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Section: Scientific Foundations

Structured population modeling

The introduction of one or several structuring variables is important when one wants to more precisely describe the evolution of populations. Besides large time behavior this concerns transient behaviors, e.g., describing epidemic curves at the onset of an epidemic or the initial development of cell growth and tumors. It also depends on the final goals of modeling, i.e., mathematical analysis, numerical simulations or experiments, or both.

Spatial structures are widely used to assess the impact of heterogeneities or variable local densities in population dynamics, cf. [40] . This leads to systems of reaction diffusion for continuous models, or to networks of systems of ordinary differential equations in the discrete case. Discrete spatial models are also in order, cf. [60] , [64] . A new set of models is dedicated towards analyzing the transmission of parasites between populations distributed over distinct spatial models.

Multimodeling techniques could be useful when the model changes from one region to another. Methods presented in section 3.3 could then be used to give interface conditions.

Structured modeling in demography and epidemiology

In demography the most significant variable is the chronological age of individuals, cf. [51] , [63] . This age-structure although already intensively studied in our team in the past, cf. [3] , [9] , [65] , will be central in our future research. Discrete age structures are also in order.

Lot of models in epidemiology couple spatial and age structures to take care of the spreading rate of individuals together with the vital dynamics of the population. This structuration can lead to complex patterns formation and waves. A new problem we would like to investigate is the propagation phenomenom that, like in the classical reaction-diffusion framework, arises due to travelling waves. More specifically the description of the wave speed in function of the demography characteristics of the population is of particular interest for biologists.

In addition to spatial and age variables, other continuous structuring variables will be considered, i.e., size of individuals (fishing), weight, age of the disease for an infected individuals, cf.[9] .

For interacting populations or subpopulations additional discrete structures can be put forth. In the study of disease propagation (microparasites) usually a structure linked to the health status or parasitic state of individuals in the host population is used, i.e., SIS, SIR, SIRS, SEIRS models.

In previous works, rather strong assumptions were made on demographic and diffusion coefficients (e.g. identical or independent of age) to obtain qualitative results. In recent works it becomes possible to weaken these conditions, cf. [2] .

With M. Iannelli, we intend to study the impact of the spatial location (developed or underdeveloped country) on the propagation of an infectious disease (tuberculosis, AIDS ...). Then we have to model the way that the infectiveness rate or the recovery rate, which are dependent on the location, influence the dynamics of the infected population.

Various ways can be experienced. In a first approach we could assume that individuals are randomly distributed in space, cf. [40] , [42] . We would obtain a reaction-diffusion system whose reaction term would depend on space. In an alternate approach we could define patches where the population dynamics is governed by ordinary differential equation yielding large size systems of ODEs, cf. [47] .

Invasion processes in fragile isolated environments

In a series of joint works with F. Courchamp and G. Sugihara, e.g., [8] , we were concerned by ecological models designed to model the fate of native species living in isolated environments after the introduction of alien predator or competitor species, cf. [57] . Isolated environments we had in mind were mostly remote islands in Southern Indian Ocean, e.g., Kerguelen Archipelageo. Native species were seabirds while purposely or accidentally introduced species were small predators, i.e., domestic cats, or small rodents, i.e., rats and rabbits.

Singular systems of ODEs with unusual dynamics were derived. Typically finite or infinite time extinction of state variables may coexist, a Hopf bifurcation being also observed. This has important ecological implications and requires a detailed mathematical and numerical analysis.

It is also important to introduce a spatial structure, spatial heterogeneities being rather frequently observed in these environments (cf. [57] ; see also [47] ).

Control problems related to overcome finite or infinite time extinction of endangered native species emerge. In collaboration with H. Malchow we deal with the impact of a virus on an invasive population which is another way of controlling an invading species [50] . A comprehensive analysis is required. More specifically for spatially distributed systems with three populations the emergence of spatial heterogeneities and pattern formations must be understood.

Indirectly transmitted diseases

Host–parasite systems have been present in our team for many years with studies on viruses of carnivorous animals (foxes Vulpes vulpes , domestic cats Felis catus ), cf. [47] , [43] , or on macroprarasites (Diplectanum aequens ) infesting sea-bass (Dicentrarchus labrax ) populations, cf. [44] . It remains a main research theme through new developments of a collaborative effort with D. Pontier for zoonosis and anthropozoonosis, of a new collaboration with A. Callonec on pathogens of vineyards and of new proposals concerning interspecific transmission of toxoplasma with E. Fromont (mainland France) and P. Silan (French Guyana).

New important problems arise occurring in the generic setting of emerging diseases, invasion and persistence of parasites. Typically a parasite is transmitted from a population 1 wherein it is benign to a population 2 wherein it is lethal. It becomes important to assess and control the impact of the parasite on the host population 2.

This involves models dedicated to indirect transmission of parasites either via vectors, or through the contaminated ground or environment, cf. [43] , [59] , or through predation. In that case spatial structuration of species yields systems of reaction-diffusion equations posed on distinct spatial domains that may be coupled to ordinary differential equations, cf. [46] .

Pathogens of vineyards and transmision of toxoplasma within multicomponent host-pathogens systems yield complex biological systems. Analyzing their actual dynamics requires a dedicated effort that is to be developed in collaborative efforts with our biologists colleagues.

Direct movement of population

The classical chemotaxis model introduced by Keller and Segel (1971) demonstrates the emergence of endogenous patterns, including travelling waves. The appearance of advection-driven heterogeneity in relation to single and multispecies ecological interactions was studied by Levin (1977), Levin and Segel, (1976), Okubo (1980), Mimura and Murray (1978), Mimura and Kawasaki (1980), Mimura and Yamaguti (1982), and many other authors. These studies form a theoretical basis for modeling complex spatio-temporal dynamics observed in real systems.

Several field studies measuring characteristics of individual movement confirm the basic hypothesis about the dependence of acceleration on a stimulus gradient. For example, acceleration vectors of individuals in fish schools (Parrish and Turchin, 1997) and in swarms of flying insects (Okubo and Chiang, 1974; Okubo et al. , 1977) are directed towards the centroid of such dynamically stable formations. The acceleration increases with distance from this point, being maximal on the edges (where the density gradient is maximal) and equal to zero at the centroid position (where the density gradient is zero).

We will focus our attention to study chemotaxis phenomena of bacteria population, we also investigate the effect of prey-taxis and the effect of pursuit-evasion in prey-predators interactions.

We will consider a system of partial differential equations describing two spatially distributed population in “predator-prey” relationship with each other. Assuming that, locally (i.e., at each point and each instant), predators attack prey following the familiar Lotka-Volterra interaction term, we intend to investigate how the heterogeneities induced by the behavioural mechanisms affect the functional relationships between the population abundances.


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