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An extension of Gompertzian growth dynamics: Weibull and Fréchet models
Competition of motile and immotile bacterial strains in a petri dish
1.  Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada, Canada 
References:
[1] 
S.Asei, B. Byers, A. Eng, N. James and J. Leto, "Bacterial Chemostat Model," 2007. 
[2] 
P. K. Brazhnik and J. J Tyson, On traveling wave solutions of fisher's equation in two spatial dimensions, SIAM. J. Appl. Math., 60 (2000), 371391. doi: 10.1137/S0036139997325497. 
[3] 
I. Chang, E. S. Gilbert, N. Eliashberg and J. D. Keasling, A threedimensional stochastic simulation of biofilm growth and transportrelated factors that affect structure, Micro. Bio., 149 (2003), 28592871. 
[4] 
M. Fontes and D. Kaiser, Myxococcus cells respond to elastic forces in their substances, Proceedings of the National Academy of Sciences of the United States of America, 96 (1999), 80528057. 
[5] 
H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Jpn., 58 (1989), 38753878. 
[6] 
H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient, J. Phys. Soc. Jpn., 60 (1991), 8894. 
[7] 
M. E. Hibbing, C. Fuqua, M. R. Parsek and B. S. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nature Reviews Microbiology, 8 (2010), 1525. 
[8] 
D. P. Hzder, R. Hemmerbach and M. Lebert, Gravity and the bacterial unicellular organisms, Developmental and Cell Biology Series, 40 (2005). 
[9] 
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. of Math. Biol., 42 (1980), 397429. doi: 10.1016/S00928240(80)800577. 
[10] 
E. Keller, Mathematical aspects of bacterial chemotaxis, Antibiotics and Chemotherapy, 19 (1974), 7993. 
[11] 
F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Micro. Biol., 16 (1988), 115131. 
[12] 
E. Khain, L. M. Sander and A. M. Stein, A model for glioma growth, Research Article, 11 (2005), 5357. doi: 10.1002/cplx.20108. 
[13] 
S. M. Krone, R. Lu, R. Fox, H. Suzuki and E. M. Top, Modelling the spatial dynamics of plasmid transfer and persistence, Micro. Biol., 153 (2007), 28032816. 
[14] 
D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations, Micro. Ecol., 7 (1981), 207227. 
[15] 
D. Lauffenburger, R. Aris and K. H. Keller, Effects of cell motility and chemotaxis on growth of bacterial populations, Biophys. J., 40 (1982), 209219. 
[16] 
D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Bio. Tech. and Bio. Eng., xxv (1983), 21032125. 
[17] 
M. Matsushita, J. Wakitaa, H. Itoha, K. Watanabea, T. Araia, T. Matsuyamab, H. Sakaguchic and M. Mimurad, Formation of colony patterns by a bacterial cell population, Physica A: Statistical Mechanics and Its Applications, 274 (1999), 190199. 
[18] 
M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamazaki and T. Matsuyama, Colony formation in bacteria: Experiments and modeling, Biofilms, 1 (2004), 305317. 
[19] 
M. Mimura, H. Sakaguchi and M. Matsushita, Reactiondiffusion modeling of bacterial colony patterns, Physica. A. Stat. Mech. Appl., 282 (2000), 283303. 
[20] 
J. D. Murray, "Murray JD," $1^{st}$, $3^{rd}$ edition, USA, 2002. 
[21] 
K. Nowaczyk, A. Juszczak A and F. Domka, Microbiological oxidation of the waste ferrous sulphate, Polish Journal of Environmental Studies, 6 (1999), 409416. 
[22] 
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311338. 
[23] 
P. T. Saunders and M. J. Bazin, On the stability of food chains, J. Theor. Biol., 52 (1975), 121142. 
[24] 
R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats, Geobiology, 8 (2010), 179190. 
[25] 
J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili, PNAS, 98 (2001), 69016904. 
[26] 
L. Simonsen, Dynamics of plasmid transfer on surfaces, J. General Microbiology, 136 (1990), 10011007. 
[27] 
R. Tokita, T. Katoh, Y. Maeda, J. I. Wakita, M. Sano, T. Matsuyama and M. Matsushita, Pattern formation of bacterial colonies by Escherichia coli, J. Phys. Soc. Jpn., 78 (2009), 074005 (6 pages). 
[28] 
Y. Wei, X. Wang, J. Liu, L. Nememan, A. H. Singh, H. Howie and B. R. Levin, The populatiion and evolutionary of bacteria in physically structured habitats: The adaptive virtues of motility, PNAS, 108 (2011), 40474052. 
[29] 
J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems," 2 1998. 
[30] 
P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria, Current Biology, 8 (1998), 408411. 
[31] 
A. Ishihara, J. E. Segall, S. M. Block and H. L Berg, Coordination of flagella on filgmentous cells of Escherichia Coli, J. Bacteriology, 155 (1983), 228237. 
[32] 
B. L. Taylor and D. E. Koshlard, Reversal of flafella rotation in Monotrichous and Peritrichous bacteria: Generation of changes in direction, J. Bacteriology, 119 (1974), 640642. 
show all references
References:
[1] 
S.Asei, B. Byers, A. Eng, N. James and J. Leto, "Bacterial Chemostat Model," 2007. 
[2] 
P. K. Brazhnik and J. J Tyson, On traveling wave solutions of fisher's equation in two spatial dimensions, SIAM. J. Appl. Math., 60 (2000), 371391. doi: 10.1137/S0036139997325497. 
[3] 
I. Chang, E. S. Gilbert, N. Eliashberg and J. D. Keasling, A threedimensional stochastic simulation of biofilm growth and transportrelated factors that affect structure, Micro. Bio., 149 (2003), 28592871. 
[4] 
M. Fontes and D. Kaiser, Myxococcus cells respond to elastic forces in their substances, Proceedings of the National Academy of Sciences of the United States of America, 96 (1999), 80528057. 
[5] 
H. Fujikawa and M. Matsushita, Fractal growth of Bacillus subtilis on agar plates, J. Phys. Soc. Jpn., 58 (1989), 38753878. 
[6] 
H. Fujikawa and M. Matsushita, Bacterial fractal growth in the concentration field of nutrient, J. Phys. Soc. Jpn., 60 (1991), 8894. 
[7] 
M. E. Hibbing, C. Fuqua, M. R. Parsek and B. S. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nature Reviews Microbiology, 8 (2010), 1525. 
[8] 
D. P. Hzder, R. Hemmerbach and M. Lebert, Gravity and the bacterial unicellular organisms, Developmental and Cell Biology Series, 40 (2005). 
[9] 
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. of Math. Biol., 42 (1980), 397429. doi: 10.1016/S00928240(80)800577. 
[10] 
E. Keller, Mathematical aspects of bacterial chemotaxis, Antibiotics and Chemotherapy, 19 (1974), 7993. 
[11] 
F. X. Kelly, K. J. Dapsis and D. Lauffenburger, Effect of bacterial chemotaxis on dynamics of microbial competition, Micro. Biol., 16 (1988), 115131. 
[12] 
E. Khain, L. M. Sander and A. M. Stein, A model for glioma growth, Research Article, 11 (2005), 5357. doi: 10.1002/cplx.20108. 
[13] 
S. M. Krone, R. Lu, R. Fox, H. Suzuki and E. M. Top, Modelling the spatial dynamics of plasmid transfer and persistence, Micro. Biol., 153 (2007), 28032816. 
[14] 
D. Lauffenburger, R. Aris and K. H. Keller, Effects of random motility on growth of bacterial populations, Micro. Ecol., 7 (1981), 207227. 
[15] 
D. Lauffenburger, R. Aris and K. H. Keller, Effects of cell motility and chemotaxis on growth of bacterial populations, Biophys. J., 40 (1982), 209219. 
[16] 
D. Lauffenburger and P. Calcagno, Competition between two microbial populations in a nonmixed environment: Effect of cell random motility, Bio. Tech. and Bio. Eng., xxv (1983), 21032125. 
[17] 
M. Matsushita, J. Wakitaa, H. Itoha, K. Watanabea, T. Araia, T. Matsuyamab, H. Sakaguchic and M. Mimurad, Formation of colony patterns by a bacterial cell population, Physica A: Statistical Mechanics and Its Applications, 274 (1999), 190199. 
[18] 
M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamazaki and T. Matsuyama, Colony formation in bacteria: Experiments and modeling, Biofilms, 1 (2004), 305317. 
[19] 
M. Mimura, H. Sakaguchi and M. Matsushita, Reactiondiffusion modeling of bacterial colony patterns, Physica. A. Stat. Mech. Appl., 282 (2000), 283303. 
[20] 
J. D. Murray, "Murray JD," $1^{st}$, $3^{rd}$ edition, USA, 2002. 
[21] 
K. Nowaczyk, A. Juszczak A and F. Domka, Microbiological oxidation of the waste ferrous sulphate, Polish Journal of Environmental Studies, 6 (1999), 409416. 
[22] 
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311338. 
[23] 
P. T. Saunders and M. J. Bazin, On the stability of food chains, J. Theor. Biol., 52 (1975), 121142. 
[24] 
R. N. D. Shepard and D. Y. Sumner, Undirected motility of filamentous cyanobacteria produces reticulate mats, Geobiology, 8 (2010), 179190. 
[25] 
J. M. Skerker and H. C. Berger, Direct observation of extension and retraction of type IV pili, PNAS, 98 (2001), 69016904. 
[26] 
L. Simonsen, Dynamics of plasmid transfer on surfaces, J. General Microbiology, 136 (1990), 10011007. 
[27] 
R. Tokita, T. Katoh, Y. Maeda, J. I. Wakita, M. Sano, T. Matsuyama and M. Matsushita, Pattern formation of bacterial colonies by Escherichia coli, J. Phys. Soc. Jpn., 78 (2009), 074005 (6 pages). 
[28] 
Y. Wei, X. Wang, J. Liu, L. Nememan, A. H. Singh, H. Howie and B. R. Levin, The populatiion and evolutionary of bacteria in physically structured habitats: The adaptive virtues of motility, PNAS, 108 (2011), 40474052. 
[29] 
J. T. Wimpenny, "CRC Handbook of Laboratory Model Systems for Microbial Ecosystems," 2 1998. 
[30] 
P. Youderian, Bacterial motility: Secretory secrets of gliding bacteria, Current Biology, 8 (1998), 408411. 
[31] 
A. Ishihara, J. E. Segall, S. M. Block and H. L Berg, Coordination of flagella on filgmentous cells of Escherichia Coli, J. Bacteriology, 155 (1983), 228237. 
[32] 
B. L. Taylor and D. E. Koshlard, Reversal of flafella rotation in Monotrichous and Peritrichous bacteria: Generation of changes in direction, J. Bacteriology, 119 (1974), 640642. 
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