Analysis of a non-linear model of populations structured by size
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Analysis of a non‑linear model of populations structured by size Anna Lo Grasso1 · Silvia Totaro1 Received: 3 March 2019 / Accepted: 14 January 2020 © The Author(s) 2020
Abstract The model we study deals with a population of marine invertebrates structured by size whose life stage is composed of adults and pelagic larvae such as barnacles contained in a local habitat. We prove existence and uniqueness of a continuous positive global mild solution and we give an estimate of it. We prove also that this solution is the strong solution of the problem. Keywords Size-structured population dynamics · Semigroup theory · Affine semigroups · Mild solution · Strong solution
1 Introduction A famous American zoologist of Swiss origin, Louis Agassiz, lived in the XIX century defined the barnacles like “little shrimps hanging from the rock with their heads, locked in a limestone house and kicked throwing food into their mouths”. They belong to a species of crustaceans, marine invertebrates whose life is composed of two stages, pelagic larvae and adult sessile. Barnacles have two larval stages: the first (nauplius) spends its time as part of zooplankton floating wherever the wind, waves, currents, and tides may take it. In this period, which lasts of about two weeks, it can eat and moult; hence the second
The authors dedicate this paper to the memory of Prof. Aldo Belleni Morante, who showed them the way of modelling and studying evolution problems by means of semigroup theory.
Communicated by Abdelaziz Rhandi. * Anna Lo Grasso [email protected] Silvia Totaro [email protected] 1
Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, University of Siena, Via Roma, 46, 53100 Siena, Italy
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stage is reached. At this point the nauplius metamorphoses into a non-feeding, more strongly swimming, cyprid larva. When an appropriate place is found, the cyprid larva cements itself head first to the surface and then undergoes metamorphosis into a juvenile barnacle. Typical barnacles develop six hard plates to surround and protect their bodies. For the rest of their lives they are cemented to the ground or even on shells of other animals, using their feathery legs, called cirrus, to capture plankton and gametes when spawning. They are usually found in the intertidal zone. Once metamorphosis is over and they have reached their adult form, barnacles will continue to grow, but not moult. Actually, they grow by adding new material to the ends of their heavily calcified plates. The model we study, following He and Wang [7], consists of a non-linear system of two equations, the first one models the density of the adults, whereas the second one involves larval evolution. The equations are connected by means of the boundary conditions of the evolution of the adults, which takes into account the larval evolution. The population dynamics of marine invertebrates such as barnacles, in which sessile adults and pelagic larvae are contained in a local area, are very much diffe
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