cycle (we fix $ S^0 = 5 $ and plot the projection of the limit cycle on the $ (S,x) $ space as a function of $ D $). (b) Two parameter bifurcation of the limit cycle, the curves (in blue) correspond to the period doubling (flip bifurcation). Matcont was used to produce both of the diagrams. PD: Period Doubling, LPC: Limit Point Cycle
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Figure 15.  Period doubling (Flip-bifurcation) before and after $ D_2 $. (a): The limit cycle for $ (D,S^0) = (0.24845,5) $. (b): The limit cycle for $ (D,S^0) = (0.24835,5) $
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Table 1.  Existence and stability of equilibria of system (2) when $ \lambda_1 <S^0 $ and $ \lambda_2 <S^0 $. The letter S (resp. U) means stable (resp. unstable) and no letter means that the equilibrium does not exist
Case Condition Equilibria and nature
$ E_1 $ $ E_2^1 $ $ E_2^2 $ $ E_c^1 $ $ E_c^2 $
$ \mu_2>S^0 $ $ \lambda_1<\widehat{\lambda}<\lambda_2<\mu_2 $ S U
$ \lambda_1 <\lambda_2 <\widehat{\lambda} <\mu_2 $ S S U
$ \lambda_2 <\lambda_1 <\widehat{\lambda} <\mu_2 $ U S
$ \mu_2<S^0 $ $ \lambda_1 <\widehat{\lambda} <\lambda_2 <\mu_2 $ S U U
$ \lambda_1 <\lambda_2 <\widehat{\lambda} <\mu_2 $ S S U U
$ \lambda_2<\lambda_1<\widehat{\lambda}<\mu_2 $ U S U
$ \lambda_1<\lambda_2<\mu_2<\widehat{\lambda} $ & $ AB >C \ \hbox{and} \ A >0 $ S S U U S
$ \lambda_1<\lambda_2<\mu_2<\widehat{\lambda} $ & $ AB<C \ \hbox{or} \ A<0 $ S S U U U
$ \lambda_2<\lambda_1<\mu_2<\widehat{\lambda} $ & $ AB >C \ \hbox{and} \ A >0 $ U S U S
$ \lambda_2<\lambda_1<\mu_2<\widehat{\lambda} $ & $ AB< C \ \hbox{or} \ A<0 $ U S U U
$ \lambda_2<\mu_2<\lambda_1<\widehat{\lambda} $ S S U
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Case Condition Equilibria and nature
$ E_1 $ $ E_2^1 $ $ E_2^2 $ $ E_c^1 $ $ E_c^2 $
$ \mu_2>S^0 $ $ \lambda_1<\widehat{\lambda}<\lambda_2<\mu_2 $ S U
$ \lambda_1 <\lambda_2 <\widehat{\lambda} <\mu_2 $ S S U
$ \lambda_2 <\lambda_1 <\widehat{\lambda} <\mu_2 $ U S
$ \mu_2<S^0 $ $ \lambda_1 <\widehat{\lambda} <\lambda_2 <\mu_2 $ S U U
$ \lambda_1 <\lambda_2 <\widehat{\lambda} <\mu_2 $ S S U U
$ \lambda_2<\lambda_1<\widehat{\lambda}<\mu_2 $ U S U
$ \lambda_1<\lambda_2<\mu_2<\widehat{\lambda} $ & $ AB >C \ \hbox{and} \ A >0 $ S S U U S
$ \lambda_1<\lambda_2<\mu_2<\widehat{\lambda} $ & $ AB<C \ \hbox{or} \ A<0 $ S S U U U
$ \lambda_2<\lambda_1<\mu_2<\widehat{\lambda} $ & $ AB >C \ \hbox{and} \ A >0 $ U S U S
$ \lambda_2<\lambda_1<\mu_2<\widehat{\lambda} $ & $ AB< C \ \hbox{or} \ A<0 $ U S U U
$ \lambda_2<\mu_2<\lambda_1<\widehat{\lambda} $ S S U
Table 2.  Existence and stability of equilibria of system (2) with respect to the operating parameters
Equilibria Existence Local exponential stability
$ E_0 $ Always $D >\!\max(f_1(S^0),(1-k)f_2(S^0)) $
$ E_1 $ $ S^0>\lambda_1(D) $ $ \lambda_1(D)<\lambda_2(D) $ or $ \lambda_1(D)>\mu_2(D) $
$ E_2^1 $ $ S^0>\lambda_2(D) $ $ S^0>F_1(D) $
$ E_2^2 $ $ S^0>\mu_2(D) $ Unstable if it exists
$ E_c^1 $ $ \lambda_1(D)<\lambda_2(D)<S^0 $ & $ S^0>F_1(D) $ Unstable if it exists
$ E_c^2 $ $ \lambda_1(D)<\mu_2(D)<S^0 $ & $ S^0>F_2(D) $ $ F_3(D,S^0)>0 \ \hbox{and} \ A(D,S^0)>0 $
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Equilibria Existence Local exponential stability
$ E_0 $ Always $D >\!\max(f_1(S^0),(1-k)f_2(S^0)) $
$ E_1 $ $ S^0>\lambda_1(D) $ $ \lambda_1(D)<\lambda_2(D) $ or $ \lambda_1(D)>\mu_2(D) $
$ E_2^1 $ $ S^0>\lambda_2(D) $ $ S^0>F_1(D) $
$ E_2^2 $ $ S^0>\mu_2(D) $ Unstable if it exists
$ E_c^1 $ $ \lambda_1(D)<\lambda_2(D)<S^0 $ & $ S^0>F_1(D) $ Unstable if it exists
$ E_c^2 $ $ \lambda_1(D)<\mu_2(D)<S^0 $ & $ S^0>F_2(D) $ $ F_3(D,S^0)>0 \ \hbox{and} \ A(D,S^0)>0 $
Table 3.  Boundaries of the regions in the operating diagram
The curve $ \Gamma_i $, $ i=1...9 $ Boundary
$ \Gamma_1=\left\{(D,S^0):S^0=\lambda_1(D)\right\} $ is the border to which $ E_1 $ exists
$ \Gamma_2=\left\{(D,S^0):S^0=\lambda_2(D)\right\} $ is the border to which $ E_2^1 $ exists
$ \Gamma_3=\left\{(D,S^0):S^0=\mu_2(D)\right\} $ is the border to which $ E_2^2 $ exists
$ \Gamma_4=\left\{(D,S^0)\!:\lambda_1(D)\!=\!\lambda_2(D), S^0\!>\!\lambda_1(\!D)\right\} $ is the border to which $ E_1 $ is stable
and at the same time $ E_c^1 $ exists
$ \Gamma_5=\left\{(D,S^0)\!:\lambda_1(D)\!=\!\mu_2(D), S^0\!>\!\lambda_1(\!D)\right\} $ is the border to which $ E_1 $ is stable
and at the same time $ E_c^2 $ exists
$ \Gamma_6=\left\{(D,S^0):S^0=F_1(D), S^0>\lambda_2(D)\right\} $ is the border to which $ E_2^1 $ is stable
and at the same time $ E_c^1 $ exists
$ \Gamma_7=\left\{(D,S^0):S^0=F_2(D), S^0>\mu_2(D)\right\} $ is the border to which $ E_c^2 $ exists
$ \Gamma_8=\left\{(D,S^0):S^0=F_5(D) \right\} $ is the border to which $ E_c^2 $ is stable
$ \Gamma_9=\left\{(D,S^0)\!: \lambda_2(D)\!=\!\mu_2(D), S^0\!>\!\lambda_2(\!D) \right\} $ Horizontal line $ D=(\!1\!-\!k\!)f_2(S_2^m) $
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The curve $ \Gamma_i $, $ i=1...9 $ Boundary
$ \Gamma_1=\left\{(D,S^0):S^0=\lambda_1(D)\right\} $ is the border to which $ E_1 $ exists
$ \Gamma_2=\left\{(D,S^0):S^0=\lambda_2(D)\right\} $ is the border to which $ E_2^1 $ exists
$ \Gamma_3=\left\{(D,S^0):S^0=\mu_2(D)\right\} $ is the border to which $ E_2^2 $ exists
$ \Gamma_4=\left\{(D,S^0)\!:\lambda_1(D)\!=\!\lambda_2(D), S^0\!>\!\lambda_1(\!D)\right\} $ is the border to which $ E_1 $ is stable
and at the same time $ E_c^1 $ exists
$ \Gamma_5=\left\{(D,S^0)\!:\lambda_1(D)\!=\!\mu_2(D), S^0\!>\!\lambda_1(\!D)\right\} $ is the border to which $ E_1 $ is stable
and at the same time $ E_c^2 $ exists
$ \Gamma_6=\left\{(D,S^0):S^0=F_1(D), S^0>\lambda_2(D)\right\} $ is the border to which $ E_2^1 $ is stable
and at the same time $ E_c^1 $ exists
$ \Gamma_7=\left\{(D,S^0):S^0=F_2(D), S^0>\mu_2(D)\right\} $ is the border to which $ E_c^2 $ exists
$ \Gamma_8=\left\{(D,S^0):S^0=F_5(D) \right\} $ is the border to which $ E_c^2 $ is stable
$ \Gamma_9=\left\{(D,S^0)\!: \lambda_2(D)\!=\!\mu_2(D), S^0\!>\!\lambda_2(\!D) \right\} $ Horizontal line $ D=(\!1\!-\!k\!)f_2(S_2^m) $
Table 4.  Definitions of the regions $ \mathcal J_k $, $ k = 0...14 $, in the operating diagrams in Figures 5, Figure 6 and 11
Region Definition
$ \mathcal J_0 $ $ S^0<\lambda_1(D) $ and $ S^0<\lambda_2(D) $
$ \mathcal J_1 $ $ S^0<\lambda_1(D) $ and $ \lambda_2(D)<S^0<\mu_2(D) $
$ \mathcal J_2 $ $ S^0<\lambda_1(D) $ and $ S^0>\mu_2(D) $
$ \mathcal J_3 $ $ S^0>\lambda_1(D) $ and $ S^0<\lambda_2(D) $
$ \mathcal J_4 $ $ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $ and $ S^0<F_1(D) $
$ \mathcal J_5 $ $ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $, $ S^0>F_1(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_6 $ $ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $, $ S^0>F_1(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_7 $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $ and $ S^0<F_1(D) $
$ \mathcal J_8 $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ F_1(D)<S^0<F_2(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_9 $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ F_1(D)<S^0<F_2(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_{10} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $ and $ \lambda_1(D)>\mu_2(D) $
$ \mathcal J_{11} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!<\!F_5(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_{12} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!>\!F_5(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal{J}_{13} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!<\!F_5(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_{14} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!>\!F_5(D) $ and $ \lambda_1(D)<\lambda_2(D) $
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Region Definition
$ \mathcal J_0 $ $ S^0<\lambda_1(D) $ and $ S^0<\lambda_2(D) $
$ \mathcal J_1 $ $ S^0<\lambda_1(D) $ and $ \lambda_2(D)<S^0<\mu_2(D) $
$ \mathcal J_2 $ $ S^0<\lambda_1(D) $ and $ S^0>\mu_2(D) $
$ \mathcal J_3 $ $ S^0>\lambda_1(D) $ and $ S^0<\lambda_2(D) $
$ \mathcal J_4 $ $ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $ and $ S^0<F_1(D) $
$ \mathcal J_5 $ $ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $, $ S^0>F_1(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_6 $ $ S^0>\lambda_1(D) $, $ \lambda_2(D)<S^0<\mu_2(D) $, $ S^0>F_1(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_7 $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $ and $ S^0<F_1(D) $
$ \mathcal J_8 $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ F_1(D)<S^0<F_2(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_9 $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ F_1(D)<S^0<F_2(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_{10} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $ and $ \lambda_1(D)>\mu_2(D) $
$ \mathcal J_{11} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!<\!F_5(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal J_{12} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!>\!F_5(D) $ and $ \lambda_2(D)<\lambda_1(D) $
$ \mathcal{J}_{13} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!<\!F_5(D) $ and $ \lambda_1(D)<\lambda_2(D) $
$ \mathcal J_{14} $ $ S^0>\lambda_1(D) $, $ S^0>\mu_2(D) $, $ S^0\!>\!F_2(D) $, $ S^0\!>\!F_5(D) $ and $ \lambda_1(D)<\lambda_2(D) $
Table 5.  Existence and stability of equilibria in the regions of the operating diagrams in Figsures 5, 6 and 11
Region $ \mathcal{J}_0 $ $ \mathcal{J}_1 $ $ \mathcal{J}_2 $ $ \mathcal{J}_3 $ $ \mathcal{J}_4 $ $ \mathcal{J}_5 $ $ \mathcal{J}_6 $ $ \mathcal{J}_7 $ $ \mathcal{J}_8 $ $ \mathcal{J}_9 $ $ \mathcal{J}_{10} $ $ \mathcal{J}_{11} $ $ \mathcal{J}_{12} $ $ \mathcal J_{13} $ $ \mathcal{J}_{14} $
$ E_0 $ S U S U U U U U U U U U U U U
$ E_1 $ S S S U S S U S U U S S
$ E_2^1 $ S S U S S U S S S S S S S
$ E_2^2 $ U U U U U U U U U
$ E_c^1 $ U U U U
$ E_c^2 $ U S U S
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Region $ \mathcal{J}_0 $ $ \mathcal{J}_1 $ $ \mathcal{J}_2 $ $ \mathcal{J}_3 $ $ \mathcal{J}_4 $ $ \mathcal{J}_5 $ $ \mathcal{J}_6 $ $ \mathcal{J}_7 $ $ \mathcal{J}_8 $ $ \mathcal{J}_9 $ $ \mathcal{J}_{10} $ $ \mathcal{J}_{11} $ $ \mathcal{J}_{12} $ $ \mathcal J_{13} $ $ \mathcal{J}_{14} $
$ E_0 $ S U S U U U U U U U U U U U U
$ E_1 $ S S S U S S U S U U S S
$ E_2^1 $ S S U S S U S S S S S S S
$ E_2^2 $ U U U U U U U U U
$ E_c^1 $ U U U U
$ E_c^2 $ U S U S
Table 6.  Biological parameters values used in the numerical computations shown in the figures
Case $ m_1 $ $ m_2 $ $ K_1 $ $ K_2 $ $ K_3 $ $ k $ $ \gamma $ Figs
1 1.0 4.0 1.0 1.0 0.5 0.2 0.3 6, 7, 8, 9, 10
2 1.5 2.7 1.0 1.0 0.08 0.2 0.3 11, 12, 14, 15.
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Case $ m_1 $ $ m_2 $ $ K_1 $ $ K_2 $ $ K_3 $ $ k $ $ \gamma $ Figs
1 1.0 4.0 1.0 1.0 0.5 0.2 0.3 6, 7, 8, 9, 10
2 1.5 2.7 1.0 1.0 0.08 0.2 0.3 11, 12, 14, 15.
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