Abstract:In this paper we study a semi-Kolmogorov type of population model, arising from a predator-prey system with indirect effects. In particular we are interested in investigating the population dynamics when the indirect effects are time dependent and periodic. We first prove the existence of a global pullback attractor. We then estimate the fractal dimension of the attractor, which is done for a subclass by using Leonov's theorem and constructing a proper Lyapunov function. To have more insights about the dynamical behavior of the system we also study the coexistence of the three species. Numerical examples are provided to illustrate all the theoretical results. (C) 2016 Elsevier Ltd. All rights reserved.

Abstract:We study the existence of global solutions of a singular ordinary differential equation arising in the construction of self similar solution for a backward-forward parabolic equation. Also we present several numerical simulations and obtain an upper bound for the blowup time.

Abstract:We investigate the long-term dynamics for a predation model of Plankton community with indirect effects, under fluctuating environments. A random version and a stochastic version with multiplicative noise of the model are discussed and compared. We prove that the solutions to both versions are nonnegative and bounded given any nonnegative positive initial conditions. We also prove that both the random system and the stochastic system possess a unique random attractor under the same set of assumptions, by using the classical theory of random dynamical systems. In addition, we provide conditions under which coexistence of species exists for the random system.

Abstract:We work on a model that has succeeded in describing real cases of coexistence of two languages within a closed community of speakers, taking into account bilingualism and incorporating a parameter to measure the distance between languages. The dynamics of this model depend on a characteristic exponent, which weighs the power of the size of a group of speakers to attract new members. So far, this model had been solved only when this characteristic exponent is greater than 1. In this article, we have managed to solve the nature of the stability of all the possible situations for this characteristic exponent, that is, when it is less or equal than 1 and covering also the situations produced when it is 0 or negative. We interpret these new situations and find that, even in such exotic scenarios, there are configurations of the resulting societies where all the languages coexist. (c) 2014 Wiley Periodicals, Inc. Complexity 21: 86-93, 2016

Abstract:We consider the effects of additive and multiplicative noise on the asymptotic behavior of a fourth order parabolic equation arising in the study of phase transitions. On account that the deterministic model presents three different time scales, in this paper we have established some conditions under which the third time scale, which encounter finite dimensional behavior of the system, is preserved under both additive and multiplicative linear noise. In particular we have proved the existence of a random attractor in both cases, and observed that the order of magnitude of the third time scale is also preserved.

Abstract:We consider a modification of the model proposed by Abrams and Strogatz to describe the death of a language when it competes with a stronger one within the same community of speakers. The modification opened the possibility of coexistence of both languages under some conditions, but so far it has not been possible to write down the expression of the equilibrium points. In this paper, we nontrivially use bifurcation theory to calculate under which conditions such coexistence arises; namely, we calculate the specific ranges of the parameters that describe the modified model to have this situation, paying special attention to the cases that yield a stable cohabitation of two monolingual populations along with a bilingual one.