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A mathematical model of HTLV-I infection with two time delays
A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain
1. | Department of Mathematics and Computer Science, University of Messina, Viale F. Stagno D'Alcontres 31, I-98166 Messina, Italy, Italy, Italy |
References:
[1] |
M. Al-Ghoul and B. C. Eu, Hyperbolic reaction-diffusion equations and irreversible thermodynamics, Physica D, 90 (1996), 119-153.
doi: 10.1016/0167-2789(95)00231-6. |
[2] |
E. Barbera, C. Currò and G. Valenti, A hyperbolic reaction-diffusion model for the hantavirus infection, Mathematical Methods in Applied Sciences, 31 (2008), 481-499.
doi: 10.1002/mma.929. |
[3] |
E. Barbera, C. Currò and G. Valenti, A hyperbolic model for the effects of urbanization on air pollution, Applied Mathematical Modelling, 34 (2010), 2192-2202.
doi: 10.1016/j.apm.2009.10.030. |
[4] |
E. Barbera, C. Currò and G. Valenti, Wave features of a hyperbolic prey-predator model, Mathematical Methods in the Applied Sciences, 33 (2010), 1504-1515.
doi: 10.1002/mma.1270. |
[5] |
E. Barbera, G. Consolo and G. Valenti, Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model, Physical Review E, 88 (2013), 052719.
doi: 10.1103/PhysRevE.88.052719. |
[6] |
K. Boushaba, O. Arino and A. Boussouar, A mathematical model for phytoplankton, Mathematical Models and Methods in Applied Sciences, 12 (2002), 871-901.
doi: 10.1142/S0218202502001945. |
[7] |
S. J. Brentnall, K. J. Richards, J. Brindley and E. Murphy, Plankton patchiness and its effect on larger-scale productivity, Journal of Plankton Research, 25 (2003), 121-140.
doi: 10.1093/plankt/25.2.121. |
[8] |
M. P. Cassinari, M. Groppi and C. Tebaldi, Effects of predation efficiencies on the dynamics of a tritrophic food chain, Mathematical Biosciences and Engineering, 4 (2007), 431-456.
doi: 10.3934/mbe.2007.4.431. |
[9] |
A. Chatterjee, S. Pal and S. Chatterjee, Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom, Applied Mathematics and Computation, 218 (2011), 3387-3398.
doi: 10.1016/j.amc.2011.08.082. |
[10] |
J. Chattopadhyay and S. Pal, Viral infection on phytoplankton-zooplankton system-a mathematical model, Ecological Modelling, 151 (2002), 15-28.
doi: 10.1016/S0304-3800(01)00415-X. |
[11] |
C. Currò, D. Fusco and G. Valenti, Nonlinear wave analysis of a dissipative hyperbolic model of interest in biodynamics, Far East Journal of Applied Mathematics, 13 (2003), 195-215. |
[12] |
S. R. Dumbar and H. G. Othmer, On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, in Nonlinear Oscillations in Biology and Chemistry (ed. H.G. Othmer), Lecture Notes in Biomathematics, Berlin, 66 (1986), 274-289.
doi: 10.1007/978-3-642-93318-9_18. |
[13] |
U. Ebert, M. Array, N. Temme, B. Sommeijer and J. Huisman, Critical Conditions for Phytoplankton Blooms, Bulletin of Mathematical Biology, 63 (2001), 1095-1124.
doi: 10.1006/bulm.2001.0261. |
[14] |
A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton ecosystem models, Bulletin of Mathematical Biology, 61 (1999), 303-339.
doi: 10.1006/bulm.1998.0082. |
[15] |
A. M. Edwards and A. Yool, The role of higher predation in plankton population models, Journal of Plankton Research, 22 (2000), 1085-1112.
doi: 10.1093/plankt/22.6.1085. |
[16] |
A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical systems approach, Journal of Plankton Research, 23 (2001), 389-413.
doi: 10.1093/plankt/23.4.389. |
[17] |
J. Fort and V. Méndez, Wavefronts in time-delayed reaction-diffusion system. Theory and comparison to experiments, Reports on Progress in Physics, 65 (2002), 895-954.
doi: 10.1088/0034-4885/65/6/201. |
[18] |
J. A. Freund, S. Mieruch, B. Scholze, K. Wiltshire and U. Feudel, Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects, Ecological complexity, 3 (2006), 129-139.
doi: 10.1016/j.ecocom.2005.11.001. |
[19] |
K. O. Friedrichs and P. D. Lax, System of Conservation Equation with a convex extension, Proceedings of the National Academy of Sciences USA, 68 (1971), 1686-1688.
doi: 10.1073/pnas.68.8.1686. |
[20] |
I. Koszalka, A. Bracco, C. Pasquero and A. Provenzale, Plankton cycles disguised by turbulent advection, Theoretical Population Biology, 72 (2007), 1-6.
doi: 10.1016/j.tpb.2007.03.007. |
[21] |
I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Archive for Rational Mechanics and Analysis, 46 (1972), 131-148.
doi: 10.1007/BF00250688. |
[22] |
H. Malchow, F. M. Hilker, R. R. Sarkar and K. Brauer, Spatiotemporal patterns in an excitable system with lysogenic viral infection, Mathematical and Computer Modelling, 42 (2005), 1035-1048.
doi: 10.1016/j.mcm.2004.10.025. |
[23] |
L. Matthews and J. Brindley, Patchiness in plankton populations, Dynamics and Stability of Systems, 12 (1997), 39-59.
doi: 10.1080/02681119708806235. |
[24] |
B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecological modelling, 198 (2006), 163-173.
doi: 10.1016/j.ecolmodel.2006.04.005. |
[25] |
B. Mukhopadhyay and R. Bhattacharyya, Role of gestation delay in a plankton-fish model under stochastic fluctuations, Mathematical Biosciences, 215 (2008), 26-34.
doi: 10.1016/j.mbs.2008.05.007. |
[26] |
I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Springer, New York, 1998.
doi: 10.1007/978-1-4612-2210-1. |
[27] |
J. D. Murray, Mathematical Biology I: An Introduction, third ed., Springer, Berlin, 2002. |
[28] |
A. Palumbo and G. Valenti, A mathematical model for a spatial predator-prey interaction, Mathematical Methods in the Applied Sciences, 25 (2002), 945-954.
doi: 10.1002/mma.322. |
[29] |
J. H. Steele and E. W. Henderson, A simple plankton model, The American Naturalist, 117 (1981), 676-691. |
[30] |
J. H. Steele and E. W. Henderson, The role of predation in plankton models, Journal of Plankton Research, 14 (1992), 157-172.
doi: 10.1093/plankt/14.1.157. |
[31] |
J. E. Truscott and J. Brindley, Ocean plankton populations as excitable media, Bulletin of Mathematical Biology, 56 (1994), 981-998.
doi: 10.1016/S0092-8240(05)80300-3. |
[32] |
J. E. Truscott and J. Brindley, Equilibria, Stability and Excitability in a General Class of Plankton Population Models, Philosophical Transactions: Physical Sciences and Engineering, Nonlinear Phenomena in Excitable Media, 347 (1994), 703-718.
doi: 10.1098/rsta.1994.0076. |
[33] |
R. K. Upadhyay, W. Wang and N. K. Thakur, Spatiotemporal dynamics in a spatial plankton system, The Mathematical Modelling of Natural Phenomena, 5 (2010), 102-122.
doi: 10.1051/mmnp/20105507. |
show all references
References:
[1] |
M. Al-Ghoul and B. C. Eu, Hyperbolic reaction-diffusion equations and irreversible thermodynamics, Physica D, 90 (1996), 119-153.
doi: 10.1016/0167-2789(95)00231-6. |
[2] |
E. Barbera, C. Currò and G. Valenti, A hyperbolic reaction-diffusion model for the hantavirus infection, Mathematical Methods in Applied Sciences, 31 (2008), 481-499.
doi: 10.1002/mma.929. |
[3] |
E. Barbera, C. Currò and G. Valenti, A hyperbolic model for the effects of urbanization on air pollution, Applied Mathematical Modelling, 34 (2010), 2192-2202.
doi: 10.1016/j.apm.2009.10.030. |
[4] |
E. Barbera, C. Currò and G. Valenti, Wave features of a hyperbolic prey-predator model, Mathematical Methods in the Applied Sciences, 33 (2010), 1504-1515.
doi: 10.1002/mma.1270. |
[5] |
E. Barbera, G. Consolo and G. Valenti, Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model, Physical Review E, 88 (2013), 052719.
doi: 10.1103/PhysRevE.88.052719. |
[6] |
K. Boushaba, O. Arino and A. Boussouar, A mathematical model for phytoplankton, Mathematical Models and Methods in Applied Sciences, 12 (2002), 871-901.
doi: 10.1142/S0218202502001945. |
[7] |
S. J. Brentnall, K. J. Richards, J. Brindley and E. Murphy, Plankton patchiness and its effect on larger-scale productivity, Journal of Plankton Research, 25 (2003), 121-140.
doi: 10.1093/plankt/25.2.121. |
[8] |
M. P. Cassinari, M. Groppi and C. Tebaldi, Effects of predation efficiencies on the dynamics of a tritrophic food chain, Mathematical Biosciences and Engineering, 4 (2007), 431-456.
doi: 10.3934/mbe.2007.4.431. |
[9] |
A. Chatterjee, S. Pal and S. Chatterjee, Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom, Applied Mathematics and Computation, 218 (2011), 3387-3398.
doi: 10.1016/j.amc.2011.08.082. |
[10] |
J. Chattopadhyay and S. Pal, Viral infection on phytoplankton-zooplankton system-a mathematical model, Ecological Modelling, 151 (2002), 15-28.
doi: 10.1016/S0304-3800(01)00415-X. |
[11] |
C. Currò, D. Fusco and G. Valenti, Nonlinear wave analysis of a dissipative hyperbolic model of interest in biodynamics, Far East Journal of Applied Mathematics, 13 (2003), 195-215. |
[12] |
S. R. Dumbar and H. G. Othmer, On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, in Nonlinear Oscillations in Biology and Chemistry (ed. H.G. Othmer), Lecture Notes in Biomathematics, Berlin, 66 (1986), 274-289.
doi: 10.1007/978-3-642-93318-9_18. |
[13] |
U. Ebert, M. Array, N. Temme, B. Sommeijer and J. Huisman, Critical Conditions for Phytoplankton Blooms, Bulletin of Mathematical Biology, 63 (2001), 1095-1124.
doi: 10.1006/bulm.2001.0261. |
[14] |
A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton ecosystem models, Bulletin of Mathematical Biology, 61 (1999), 303-339.
doi: 10.1006/bulm.1998.0082. |
[15] |
A. M. Edwards and A. Yool, The role of higher predation in plankton population models, Journal of Plankton Research, 22 (2000), 1085-1112.
doi: 10.1093/plankt/22.6.1085. |
[16] |
A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical systems approach, Journal of Plankton Research, 23 (2001), 389-413.
doi: 10.1093/plankt/23.4.389. |
[17] |
J. Fort and V. Méndez, Wavefronts in time-delayed reaction-diffusion system. Theory and comparison to experiments, Reports on Progress in Physics, 65 (2002), 895-954.
doi: 10.1088/0034-4885/65/6/201. |
[18] |
J. A. Freund, S. Mieruch, B. Scholze, K. Wiltshire and U. Feudel, Bloom dynamics in a seasonally forced phytoplankton-zooplankton model: Trigger mechanisms and timing effects, Ecological complexity, 3 (2006), 129-139.
doi: 10.1016/j.ecocom.2005.11.001. |
[19] |
K. O. Friedrichs and P. D. Lax, System of Conservation Equation with a convex extension, Proceedings of the National Academy of Sciences USA, 68 (1971), 1686-1688.
doi: 10.1073/pnas.68.8.1686. |
[20] |
I. Koszalka, A. Bracco, C. Pasquero and A. Provenzale, Plankton cycles disguised by turbulent advection, Theoretical Population Biology, 72 (2007), 1-6.
doi: 10.1016/j.tpb.2007.03.007. |
[21] |
I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Archive for Rational Mechanics and Analysis, 46 (1972), 131-148.
doi: 10.1007/BF00250688. |
[22] |
H. Malchow, F. M. Hilker, R. R. Sarkar and K. Brauer, Spatiotemporal patterns in an excitable system with lysogenic viral infection, Mathematical and Computer Modelling, 42 (2005), 1035-1048.
doi: 10.1016/j.mcm.2004.10.025. |
[23] |
L. Matthews and J. Brindley, Patchiness in plankton populations, Dynamics and Stability of Systems, 12 (1997), 39-59.
doi: 10.1080/02681119708806235. |
[24] |
B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity, Ecological modelling, 198 (2006), 163-173.
doi: 10.1016/j.ecolmodel.2006.04.005. |
[25] |
B. Mukhopadhyay and R. Bhattacharyya, Role of gestation delay in a plankton-fish model under stochastic fluctuations, Mathematical Biosciences, 215 (2008), 26-34.
doi: 10.1016/j.mbs.2008.05.007. |
[26] |
I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Springer, New York, 1998.
doi: 10.1007/978-1-4612-2210-1. |
[27] |
J. D. Murray, Mathematical Biology I: An Introduction, third ed., Springer, Berlin, 2002. |
[28] |
A. Palumbo and G. Valenti, A mathematical model for a spatial predator-prey interaction, Mathematical Methods in the Applied Sciences, 25 (2002), 945-954.
doi: 10.1002/mma.322. |
[29] |
J. H. Steele and E. W. Henderson, A simple plankton model, The American Naturalist, 117 (1981), 676-691. |
[30] |
J. H. Steele and E. W. Henderson, The role of predation in plankton models, Journal of Plankton Research, 14 (1992), 157-172.
doi: 10.1093/plankt/14.1.157. |
[31] |
J. E. Truscott and J. Brindley, Ocean plankton populations as excitable media, Bulletin of Mathematical Biology, 56 (1994), 981-998.
doi: 10.1016/S0092-8240(05)80300-3. |
[32] |
J. E. Truscott and J. Brindley, Equilibria, Stability and Excitability in a General Class of Plankton Population Models, Philosophical Transactions: Physical Sciences and Engineering, Nonlinear Phenomena in Excitable Media, 347 (1994), 703-718.
doi: 10.1098/rsta.1994.0076. |
[33] |
R. K. Upadhyay, W. Wang and N. K. Thakur, Spatiotemporal dynamics in a spatial plankton system, The Mathematical Modelling of Natural Phenomena, 5 (2010), 102-122.
doi: 10.1051/mmnp/20105507. |
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