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

Primary: 35L60, 92D25, 35C07; Secondary: 35Q92.

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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

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 aquaticfood 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.
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Keywords Aquatic food chain; hyperbolic reaction-diffusion model; traveling wave solutions.; Hopf-bifurcation

Citation: Elvira Barbera, Giancarlo Consolo, Giovanna Valenti. A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain. Mathematical Biosciences and Engineering, 2015, 12(3): 451-472. doi: 10.3934/mbe.2015.12.451

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