Cells, the basic units of organisms, consist of multiple essential elements such as carbon, nitrogen, and phosphorus. The scarcity of any of these elements can strongly restrict cellular and organismal growth. During recent years, ecological models incorporating multiple elements have been rapidly developed in many studies, which form a new research field of mathematical and theoretical biology...

Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrop...

We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod's model for the dependence of the specific growth rate of the predator on the concentrations...

One of the main virtues of trees is the representation of formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in power series rings. When analyzed in terms of combinatorial Hopf algebras, the simplest examples yield interesting algebraic identities or enumerative...

The dynamics is investigated on the delayed regulation model under period-2 perturbation described as x(t+ 1) = {A + (−1)te}x(t){1 − x(t − 1)}, t = 1, 2, 3 . . . The edependences of the bifurcation points are analyzed through usual stability analysis of fixed point and periodic solution, however one of them is derived through the stability analysis of the “virtual” period-2 solution. key words:...

A class of polynomial systems of odd degree with limit cycles, invariant parabolas and invariant straight lines, is examined. The limit cycles can be obtain as a bifurcation of a non hyperbolic focus at the origin as Hopf bifurcations. We will also obtain the necessary and sufficient conditions for the critical point at the interior of bounded region to be a center. 2010 Mathematics Subject Cla...

In this paper we apply the generalized degree introduced by Geba, Massabo and Vignoli, in [3], to extend the notion of complementing maps defined by Fitzpatrick, Massabo and Pejsachowicz, in [1] and [2]. On the other hand, we obtain, in low dimension, a bifurcation result in terms of the linking number of some 1-dimensional manifolds. We also present a global theorem that improves a Rabinowitz’...

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particu...

This paper develops the Liapunov-Schmidt procedure for systems with additional constraints such as having a first integral, being Hamiltonian, or being a gradient system. Similar developments for systems with symmetry, including reversibility, are well known, and the method of this paper augments and is consistent with that approach. One of the results states that the bifurcation equation for H...

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert’s 16th problem [Hilbert, 1900, Hilbert, 1902], are still open for this family. In this article we make a global study of the familyQTN ...