M. Marvá | Universidad de Alcala (original) (raw)

Papers by M. Marvá

Research paper thumbnail of Group defense promotes coexistence in interference competition: The Holling type IV competitive response

Mathematics and Computers in Simulation, 2022

Research paper thumbnail of Age-structure density-dependent fertility and individuals dispersal in a population model

Research paper thumbnail of Discrete Models of Disease and Competition

Discrete Dynamics in Nature and Society

The aim of this work is to analyze the influence of the fast development of a disease on competit... more The aim of this work is to analyze the influence of the fast development of a disease on competition dynamics. To this end we present two discrete time ecoepidemic models. The first one corresponds to the case of one parasite affecting demography and intraspecific competition in a single host, whereas the second one contemplates the more complex case of competition between two different species, one of which is infected by the parasite. We carry out a complete mathematical analysis of the asymptotic behavior of the solutions of the corresponding systems of difference equations and derive interesting ecological information about the influence of a disease in competition dynamics. This includes an assessment of the impact of the disease on the equilibrium population of both species as well as some counterintuitive behaviors in which although we would expect the outbreak of the disease to negatively affect the infected species, the contrary happens.

Research paper thumbnail of A Discrete Predator-Prey Ecoepidemic Model

Mathematical Modelling of Natural Phenomena

In this work we present a discrete predator-prey ecoepidemic model. The predatorprey interactions... more In this work we present a discrete predator-prey ecoepidemic model. The predatorprey interactions are represented by a discrete Leslie-Gower model with prey intra-specific competition. The disease dynamics follows a discrete SIS epidemic model with frequency-dependent transmission. We focus on the case of disease only affecting prey though the case of a parasite of the predators is also presented. We assume that parasites provoke density-and trait-mediated indirect interactions in the predator-prey community that occur on a shorter time scale. This is included in the model considering that in each time unit t here exist a number k of episodes of epidemic changes followed by a single episode of demographic change, all of them occurring separately. The aim of this work is examining the effects of parasites on the long-term prey-predators interactions. These interactions in the absence of disease are governed by the Leslie-Gower model. In the case of endemic disease they can be analyzed through a reduced predator-prey model which summarizes the disease dynamics in its parameters. Conditions for the disease to drive extinct the whole community are obtained. When the community keeps stabilized different cases of the influence of disease on populations sizes are presented.

Research paper thumbnail of Fast Dispersal in Semelparous Populations

Mathematical Modelling of Natural Phenomena, 2016

Research paper thumbnail of Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models

Mathematical Modelling of Natural Phenomena, 2013

In this work we review the aggregation of variables method for discrete dynamical systems. These ... more In this work we review the aggregation of variables method for discrete dynamical systems. These methods consist of describing the asymptotic behaviour of a complex system involving many coupled variables through the asymptotic behaviour of a reduced system formulated in terms of a few global variables. We consider population dynamics models including two processes acting at different time scales. Each process has associated a map describing its effect along its specific time unit. The discrete system encompassing both processes is expressed in the slow time scale composing the map associated to the slow one and the k-th iterate of the map associated to the fast one. In the linear case a result is stated showing the relationship between the corresponding asymptotic elements of both systems, initial and reduced. In the nonlinear case, the reduction result establishes the existence, stability and basins of attraction of steady states and periodic solutions of the original system with the help of the same elements of the corresponding reduced system. Several models looking over the main applications of the method to populations dynamics are collected to illustrate the general results.

Research paper thumbnail of Reduction of slow-fast asymptotically autonomous systems with applications to gradostat models

Ecological Complexity, 2013

ABSTRACT Two distinguishing features characterize the population dynamic models considered in the... more ABSTRACT Two distinguishing features characterize the population dynamic models considered in the present work. On the one hand, we consider several interacting organization levels associated to different time scales. On the other hand, the environment tends to be constant in the long term. The mathematical representation of these properties leads to slow-fast asymptotically autonomous systems. These characteristics add some realism in the models. However, the analytical study of this class of systems is generally hard to perform.Here we present a reduction technique that can be included among the so-called approximate aggregation methods. The existence of different time scales, together with the long term features, are used to build up a simpler system, which can be described by means of a lower number of state variables. The asymptotic behavior of the simplified model helps to study the original one.The reduction procedure is formulated in a general way. Following, two illustrations of asymptotically autonomous models with two time scales, in a gradostat, are given: a consumer–resource model and a competition model. Finally, a wider range of applications is suggested.

Research paper thumbnail of Reduction of Slow–Fast Periodic Systems with Applications to Population Dynamics Models

This work deals with the approximate reduction of a non-autonomous two time scales ordinary diffe... more This work deals with the approximate reduction of a non-autonomous two time scales ordinary differential equations system with periodic coefficients. We illustrate this technique with the analysis of a two patches periodic Lotka-Volterra predator-prey type model with a refuge for prey. Considering migrations between patches to be faster than local interaction allows us to study a three dimensional system by means of a two dimensional one.

Research paper thumbnail of Software tools used in math refresher courses at the University of Alcalá, Spain

... Chapter 16 DOI: 10.4018/978-1-60960-875-0.ch016 JG Alcázar University of Alcalá, Spain M. Mar... more ... Chapter 16 DOI: 10.4018/978-1-60960-875-0.ch016 JG Alcázar University of Alcalá, Spain M. Marvá University of Alcalá, Spain ... Related Content Preparing Students for PBL Lorna Uden, and Chris Beaumont (2006). Technology and Problem-Based Learning (pp. 87-103). ...

Research paper thumbnail of Reduction of slow–fast discrete models coupling migration and demography

Journal of Theoretical Biology, 2009

Cite this article as: M. Marvá, E. Sánchez, R. Bravo de la Parra and L. Sanz, Reduction of slow-f... more Cite this article as: M. Marvá, E. Sánchez, R. Bravo de la Parra and L. Sanz, Reduction of slow-fast discrete models coupling migration and demography

Research paper thumbnail of A density-dependent model describing age-structured population dynamics using hawk–dove tactics

Journal of Difference Equations and Applications, 2013

In this paper we deal with a nonlinear two-timescale discrete population model that couples age-s... more In this paper we deal with a nonlinear two-timescale discrete population model that couples age-structured demography with individual competition for resources. Individuals are divided into juvenile and adult classes, and demography is described by means of a density-dependent Leslie matrix. Adults compete to access resources; every time two adults meet, they choose either being aggressive (hawk) or non-aggressive (dove) to

Research paper thumbnail of Approximate aggregation of a two time scales periodic multi-strain SIS epidemic model: A patchy environment with fast migrations

Ecological Complexity, 2012

Research paper thumbnail of Reproductive Numbers for Nonautonomous Spatially Distributed Periodic SIS Models Acting on Two Time Scales

Acta Biotheoretica, 2012

In this work we deal with a general class of spatially distributed periodic SIS epidemic models w... more In this work we deal with a general class of spatially distributed periodic SIS epidemic models with two time scales. We let susceptible and infected individuals migrate between patches with periodic time dependent migration rates. The existence of two time scales in the system allows to describe certain features of the asymptotic behavior of its solutions with the help of a less dimensional, aggregated, system. We derive global reproduction numbers governing the general spatially distributed nonautonomous system through the aggregated system. We apply this result when the mass action law and the frequency dependent transmission law are considered. Comparing these global reproductive numbers to their non spatially distributed counterparts yields the following: adequate periodic migration rates allow global persistence or eradication of epidemics where locally, in absence of migrations, the contrary is expected.

Research paper thumbnail of A simple geometrical condition for the existence of periodic solutions of planar periodic systems. Applications to some biological models

Using invariant regions for proving the existence of periodic solutions of periodic ordinary diff... more Using invariant regions for proving the existence of periodic solutions of periodic ordinary differential equations is a common tool. However, describing such a region is, in general, far from trivial. In this paper we provide sufficient conditions for the existence of an invariant region for certain planar systems. Our method locates the solution, in the sense that the region we determine evolves with time around the solution in the phase plane. Also, unlike other approaches, the construction does not depend on upper or lower bounds with respect to time of the functions involved in the system. The criterion is formulated for a general planar periodic ODEs system, and therefore it can be applied in very different contexts. In particular, we use the criterion to improve on previously known results on Holling's type II predator-prey periodic model, and on the classic periodic competition model.

Research paper thumbnail of Group defense promotes coexistence in interference competition: The Holling type IV competitive response

Mathematics and Computers in Simulation, 2022

Research paper thumbnail of Age-structure density-dependent fertility and individuals dispersal in a population model

Research paper thumbnail of Discrete Models of Disease and Competition

Discrete Dynamics in Nature and Society

The aim of this work is to analyze the influence of the fast development of a disease on competit... more The aim of this work is to analyze the influence of the fast development of a disease on competition dynamics. To this end we present two discrete time ecoepidemic models. The first one corresponds to the case of one parasite affecting demography and intraspecific competition in a single host, whereas the second one contemplates the more complex case of competition between two different species, one of which is infected by the parasite. We carry out a complete mathematical analysis of the asymptotic behavior of the solutions of the corresponding systems of difference equations and derive interesting ecological information about the influence of a disease in competition dynamics. This includes an assessment of the impact of the disease on the equilibrium population of both species as well as some counterintuitive behaviors in which although we would expect the outbreak of the disease to negatively affect the infected species, the contrary happens.

Research paper thumbnail of A Discrete Predator-Prey Ecoepidemic Model

Mathematical Modelling of Natural Phenomena

In this work we present a discrete predator-prey ecoepidemic model. The predatorprey interactions... more In this work we present a discrete predator-prey ecoepidemic model. The predatorprey interactions are represented by a discrete Leslie-Gower model with prey intra-specific competition. The disease dynamics follows a discrete SIS epidemic model with frequency-dependent transmission. We focus on the case of disease only affecting prey though the case of a parasite of the predators is also presented. We assume that parasites provoke density-and trait-mediated indirect interactions in the predator-prey community that occur on a shorter time scale. This is included in the model considering that in each time unit t here exist a number k of episodes of epidemic changes followed by a single episode of demographic change, all of them occurring separately. The aim of this work is examining the effects of parasites on the long-term prey-predators interactions. These interactions in the absence of disease are governed by the Leslie-Gower model. In the case of endemic disease they can be analyzed through a reduced predator-prey model which summarizes the disease dynamics in its parameters. Conditions for the disease to drive extinct the whole community are obtained. When the community keeps stabilized different cases of the influence of disease on populations sizes are presented.

Research paper thumbnail of Fast Dispersal in Semelparous Populations

Mathematical Modelling of Natural Phenomena, 2016

Research paper thumbnail of Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models

Mathematical Modelling of Natural Phenomena, 2013

In this work we review the aggregation of variables method for discrete dynamical systems. These ... more In this work we review the aggregation of variables method for discrete dynamical systems. These methods consist of describing the asymptotic behaviour of a complex system involving many coupled variables through the asymptotic behaviour of a reduced system formulated in terms of a few global variables. We consider population dynamics models including two processes acting at different time scales. Each process has associated a map describing its effect along its specific time unit. The discrete system encompassing both processes is expressed in the slow time scale composing the map associated to the slow one and the k-th iterate of the map associated to the fast one. In the linear case a result is stated showing the relationship between the corresponding asymptotic elements of both systems, initial and reduced. In the nonlinear case, the reduction result establishes the existence, stability and basins of attraction of steady states and periodic solutions of the original system with the help of the same elements of the corresponding reduced system. Several models looking over the main applications of the method to populations dynamics are collected to illustrate the general results.

Research paper thumbnail of Reduction of slow-fast asymptotically autonomous systems with applications to gradostat models

Ecological Complexity, 2013

ABSTRACT Two distinguishing features characterize the population dynamic models considered in the... more ABSTRACT Two distinguishing features characterize the population dynamic models considered in the present work. On the one hand, we consider several interacting organization levels associated to different time scales. On the other hand, the environment tends to be constant in the long term. The mathematical representation of these properties leads to slow-fast asymptotically autonomous systems. These characteristics add some realism in the models. However, the analytical study of this class of systems is generally hard to perform.Here we present a reduction technique that can be included among the so-called approximate aggregation methods. The existence of different time scales, together with the long term features, are used to build up a simpler system, which can be described by means of a lower number of state variables. The asymptotic behavior of the simplified model helps to study the original one.The reduction procedure is formulated in a general way. Following, two illustrations of asymptotically autonomous models with two time scales, in a gradostat, are given: a consumer–resource model and a competition model. Finally, a wider range of applications is suggested.

Research paper thumbnail of Reduction of Slow–Fast Periodic Systems with Applications to Population Dynamics Models

This work deals with the approximate reduction of a non-autonomous two time scales ordinary diffe... more This work deals with the approximate reduction of a non-autonomous two time scales ordinary differential equations system with periodic coefficients. We illustrate this technique with the analysis of a two patches periodic Lotka-Volterra predator-prey type model with a refuge for prey. Considering migrations between patches to be faster than local interaction allows us to study a three dimensional system by means of a two dimensional one.

Research paper thumbnail of Software tools used in math refresher courses at the University of Alcalá, Spain

... Chapter 16 DOI: 10.4018/978-1-60960-875-0.ch016 JG Alcázar University of Alcalá, Spain M. Mar... more ... Chapter 16 DOI: 10.4018/978-1-60960-875-0.ch016 JG Alcázar University of Alcalá, Spain M. Marvá University of Alcalá, Spain ... Related Content Preparing Students for PBL Lorna Uden, and Chris Beaumont (2006). Technology and Problem-Based Learning (pp. 87-103). ...

Research paper thumbnail of Reduction of slow–fast discrete models coupling migration and demography

Journal of Theoretical Biology, 2009

Cite this article as: M. Marvá, E. Sánchez, R. Bravo de la Parra and L. Sanz, Reduction of slow-f... more Cite this article as: M. Marvá, E. Sánchez, R. Bravo de la Parra and L. Sanz, Reduction of slow-fast discrete models coupling migration and demography

Research paper thumbnail of A density-dependent model describing age-structured population dynamics using hawk–dove tactics

Journal of Difference Equations and Applications, 2013

In this paper we deal with a nonlinear two-timescale discrete population model that couples age-s... more In this paper we deal with a nonlinear two-timescale discrete population model that couples age-structured demography with individual competition for resources. Individuals are divided into juvenile and adult classes, and demography is described by means of a density-dependent Leslie matrix. Adults compete to access resources; every time two adults meet, they choose either being aggressive (hawk) or non-aggressive (dove) to

Research paper thumbnail of Approximate aggregation of a two time scales periodic multi-strain SIS epidemic model: A patchy environment with fast migrations

Ecological Complexity, 2012

Research paper thumbnail of Reproductive Numbers for Nonautonomous Spatially Distributed Periodic SIS Models Acting on Two Time Scales

Acta Biotheoretica, 2012

In this work we deal with a general class of spatially distributed periodic SIS epidemic models w... more In this work we deal with a general class of spatially distributed periodic SIS epidemic models with two time scales. We let susceptible and infected individuals migrate between patches with periodic time dependent migration rates. The existence of two time scales in the system allows to describe certain features of the asymptotic behavior of its solutions with the help of a less dimensional, aggregated, system. We derive global reproduction numbers governing the general spatially distributed nonautonomous system through the aggregated system. We apply this result when the mass action law and the frequency dependent transmission law are considered. Comparing these global reproductive numbers to their non spatially distributed counterparts yields the following: adequate periodic migration rates allow global persistence or eradication of epidemics where locally, in absence of migrations, the contrary is expected.

Research paper thumbnail of A simple geometrical condition for the existence of periodic solutions of planar periodic systems. Applications to some biological models

Using invariant regions for proving the existence of periodic solutions of periodic ordinary diff... more Using invariant regions for proving the existence of periodic solutions of periodic ordinary differential equations is a common tool. However, describing such a region is, in general, far from trivial. In this paper we provide sufficient conditions for the existence of an invariant region for certain planar systems. Our method locates the solution, in the sense that the region we determine evolves with time around the solution in the phase plane. Also, unlike other approaches, the construction does not depend on upper or lower bounds with respect to time of the functions involved in the system. The criterion is formulated for a general planar periodic ODEs system, and therefore it can be applied in very different contexts. In particular, we use the criterion to improve on previously known results on Holling's type II predator-prey periodic model, and on the classic periodic competition model.