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We develop the qualitative theory of the solutions of the McKendrick partialdifferential equation of population dynamics. We calculate explicitly the weaksolutions of the McKendrick equation and of the Lotka renewal integral equationwith time and age dependent birth rate. Mortality modulus is considered agedependent. We show the existence of demography cycles. For a population withonly one reproductive age class, independently of the stability of the weaksolutions and after a transient time, the temporal evolution of the number ofindividuals of a population is always modulated by a time periodic function.The periodicity of the cycles is equal to the age of the reproductive ageclass, and a population retains the memory from the initial data through theamplitude of oscillations. For a population with a continuous distribution ofreproductive age classes, the amplitude of oscillation is damped. Theperiodicity of the damped cycles is associated with the age of the firstreproductive age class. Damping increases as the dispersion of the fertilityfunction around the age class with maximal fertility increases. In general, theperiod of the demography cycles is associated with the time that a speciestakes to reach the reproductive maturity.

On the Weak Solutions of the McKendrick Equation: Existence of Demography Cycles