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

We develop the qualitative theory of the solutions of the 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. Mortality 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 a 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 a population retains the memory from the initial data through the amplitude of oscillations. For a population with a continuous distribution of reproductive age classes, the amplitude of oscillation is damped. 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 demography cycles is associated with the time that a species takes to reach the reproductive maturity

Preface

The workshop of mathematics applied to life sciences (WMLS) was organized by the Laboratory of Biomathematics LBIOMATH and Mathematics Department of Djillali LIABES University (September 14–16th 2014, Sidi Bel-Abbes, Algeria). For our university, it was the first event dedicated to biomathematics. It was also an opportunity for a hundred participants to animate this multidisciplinary event, which could bring together mathematicians, statisticians, physicists, chemists, biologists and computer scientists as they worked together on natural phenomena. This volume contains selected works presented in the workshop WMLS. It concerns theoretical analysis and numerical simulations of various biological phenomena in morphogenesis, population dynamics and human diseases. The first work by R. Dilo is devoted to a reaction-diffusion model describing the development of fruit fly Drosophila. In the next paper A. Bouchnita et al. develop a hybrid discrete-continuous multi-scale model in order to study red blood cell production in the bone marrow. L. Matar Tine studies a size structured cell population model where the inverse problem for cell division rate in population dynamics is analyzed. Pulsed chemotherapy for heterogeneous tumor is considered by A. Lakmeche with coauthors. Conditions for disease eradication and persistence are found. The next two papers deal with modeling of leukemia. In the first one is a new model is suggested. This study is focused on the influence of death rates on the stability of equilibrium steady states of the system under consideration. The second of these papers treats a model inspired by those developed by Michor. In the next work, F. Boukhalfa et al. study existence and stability of solutions in a model of leishmania disease. A predator-prey model with state dependent impulse effects is analyzed by F. Charif with coauthors. Existence and stability of periodic solutions is proved. Finally, the last paper concerns a model of prion diseases constituted by impulsive differential equation and partial differential equation describing the production of monomers and evolution of polymers, respectively. The authors use the theory of evolution semi-group to prove existence of solutions

Mathematical Analysis of Visceral Leishmaniasis Model

In this work, we consider a mathematical model describing the dynamics of visceral leishmaniasis in a population of dogs D. First, we consider the case of constant total population D, this is the case where birth and death rates are equal, in this case transcritical bifurcation occurs when the basic reproduction number ℛ0 is equal to one, and global stability is shown by the mean of suitable Lyapunov functions. After that, we consider the case where the birth and death rates are different, if the birth rate is great than death rate the total dog population increases exponentially, while the infectious dogs I dies out if the basic reproduction number is less than one, if it is great than one then D goes to infinity. We also prove that the total population D will extinct for birth rate less than death rate. Finally we give numerical simulations

Periodicity and stability in a single-species model governed by impulsive differential equation

A periodic single-species model with periodic impulsive perturbations was investigated. By using Brouwer’s fixed point theorem and the Lyapunov function, sufficient conditions for the existence and global asymptotic stability of positive periodic solutions of the system were derived. Numerical simulations were presented to verify the feasibilities of our main results