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Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses
Steady-state solutions and stability for a cubic autocatalysis model
1. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China |
2. | College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119 |
3. | Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China |
References:
[1] |
J. C. Tsai, Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay, Quart. Appl. Math., 69 (2011), 123-146. |
[2] |
R. Peng and F. Yi, On spatiotemporal pattern formation in a diffusive bimolecular model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 217-230.
doi: 10.3934/dcdsb.2011.15.217. |
[3] |
Y. You, Dynamics of three-component reversible Gray-Scott model, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1671-1688.
doi: 10.3934/dcdsb.2010.14.1671. |
[4] |
X. F. Chen and Y. W. Qi, Propagation of local disturbances in reaction diffusion systems modeling quadratic autocatalysis, SIAM J. Appl. Math., 69 (2008), 273-282.
doi: 10.1137/07070276X. |
[5] |
A. L. Kay, D. J. Needham and J. A. Leach, Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay. I. Permanent form travelling waves, Nonlinearity, 16 (2003), 735-770.
doi: 10.1088/0951-7715/16/2/322. |
[6] |
J. A. Leach and J. C. Wei, Pattern formation in a simple chemical system with general orders of autocatalysis and decay. I. Stability analysis, Phys. D, 180 (2003), 185-209.
doi: 10.1016/S0167-2789(03)00065-4. |
[7] |
P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci., 39 (1983), 29-43. |
[8] |
P. Gray and S. K. Scott, Autocatalytic reactions in the CSTR: oscillations and instabilities in the system $A + 2B\rightarrow 3B$; $B\rightarrow C$, Chem. Eng. Sci., 39 (1984), 1087-1097. |
[9] |
A. D'Anna, P. G. Lignola and S. K. Scott, The application of singularity theory to isothermal autocatalytic open systems, Proc. Roy. Soc. A, 403 (1986), 341-363. |
[10] |
B. Peng, S. K. Scott and K. Showalter, Period doubling and chaos in a three variable autocatalator, J. Phys. Chem., 94 (1990), 5243-5246. |
[11] |
D. T. Lynch, Chaotic behavior of reactions systems: mixed cubic and quadratic autocatalysis, Chem. Eng. Sci., 47 (1992), 4435-4444. |
[12] |
K. Alhumaizi and R. Aris, Chaos in a simple two-phase reactor, Chaos Solitons Fractals, 4 (1994), 1985-2014. |
[13] |
H. I. Abdel-Gawad and A. M. El-Shrae, Approximate solutions to the two-cell cubic autocatalytic reaction model, Kyungpook Math. J., 44 (2004), 187-211. |
[14] |
E. A. Elrifai, On cubic autocatalytic chemical reaction model, CSTR and invariants of knots, Far East J. Appl. Math., 32 (2008), 435-443. |
[15] |
J. H. Merkin, D. J. Needham and S. K. Scott, Oscillatory chemical reactions in closed vessels, Proc. Roy. Soc. London Ser. A, 406 (1986), 299-323. |
[16] |
A. B. Finlayson and J. H. Merkin, Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system, J. Engrg. Math., 38 (2000), 279-296.
doi: 10.1023/A:1004799200173. |
[17] |
L. S. Chen and D. D. Wang, A biochemical oscillation, Acta Math. Sci. Ser. B Engl. Ed., 5 (1985), 261-266. |
[18] |
J. H. Merkin, D. J. Needham and S. K. Scott, On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme, SIAM J. Appl. Math., 47 (1987), 1040-1060.
doi: 10.1137/0147068. |
[19] |
J. H. Merkin and D. J. Needham, Reaction-diffusion in a simple pooled chemical system, Dyn. Stab. Syst., 4 (1989), 141-167.
doi: 10.1080/02681118908806069. |
[20] |
D. J. Needham and J. H. Merkin, Pattern formation through reaction and diffusion in a simple pooled-chemical system, Dyn. Stab. Syst., 4 (1989), 259-284.
doi: 10.1080/02681118908806076. |
[21] |
R. Hill, J. H. Merkin and D. J. Needham, Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme, J. Engrg. Math., 29 (1995), 413-436.
doi: 10.1007/BF00043976. |
[22] |
J. Jang, W. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2005), 297-320.
doi: 10.1007/s10884-004-2782-x. |
[23] |
M. H. Wei, J. H. Wu and G. H. Guo, Turing structures and stability for the 1-D Lengyel-Epstein system, J. Math. Chem., 50 (2012), 2374-2396.
doi: 10.1007/s10910-012-0037-3. |
[24] |
M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalue, J. Funct. Anal., 8 (1971), 321-340. |
[25] |
K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.
doi: 10.1016/j.jde.2007.05.013. |
[26] |
D. Schaeffer and M. Golubitsky, Bifurcation analysis near a double eigenvalue of a model chemical reaction, Arch. Rational Mech. Anal., 75 (1981), 315-347.
doi: 10.1007/BF00256382. |
[27] |
M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer, New York, 1985.
doi: 10.1007/978-1-4612-5034-0. |
[28] |
M. Golubitsky and D. Schaeffer, Imperfect bifurcation in the presence of symmetry, Comm. Math. Phys., 67 (1979), 205-232. |
[29] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. |
[30] |
J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.
doi: 10.1016/S0362-546X(98)00250-8. |
[31] |
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98.
doi: 10.1002/cpa.3160320103. |
show all references
References:
[1] |
J. C. Tsai, Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay, Quart. Appl. Math., 69 (2011), 123-146. |
[2] |
R. Peng and F. Yi, On spatiotemporal pattern formation in a diffusive bimolecular model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 217-230.
doi: 10.3934/dcdsb.2011.15.217. |
[3] |
Y. You, Dynamics of three-component reversible Gray-Scott model, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1671-1688.
doi: 10.3934/dcdsb.2010.14.1671. |
[4] |
X. F. Chen and Y. W. Qi, Propagation of local disturbances in reaction diffusion systems modeling quadratic autocatalysis, SIAM J. Appl. Math., 69 (2008), 273-282.
doi: 10.1137/07070276X. |
[5] |
A. L. Kay, D. J. Needham and J. A. Leach, Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay. I. Permanent form travelling waves, Nonlinearity, 16 (2003), 735-770.
doi: 10.1088/0951-7715/16/2/322. |
[6] |
J. A. Leach and J. C. Wei, Pattern formation in a simple chemical system with general orders of autocatalysis and decay. I. Stability analysis, Phys. D, 180 (2003), 185-209.
doi: 10.1016/S0167-2789(03)00065-4. |
[7] |
P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci., 39 (1983), 29-43. |
[8] |
P. Gray and S. K. Scott, Autocatalytic reactions in the CSTR: oscillations and instabilities in the system $A + 2B\rightarrow 3B$; $B\rightarrow C$, Chem. Eng. Sci., 39 (1984), 1087-1097. |
[9] |
A. D'Anna, P. G. Lignola and S. K. Scott, The application of singularity theory to isothermal autocatalytic open systems, Proc. Roy. Soc. A, 403 (1986), 341-363. |
[10] |
B. Peng, S. K. Scott and K. Showalter, Period doubling and chaos in a three variable autocatalator, J. Phys. Chem., 94 (1990), 5243-5246. |
[11] |
D. T. Lynch, Chaotic behavior of reactions systems: mixed cubic and quadratic autocatalysis, Chem. Eng. Sci., 47 (1992), 4435-4444. |
[12] |
K. Alhumaizi and R. Aris, Chaos in a simple two-phase reactor, Chaos Solitons Fractals, 4 (1994), 1985-2014. |
[13] |
H. I. Abdel-Gawad and A. M. El-Shrae, Approximate solutions to the two-cell cubic autocatalytic reaction model, Kyungpook Math. J., 44 (2004), 187-211. |
[14] |
E. A. Elrifai, On cubic autocatalytic chemical reaction model, CSTR and invariants of knots, Far East J. Appl. Math., 32 (2008), 435-443. |
[15] |
J. H. Merkin, D. J. Needham and S. K. Scott, Oscillatory chemical reactions in closed vessels, Proc. Roy. Soc. London Ser. A, 406 (1986), 299-323. |
[16] |
A. B. Finlayson and J. H. Merkin, Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system, J. Engrg. Math., 38 (2000), 279-296.
doi: 10.1023/A:1004799200173. |
[17] |
L. S. Chen and D. D. Wang, A biochemical oscillation, Acta Math. Sci. Ser. B Engl. Ed., 5 (1985), 261-266. |
[18] |
J. H. Merkin, D. J. Needham and S. K. Scott, On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme, SIAM J. Appl. Math., 47 (1987), 1040-1060.
doi: 10.1137/0147068. |
[19] |
J. H. Merkin and D. J. Needham, Reaction-diffusion in a simple pooled chemical system, Dyn. Stab. Syst., 4 (1989), 141-167.
doi: 10.1080/02681118908806069. |
[20] |
D. J. Needham and J. H. Merkin, Pattern formation through reaction and diffusion in a simple pooled-chemical system, Dyn. Stab. Syst., 4 (1989), 259-284.
doi: 10.1080/02681118908806076. |
[21] |
R. Hill, J. H. Merkin and D. J. Needham, Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme, J. Engrg. Math., 29 (1995), 413-436.
doi: 10.1007/BF00043976. |
[22] |
J. Jang, W. M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2005), 297-320.
doi: 10.1007/s10884-004-2782-x. |
[23] |
M. H. Wei, J. H. Wu and G. H. Guo, Turing structures and stability for the 1-D Lengyel-Epstein system, J. Math. Chem., 50 (2012), 2374-2396.
doi: 10.1007/s10910-012-0037-3. |
[24] |
M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalue, J. Funct. Anal., 8 (1971), 321-340. |
[25] |
K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem, J. Differential Equations, 239 (2007), 296-310.
doi: 10.1016/j.jde.2007.05.013. |
[26] |
D. Schaeffer and M. Golubitsky, Bifurcation analysis near a double eigenvalue of a model chemical reaction, Arch. Rational Mech. Anal., 75 (1981), 315-347.
doi: 10.1007/BF00256382. |
[27] |
M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer, New York, 1985.
doi: 10.1007/978-1-4612-5034-0. |
[28] |
M. Golubitsky and D. Schaeffer, Imperfect bifurcation in the presence of symmetry, Comm. Math. Phys., 67 (1979), 205-232. |
[29] |
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. |
[30] |
J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.
doi: 10.1016/S0362-546X(98)00250-8. |
[31] |
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math., 32 (1979), 21-98.
doi: 10.1002/cpa.3160320103. |
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