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A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics
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Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion
1. | Laboratoire MAE2D, Université A. ESSAADI, Faculté Polydisciplinaire - B.P 745, Larache, Maroc |
2. | Laboratoire LAREFAG, Ecole Nationale de Commerce et de Gestion, B.P. 1255, Tanger, Maroc |
3. | Sorbonne University, IRD, UMMISCO, F-93143, Bondy, France, LMDP, Université Cadi Ayyad, Faculté des Sciences de Semlalia, Marrakech, Maroc |
4. | IXXI, ENS Lyon, 46 allée d'Italie, 69364 Lyon cedex 7, France |
5. | Sorbonne University, IRD, UMMISCO, F-93143, Bondy, France |
The main goal of this paper is to adapt a class of complexity reduction methods called aggregation of variables methods to the construction of reduced models of two-time reaction-diffusion-chemotaxis models of spatially structured populations and to provide an error bound of the approximate dynamics. Aggregation of variables methods are general techniques that allow reducing the dimension of a mathematical dynamical system. Here we reduce a system of Partial Differential Equations to a simpler Ordinary Differential Equation system, provided that the evolution processes occur at two different time scales: a slow one for the demography and a fast one for migrations and chemotaxis, with a ratio $ \varepsilon>0 $ small enough. We give an approximation of the error between solutions of both original and reduced model for a generic function representing the demography. Finally, we provide an optimization of the error bound and validate numerically this result for a spatial inter-specific model with constant diffusion and population growth given by a logistic law in population dynamics.
References:
[1] |
W. Arendt, A. Grabosch, G. Greiner, U, MoustakasRainer, N. Schlotterbeck, U. Groh, H. P. Lotz and F. Neubrander, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, Springer, 2006.
doi: 10.1007/BFb0074922. |
[2] |
P. Auger and R. Bravo de la Parra, Methods of aggregation of variables in population dynamics Méthodes d'agrégation de variables en dynamique de population, C. R. Acad. Sci. Paris Sér. Ⅲ Sci. Vie, 323 (2000), 665 – 674.
doi: 10.1016/S0764-4469(00)00182-7. |
[3] |
P. Auger and J.-C. Poggiale,
Aggregation and emergence in systems of ordinary differential equations, Math. Comput. Modelling, 27 (1998), 1-21.
doi: 10.1016/S0895-7177(98)00002-8. |
[4] |
P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics, Lect. Notes Math., Springer Berlin Heidelberg, (2008), 209–263.
doi: 10.1007/978-3-540-78273-5_5. |
[5] |
P. M. Auger and R. Roussarie,
Complex ecological models with simple dynamics: From individuals to populations, Acta Biotheor., 42 (1994), 111-136.
doi: 10.1007/BF00709485. |
[6] |
T. Brochier, P. Auger, D. Thiao, A. Bah, S. Ly, T. Nguyen-Huu and P. Brehmer,
Can overexploited fisheries recover by self-organization? reallocation of fishing effort as an emergent form of governance, Mar. Policy, 95 (2018), 46-56.
doi: 10.1016/j.marpol.2018.06.009. |
[7] |
M. Eisenbach, Chemotaxis, World Scientific Publishing Company, 2004. |
[8] |
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handb. Numer. Anal., North-Holland, Amsterdam, VII (2000), 713–1020.
doi: 10.1086/phos.67.4.188705. |
[9] |
S. D. Fretwell, Populations in a Seasonal Environment, Princeton University Press, 1972.
![]() |
[10] |
M. F. Goy, M. S. Springer and G. L. Hazelbauer, Taxis and behaviour, elementary sensory systems in biology, in Receptors and Recognition, Ser. B, Chapman and Hall London, 5 (1978), 1–34. |
[11] |
W. Gurney and R. M. Nisbet, Ecological Dynamics, Oxford University Press, 1998.
![]() |
[12] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[13] |
X. Mora,
Semilinear parabolic problems define semiflows on $ {C}^k $ spaces, Trans. Amer. Math. Soc., 278 (1983), 21-55.
doi: 10.2307/1999300. |
[14] |
K. W. Morton and E. Sūli,
Finite volume methods and their analysis, IMA J. Numer. Anal., 11 (1991), 241-260.
doi: 10.1093/imanum/11.2.241. |
[15] |
V. N. Mose, T. Nguyen-Huu, P. Auger and D. Western,
Modelling herbivore population dynamics in the amboseli national park, kenya: Application of spatial aggregation of variables to derive a master model, Ecol. Complex., 10 (2012), 42-51.
doi: 10.1016/j.ecocom.2012.02.002. |
[16] |
A. Moussaoui, M. Bensenane, P. Auger and A. Bah,
On the optimal size and number of reserves in a multi-site fishery model. journal of biological systems, J. Biol. Systems, 23 (2015), 31-47.
doi: 10.1142/S0218339015500023. |
[17] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[18] |
T. H. Nguyen, T. Brochier, P. Auger, V. D. Trinh and P. Brehmer,
Competition or cooperation in transboundary fish stocks management: Insight from a dynamical model, J. Theoret. Biol., 447 (2018), 1-11.
doi: 10.1016/j.jtbi.2018.03.017. |
[19] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[20] |
E. Sánchez, P. Auger and J. C. Poggiale,
Two-time scales in spatially structured models of population dynamics: A semigroup approach, J. Math. Anal. Appl., 375 (2011), 149-165.
doi: 10.1016/j.jmaa.2010.08.014. |
[21] |
E. Sánchez, O. Arino, P. Auger and R. Bravo de la Parra,
A singular perturbation in an age-structured population model, SIAM J. Appl. Math., 60 (2000), 408-436.
doi: 10.1137/S0036139998337966. |
[22] |
E. Sánchez, R. Bravo de la Parra, P. Auger and P. Gómez-Mourelo,
Time scales in linear delayed differential equations, J. Math. Anal. Appl., 323 (2006), 680-699.
doi: 10.1016/j.jmaa.2005.10.074. |
[23] |
D. Tilman and P. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (MPB-30), Princeton University Press, 1997.
doi: 10.1515/9780691188362.![]() ![]() |
[24] |
R. Vilsmeier, F. Benkhaldoun and D. Hänel, Finite Volumes for Complex Applications II, Hermes Science Publications, Paris, 1999. |
show all references
References:
[1] |
W. Arendt, A. Grabosch, G. Greiner, U, MoustakasRainer, N. Schlotterbeck, U. Groh, H. P. Lotz and F. Neubrander, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, Springer, 2006.
doi: 10.1007/BFb0074922. |
[2] |
P. Auger and R. Bravo de la Parra, Methods of aggregation of variables in population dynamics Méthodes d'agrégation de variables en dynamique de population, C. R. Acad. Sci. Paris Sér. Ⅲ Sci. Vie, 323 (2000), 665 – 674.
doi: 10.1016/S0764-4469(00)00182-7. |
[3] |
P. Auger and J.-C. Poggiale,
Aggregation and emergence in systems of ordinary differential equations, Math. Comput. Modelling, 27 (1998), 1-21.
doi: 10.1016/S0895-7177(98)00002-8. |
[4] |
P. Auger, R. Bravo de la Parra, J.-C. Poggiale, E. Sánchez and T. Nguyen-Huu, Aggregation of variables and applications to population dynamics, Lect. Notes Math., Springer Berlin Heidelberg, (2008), 209–263.
doi: 10.1007/978-3-540-78273-5_5. |
[5] |
P. M. Auger and R. Roussarie,
Complex ecological models with simple dynamics: From individuals to populations, Acta Biotheor., 42 (1994), 111-136.
doi: 10.1007/BF00709485. |
[6] |
T. Brochier, P. Auger, D. Thiao, A. Bah, S. Ly, T. Nguyen-Huu and P. Brehmer,
Can overexploited fisheries recover by self-organization? reallocation of fishing effort as an emergent form of governance, Mar. Policy, 95 (2018), 46-56.
doi: 10.1016/j.marpol.2018.06.009. |
[7] |
M. Eisenbach, Chemotaxis, World Scientific Publishing Company, 2004. |
[8] |
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handb. Numer. Anal., North-Holland, Amsterdam, VII (2000), 713–1020.
doi: 10.1086/phos.67.4.188705. |
[9] |
S. D. Fretwell, Populations in a Seasonal Environment, Princeton University Press, 1972.
![]() |
[10] |
M. F. Goy, M. S. Springer and G. L. Hazelbauer, Taxis and behaviour, elementary sensory systems in biology, in Receptors and Recognition, Ser. B, Chapman and Hall London, 5 (1978), 1–34. |
[11] |
W. Gurney and R. M. Nisbet, Ecological Dynamics, Oxford University Press, 1998.
![]() |
[12] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[13] |
X. Mora,
Semilinear parabolic problems define semiflows on $ {C}^k $ spaces, Trans. Amer. Math. Soc., 278 (1983), 21-55.
doi: 10.2307/1999300. |
[14] |
K. W. Morton and E. Sūli,
Finite volume methods and their analysis, IMA J. Numer. Anal., 11 (1991), 241-260.
doi: 10.1093/imanum/11.2.241. |
[15] |
V. N. Mose, T. Nguyen-Huu, P. Auger and D. Western,
Modelling herbivore population dynamics in the amboseli national park, kenya: Application of spatial aggregation of variables to derive a master model, Ecol. Complex., 10 (2012), 42-51.
doi: 10.1016/j.ecocom.2012.02.002. |
[16] |
A. Moussaoui, M. Bensenane, P. Auger and A. Bah,
On the optimal size and number of reserves in a multi-site fishery model. journal of biological systems, J. Biol. Systems, 23 (2015), 31-47.
doi: 10.1142/S0218339015500023. |
[17] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[18] |
T. H. Nguyen, T. Brochier, P. Auger, V. D. Trinh and P. Brehmer,
Competition or cooperation in transboundary fish stocks management: Insight from a dynamical model, J. Theoret. Biol., 447 (2018), 1-11.
doi: 10.1016/j.jtbi.2018.03.017. |
[19] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[20] |
E. Sánchez, P. Auger and J. C. Poggiale,
Two-time scales in spatially structured models of population dynamics: A semigroup approach, J. Math. Anal. Appl., 375 (2011), 149-165.
doi: 10.1016/j.jmaa.2010.08.014. |
[21] |
E. Sánchez, O. Arino, P. Auger and R. Bravo de la Parra,
A singular perturbation in an age-structured population model, SIAM J. Appl. Math., 60 (2000), 408-436.
doi: 10.1137/S0036139998337966. |
[22] |
E. Sánchez, R. Bravo de la Parra, P. Auger and P. Gómez-Mourelo,
Time scales in linear delayed differential equations, J. Math. Anal. Appl., 323 (2006), 680-699.
doi: 10.1016/j.jmaa.2005.10.074. |
[23] |
D. Tilman and P. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (MPB-30), Princeton University Press, 1997.
doi: 10.1515/9780691188362.![]() ![]() |
[24] |
R. Vilsmeier, F. Benkhaldoun and D. Hänel, Finite Volumes for Complex Applications II, Hermes Science Publications, Paris, 1999. |



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