arXiv:q-bio/0604035v1 [q-bio.PE] 27 Apr 2006 ON THE WEAK SOLUTIONS OF THE MCKENDRICK EQUATION ∗ RUI DIL ˜ AO † AND ABDELKADER LAKMECHE † † Nonlinear Dynamics Group, Instituto Superior T´ ecnico, Av. Rovisco Pais, 1049- 001 Lisbon, Portugal. (rui@sd.ist.utl.pt) Abstract. We develop the qualitative theory of the solutions of McKendrick partial differential equation of population dynamics. We calculate explicitly the weak solutions of the McKendrick equation and of the Lotka renewal integral equation with time and age dependent birth rate. Mor- tality modulus is considered age dependent. We show the existence of demography cycles. For a population with only one reproductive age class, independently of the stability of the weak solutions and after a transient time, the temporal evolution of the number of individuals of the population is always modulated by a time periodic function. The periodicity of the cycles is equal to the age of the reproductive age class, and the population retains the memory from the initial data through the amplitude of oscillations of the cycles. For a population with a continuous distribution of reproduc- tive age classes, we prove the existence of damped cycles. The periodicity of the damped cycles is associated with the age of the first reproductive age class. Damping increases as the dispersion of the fertility function around the age class with maximal fertility increases. In general, the period of the demographic cycles is associated with the time that a species takes to reach the reproductive maturity. Key words. McKendrick equation, renewal equation, demography cycles, periodic solutions, age-structure. AMS subject classifications. 92B05, 92D25 1. Introduction. The McKendrick equation describes the time evolution of a population structured in age. The first time it appears explicitly in the literature of population dynamics was in 1926 in a paper by McKendrick [18]. The McKendrick equation is a first order hyperbolic partial differential equation, with time and age as independent variables, together with a boundary condition that takes into account the births in a population. The existence of classical solutions of the McKendrick equation and their asymptotic time behavior is well established, and there exists in the literature of population dynamics a large number of surveys. See for example the books of Cushing [3], Webb [23], Iannelli [11], Keyfitz [13], Kot [15], Charlesworth [1], Metz and Diekman [19], Farkas [8] and Chu [2]. Despite the fact that the existence of classical solutions of the McKendrick equa- tion is well established, there is a lack of specific examples and no explicit solutions are known. This is due to the particular form of the boundary condition which is difficult to handle analytically, [15] and [8]. The McKendrick modeling approach is an attempt to overcome the deficiencies shown by the Malthusian or exponential growth law of population dynamics, intro- ducing the dependence on age into the mortality and fertility of a population. In its simpler form, the McKendrick model does not describe overcrowding effects, depen- dence on resources or, in human populations, economic and intraspecific interactions. To include these effects, several other models have been introduced and analyzed from the mathematical and numerical point of view, [23], [19], [4] and [22]. In demography, in order to make predictions about population growth, another approach is in general followed. After measuring birth and death rates by age classes or cohorts, demographers use the Leslie model [16], a discrete analog of the McKendrick equation, [13] and [14]. * This work has been partially supported by the POCTI Project /FIS/13161/1998 (Portugal) 1