• Nessuna parola chiave trovata

In the last decades the mathematical interest in modelling biological structures and phenomena have seen an unusual development, mostly due to the increase of themes coming also from the medical field, hence the classical models experienced various improvements to reproduce more specific situations.
In the present work we deal with some structured models for dynamics of two Corallium rubrum populations.
Red Coral (Corallium rubrum L 1758) is a colonial anthozoan endemic to the Mediter-
ranean sea. Due to high economical value of its carbonate skeleton (used in jewelry),
this species has been harvested since ancient times. For this reason in the last two
decades a reduction of the overall fishing yield by 2/3 has been recorded (Santangelo and Abbiati, 2001). Therefore in the last years two works (Iannelli, Santangelo, Bramanti, 2007 2009) has been conducted to modelling a red coral population located in Calafuria (LI, Italy) and its sensitivity to environmental factors. concerning two different populations, located in Portofino (Ligurian coast, Italy) and Cap de Creus (Costa Brava, Spain), reveal some differences between these populations and the old one. Then another demographic study (Vielmini, Santangelo 2009)
The aim of this study is to develop several models that result more suitable for the data coming from Portofino and Cap de Creus. These datasets reveal an high variability concerning the noticed growth rates of the colonies. Furthermore they appear not homogeneous (probably for the few quantity of samples) and present a lacking of information at different levels. In particular for several colonies we do not have fertility parameters and there are none informations about several age classes, indeed, during the samples of the bigger colonies, the smaller ones get lost. Hence we used this data as best as we could and try to follow the noticed variability towards the modelling process.
The first chapter is devoted to present the most part of data and its analysis, performed with R program. We have several type of data, certain containing growth informations, in term of diameter or circular annual crown, and others about reproduction (i.e. number of planulae produced by a colony). Concerning the growth of the basal section, we infer, analyzing a collection of few selected data, that is nearly linear with the year. Furthermore we notice a peculiar variability, indeed we report some colonies (from both locations) that present a greatly bigger growth rate. However we construct an age distribution for the two populations using the entire datasets, in this
way we suppose to obtain the surviving parameters, unfortunately these distributions appear affected by some noise, caused by a lack of data for certain age classes. Hence the obtained distributions result not satisfactory and need to be fitted to a survival function. On the other hand reproductive parameters presented several lacks and we use them as much as possible to determine the required parameters.
In the second chapter several models are presented and analyzed: we begin with discrete ones that should result more suitable and are used in the two previous works of Iannelli, Santangelo and Bramanti; indeed reproduction of red coral occur once for year in early summer and a discrete model with a step of one year reproduce this phenomena as the best. In this framework we improve the older models, to follow the high variability of the income data, indeed we add another parameter, to take account of the growth level of a colony within its age class.
Hence we construct a continuous model that results equivalent to a system of Volterra equations (Iannelli , 1994), even in this case we add a size parameter. Hence we prove an existence and unicity result for integral equation solutions, using a fixed point ar- gument. The analysis of the continuous model is conducted with respect to Volterra equations, their solutions behaviour appear connected to the Laplace transform of the integral kernel through the Paley-Wiener theorem (Paley Wiener, 1934). Hence we re- conduct the previous problem to several equations involving the Laplace transform and prove a stability result.
Finally we construct a simple spatial model, considering the spread of the planulae among a limited interval or a two dimensional disk, this lead to a discrete dynamical system that contain a diffusion equation throughout the steps. We find a general solutions of the diffusion equation (through Fourier and Fourier-Bessel series) with Neumann and Dirichlet boundary conditions and therefore the solutions of the dis- crete system.
In the last chapter we perform some simulations and computations concerning the discrete model defined above and the stability of its steady states. In this framework the non homogeneus of the data caused a loss of accuracy in the determination of survival parameters. Moreover the lack of complete and organic informations concerning the reproduction process, compel us to keep cutoff function and some reproduction parameters, from the old model concerning Calafuria populations. Hence the obtained results are not so explanatory.
In conclusion, we have performed several new models, that generalized those used in the previous works (Santangelo Iannelli Bramanti 2007 2009). These should reproduce better the variability of these two red coral populations; unfortu- nately due to incompleteness and non homogeneous of the income data we didn’t have the instrument to confirm this in full.