May 2019, 39(5): 2933-2960. doi: 10.3934/dcds.2019122

Generalized linear models for population dynamics in two juxtaposed habitats

1. 

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France

2. 

USTHB, LAMNEDP, Faculté de Mathématiques, BP.32, El Alia, Bab Ezzouar, 16111 Alger, Algérie

3. 

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France

* Corresponding author: Alexandre Thorel

Received  September 2018 Revised  October 2018 Published  January 2019

Fund Project: The last author is supported by CIFRE contract 2014/1307 with Qualiom Eco company

In this work we introduce a generalized linear model regulating the spread of population displayed in a $ d $-dimensional spatial region $ \Omega $ of $ \mathbb{R}^{d} $ constituted by two juxtaposed habitats having a common interface $ \Gamma $. This model is described by an operator $ \mathcal{L} $ of fourth order combining the Laplace and Biharmonic operators under some natural boundary and transmission conditions. We then invert explicitly this operator in $ L^{p} $-spaces using the $ H^{\infty } $-calculus and the Dore-Venni sums theory. This main result will lead us in a later work to study the nature of the semigroup generated by $ \mathcal{L} $ which is important for the study of the complete nonlinear generalized diffusion equation associated to it.

Citation: Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Alexandre Thorel. Generalized linear models for population dynamics in two juxtaposed habitats. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2933-2960. doi: 10.3934/dcds.2019122
References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, second Edition, Monographs in Mathematics, 96, Birkhauser, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.

[3]

D. L. Burkholder, A geometrical characterisation of Banach spaces in which martingale dif-ference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. doi: 10.1214/aop/1176994270.

[4]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in po-pulation, Journal of Mathematical Biology, 12 (1981), 237-249. doi: 10.1007/BF00276132.

[5]

G. Da Prato and P. Grisvard, Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures et Appl., 54 (1975), 305-387.

[6]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.

[7]

A. FaviniR. LabbasS. MaingotK. Lemrabet and H. Sidibé, Resolution and Optimal Re-gularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Gene-ralizations, Discrete and Continuous Dynamical Systems, 33 (2013), 4991-5014. doi: 10.3934/dcds.2013.33.4991.

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[9]

P. Grisvard, Équations différentielles abstraites, Extrait des Annales scientifiques de l'École Normale Supérieure, 2 (1969), 311-395. doi: 10.24033/asens.1178.

[10]

P. Grisvard, Spazi di tracce e applicazioni, Rendiconti di Matematica, 5 (1972), 657-729.

[11]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhauser, 2006. doi: 10.1007/3-7643-7698-8.

[12]

H. Komatsu, Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.

[13]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376. doi: 10.1016/j.jmaa.2017.01.026.

[14]

K. LimamR. LabbasK. LemrabetA. Medeghri and M. Meisner, On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution, Journal of Differential Equations, 259 (2015), 2695-2731. doi: 10.1016/j.jde.2015.04.002.

[15]

J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications Mathé-matiques de l'I.H.É.S., 19 (1964), 5-68.

[16]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, Boston, Berlin, 1995.

[17]

S. G. Mikhlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 109 (1956), 701-703.

[18]

F. L. Ochoa, A generalized reaction-diffusion model for spatial structures formed by motile cells, BioSystems, 17 (1984), 35-50. doi: 10.1016/0303-2647(84)90014-5.

[19]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathe-matische Zeitschrift, 203 (1990), 429-452. doi: 10.1007/BF02570748.

[20]

J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in Lp-spaces, Hiroshima Math. J., 23 (1993), 161-192. doi: 10.32917/hmj/1206128381.

[21]

J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, J. Bastero, M. San Miguel (Eds.), Probability and Banach Spaces, Zaragoza, 1221 (1986), 195-222. doi: 10.1007/BFb0099115.

[22]

H. Triebel, Interpolation theory, function Spaces, differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.

show all references

References:
[1]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, second Edition, Monographs in Mathematics, 96, Birkhauser, 2011. doi: 10.1007/978-3-0348-0087-7.

[2]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.

[3]

D. L. Burkholder, A geometrical characterisation of Banach spaces in which martingale dif-ference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011. doi: 10.1214/aop/1176994270.

[4]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in po-pulation, Journal of Mathematical Biology, 12 (1981), 237-249. doi: 10.1007/BF00276132.

[5]

G. Da Prato and P. Grisvard, Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures et Appl., 54 (1975), 305-387.

[6]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.

[7]

A. FaviniR. LabbasS. MaingotK. Lemrabet and H. Sidibé, Resolution and Optimal Re-gularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Gene-ralizations, Discrete and Continuous Dynamical Systems, 33 (2013), 4991-5014. doi: 10.3934/dcds.2013.33.4991.

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[9]

P. Grisvard, Équations différentielles abstraites, Extrait des Annales scientifiques de l'École Normale Supérieure, 2 (1969), 311-395. doi: 10.24033/asens.1178.

[10]

P. Grisvard, Spazi di tracce e applicazioni, Rendiconti di Matematica, 5 (1972), 657-729.

[11]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhauser, 2006. doi: 10.1007/3-7643-7698-8.

[12]

H. Komatsu, Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.

[13]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376. doi: 10.1016/j.jmaa.2017.01.026.

[14]

K. LimamR. LabbasK. LemrabetA. Medeghri and M. Meisner, On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution, Journal of Differential Equations, 259 (2015), 2695-2731. doi: 10.1016/j.jde.2015.04.002.

[15]

J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications Mathé-matiques de l'I.H.É.S., 19 (1964), 5-68.

[16]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, Boston, Berlin, 1995.

[17]

S. G. Mikhlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 109 (1956), 701-703.

[18]

F. L. Ochoa, A generalized reaction-diffusion model for spatial structures formed by motile cells, BioSystems, 17 (1984), 35-50. doi: 10.1016/0303-2647(84)90014-5.

[19]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathe-matische Zeitschrift, 203 (1990), 429-452. doi: 10.1007/BF02570748.

[20]

J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in Lp-spaces, Hiroshima Math. J., 23 (1993), 161-192. doi: 10.32917/hmj/1206128381.

[21]

J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, J. Bastero, M. San Miguel (Eds.), Probability and Banach Spaces, Zaragoza, 1221 (1986), 195-222. doi: 10.1007/BFb0099115.

[22]

H. Triebel, Interpolation theory, function Spaces, differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.

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