Home, Research, Editions, Vita, MiscLinks
PUBLICATIONS:
Rafael Bravo de la Parra
Catedrático de Universidad de
Matemática Aplicada
U.D.
Matemáticas, Universidad de Alcalá,
28871 Alcalá de Henares, Madrid, Spain
rafael.bravo@uah.es
BOOK
CHAPTERS
- P. Auger, R. Bravo de la Parra, J.C. Poggiale,
E. Sánchez, T. Nguyen Huu, Aggregation of variables and applications to
population dynamics. In: P. Magal, S. Ruan (Eds.), Structured Population Models in
Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936,
Mathematical Biosciences Subseries. Springer, Berlin, 2008, p. 209-263.
JOURNAL
ARTICLES
- R. Bravo de la Parra, L. Sanz, Discrete epidemic models with two time scales, Advances in Difference Equations, 2021:478, 2021. doi.org/10.1186/s13662-021-03633-0. (PDF)
- L. Sanz, R. Bravo de la Parra, Stochastic matrix metapopulation
models with fast migration: re-scaling survival to the fast scale, Ecological Modelling, 418, 2020, 108829. doi.org/10.1016/j.ecolmodel.2019.108829. (PDF)
- R. Bravo de la Parra, L. Sanz, A Discrete Model of Competing Species
Sharing a Parasite,
Discrete and Continuous Dynamical
Systems Series B, 25(6):2121-2142, 2020. Doi:10.3934/dcdsb.2019204.
(PDF)
- L. Sanz, R. Bravo de la Parra,
M. Marvá, E. Sánchez, Non-linear Population Discrete Models with Two Time Scales: Re-scaling
of Part of the Slow Process, Advances in Difference Equations, 2019:401, 2019. doi.org/10.1186/s13662-019-2303-1.
(PDF)
- M. Marvá, R. Bravo
de la Parra, E. Venturino, Modelling the Role of Opportunistic
Diseases in Coinfection, Mathematical Modelling of Natural Phenomena, 13(3): 28, 2018. doi.org/10.1051/mmnp/2018034. (PDF)
- R. Bravo de la Parra, M. Marvá, E. Sánchez,
L. Sanz, Discrete
Models of Disease and Competition, Discrete Dynamics in Nature and Society,
2017: 5310837, 2017. doi.org/10.1155/2017/5310837. (PDF)
- R. Bravo de la Parra, M. Marvá,
E. Sánchez, L. Sanz, A
Discrete Predator-Prey Ecoepidemic Model, Mathematical Modelling of
Natural Phenomena, 12(2):116-132, 2017. (PDF)
- R. Bravo de la
Parra, M. Marvá, F. Sansegundo, Fast Dispersal
in Semelparous Populations, Mathematical Modelling
of Natural Phenomena, 11(4):121-135,
2016. (PDF)
- M. Marvá, E. Venturino, R. Bravo de la Parra, A Time Scales Approach to Coinfection
by Opportunistic Diseases, Journal of Applied
Mathematics, 2015:275485, 2015. (PDF)
- M. Marvá, R. Bravo
de la Parra, Coexistence and superior competitor exclusion in
the Leslie–Gower competition model with fast dispersal, Ecological Modelling, 306:247-256, 2015. (PDF)
- M. Marvá, J.G. Alcázar, J.-C. Poggiale, R.
Bravo de la Parra, A simple geometrical condition for the existence
of periodic solutions of planar periodic systems. Applications to some
biological models, Journal of Mathematical Analysis and
Applications, 423(2):1469-1479, 2015. (PDF)
- M. Marvá, R. Bravo
de la Parra, Reduction
of nonautonomous population dynamics models with two
time scales, Acta Biotheoretica,
62(3): 285-303, 2014. (PDF)
- E. Sánchez, H. Hbid, R.
Bravo de la Parra, Mathematical
Analysis of a Population Model with an Age-weight Structured Two-stage
Life History: Asymptotic Behaviour of Solutions, Journal of Evolution
Equations, 14(3): 603-616,
2014. (PDF)
- R. Bravo de la Parra, M. Marvá, E. Sánchez, L. Sanz, Reduction of Discrete Dynamical
Systems with Applications to Dynamics Population Models, Mathematical
Models of Natural Phenomena, 8(6): 107–129, 2013. (PDF)
- O. Angulo, R. Bravo de la Parra, J.C.
López-Marcos, M.A. Zavala, Stand dynamics
and tree coexistence in an analytical structured model: the role of
recruitment, Journal of Theoretical Biology, 333(1):91-101, 2013. (PDF)
- M. Marvá, R. Bravo
de la Parra, J.-C. Poggiale, Reduction of slow-fast asymptotically
autonomous systems with applications to gradostat
models, Ecological
Complexity, 14(1):75-84,
2013. (PDF)
- M. Marvá, A. Moussaoui,
R. Bravo de la Parra, P. Auger, A density dependent model describing age
structured population dynamics using hawk-dove tactics, Journal of Difference Equations and Applications, 19(6):1022-1034,
2013. (PDF)
- M. Marvá, J.-C. Poggiale, R. Bravo de la Parra, Reduction of slow-fast periodic
systems with applications to population dynamics models, Mathematical
Models and Methods in Applied Sciences, 22(12):1250025, 2012. (PDF)
- M. Marvá, R. Bravo
de la Parra, P. Auger, Reproductive
Numbers for Nonautonomous Spatially Distributed Periodic SIS Models Acting
on Two Time Scales, Acta Biotheoretica,
60(1-2):139-154, 2012. (PDF)
- M. Marvá, R. Bravo de
la Parra, J.-C. Poggiale, Approximate aggregation of a two time scales periodic multi-strain SIS epidemic
model: A patchy environment with fast migrations, Ecological Complexity, 10(1):34-41, 2012. (PDF)
- T. Nguyen Huu, R.
Bravo de la Parra, P. Auger, Approximate
aggregation of linear discrete models with two time-scales:
re-scaling slow processes to the fast scale, Journal of Difference
Equations and Applications, 17(4):621-635,
2011. (PDF)
- P. Ruiz Benito, J.A. Cuevas, R. Bravo de la
Parra, J.M. García del Barrio, M.A. Zavala, Land use change in a Mediterranean metropolitan region
and its periphery: assessment of conservation policies through CORINE Land
Cover data and Markov models, Forest Systems, 19(3):315-328, 2010. (PDF)
- D. Nguyen Ngoc, R. Bravo de la Parra, M.A.
Zavala, P. Auger, Competition
and species coexistence in a metapopulation model: Can fast asymmetric
migration reverse the outcome of competition in a homogeneous environment?, Journal of
Theoretical Biology, 266:256-263,
2010. (PDF)
- A. Kebir, S. Ben Miled ,
M.L. Hbid, R. Bravo de la Parra, Effects of density dependent sex
allocation on the dynamics of a simultaneous hermaphroditic population:
Modelling and analysis, Journal of Theoretical Biology, 263:521–529, 2010. (PDF)
- M. Marvá, E. Sánchez, R.
Bravo de la Parra, L. Sanz, Reduction
of slow–fast discrete models coupling migration and demography, Journal
of Theoretical Biology, 258(3):371-379,
2009. (PDF)
- H. Hbid, A. Kacha, R. Bravo de la Parra, Mathematical study of a bacteria-fish
model with level of infection structure, Nonlinear Analysis Series
B: Real World Applications, 10(3):1662-1678,
2009. (PDF)
- P. Auger, R. Bravo de la Parra, J.C. Poggiale, E. Sánchez, L. Sanz, Aggregation methods in dynamical
systems variables and applications in population and community dynamics, Physics
of Life Reviews, 5(2):79-105,
2008. (PDF)
- L. Sanz, R. Bravo de la Parra, E. Sánchez,
Approximate Reduction of Non-Linear
Discrete Models with Two Time Scales, Journal of Difference
Equations and Applications, 14(6):607-627,
2008. (PDF)
- H. Hbid, E. Sánchez, R.
Bravo de la Parra, State-Dependent
Delays Associated to Threshold Phenomena in Structured Population Dynamics, Mathematical
Models and Methods in Applied Sciences, 17(6):877-900, 2007. (PDF)
- El Abdllaoui, P. Auger,
B.W. Kooi, R. Bravo de la Parra, R. Mchich, Effects
of density-dependent migrations on stability of a two-patch predator-prey
model, Mathematical
Biosciences, 210(1):335-354, 2007. (PDF)
- L. Sanz, R. Bravo de la Parra, Approximate reduction of multiregional
models with environmental stochasticity, Mathematical Biosciences,
206(1):134-154, 2007. (PDF)
- M.A. Zavala, O. Angulo, R. Bravo de la Parra,
J.C. López-Marcos, An analytical model of stand dynamics as a function
of tree growth, mortality and recruitment: The shade tolerance-stand
structure hypothesis revisited, Journal of Theoretical Biology, 244(3):440-450, 2007. (PDF)
- E. Sánchez, R. Bravo de la Parra, P.
Auger, P. Gómez Mourelo, Time scales in linear delayed
differential equations, Journal of Mathematical Analysis and
Applications, 323(1):680-699,
2006. (PDF)
- P. Auger, B.W. Kooi, R.
Bravo de la Parra, J.C. Poggiale, Bifurcation analysis of a predator-prey
model with predators using hawk and dove tactics, Journal of Theoretical
Biology, 238(3):587-607,
2006. (PDF)
- M.A. Zavala, I.R.
Urbieta, R. Bravo de la Parra, O. Angulo, Modelos
de proceso de la producción y dinámica del bosque mediterráneo, Invest Agrar: Sist Recur For, 14(3):482-496,
2005. (PDF)
- M.A. Zavala, R. Bravo de la Parra, A mechanistic model of tree
competition and facilitation for Mediterranean forests: scaling from leaf
physiology to stand dynamics, Ecological Modelling, 188(1):76-92, 2005. (PDF)
- Lett, P. Auger, R. Bravo de la Parra, Migration Frequency and the
Persistence of Host-Parasitoid Interactions, Journal
of Theoretical Biology, 221:639-654,
2003. (PDF)
- L. Sanz, A. Blasco, R.
Bravo de la Parra, Approximate
reduction of multi-type Galton-Watson processes with two
time scales, Mathematical Models and Methods in
Applied Sciences, 13(4):491-525,
2003. (PDF)
- L. Sanz, R. Bravo de la Parra, The Reliability of Approximate
Reduction Techniques in Population Models with Two Time Scales, Acta Biotheoretica, 50(4):297-322, 2002. (PDF)
- R. Mchich, P. Auger, R.
Bravo de la Parra, N. Raïssi, Dynamics of a fishery on two fishing
zones with fishing stock dependent migrations: aggregation and control, Ecological
Modelling, 158(1-2):51-62,
2002. (PDF)
- A. Blasco, L. Sanz, R.
Bravo de la Parra, Approximate
Reduction of Multiregional Birth-Death Models with Fast Migration, Mathematical
and Computer Modelling, 36:47-65,
2002. (PDF)
- P. Auger, R. Bravo de la Parra, S. Morand, E. Sánchez, A predator-prey model with predators using hawk
and dove tactics, Mathematical Biosciences, 177&178: 185-200, 2002. (PDF)
- A. Blasco, L. Sanz, P.
Auger, R. Bravo de la Parra, Linear Discrete Population Models with Two Time Scales
in Fast Changing Environments II: Non-Autonomous Case, Acta Biotheoretica, 50(1):15-38, 2002. (PDF)
- A. Blasco, L. Sanz, P.
Auger, R. Bravo de la Parra, Linear Discrete Population Models with Two Time
Scales in Fast Changing Environments I: Autonomous Case, Acta Biotheoretica, 49:261-276, 2001. (PDF)
- L. Sanz, R. Bravo de la Parra, Time scales in a non-autonomous linear
discrete model, Mathematical Models and Methods in Applied
Sciences, 11(7):1203-1235, 2001. (PDF)
- P. Auger, R. Bravo de la Parra, E.
Sánchez, Behavioural dynamics of two
interacting hawk-dove populations, Mathematical Models and Methods in
Applied Sciences, 11(4):1-18,
2001. (PDF)
- L. Sanz, R. Bravo de la Parra, Time scales in stochastic
multiregional models, Nonlinear Analysis: Real World
Applications, 1:89-122,
2000. (PDF)
- P. Auger, R. Bravo de la Parra, Methods of aggregation of variables in
population dynamics, C. R. Acad. Sci. Paris, Sciences de la
vie, 323:665-674,
2000. (PDF)
- Pontier, P. Auger, R. Bravo de la Parra, E. Sánchez, The impact of behavioural plasticity
at individual level on domestic cat population dynamics, Ecological
Modelling, 133:117-124,
2000. (PDF)
- S. Charles, R. Bravo de la Parra, J. P.
Mallet, H. Persat, P. Auger, Annual spawning migrations in modeling brown trout population dynamics inside an
arborescent river network, Ecological Modelling, 133:15-31, 2000. (PDF)
- R. Bravo de la Parra, O. Arino, E. Sánchez, P. Auger, A Model for an Age-Structured
Population with Two Time Scales, Mathematical and Computer Modelling,
31:17-26, 2000. (PDF)
- O. Arino, E. Sánchez, R.
Bravo de la Parra, P. Auger, A
Singular Perturbation in an Age-Structured Population Model, SIAM
Journal on Applied Mathematics, 60:408-436,
1999. (PDF)
- R. Bravo de la Parra, E. Sánchez, O. Arino, P. Auger, A discrete model with density
dependent fast migration, Mathematical Biosciences, 157(1-2):111-146, 1999. (PDF)
- L. Sanz, R. Bravo de la Parra, Variables aggregation in a time discrete linear
model, Mathematical
Biosciences, 157(1-2):91-109,
1999. (PDF)
- S. Charles, R. Bravo de la Parra, J. P.
Mallet, H. Persat, P. Auger, A Density Dependent Model Describing
Salmo trutta Population Dynamics in an arborescent river network: effects
of dams and channelling, C. R. Acad. Sci. Paris, Sciences de la
vie, 321:979-990, 1998. (PDF)
- L. Sanz, R. Bravo de la Parra, Variables Aggregation in Time Varying
Discrete Systems, Acta Biotheoretica,
46:273-297, 1998. (PDF)
- S. Charles, R. Bravo de la Parra, J. P.
Mallet, H. Persat, P. Auger, Population Dynamics Modelling in an Hierarchical Arborescent River Network: An Attempt
with Salmo trutta, Acta Biotheoretica,
46(3):223-234, 1998. (PDF)
- O. Arino, M. L. Hbid, R. Bravo de la Parra, A mathematical model of growth of
population of fish in the larval stage: Density-dependent effects, Mathematical
Biosciences, 150:1-20,
1998. (PDF)
- O. Arino, E. Sánchez, R.
Bravo de la Parra, A Model
of an Age-Structured Population in a Multipatch
Environment, Mathematical and Computer Modelling, 27(4):137-150, 1998. (PDF)
- P. Auger, R. Bravo de la Parra, E.
Sánchez, Hawk-Dove Game and Competition
Dynamics, Mathematical
and Computer Modelling, 27(4):89-98,
1998. (PDF)
- R. Bravo de la Parra, E. Sánchez, Aggregation
Methods in Population Dynamics Discrete Models, Mathematical and Computer
Modelling, 27(4):23-39,
1998. (PDF)
- Sánchez, P. Auger, R. Bravo de la Parra, Influence of Individual Agressiveness on the Dynamics of Competitive
Populations, Acta Biotheoretica, 45:321-333, 1997. (PDF)
- R. Bravo de la Parra, E. Sánchez, P. Auger, Time
Scales in Density Dependent Discrete Models, Journal of Biological
Systems, 5:111-129, 1997. (PDF)
- Sánchez, R. Bravo de la Parra, P. Auger, Linear Discrete Models with Different Time-Scales, Acta Biotheoretica,
43:465-479, 1995. (PDF)
- R. Bravo de la Parra, P. Auger, E. Sánchez, Aggregation
Methods in Discrete Models, Journal of Biological Systems, 3:603-612, 1995. (PDF)
- R. Bravo de la Parra, A. Fernández, On the Valiron Deficiencies of Oriented Functions, Mathematica
Scandinavica, 69:31-44,
1991. (PDF)
- R. Bravo de la
Parra,
P. Jiménez-Guerra, Linear Operators and Vector Integrals, Mathematica Japonica, 36:255-262,
1991. (PDF)
- R. Bravo de la Parra, Un théorème de la valeur moyenne pour des intégrales
bilinéaires, Les Annales
des Sciences Mathématiques du Québec, 12:179-187, 1988. (PDF)
- R. Bravo de la
Parra,
Espacios Lp y teoremas de
convergencia para integrales vectoriales, Collectanea Mathematica,
38:65-76, 1987. (PDF)