Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler ʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence...

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

We show that the result of Kato on the existence of a semigroup solving the Kolmogorov system of equations in l₁ can be generalized to a larger class of the so-called Kantorovich-Banach spaces. We also present a number of related generation results that can be proved using positivity methods, as well as some examples.

Salmonella is one of the major sources of toxi-infection in humans in France and United States. The incidence of human salmonellosis has considerably increased over the past 20 years and this can be largely attributed to epidemics of S. enteritidis phage type 4 in poultry in numerous countries. In this article, we formulate and analyse a model in which the transmission of the disease is determined by contact between hens and Salmonella in the environment.

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f(x,y)=px+(1-p)y-{\sum }_{s=m}^{\infty }s(x,y)$ uniformly in a neighborhood of the origin, where m > 1, ${}_s(x,y)={\sum }_{i=0}^s{a}_{i,s}{x}^{s-i}{y}^i$; (c) $ₘ(1,1)={\sum }_{i=0}^m{a}_{i,m}>0$. Let x₀,x₁ ∈ (0,α) and $x_{n+1}=f(xₙ,x_{n-1})$, n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $xₙ\sim {((2-p)/\left((m-1){\sum }_{i=0}^m{a}_{i,m}\right))}^{1/(m-1)}1/\sqrt[m-1]{n}$.

A simple model of phenotypic evolution is introduced and analysed in a space of population states. The expected values of the population states generate a discrete dynamical system. The asymptotic behaviour of the system is studied with the use of classical tools of dynamical systems. The number, location and stability of fixed points of the system depend on parameters of a fitness function and the parameters of the evolutionary process itself. The influence of evolutionary process parameters on...

Asymptotic convergence theorems for semigroups of nonnegative operators on a Banach lattice, on C(X) and on $L^p\left(X\right)$ (1 ≤ p ≤ ∞) are proved. The general results are applied to a class of semigroups generated by some differential equations.

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a $𝛬$-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to $\infty$. Some of our results hold in the case of a general $𝛬$-coalescent...