A key points of new approach for modeling of the population dynamics in reactive environment are presented here: 1) generalization of the Logistic equation to the case of reactive environment for modeling of population dynamics (or for the fulfilling of the ecological niches); 2) new type of asymptotic solution for such equation (which is tested on human population growth); 3) reduction to the ...

An Abel differential equation y′ = p(x)y + q(x)y is said to have a center at a pair of complex numbers (a, b) if y(a) = y(b) for every solution y(x) with the initial value y(a) small enough. This notion is closely related to the classical center-focus problem for plane vector fields. Recently, conditions for the Abel equation to have a center have been related to the composition factorization o...

(an Abel equation in canonical form, see [1, p. 24]) is a useful instrument to display rich qualitative properties such as extendability and finite escape time, stability and instability, boundedness, and periodicity. We show the following: (a) The equation has a unique solution that is defined on (−∞,∞). This solution is periodic. (b) As t increases, each solution is attracted into the strip |...

We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems in Goncharov-Kenyon 2013. The dynamic on the graph is an urban renewal together with shrinking all 2-valent vertices, while it is a cluster transformation on the weight. The graph is not necessary obtained from an integral polygon. We define the Hamiltonians of a weighted graph as partitio...

If the limit of this as t → 0 exists, then we say that h(x) is Abel summable with the limit as the sum. If h(x) is integrable on R, then h is Abel summable with the usual integral of h being the Abel sum. One can show that the function At(x) can be expressed as an average of Gaussians. This is a quite useful fact, and basically it is a way of saying that Gaussians are more concentrated. For ins...

Abstract: In this article, the approximate analytical solutions of the linear and nonlinear Abel type integral equation are obtained with the help of analytical method of linear and nonlinear problems called the Modified New Iterative Method (MNIM). By choosing initial value, the explicit solution of the integral equation for different examples have been solved which demonstrate the effectivene...