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July  2021, 14(7): 2163-2181. doi: 10.3934/dcdss.2021055

Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion

Anouar El Harrak 1,2,, , Amal Bergam 1, , Tri Nguyen-Huu 3,4, , Pierre Auger 5, and  Rachid Mchich 2,

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

* Corresponding author: anouarelharrak1@gmail.com

Received  July 2019 Revised  September 2020 Published  July 2021 Early access  May 2021

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.

Keywords: Aggregation of variables, two time scales, approximate dynamics, spatially structured population dynamics, reaction-diffusion-chemotaxis equations.
Mathematics Subject Classification: Primary: 37M99, 37N25; Secondary: 35K55.
Citation: Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger, Rachid Mchich. Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2163-2181. doi: 10.3934/dcdss.2021055
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.

Figure 1.  Plot of the numerical solution of the perturbed problem (11), (A) and (C), and its approximate solution using aggregation of variables methods, (B) and (D), at two times $ t = 0.2 $ and $ t = 25 $ for $ \varepsilon = 1e-1 $
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Figure 2.  Plot of total population of the global model and its approximate solution using aggregation of variables methods for for $ \varepsilon = 1e-1 $. $ K_T: = \int_{\Omega}K(x)dx $ stands for the total carrying capacity of the environment and $ K^\ast $ for the new homogeneous one
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Figure 3.  Plot of errors, $ |N_{\varepsilon}(t)-N(t)| $, (A), and $ {\Vert n_{\varepsilon}(.,t)-\lambda(.)N(t) \Vert_\infty} $ with $ t\geq 1 $, (B), with respect to time $ t $ for different values of $ \varepsilon $; $ \varepsilon = 1e-1 $, $ \varepsilon = 1e-2 $, $ \varepsilon = 1e-3 $, and $ \varepsilon = 1e-4 $
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