# American Institute of Mathematical Sciences

2012, 9(2): 259-279. doi: 10.3934/mbe.2012.9.259

## Qualitative analysis of a model for co-culture of bacteria and amoebae

 1 Center for Information Technology, Bruno Kessler Foundation, via Sommarive 18, I-38123 Trento Povo, Italy 2 Institut de Mathématiques de Bordeaux, UMR CNRS 5251 - Case 36, Université Victor Segalen Bordeaux 2, 3ter place de la Victoire 33076 Bordeaux Cedex, France 3 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received  February 2011 Revised  November 2011 Published  March 2012

In this article we analyze a mathematical model presented in [11]. The model consists of two scalar ordinary differential equations, which describe the interaction between bacteria and amoebae. We first give the sufficient conditions for the uniform persistence of the model, then we prove that the model can undergo Hopf bifurcation and Bogdanov-Takens bifurcation for some parameter values, respectively.
Citation: Laura Fumanelli, Pierre Magal, Dongmei Xiao, Xiao Yu. Qualitative analysis of a model for co-culture of bacteria and amoebae. Mathematical Biosciences & Engineering, 2012, 9 (2) : 259-279. doi: 10.3934/mbe.2012.9.259
##### References:
 [1] A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, "Theory of Bifurcations of Dynamical Systems on a Plane,'' Israel Program for Scientific Translations, Jerusalem, 1971. [2] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta. Math. Soviet., 1 (1981), 373-388. [3] R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta. Math. Soviet., 1 (1981), 389-421. [4] S.-N. Chow and J. K. Hale, "Methods of Bifurcation Theory,'' Springer-Verlag, New York-Heidelberg-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4. [5] P. Cosson, L. Zulianello, O. Join-Lambert, F. Faurisson, L. Gebbie, M. Benghezal, C. Van Delden, L. K. Curty and T. Khler, Pseudomonas aeruginosa virulence analyzed in a Dictyostelium discoideum host system, J. Bacteriol., 184 (2002), 3027-3033. doi: 10.1128/JB.184.11.3027-3033.2002. [6] F. Dumortier, R. Roussarie and J. Sotomayor, Generic $3$-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension $3$, Ergodic Theory Dynam. Systems, 7 (1987), 375-413. doi: 10.1017/S0143385700004119. [7] F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, "Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals,'' Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. [8] E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theor. Biol., 249 (2007), 487-499. doi: 10.1016/j.jtbi.2007.08.011. [9] E. M. C. D'Agata, M. Dupont-Rouzeyrol, P. Magal, D. Olivier and S. Ruan, The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria, PLoS ONE, 3 (2008), 1-9. [10] R. Froquet, N. Cherix, S. E. Burr, J. Frey, S. Vilches, J. M. Tomas and P. Cosson, Alternative host model to evaluate Aeromonas virulence, Appl. Environ. Microbiol., 73 (2007), 5657-5659. doi: 10.1128/AEM.00908-07. [11] L. Fumanelli, M. Iannelli, H. A. Janjua and O. Jousson, Mathematical modeling of bacterial virulence and host-pathogen interactions in the Dictyostelium/Pseudomonas system, J. Theor. Biol., 270 (2011), 19-24. doi: 10.1016/j.jtbi.2010.11.018. [12] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [13] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [14] E. Kipnis, T. Sawa and J. Wiener-Kronish, Targeting mechanisms of Pseudomonas aeruginosa pathogenesis, Medecine et Maladies Infectieuses, 36 (2006), 78-91. doi: 10.1016/j.medmal.2005.10.007. [15] C. L. Kurz and J. J. Ewbank, Infection in a dish: High-throughput analyses of bacterial pathogenesis, Curr. Opin. Microbiol., 10 (2007), 10-16. doi: 10.1016/j.mib.2006.12.001. [16] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472. doi: 10.1137/S0036139999361896. [17] G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Natl. Acad. Sci. USA, 102 (2005), 13343-13348. doi: 10.1073/pnas.0504053102. [18] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology" (Halifax, NS, 1997), Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, (1999), 493-506. [19] Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, "Qualitative Theory of Differential Equations,'' Transl. Math. Monogr., 101, American Mathematical Society, Providence, RI, 1992.

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##### References:
 [1] A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, "Theory of Bifurcations of Dynamical Systems on a Plane,'' Israel Program for Scientific Translations, Jerusalem, 1971. [2] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta. Math. Soviet., 1 (1981), 373-388. [3] R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta. Math. Soviet., 1 (1981), 389-421. [4] S.-N. Chow and J. K. Hale, "Methods of Bifurcation Theory,'' Springer-Verlag, New York-Heidelberg-Berlin, 1982. doi: 10.1007/978-1-4613-8159-4. [5] P. Cosson, L. Zulianello, O. Join-Lambert, F. Faurisson, L. Gebbie, M. Benghezal, C. Van Delden, L. K. Curty and T. Khler, Pseudomonas aeruginosa virulence analyzed in a Dictyostelium discoideum host system, J. Bacteriol., 184 (2002), 3027-3033. doi: 10.1128/JB.184.11.3027-3033.2002. [6] F. Dumortier, R. Roussarie and J. Sotomayor, Generic $3$-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension $3$, Ergodic Theory Dynam. Systems, 7 (1987), 375-413. doi: 10.1017/S0143385700004119. [7] F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, "Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals,'' Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. [8] E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theor. Biol., 249 (2007), 487-499. doi: 10.1016/j.jtbi.2007.08.011. [9] E. M. C. D'Agata, M. Dupont-Rouzeyrol, P. Magal, D. Olivier and S. Ruan, The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria, PLoS ONE, 3 (2008), 1-9. [10] R. Froquet, N. Cherix, S. E. Burr, J. Frey, S. Vilches, J. M. Tomas and P. Cosson, Alternative host model to evaluate Aeromonas virulence, Appl. Environ. Microbiol., 73 (2007), 5657-5659. doi: 10.1128/AEM.00908-07. [11] L. Fumanelli, M. Iannelli, H. A. Janjua and O. Jousson, Mathematical modeling of bacterial virulence and host-pathogen interactions in the Dictyostelium/Pseudomonas system, J. Theor. Biol., 270 (2011), 19-24. doi: 10.1016/j.jtbi.2010.11.018. [12] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [13] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [14] E. Kipnis, T. Sawa and J. Wiener-Kronish, Targeting mechanisms of Pseudomonas aeruginosa pathogenesis, Medecine et Maladies Infectieuses, 36 (2006), 78-91. doi: 10.1016/j.medmal.2005.10.007. [15] C. L. Kurz and J. J. Ewbank, Infection in a dish: High-throughput analyses of bacterial pathogenesis, Curr. Opin. Microbiol., 10 (2007), 10-16. doi: 10.1016/j.mib.2006.12.001. [16] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472. doi: 10.1137/S0036139999361896. [17] G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Natl. Acad. Sci. USA, 102 (2005), 13343-13348. doi: 10.1073/pnas.0504053102. [18] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology" (Halifax, NS, 1997), Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, (1999), 493-506. [19] Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, "Qualitative Theory of Differential Equations,'' Transl. Math. Monogr., 101, American Mathematical Society, Providence, RI, 1992.
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