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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), 119-153. 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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