A discrete model of competing species sharing a parasite

In this work we develop a discrete model of competing species affected by a common parasite. We analyze the influence of the fast development of the shared disease on the community dynamics. The model is presented under the form of a two time scales discrete system with four variables. Thus, it becomes analytically tractable with the help of the appropriate reduction method. The 2-dimensional reduced system, that has the same the asymptotic behaviour of the full model, is a generalization of the Leslie-Gower competition model. It has the unfrequent property in this kind of models of including multiple equilibrium attractors of mixed type. The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number R0. In some cases, initial conditions decide among several exclusion or coexistence scenarios

Discrete-time staged progression epidemic models

In the Staged Progression (SP) epidemic models, infected individuals are classified into a suitable number of states. The goal of these models is to describe as closely as possible the effect of differences in infectiousness exhibited by individuals going through the different stages. The main objective of this work is to study, from the methodological point of view, the behavior of solutions of the discrete time SP models without reinfection and with a general incidence function. Besides calculating $\mathcal{R}_{0}$, we find bounds for the epidemic final size, characterize the asymptotic behavior of the infected classes, give results about the final monotonicity of the infected classes, and obtain results regarding the initial dynamics of the prevalence of the disease. Moreover, we incorporate into the model the probability distribution of the number of contacts in order to make the model amenable to study its effect in the dynamics of the disease

Discrete epidemic models with two time scales

The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviours, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time Susceptible-Exposed-Infectious-Recovered-Susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number R0. Then, we propose a general full model that includes sequentially the two processes at different time scales, and proceed to its analysis through a reduced model. The basic reproduction number R0 of the reduced system gives a good approximation of the R0 of the full model since it serves at analyzing its asymptotic behaviour. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches

Non-linear population discrete models with two time scales: re-scaling of part of the slow process

In this work we present a reduction result for discrete time systems with two time scales. In order to be valid, previous results in the field require some strong hypotheses that are difficult to check in practical applications. Roughly speaking, the iterates of a map as well as their differentials must converge uniformly on compact sets. Here, we eliminate the hypothesis of uniform convergence of the differentials at no significant cost in the conclusions of the result. This new result is then used to extend to nonlinear cases the reduction of some population discrete models involving processes acting at different time scales. In practical cases, some processes that occur at a fast time scale are often only measured at slow time intervals, notably mortality. For a general class of linear models that include such kind of processes, it has been shown that a more realistic approach requires the re-scaling of those processes to be considered at the fast time scale. We develop the same type of re-scaling in some nonlinear models and prove the corresponding reduction results. We also provide an application to a particular model of a structured population in a two-patch environment

Approximate reduction of nonlinear discrete models with two time scales

The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system

Aggregation methods in dynamical systems and applications in population and community dynamics

Approximate aggregation techniques allow one to transform a complex system involving many coupled variables into a simpler reduced model with a lesser number of global variables in such a way that the dynamics of the former can be approximated by that of the latter. In ecology, as a paradigmatic example, we are faced with modelling complex systems involving many variables corresponding to various interacting organization levels. This review is devoted to approximate aggregation methods that are based on the existence of different time scales, which is the case in many real systems as ecological ones where the different organization levels (individual, population, community and ecosystem) possess a different characteristic time scale. Two main goals of variables aggregation are dealt with in this work. The first one is to reduce the dimension of the mathematical model to be handled analytically and the second one is to understand how different organization levels interact and which properties of a given level emerge at other levels. The review is organized in three sections devoted to aggregation methods associated to different mathematical formalisms: ordinary differential equations, infinite-dimensional evolution equations and difference equations

Discrete Models of Disease and Competition

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

Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models

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

Land use change in a Mediterranean metropolitan region and its periphery: Assessment of conservation policies through CORINE land cover data and Markov models

Sustainable territorial management requires reliable assessment of the impact of conservation policies on landscape structure and dynamics. Euro-Mediterranean regions present a remarkable biodiversity which is linked in part to traditional land use practices and which is currently threatened by global change. The effectiveness of one-decade conservation policies against land use changes was examined in Central Spain (Madrid Autonomous Community). A Markov model of landscape dynamics was parameterized with CORINE Land Cover information and transition matrices were obtained. The methods were applied in both protected and unprotected areas to examine whether the intensity and direction of key land use changes —urbanisation, agricultural intensification and land abandonment— differed significantly depending on the protection status of those areas. Protected areas experienced slower rates of agricultural intensification processes and faster rates of land abandonment, with respect to those which occurred in unprotected areas. It illustrates how simple mathematical tools and models —parameterized with available data— can provide to managers and policy makers useful indicators for conservation policy assessment and identification of land use transitions

Evaluation of Bacillus thuringiensis Pathogenicity for a Strain of the Tick, Rhipicephalus microplus, Resistant to Chemical Pesticides

The pathogenicity of four native strains of Bacillus thuringiensis against Rhipicephalus (Boophilus) microplus (Canestrine) (Acari: Ixodidae) was evaluated. A R. microplus strain that is resistant to organophosphates, pyrethroids, and amidines, was used in this study. Adult R. microplus females were bioassayed using the immersion test of Drummond against 60 B. thuringiensis strains. Four strains, GP123, GP138, GP130, and GP140, were found to be toxic. For the immersion test, the total protein concentration for each bacterial strain was 1.25 mg/ml. Mortality, oviposition, and egg hatch were recorded. All of the bacterial strains had significant effects compared to the controls, but no significant differences were seen between the 4 strains. It is evident that these B. thuringiensis strains have a considerable detrimental effect on the R. microplus strain that is resistant to pesticides