Data collapse, scaling functions, and analytical solutions of generalized growth models

Single-species fragmentation: The role of density-dependent feedback

Internal feedback is commonly present in biological populations and can play a crucial role in the emergence of collective behavior. To describe the temporal evolution of the distribution of a single-species population, we consider a generalization of the Fisher-KPP equation. This equation includes the elementary processes of random motion, reproduction, and, importantly, nonlocal interspecific competition, which introduces a spatial scale of interaction. In addition, we take into account feedback mechanisms in diffusion and growth processes, mimicked by power-law density dependencies. This feedback includes, for instance, anomalous diffusion, reaction to overcrowding or to the rarefaction of the population, as well as Allee-like effects. We show that, depending on the kind of feedback that takes place, the population can self-organize splitting into disconnected subpopulations, in the absence of external constraints. Through extensive numerical simulations, we investigate the temporal evolution and the characteristics of the stationary population distribution in the one-dimensional case. We discuss the crucial role that density-dependence has on pattern formation, particularly on fragmentation, which can bring important consequences to processes such as epidemic spread and speciation.

Fractal dimension and universality in avascular tumor growth

For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation-the Bertalanffy-Richards model-even if they do not necessarily share the same biological properties.

Models for Allee effect based on physical principles

We propose some models of single species with Allee effect based on physical principles. A method is used to obtain the expression for the per capita growth rate (a macroscopic information) starting from the characteristics of interactions between the individuals (a microscopic information). We assume that the agents in a model of a single species interact according to the distance between them. Moreover these agents must (i) cooperate with their nearest neighbors, (ii) compete with neighbors at an intermediate distance, and (iii) being indifferent to those who are far away. Using these assumptions and based on fundamental physical principles, we find what appears to be a new way of establishing models of single species with Allee effect.

Analytical properties of a three-compartmental dynamical demographic model

The three-compartmental demographic model by Korotaeyv-Malkov-Khaltourina, connecting population size, economic surplus, and education level, is considered from the point of view of dynamical systems theory. It is shown that there exist two integrals of motion, which enables the system to be reduced to one nonlinear ordinary differential equation. The study of its structure provides analytical criteria for the dominance ranges of the dynamics of Malthus and Kremer. Additionally, the particular ranges of parameters enable the derived general ordinary differential equations to be reduced to the models of Gompertz and Thoularis-Wallace.

A non-phenomenological model of competition and cooperation to explain population growth behaviors

This paper is an extension of a previous work which proposes a non-phenomenological model of population growth that is based on the interactions among the individuals of a population. In addition to what had already been studied—that the individuals interact competitively—in the present work it is also considered that the individuals interact cooperatively. As a consequence of this new consideration, a richer dynamics is observed. For instance, besides getting the population models already reached from the original version of the model (as the Malthus, Verhulst, Gompertz, Richards, Bertalanffy and power-law growth models), the new formulation also reaches the von Foerster growth model and also a regime of divergence of the population at a finite time. An agent-based model is also presented in order to give support to the analytical results. Moreover, this new approach of the model explains the Allee effect as an emergent behavior of the cooperative and competitive interactions among the individuals. The Allee effect is the characteristic of some populations of increasing the population growth rate in a small-sized population. Whereas the models presented in the literature explain the Allee effect with phenomenological ideas, the model presented here explains this effect by the interactions between the individuals. The model is tested with empirical data to justify its formulation. Another interesting macroscopic emergent behavior from the model proposed is the observation of a regime of population divergence at a finite time. It is interesting that this characteristic is observed in humanity's global population growth. It is shown that in a regime of cooperation, the model fits very well to the human population growth data from 1000 AD to nowadays.

Generalized Allee effect model

The Allee effect consists of a positive correlation between very small population size and fitness. Offering a new view point on the weak and strong demographic Allee effect, we propose to combine them with the Richards growth model. In particular, a peculiar manifestation of the Allee effect is analytically predicted and still not validated by experiments. Model validation with ecological data is presented for some special situations.

Generalized Metropolis dynamics with a generalized master equation: an approach for time-independent and time-dependent Monte Carlo simulations of generalized spin systems

The extension of Boltzmann-Gibbs thermostatistics, proposed by Tsallis, introduces an additional parameter q to the inverse temperature β. Here, we show that a previously introduced generalized Metropolis dynamics to evolve spin models is not local and does not obey the detailed energy balance. In this dynamics, locality is only retrieved for q=1, which corresponds to the standard Metropolis algorithm. Nonlocality implies very time-consuming computer calculations, since the energy of the whole system must be reevaluated when a single spin is flipped. To circumvent this costly calculation, we propose a generalized master equation, which gives rise to a local generalized Metropolis dynamics that obeys the detailed energy balance. To compare the different critical values obtained with other generalized dynamics, we perform Monte Carlo simulations in equilibrium for the Ising model. By using short-time nonequilibrium numerical simulations, we also calculate for this model the critical temperature and the static and dynamical critical exponents as functions of q. Even for q≠1, we show that suitable time-evolving power laws can be found for each initial condition. Our numerical experiments corroborate the literature results when we use nonlocal dynamics, showing that short-time parameter determination works also in this case. However, the dynamics governed by the new master equation leads to different results for critical temperatures and also the critical exponents affecting universality classes. We further propose a simple algorithm to optimize modeling the time evolution with a power law, considering in a log-log plot two successive refinements.