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This paper aims to present a comprehensive framework for coupling tumor-bone remodeling processes in a 2-dimensional geometry. This is achieved by introducing a bio-inspired damage that represents the growing tumor, which subsequently affects the main populations involved in the remodeling process, namely, osteoclasts, osteoblasts, and bone tissue. The model is constructed using a set of differential equations based on the Komarova's and Ayati's models, modified to incorporate the bio-inspired damage that may result in tumor mass formation. Three distinct models were developed. The first two models are based on the Komarova's governing equations, with one demonstrating an osteolytic behavior and the second one an osteoblastic model. The third model is a variation of Ayati's model, where the bio-inspired damage is induced through the paracrine and autocrine parameters, exhibiting an osteolytic behavior. The obtained results are consistent with existing literature, leading us to believe that our in-silico experiments will serve as a cornerstone for paving the way towards targeted interventions and personalized treatment strategies, ultimately improving the quality of life for those affected by these conditions. [ABSTRACT FROM AUTHOR]

• Philip Maini has made multiple significant contributions to the field of Mathematical Enzyme Kinetics. • Philip Maini has also contributed to the field by mentoring young researchers. • The Quasi-Steady State Assumption has now been expanded to include just about every possible initial condition. Modelling enzyme kinetics has a long history, as one of the earliest areas where mathematics was utilized to understand biological (in this case, biochemical) phenomena. By using differential equations and the law of mass action, many researchers have discovered information about the mechanisms of enzyme workings in the more than 100 years since Leonor Michaelis and Maud Menten published their classic paper in 1913. This work uses he laws obeyed by molecules during a reaction, translates them into mathematical equations, and then compares the results predicted by the equations to data obtained in laboratory conditions. This can inform whether the mechanism used to create the mathematical model is consistent with experimental results. Philip K. Maini has contributed to several significant results in the field of mathematical modelling in enzyme kinetics. Along with his collaborators, he has published results analyzing and extending the Quasi-Steady State Assumption, applying it to special enzyme reactions, and considering special conditions. This review paper will provide an overview of the contributions made by PK Maini to the field of modelling enzyme kinetics, summarize the main results and discuss their significance. [ABSTRACT FROM AUTHOR]

The paper is aimed at the methodological development of ecological monitoring in discrete-time dynamic models. In earlier papers, in the framework of continuous-time models, we have shown how a systems-theoretical methodology can be applied to the monitoring of the state process of a system of interacting populations, also estimating certain abiotic environmental changes such as pollution, climatic or seasonal changes. In practice, however, there may be good reasons to use discrete-time models. (For instance, there may be discrete cycles in the development of the populations, or observations can be made only at discrete time steps.) Therefore the present paper is devoted to the development of the monitoring methodology in the framework of discrete-time models of population ecology. By monitoring we mean that, observing only certain component(s) of the system, we reconstruct the whole state process. This may be necessary, e.g., when in a complex ecosystem the observation of the densities of certain species is impossible, or too expensive. For the first presentation of the offered methodology, we have chosen a discrete-time version of the classical Lotka–Volterra prey–predator model. This is a minimal but not trivial system where the methodology can still be presented. We also show how this methodology can be applied to estimate the effect of an abiotic environmental change, using a component of the population system as an environmental indicator. Although this approach is illustrated in a simplest possible case, it can be easily extended to larger ecosystems with several interacting populations and different types of abiotic environmental effects. [ABSTRACT FROM AUTHOR]

Drug resistance and strong contacts actually play crucial roles in epidemic spread in complex systems. Nevertheless, neither theoretical model or methodology is proposed to address this. We thus consider an edge-based epidemic spread model considering the two key ingredients, in which the contacts are grouped into two classes: strong contacts and normal ones. Next, we present a unified edge-based compartmental approach to the spread dynamics on Erdös–Rényi (ER) networks and validate its results by extensive numerical simulations. In case that epidemic is totally drug-resistant, we both numerically and theoretically show a continuous growth of epidemics with infection probability when number of strong contacts is not enough for the emergence of null threshold. If the epidemic owns partial resistance, we would observe evident discontinuous growth with infection probability (discontinuous transitions) and larger final epidemic sizes for few strong contacts, instead of emergence of null threshold with increase of strong contacts. Inhibiting effect of infection threshold, positive roles of strong contacts and strength of strong contacts in promoting outbreaks are also approved. Throughout this paper, we could drive exact predictions through the analytical approach, showing good agreements with numerical simulations. [ABSTRACT FROM AUTHOR]

As the battle continues against hospital-acquired infections and the concurrent rise in antibiotic resistance among many of the major causative pathogens, there is a dire need to conduct controlled experiments, in order to compare proposed control strategies. However, cost, time, and ethical considerations make this evaluation strategy either impractical or impossible to implement with living patients. This paper presents a multi-scale model that offers promise as the basis for a tool to simulate these (and other) controlled experiments. This is a “unified” model in two important ways: (i) It combines inter-host and in-host dynamics into a single model, and (ii) it links two very different modeling approaches - agent-based modeling and differential equations - into a single model. The potential of this model as an instrument to combat antibiotic resistance in hospitals is demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: This paper extends the classical result of Fisher (1937) from the case of two alleles to the case of multiple alleles. Consider a population living in a homogeneous one-dimensional infinite habitat. Individuals in this population carry a gene that occurs in k forms, called alleles. Under the joint action of migration and selection and some additional conditions, the frequencies of the alleles, , satisfy a system of differential equations of the form (1.2). In this paper, we first show that under the conditions is the most fit among the homozygotes, (1.2) is cooperative, the state that only allele A 1 is present in the population is stable, and the state that allele A 1 is absent and all other alleles are present in the population is unstable, then there exists a positive constant, c ⁎, such that allele A 1 propagates asymptotically with speed c ⁎ in the population as . We then show that traveling wave solutions connecting these two states exist for . Finally, we show that under certain additional conditions, there exists an explicit formula for c ⁎. These results allow us to estimate how fast an advantageous gene propagates in a population under selection and migration forces as . Selection is one of the major evolutionary forces and understanding how it works will help predict the genetic makeup of a population in the long run. [Copyright &y& Elsevier]

Abstract: Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mechanistic mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. The model generates successive estrous cycles of 21 days, with three waves of follicle growth per cycle. The model contains 12 differential equations and 54 parameters. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included in the model are based on relations that in the literature have only been described qualitatively (i.e. stimulation and inhibition), the output of the model is surprisingly well in line with empirical data. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences. [Copyright &y& Elsevier]

Abstract: We present a two-sided search model in which individuals from two groups (males and females, employers and workers) would like to form a long-term relationship with a highly ranked individual of the other group, but are limited to individuals who they randomly encounter and to those who also accept them. This article extends the research program, begun in Alpern and Reyniers [1999. J. Theor. Biol. 198, 71–88], of providing a game theoretic analysis for the Kalick–Hamilton [1986. J. Personality Soc. Psychol. 51, 673–682] mating model in which a cohort of males and females of various ‘fitness’ or ‘attractiveness’ levels are randomly paired in successive periods and mate if they accept each other. Their model compared two acceptance rules chosen to represent homotypic (similarity) preferences and common (or ‘type’) preferences. Our earlier paper modeled the first kind by assuming that if a level x male mates with a level y female, both get utility , whereas this paper models the second kind by giving the male utility y and the female utility x. Our model can also be seen as a continuous generalization of the discrete fitness-level game of Johnstone [1997. Behav. Ecol. Sociobiol. 40, 51–59]. We establish the existence of equilibrium strategy pairs, give examples of multiple equilibria, and conditions guaranteeing uniqueness. In all equilibria individuals become less choosy over time, with high fitness individuals pairing off with each other first, leaving the rest to pair off later. This route to assortative mating was suggested by Parker [1983. Mate Choice, Cambridge University Press, Cambridge, pp. 141–164]. If the initial fitness distributions have atoms, then mixed strategy equilibria may also occur. If these distributions are unknown, there are equilibria in which only individuals in the same fitness band are mated, as in the steady-state model of MacNamara and Collins [1990. J. Appl. Prob. 28, 815–827] for the job search problem. [Copyright &y& Elsevier]

While most aspects of virus dynamics are well understood in standard models, the phenomenon of multiple infection (or coinfection) can change the properties of the dynamics, and this has so far not been fully explored. An important parameter in determining the properties of the model is the virus output from multiply infected cells compared to that from singly infected cells. If the amount of virus produced by infected cells during their life-span is independent of the infection multiplicity, then multiple infection does not change the dynamics. If, however, multiply infected cells produce more virus during their life-span than singly infected cells, then the properties of the dynamics can change fundamentally. This paper presents a detailed mathematical analysis of this scenario. We demonstrate that under some realistic conditions, the equilibrium structure of the solutions acquires novel properties. In particular, infection can persist even for values of the basic reproductive number, R 0 , smaller than unity. In this regime, we observe the phenomenon of bistability, when two stable equilibria are present simultaneously, and the outcome is determined by the initial conditions. The two possible solutions are the virus-free equilibrium, which is exactly the same as the one observed in the absence of multiple infection, and a novel infection equilibrium. In the presence of this outcome, it is clear that the meaning of the parameter R 0 changes, as it no longer simply indicates the possibility of successful infection. This adds to our understanding and interpretation of R 0 in virus dynamics models, and also provides further insights about conditions that can lead to virus extinction rather than persistence. It turns out that conditions for bistability depend (in a fully specified way) on the model structure, particularly on the way the infection term is formulated. We provide a general condition that informs us whether or not bistability occurs, and define what needs to be measured when examining the dynamics of multiple infection in specific biological systems. [ABSTRACT FROM AUTHOR]

Intermolecular autophosphorylation refers to the process whereby a molecule of an enzyme phosphorylates another molecule of the same enzyme. The enzyme thereby catalyses its own phosphorylation. In the present paper, we develop two generic models of intermolecular autophosphorylation that also include dephosphorylation by a phosphatase of constant concentration. The first of these, a solely time-dependent model, is written as one ordinary differential equation that relies upon mass-action and Michaelis–Menten kinetics. Beginning with the enzyme in its dephosphorylated state, it predicts a lag before the enzyme becomes significantly phosphorylated, for suitable parameter values. It also predicts that there exists a threshold concentration for the phosphorylation of enzyme and that for suitable parameter values, a continuous or discontinuous switch in the phosphorylation of enzyme are possible. The model developed here has the advantage that it is relatively easy to analyse compared with most existing models for autophosphorylation and can qualitatively describe many different systems. We also extend our time-dependent model of autophosphorylation to include a spatial dependence, as well as localised binding reactions. This spatio-temporal model consists of a system of partial differential equations that describe a soluble autophosphorylating enzyme in a spherical geometry. We use the spatio-temporal model to describe the phosphorylation of an enzyme throughout the cell due to an increase in local concentration by binding. Using physically realistic values for model parameters, our results provide a proof-of-concept of the process of activation by local concentration and suggest that, in the presence of a phosphatase, this activation can be irreversible. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new model of in vitro cell electropermeabilization, which describes separately the conducting state and the permeable state of the membrane submitted to high voltage pulses. We first derive the model based on the experimental observations and we present the numerical methods to solve the non-linear partial differential equations. We then present numerical simulations that corroborate qualitatively the experimental data dealing with the uptake of propidium iodide (PI) after millipulses. This tends to justify the validity of our modeling. Forthcoming work will be to calibrate the parameters of the model for quantitative description of the uptake. [ABSTRACT FROM AUTHOR]

Abstract: Deregulation of signaling pathways and subsequent abnormal interactions of downstream genes very often results in carcinogenesis. In this paper, we propose a two-compartment model describing intricate dynamics of the target genes of the Wnt signaling pathway in medulloblastoma. The system of nine nonlinear ordinary differential equations accounts for the formation and dissociation of complexes as well as for the transcription, translation and transport between the cytoplasm and the nucleus. We focus on the interplay between MYC and SGK1 (serum and glucocorticoid-inducible kinase 1), which are the products of Wnt/β-catenin signaling pathway, and GSK3β (glycogen synthase kinase). Numerical simulations of the model solutions yield a better understanding of the process and indicate the importance of the SGK1 gene in the development of medulloblastoma, which has been confirmed in our recent experiments. The model is calibrated based on the gene expression microarray data for two types of medulloblastoma, characterized by monosomy and trisomy of chromosome 6q to highlight the difference between diagnoses. [Copyright &y& Elsevier]

• New analysis of the ligand-induced dimerisation process for VEGFR is presented. • A key idea in the mathematical model and analysis is kinetic binding cooperativity. • The nonlinear model is in contrast with the corresponding linear model for GPCRs. • Numerical and asymptotic analysis reveal insights into the underlying LID dynamics. • Our simulations show excellent fit to recent experimental data for VEGFR. The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and many other diseases. It is widely accepted that VEGFR receptors dimerise in response to VEGF binding. However, analysis of these mechanisms and their implications for drug development still requires further exploration. In this paper, we present a mathematical model representing the binding of VEGF to VEGFR and the subsequent ligand-induced dimerisation. A key factor in this work is the qualitative and quantitative effect of binding cooperativity, which describes the effect that the binding of a ligand to a receptor has on the binding of that ligand to a second receptor, and the dimerisation of these receptors. We analyse the ordinary differential equation system at equilibrium, giving analytical solutions for the total amount of ligand bound. For time-course dynamics, we use numerical methods to explore possible behaviours under various parameter regimes, while perturbation analysis is used to understand the intricacies of these behaviours. Our simulation results show an excellent fit to experimental data, towards validating the model. [ABSTRACT FROM AUTHOR]

Abstract: An n species stochastic impulsive migration Lotka–Volterra model with Markovian switching in N different patches is presented and studied in this paper. By constructing appropriate Lyapunov functions, some sufficient conditions on the global positivity, the ultimate boundedness in mean and the extinction in mean are established. Real world examples are provided to illustrate the validity of our results. A discussion is given in the end. [Copyright &y& Elsevier]

Abstract: In the present paper, a model for the early stages of peri-implant bone regeneration is developed. This model is able to capture some important characteristics of endosseous healing, which were not incorporated in the existing models. It is a well known fact, that during peri-implant osseointegration, bone forms only by apposition on the pre-existing rigid surface, which initially consists of the implant surface and the old bone surface. In order to track the movement of the front of the newly formed bone, a moving boundary problem is formulated. Another important feature of the current model, is that the cell differentiation is considered as a gradual process, evolving in time and being influenced by the presence of growth factors. Hence, the evolution of cell differentiation level is captured in the present approach. Numerical methods, used to solve the set of partial differential equations with hyperbolic terms, defined within the domain with the moving boundary, are described. [Copyright &y& Elsevier]

Abstract: The number of temporary migrant workers from rural areas to urban areas in emerging market economies like China has increased dramatically since the early 1980s. Temporary migrant workers have been labeled as the major driving force for the rising incidence of infectious diseases in cities. However, it has not been well recognized that temporary migration indeed may have tremendous impacts on the spread of infectious diseases in migrants'' home villages. In this paper, by proposing a delay differential equation model, we provide a framework to study the influence of temporary migration on the transmission of infectious diseases in a migrant workers'' home village. The model is shown to admit a unique positive equilibrium which is locally asymptotically stable and is globally asymptotically stable under certain conditions. This implies that the disease always persists at a constant level. Considering tuberculosis as an example, we explore various disease prevention and control strategies numerically to demonstrate how migration related parameters affect the early outbreak of the disease. We find that a single control strategy, such as reducing the migration time period alone, has little effect on reducing the disease endemic level. For disease prevention and control, temporary migrant workers should be identified as the top target group, and a combination of several prevention strategies should be implemented. [Copyright &y& Elsevier]

Abstract: Iron is a metal essential for cellular metabolism. However, excess iron available for reactions contributes to the formation of dangerous reactive oxygen species, such as the hydroxyl radical, via the Fenton reaction. Therefore, intracellular iron levels are tightly constrained by a control system of proteins. This paper contains a mathematical model, in the form of a system of five ordinary differential equations, of the core of this control system, including the labile iron pool as well as proteins that regulate uptake, storage, and export and are connected through negative feedback loops. The model is validated using data from an overexpression experiment with cultured human breast epithelial cells. The parameters in the mathematical model are not known for this particular cell culture system, so the analysis of the model was done for a generic choice of parameters. Through a mixture of analytical arguments and extensive simulations it is shown that for any choice of parameters the model reaches a unique stable steady state, thereby ruling out oscillatory behavior. It is shown furthermore that the model parameters are identifiable through suitable experiments. [Copyright &y& Elsevier]

Abstract: All animals and plants are, to some extent, susceptible to disease caused by varying combinations of parasites, viruses and bacteria. In this paper, we develop a mathematical model of contact spread infection to investigate the effect of introducing a parasitoid-vectored infection into a one-host–two-parasitoid competition model. We use a system of ordinary differential equations to investigate the separate influences of horizontal and vertical pathogen transmission on a model system appropriate for a variety of competitive situations. Computational simulations and steady-state analysis show that the transient and long-term dynamics exhibited under contact spread infection are highly complex. Horizontal pathogen transmission has a stabilising effect on the system whilst vertical transmission can destabilise it to the point of chaotic fluctuations in population levels. This has implications when considering the introduction of host pathogens for the control of insect vectored diseases such as bovine tuberculosis or yellow fever. [Copyright &y& Elsevier]

Abstract: A fundamental step in synthetic biology and systems biology is to derive appropriate mathematical models for the purposes of analysis and design. For example, to synthesize a gene regulatory network, the derivation of a mathematical model is important in order to carry out in silico investigations of the network dynamics and to investigate parameter variations and robustness issues. Different mathematical frameworks have been proposed to derive such models. In particular, the use of sets of nonlinear ordinary differential equations (ODEs) has been proposed to model the dynamics of the concentrations of mRNAs and proteins. These models are usually characterized by the presence of highly nonlinear Hill function terms. A typical simplification is to reduce the number of equations by means of a quasi-steady-state assumption on the mRNA concentrations. This yields a class of simplified ODE models. A radically different approach is to replace the Hill functions by piecewise-linear approximations [Casey, R., de Jong, H., Gouz, J.-L., 2006. Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol. 52 (1), 27–56]. A further modelling approach is the use of discrete-time maps [Coutinho, R., Fernandez, B., Lima, R., Meyroneinc, A., 2006. Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52, 524–570] where the evolution of the system is modelled in discrete, rather than continuous, time. The aim of this paper is to discuss and compare these different modelling approaches, using a representative gene regulatory network. We will show that different models often lead to conflicting conclusions concerning the existence and stability of equilibria and stable oscillatory behaviours. Moreover, we shall discuss, where possible, the viability of making certain modelling approximations (e.g. quasi-steady-state mRNA dynamics or piecewise-linear approximations of Hill functions) and their effects on the overall system dynamics. [Copyright &y& Elsevier]

• Bottom-up processes have regulatory effect on capelin (predator) biomass dynamics. • Identified time-frames for predator-prey overlap, optimal for predator growth. • Differences in overlap time-frames reflect age-dependent prey choices (composition and type). • Extended original match-mismatch hypothesis to capelin and its prey. Capelin (Mallotus villosus) is a short-lived (1–4 years) fish species, that plays a crucial role by dominating the intermediate trophic level in the Barents Sea. Several episodes of extreme biomass decline (collapse) have been observed during the last three decades. We postulate that these collapses might be regulated by food availability (bottom-up effect) and/or by time discrepancy between capelin feeding and abundance of its prey (match-mismatch hypothesis). This paper investigates our postulate using a model consisting of a set of coupled differential equations to describe the predator-prey system, with a single delay term, τ , in description of the predator dynamics. We derive theoretical conditions on τ , as well as determine how changes in these conditions define different stability regimes of the system. Unconstrained optimization is used to calculate optimal model parameters by fitting the predator-prey model to empirical data. The optimization results are combined with those from the theoretical analysis, to make inference about the empirical system stability. Our results show that Hopf bifurcation occurs in the predatory-prey system when τ exceeds a theoretically derived value τ ∗ > 0. This value represents the critical time for prey availability in advance of the optimal predator growth period.Set into an ecological context, our findings provide mathematical evidence for validity of the match-mismatch hypothesis and a bottom-up effect for capelin. [ABSTRACT FROM AUTHOR]

• Demographic approach to the evolutionary games is combined with state based models. • Replicator equations are completed by the additional dynamics of state changes. • Stage and age based models can be regarded as the special cases of the new framework. • New approach is illustrated by the mechanistic version of the Owner-Intruder game. • New version explicitly describes the fluxes between Owner and Intruder subpopulations. The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describe energy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game, with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the coevolution of phenotypic traits in a community comprising two competitive species subject to strong Allee effects. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy under symmetric competition. Secondly, we find that evolutionary suicide is impossible when the two species undergo symmetric competition, however, evolutionary suicide can occur in an asymmetric competition model with strong Allee effects. Thirdly, it is found that evolutionary bistability is a likely outcome of the process under both symmetric and asymmetric competitions, which depends on the properties of symmetric and asymmetric competitions. Fourthly, under asymmetric competition, we find that evolutionary cycle is a likely outcome of the process, which depends on the properties of both intraspecific and interspecific competition. When interspecific and intraspecific asymmetries vary continuously, we also find that the evolutionary dynamics may admit a stable equilibrium and two limit cycles or two stable equilibria separated by an unstable limit cycle or a stable equilibrium and a stable limit cycle. [Copyright &y& Elsevier]

Abstract: A general sensitivity and control analysis of periodically forced reaction networks with respect to small perturbations in arbitrary network parameters is presented. A well-known property of sensitivity coefficients for periodic processes in dynamical systems is that the coefficients generally become unbounded as time tends to infinity. To circumvent this conceptual obstacle, a relative time or phase variable is introduced so that the periodic sensitivity coefficients can be calculated. By employing the Green''s function method, the sensitivity coefficients can be defined using integral control operators that relate small perturbations in the network''s parameters and forcing frequency to variations in the metabolite concentrations and reaction fluxes. The properties of such operators do not depend on a particular parameter perturbation and are described by the summation and connectivity relationships within a control-matrix operator equation. The aim of this paper is to derive such a general control-matrix operator equation for periodically forced reaction networks, including metabolic pathways. To illustrate the general method, the two limiting cases of high and low forcing frequency are considered. We also discuss a practically important case where enzyme activities and forcing frequency are modulated simultaneously. We demonstrate the developed framework by calculating the sensitivity and control coefficients for a simple two reaction pathway where enzyme activities enter reaction rates linearly and specifically. [Copyright &y& Elsevier]

Abstract: This paper studies a non-autonomous Lotka–Volterra almost periodic predator–prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are “profitless”. Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka–Volterra systems. [Copyright &y& Elsevier]

Abstract: This paper describes some experiments on the manner in which external pressure affects cell colony growth in general, and tumour growth in particular. More precisely, our results show that cell colony borders growing under high-pressure conditions have geometrical and dynamical properties that are markedly different from those corresponding to growth under homeostatic, normal pressure conditions. These behaviours are characterized by means of the so-called dynamical exponents of each type of growth. These are shown to correspond to statistical properties of solutions of some stochastic partial differential equations that account for the evolution of the interface between the expanding colony and the surrounding medium. [Copyright &y& Elsevier]

Abstract: We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called ‘birth–death’, ‘death–birth’ and ‘imitation’. A fourth update rule, ‘pairwise comparison’, is shown to be equivalent to birth–death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner''s Dilemma, the Snow-Drift game, a coordination game and the Rock–Scissors–Paper game. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects. By applying the Floquet theory of linear periodic impulsive equation, we show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, that is, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, and thus, we further prove that the system is uniformly permanent if the impulsive period is larger than some critical value, and meanwhile we also give the conditions for the extinction of one of the two preys and permanence of the remaining species. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term into the prey population''s intrinsic growth rate. The resulting bifurcation diagrams show that with the varying of parameters, the system experiences process of cycles, periodic windows, periodic-doubling cascade, symmetry breaking bifurcation as well as chaos. [Copyright &y& Elsevier]

Abstract: Metapopulation models that incorporate both spatial and temporal structure are studied in this paper. The existence and stability of equilibria are provided, and an extinction threshold condition is derived which depends on patch dynamics (patch destruction and creation) and metapopulation dynamics (patch colonization and extinction). These results refine threshold conditions given by previous metapopulation models. By comparing landscapes with different spatial heterogeneities with respect to weighted long-term patch occupancies, we conclude that the pattern of a landscape is of overwhelming importance in determining metapopulation persistence and patch occupancy. We show that the same conclusion holds when a rescue effect is considered. We also derive a stochastic differential equations (SDE) model of the Itô type based on our deterministic model. Our simulations reveal good agreement between the deterministic model and the SDE model. [Copyright &y& Elsevier]

Abstract: Biochemical systems theory (BST) characterizes a given biochemical system based on the logarithmic gains, rate-constant sensitivities and kinetic-order sensitivities defined at a steady state. This paper describes an efficient method for calculation of the time courses of logarithmic gains, i.e. dynamic logarithmic gains L(Xi , Xj ; t), which expresses the percentage change in the value of a dependent variable Xi at a time t in response to an infinitesimal percentage change in the value of an independent variable Xj at . In this method, one first recasts the ordinary differential equations for the dependent variables into an exact canonical nonlinear representation (GMA system) through appropriate transformations of variables. Owing to the structured mathematical form of this representation, the recast system can be fully described by a set of numeric parameters, and the differential equations for the dynamic logarithmic gains can be set up automatically without resource to computer algebra. A simple general-purpose computer program can thus be written that requires only the relevant numeric parameters as input to calculate the time courses of the variables and of the dynamic logarithmic gains for both concentrations and fluxes. Unlike other methods, the proposed method does not require to derive any expression for the partial differentiation of flux expressions with respect to each independent variable. The proposed method has been applied to two kinds of reaction models to elucidate its usefulness. [Copyright &y& Elsevier]

Abstract: Cancer is viewed as a multistep process whereby a normal cell is transformed into a cancer cell through the acquisition of mutations. We reduce the complexities of cancer progression to a simple set of underlying rules that govern the transformation of normal cells to malignant cells. In doing so, we derive an ordinary differential equation model that explores how the balance of angiogenesis, cell death rates, genetic instability, and replication rates give rise to different kinetics in the development of cancer. The key predictions of the model are that cancer develops fastest through a particular ordering of mutations and that mutations in genes that maintain genomic integrity would be the most deleterious type of mutations to inherit. In addition, we perform a sensitivity analysis on the parameters included in the model to determine the probable contribution of each. This paper presents a novel approach to viewing the genetic basis of cancer from a systems biology perspective and provides the groundwork for other models that can be directly tied to clinical and molecular data. [Copyright &y& Elsevier]

In this paper, we propose and analyse a mathematical model for chronic myelogenous leukemia (CML), a cancer of the blood. We model the interaction between naive T cells, effector T cells, and CML cancer cells in the body, using a system of ordinary differential equations which gives rates of change of the three cell populations. One of the difficulties in modeling CML is the scarcity of experimental data which can be used to estimate parameters values. To compensate for the resulting uncertainties, we use Latin hypercube sampling (LHS) on large ranges of possible parameter values in our analysis. A major goal of this work is the determination of parameters which play a critical role in remission or clearance of the cancer in the model. Our analysis examines 12 parameters, and identifies two of these, the growth and death rates of CML, as critical to the outcome of the system. Our results indicate that the most promising research avenues for treatments of CML should be those that affect these two significant parameters (CML growth and death rates), while altering the other parameters should have little effect on the outcome. [Copyright &y& Elsevier]

A novel method is described for solving systems of differential equations pertaining to organism development, where this development is assumed to be directly influenced by fluctuation in measurable environmental variables. The system parameters are written as functions of these variables and, because these functions involve the accumulation of “environment time” (e.g., “day-degrees”), the system is therefore regulated by the prevailing environmental conditions. This method contrasts with the more usual descriptions of development along a time-line. The parameters of the differential equations involved are estimated by modelling data, which show evidence of changes in the dependent variable(s), i.e. the components of the system. They are expressed in terms of their response to continuous fluctuations in one or more independent, environmental variables. Accumulated thermal time (including day-degrees) or more complex units may be derived by using either linear or nonlinear functions. Critical environmental parameters such as the basal thresholds of a given developmental process or parameters describing a nonlinear relationship with the environment may then be estimated. This paper develops the methodology of this environmentally driven approach to describing organism development in general terms, and gives a specific example of its application with reference to the cellular development within the secondary vascular tissues in the stems of young hybrid aspen trees. [Copyright &y& Elsevier]