Development of a macro-scale model from a meso-scale model for cell culture population dynamics

In this paper, we present a novel macro-scale analytical model that allows the prediction of how the population size will change in a cell culture starting from an arbitrary initial value. General biological knowledge and some empirical observations are used to design an agent-based discrete-time model at the meso-scale, which then serves as a simulation environment and provides the necessary insights for the development of the continuous-time, differential equation-based, compact macro-scale model. This model can be parameter-tuned and employed for predicting how the population size changes. The paper gives a procedure for the estimation of parameter values of the macro-scale model via some simple tests to be conducted on the cell culture at hand. The performance of the macro-scale model is validated via simulation results that show how well the macro-scale model captures the population dynamics as obtained from the meso-scale model, while the biological plausibility of the meso-scale model is taken for granted.     

A Note on Attenuant Cycles of Population Models with Periodic Carrying Capacity

In this short note, we consider attenuant cycles of population models. This study concerns the second conjecture of Cushing and Henson [A periodically forced Beverton-Holt equation, J. Diff. Eq. Appl., 8 (2002), pp. 1119–1120], which was recently resolved affirmatively by Elaydi and Sacker [Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures, Proc. 8th Inter. Conf. Diff. Eq., Brno, (in press)]. They showed that the periodic fluctuations in the carrying capacity always reduce the average of population densities in the Beverton-Holt equation. We extend this result and give a class of population models in which the periodic fluctuations in the carrying capacity always reduce the average of population densities.     

Approximate reduction of non-linear discrete models with two time scales

The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system.     

LAGRANGIANS FOR BIOLOGICAL MODELS

We show that a method presented in [S. L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974) 299–324] and later in [G. H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749–760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation that can be derived from them, even in the case where those authors failed such as the host-parasite model. Also we show that the Lagrangians of certain second-order ordinary differential equations derived by Volterra in [V. Volterra, Calculus of variations and the logistic curve, Hum. Biol. 11 (1939) 173–178] are particular cases of the Lagrangians that can be obtained by means of the Jacobi Last Multiplier. Actually we provide more than one Lagrangian for those Volterra's equations.     

Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations with Non-uniform Interaction Rate

Traditional evolutionary games assume uniform interaction rate, which means that the rate at which individuals meet and interact is independent of their strategies. But in some systems, especially biological systems, the players interact with each other discriminately. Taylor and Nowak (2006) were the first to establish the corresponding non-uniform interaction rate model by allowing the interaction rates to depend on strategies. Their model is based on replicator dynamics which assumes an infinite size population. But in reality, the number of individuals in the population is always finite, and there will be some random interference in the individuals' strategy selection process. Therefore, it is more practical to establish the corresponding stochastic evolutionary model in finite populations. In fact, the analysis of evolutionary games in a finite size population is more difficult. Just as Taylor and Nowak said in the outlook section of their paper, "The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations." In this paper, we are exactly doing this work. We extend Taylor and Nowak's model from infinite to finite case, especially focusing on the influence of non-uniform connection characteristics on the evolutionary stable state of the system. We model the strategy evolutionary process of the population by a continuous ergodic Markov process. Based on the limit distribution of the process, we can give the evolutionary stable state of the system. We make a complete classification of the symmetric 2×2 games. For each case game, the corresponding limit distribution of the Markov-based process is given when noise intensity is small enough. In contrast with most literatures in evolutionary games using the simulation method, all our results obtained are analytical. Especially, in the dominant-case game, coexistence of the two strategies may become evolutionary stable states in our model. This result can be used to explain the emergence of cooperation in the Prisoner is Dilemma Games to some extent. Some specific examples are given to illustrate our results.     

A numerical approximation for Volterra’s population growth model with fractional order

This paper presents a numerical scheme for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. In fact, the Bessel collocation method is extended by using the time-fractional derivative in the Caputo sense to give solutions for the mentioned model problem. In this extended of the method, a generalization of the Bessel functions of the first kind is used and its matrix form is constructed. And then, the matrix form based on the collocation points is formed for the each term of this model problem. Hence, the method converts the model problem into a system of nonlinear algebraic equations. We give some numerical applications to show efficiency and accuracy of the method. In applications, the reliability of the technique is demonstrated by the error function based on accuracy of the approximate solution.     

Approximation of a population dynamics model by parabolic regularization

The basic linear model for describing an age structured population spreading in a limited habitat is considered with the purpose of investigating an approximation procedure based on parabolic regularization. In fact, a viscosity model is introduced by considering an appropriate approximating regularized parabolic problem and it is proved that the sequence of the approximating solutions tends to the solution to the original problem. The advantage of this approach is that it leads to the numerical solution of a parabolic problem that has more stable solutions than the hyperbolic‐parabolic original problem and avoids the restrictions (compatibility conditions) needed to treat the latter. Moreover, for the solution of the approximating problem, it is possible to take advantage of established software packages dedicated to parabolic problems. Some examples of the approach are provided using COMSOL Multiphysics. Copyright © 2012 John Wiley & Sons, Ltd.     

Population transfer by femtosecond laser pulses in a ladder-type atomic system

The population transfer in a ladder-type atomic system driven by linearly polarized sech-shape femtosecond laser pulses is investigated by numerically solving Schr6dinger equation without including the rotating wave approximation （RWA）. It is shown that population transfer is mainly determined by the Rabi frequency （strength） of the driving laser field and the chirp rate, and that the ratio of the dipole moments and the pulse width also have a prominent effect on the population transfer. By choosing appropriate values of the above parameters, complete population transfer can be realized.