By O. Diekmann R. Durrett K. P. Hadeler

The summer time university on arithmetic encouraged through Biology used to be held at Martina Franca, Apulia, Italy in 1997. This quantity offers 5 sequence of six lectures each one. the typical topic is the position of constitution in shaping temporary and supreme dynamics. however the form of constitution levels from spatial (hadeler and maini within the deterministic atmosphere, Durrett within the stochastic surroundings) to physiological (Diekmann) and order (Smith). each one contribution sketches the current scenario whereas, via together with a few wishful considering, pointing at open difficulties that deserve cognizance.

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2] CALSINA~ i ~ J. 1997. Asymptotic behaviour of a model of hierarchically structured population dynamics. J. Math. Biol. 35: 967987. ~ O. DIEKMANN, M. M. R. THIEME. 1989. Perturbation theory for dual semigroups. III Nonlinear Lipschitz continuous perturbations in the sun-reflexive case. Volterra Integro-DiJ]er ential Equations in Banach Spaces and Applications~ G. DA PRATO ~ M. |ANELLI,Pitman Research Notes in Maths 190: 67-89. [4] DIEKMANN, 0 . , M. J. R. THIEME. 1998. On the formulation and analysis of general deterministic structured population models.

X(~;~11f(x(~;~);~)x~,(x(~;~1) dT = R+ which~ by (70), again factors as the product of the function x, ) e ~(~) F ( ~ ; x ) and a measure with density aft) The condition that tile corresponding operator has spectral radius one therefore leads to the characteristic equation j~a1 2 e-~C~CW~C~)) ~(~) J:(~; ~) - 1 (73) Unless G ( i ) - G(~) --- constant, there is a strongly dominant real rootv. But when actua]ly this function is identically constant, we have a lattic:e case and periodic behaviour occurs.

T, to) with the imphcit convention that we think of I as defined on, at least, the interval specified by the two arguments of U. We assume that all initial time-dependence derives from the dependence on [, which we express mathematically in tile form of the assumption that for all. l(t, to) (91) (that is, if we translate both the input I and the time interval, nothing changes). )(t,0) m0) (d,) 11 (92) iJ where 7 is a t h i r d m o d e l i n g i n g r e d i e n t . Under appropriate conditions on A, u and 7 this equation (which is like a Volterra integral equation) will have for each positive measure mo a unique solution I, which we will denote by l(t;mo) to stress the dependence on m0.

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