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2015, 12(3): 451-472. doi: 10.3934/mbe.2015.12.451

A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain

Elvira Barbera 1, , Giancarlo Consolo 1, and  Giovanna Valenti 1,

1. 

Department of Mathematics and Computer Science, University of Messina, Viale F. Stagno D'Alcontres 31, I-98166 Messina, Italy, Italy, Italy

Received  February 2014 Revised  November 2014 Published  January 2015

Two hyperbolic reaction-diffusion models are built up in the framework of Extended Thermodynamics in order to describe the spatio-temporal interactions occurring in a two or three compartments aquatic food chain. The first model focuses on the dynamics between phytoplankton and zooplankton, whereas the second one accounts also for the nutrient. In these models, infections and influence of illumination on photosynthesis are neglected. It is assumed that the zooplankton predation follows a Holling type-III functional response, while the zooplankton mortality is linear. Owing to the hyperbolic structure of our equations, the wave processes occur at finite velocity, so that the paradox of instantaneous diffusion of biological quantities, typical of parabolic systems, is consequently removed. The character of steady states and travelling waves, together with the occurrence of Hopf bifurcations, is then discussed through linear stability analysis. The governing equations are also integrated numerically to validate the analytical results herein obtained and to extract additional information on the population dynamics.
Keywords: Aquatic food chain, hyperbolic reaction-diffusion model, traveling wave solutions., Hopf-bifurcation.
Mathematics Subject Classification: Primary: 35L60, 92D25, 35C07; Secondary: 35Q9.
Citation: Elvira Barbera, Giancarlo Consolo, Giovanna Valenti. A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain. Mathematical Biosciences & Engineering, 2015, 12 (3) : 451-472. doi: 10.3934/mbe.2015.12.451
References:
[1]

M. Al-Ghoul and B. C. Eu, Hyperbolic reaction-diffusion equations and irreversible thermodynamics,, Physica D, 90 (1996), 119.  doi: 10.1016/0167-2789(95)00231-6.  Google Scholar

[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.  doi: 10.1002/mma.929.  Google Scholar

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E. Barbera, C. Currò and G. Valenti, Wave features of a hyperbolic prey-predator model,, Mathematical Methods in the Applied Sciences, 33 (2010), 1504.  doi: 10.1002/mma.1270.  Google Scholar

[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).  doi: 10.1103/PhysRevE.88.052719.  Google Scholar

[6]

K. Boushaba, O. Arino and A. Boussouar, A mathematical model for phytoplankton,, Mathematical Models and Methods in Applied Sciences, 12 (2002), 871.  doi: 10.1142/S0218202502001945.  Google Scholar

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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.  doi: 10.1093/plankt/25.2.121.  Google Scholar

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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.  doi: 10.3934/mbe.2007.4.431.  Google Scholar

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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.  doi: 10.1016/j.amc.2011.08.082.  Google Scholar

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J. Chattopadhyay and S. Pal, Viral infection on phytoplankton-zooplankton system-a mathematical model,, Ecological Modelling, 151 (2002), 15.  doi: 10.1016/S0304-3800(01)00415-X.  Google Scholar

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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.   Google Scholar

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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), 66 (1986), 274.  doi: 10.1007/978-3-642-93318-9_18.  Google Scholar

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U. Ebert, M. Array, N. Temme, B. Sommeijer and J. Huisman, Critical Conditions for Phytoplankton Blooms,, Bulletin of Mathematical Biology, 63 (2001), 1095.  doi: 10.1006/bulm.2001.0261.  Google Scholar

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A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton ecosystem models,, Bulletin of Mathematical Biology, 61 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

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A. M. Edwards and A. Yool, The role of higher predation in plankton population models,, Journal of Plankton Research, 22 (2000), 1085.  doi: 10.1093/plankt/22.6.1085.  Google Scholar

[16]

A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical systems approach,, Journal of Plankton Research, 23 (2001), 389.  doi: 10.1093/plankt/23.4.389.  Google Scholar

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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.  doi: 10.1088/0034-4885/65/6/201.  Google Scholar

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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.  doi: 10.1016/j.ecocom.2005.11.001.  Google Scholar

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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.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[20]

I. Koszalka, A. Bracco, C. Pasquero and A. Provenzale, Plankton cycles disguised by turbulent advection,, Theoretical Population Biology, 72 (2007), 1.  doi: 10.1016/j.tpb.2007.03.007.  Google Scholar

[21]

I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle,, Archive for Rational Mechanics and Analysis, 46 (1972), 131.  doi: 10.1007/BF00250688.  Google Scholar

[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.  doi: 10.1016/j.mcm.2004.10.025.  Google Scholar

[23]

L. Matthews and J. Brindley, Patchiness in plankton populations,, Dynamics and Stability of Systems, 12 (1997), 39.  doi: 10.1080/02681119708806235.  Google Scholar

[24]

B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity,, Ecological modelling, 198 (2006), 163.  doi: 10.1016/j.ecolmodel.2006.04.005.  Google Scholar

[25]

B. Mukhopadhyay and R. Bhattacharyya, Role of gestation delay in a plankton-fish model under stochastic fluctuations,, Mathematical Biosciences, 215 (2008), 26.  doi: 10.1016/j.mbs.2008.05.007.  Google Scholar

[26]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics,, Springer, (1998).  doi: 10.1007/978-1-4612-2210-1.  Google Scholar

[27]

J. D. Murray, Mathematical Biology I: An Introduction,, third ed., (2002).   Google Scholar

[28]

A. Palumbo and G. Valenti, A mathematical model for a spatial predator-prey interaction,, Mathematical Methods in the Applied Sciences, 25 (2002), 945.  doi: 10.1002/mma.322.  Google Scholar

[29]

J. H. Steele and E. W. Henderson, A simple plankton model,, The American Naturalist, 117 (1981), 676.   Google Scholar

[30]

J. H. Steele and E. W. Henderson, The role of predation in plankton models,, Journal of Plankton Research, 14 (1992), 157.  doi: 10.1093/plankt/14.1.157.  Google Scholar

[31]

J. E. Truscott and J. Brindley, Ocean plankton populations as excitable media,, Bulletin of Mathematical Biology, 56 (1994), 981.  doi: 10.1016/S0092-8240(05)80300-3.  Google Scholar

[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, 347 (1994), 703.  doi: 10.1098/rsta.1994.0076.  Google Scholar

[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.  doi: 10.1051/mmnp/20105507.  Google Scholar

show all references

References:
[1]

M. Al-Ghoul and B. C. Eu, Hyperbolic reaction-diffusion equations and irreversible thermodynamics,, Physica D, 90 (1996), 119.  doi: 10.1016/0167-2789(95)00231-6.  Google Scholar

[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.  doi: 10.1002/mma.929.  Google Scholar

[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.  doi: 10.1016/j.apm.2009.10.030.  Google Scholar

[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.  doi: 10.1002/mma.1270.  Google Scholar

[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).  doi: 10.1103/PhysRevE.88.052719.  Google Scholar

[6]

K. Boushaba, O. Arino and A. Boussouar, A mathematical model for phytoplankton,, Mathematical Models and Methods in Applied Sciences, 12 (2002), 871.  doi: 10.1142/S0218202502001945.  Google Scholar

[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.  doi: 10.1093/plankt/25.2.121.  Google Scholar

[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.  doi: 10.3934/mbe.2007.4.431.  Google Scholar

[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.  doi: 10.1016/j.amc.2011.08.082.  Google Scholar

[10]

J. Chattopadhyay and S. Pal, Viral infection on phytoplankton-zooplankton system-a mathematical model,, Ecological Modelling, 151 (2002), 15.  doi: 10.1016/S0304-3800(01)00415-X.  Google Scholar

[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.   Google Scholar

[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), 66 (1986), 274.  doi: 10.1007/978-3-642-93318-9_18.  Google Scholar

[13]

U. Ebert, M. Array, N. Temme, B. Sommeijer and J. Huisman, Critical Conditions for Phytoplankton Blooms,, Bulletin of Mathematical Biology, 63 (2001), 1095.  doi: 10.1006/bulm.2001.0261.  Google Scholar

[14]

A. M. Edwards and J. Brindley, Zooplankton mortality and the dynamical behaviour of plankton ecosystem models,, Bulletin of Mathematical Biology, 61 (1999), 303.  doi: 10.1006/bulm.1998.0082.  Google Scholar

[15]

A. M. Edwards and A. Yool, The role of higher predation in plankton population models,, Journal of Plankton Research, 22 (2000), 1085.  doi: 10.1093/plankt/22.6.1085.  Google Scholar

[16]

A. M. Edwards, Adding detritus to a nutrient-phytoplankton-zooplankton model: A dynamical systems approach,, Journal of Plankton Research, 23 (2001), 389.  doi: 10.1093/plankt/23.4.389.  Google Scholar

[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.  doi: 10.1088/0034-4885/65/6/201.  Google Scholar

[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.  doi: 10.1016/j.ecocom.2005.11.001.  Google Scholar

[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.  doi: 10.1073/pnas.68.8.1686.  Google Scholar

[20]

I. Koszalka, A. Bracco, C. Pasquero and A. Provenzale, Plankton cycles disguised by turbulent advection,, Theoretical Population Biology, 72 (2007), 1.  doi: 10.1016/j.tpb.2007.03.007.  Google Scholar

[21]

I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle,, Archive for Rational Mechanics and Analysis, 46 (1972), 131.  doi: 10.1007/BF00250688.  Google Scholar

[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.  doi: 10.1016/j.mcm.2004.10.025.  Google Scholar

[23]

L. Matthews and J. Brindley, Patchiness in plankton populations,, Dynamics and Stability of Systems, 12 (1997), 39.  doi: 10.1080/02681119708806235.  Google Scholar

[24]

B. Mukhopadhyay and R. Bhattacharyya, Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity,, Ecological modelling, 198 (2006), 163.  doi: 10.1016/j.ecolmodel.2006.04.005.  Google Scholar

[25]

B. Mukhopadhyay and R. Bhattacharyya, Role of gestation delay in a plankton-fish model under stochastic fluctuations,, Mathematical Biosciences, 215 (2008), 26.  doi: 10.1016/j.mbs.2008.05.007.  Google Scholar

[26]

I. Müller and T. Ruggeri, Rational Extended Thermodynamics,, Springer, (1998).  doi: 10.1007/978-1-4612-2210-1.  Google Scholar

[27]

J. D. Murray, Mathematical Biology I: An Introduction,, third ed., (2002).   Google Scholar

[28]

A. Palumbo and G. Valenti, A mathematical model for a spatial predator-prey interaction,, Mathematical Methods in the Applied Sciences, 25 (2002), 945.  doi: 10.1002/mma.322.  Google Scholar

[29]

J. H. Steele and E. W. Henderson, A simple plankton model,, The American Naturalist, 117 (1981), 676.   Google Scholar

[30]

J. H. Steele and E. W. Henderson, The role of predation in plankton models,, Journal of Plankton Research, 14 (1992), 157.  doi: 10.1093/plankt/14.1.157.  Google Scholar

[31]

J. E. Truscott and J. Brindley, Ocean plankton populations as excitable media,, Bulletin of Mathematical Biology, 56 (1994), 981.  doi: 10.1016/S0092-8240(05)80300-3.  Google Scholar

[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, 347 (1994), 703.  doi: 10.1098/rsta.1994.0076.  Google Scholar

[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.  doi: 10.1051/mmnp/20105507.  Google Scholar

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