A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.     

Evolution of dispersal distance

The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. Ezoe (J Theor Biol 190:287–293, 1998) and Levin and Muller-Landau (Evol Ecol Res 2:409–435, 2000) approached this problem using models that were essentially deterministic, and used calculus to find optimal dispersal parameters. Here we follow Hiebeler (Theor Pop Biol 66:205–218, 2004) and use a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems is done by simulation or nonrigorous methods such as pair approximation. Here, we use machinery developed by Cox et al. (Voter model perturbations and reaction diffusion equations 2011) to rigorously and explicitly compute evolutionarily stable strategies.     

Daphnias: from the individual based model to the large population equation

The class of deterministic ‘Daphnia’ models treated by Diekmann et al. (J Math Biol 61:277–318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23:114–135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4:82–109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population (‘Daphnia’) and an unstructured resource (‘algae’). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and through their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., loc. cit.).     

How T-cells use large deviations to recognize foreign antigens

A stochastic model for the activation of T-cells is analysed. T-cells are part of the immune system and recognize foreign antigens against a background of the body's own molecules. The model under consideration is a slight generalization of a model introduced by Van den Berg et al. (J Theor Biol 209:465-486, 2001), and is capable of explaining how this recognition works on the basis of rare stochastic events. With the help of a refined large deviation theorem and numerical evaluation it is shown that, for a wide range of parameters, T-cells can distinguish reliably between foreign antigens and self-antigens.     

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450-463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173-205].     

Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II

This is a continuation of our paper [Liu, M., Wang, K., 2010. Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol. 264, 934-944]. Taking both white noise and colored noise into account, a stochastic single-species model under regime switching in a polluted environment is studied. Sufficient conditions for extinction, stochastic nonpersistence in the mean, stochastic weak persistence and stochastic permanence are established. The threshold between stochastic weak persistence and extinction is obtained. The results show that a different type of noise has a different effect on the survival results.     

The role of host population heterogeneity in the evolution of virulence

I examine here the effects of host heterogeneity in the growth of immune response on the evolution and co-evolution of virulence. The analysis is based on an extension of the 'nested model' by Gilchrist and Sasaki [Modeling host-parasite coevolution, J. Theor. Biol. 218 (2002), pp. 289-308]; the criteria for host and parasite evolution, in the paradigm of adaptive dynamics, for that model are derived in generality. Host heterogeneity is assumed to be fixed at birth according to a lognormal distribution or to the presence of two discrete types. In both cases, it is found that host heterogeneity determines a dramatic decrease in pathogen virulence, since pathogens will tune to the 'weakest' hosts. Finally we clarify how contrasting results present in the literature are due to different modelling assumptions.     

Phenotype spaces

The topological viewpoint on spaces of phenotypes presented in Stadler et al. (J Theor Biol 213:241–274, 2001) is revisited, and a quantified version is proposed. While necessary probabilistic information can be encoded in a topological- like fashion, it turns out that it is not reflected adequately by the concept of continuity. We propose alternative models, but the behavior of maps make these models non-topological in fundamental ways.     

One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death III: diffusion pulse death zones

This is the third of three papers in which we study a mathematical model of cytoskeleton-induced neuron death. In the first two papers of this suite [Lomasko, T., Clarke, G., Lumsden, C., 2007a. One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death I: cell fate arrival times. J. Theor. Biol. 249, 1-17, doi:10.1016/j.jtbi.2007.05.031; Lomasko, T., Clarke, G., Lumsden, C., 2007b. One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death II: transition state metastability. J. Theor. Biol. 249, 18-28, doi:10.1016/j.jtbi.2007.05.032], we established that the mean-field limit of our model relates the known patterns of neuron decline to specific scales of cytoskeleton reorganization and cell-cell interaction by diffusible death factors. In the mean-field limit, the spatially variable concentration of diffusing death factor is replaced by a constant average value. Recent empirical advances now permit the actual diffusion of such factors to be followed in intact neuropil. In this paper we therefore extend the model beyond the mean-field limit, to include the diffusion dynamics of death factor bursts released from dying neurons. A range of novel tissue degeneration patterns is observed, for which we confirm and extend the mean-field prediction that sigmoidal patterns of neuron population decay are a principal hallmark of cell death in the presence of death factor release.     

Stochastic models of telomere shortening

Shortening of chromosome ends, known as telomeres, is one of the supposed mechanisms of cellular aging and death. We provide a probabilistic analysis of the process of loss of telomere ends. The first work concerned with that issue is the paper by Levy et al. [J. Molec. Biol. 225 (1992) 951-960]. Their deterministic model reproduced the observed frequencies of viable cells in the in vitro experiments. Arino et al. [J. Theor. Biol. 177 (1995) 45-57] reformulated the model of Levy et al. (1992) in the terms of branching processes with denumerable type space. In the present paper, the mathematical results of Arino et al. (1995) are extended to the case in which cell death is present, in cells with telomeres above and below the critical threshold of length, generally with differing probabilities. Both exact and asymptotic results are provided, as well as a discussion of biological relevance of the results.     

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261-269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.     

Studies of protein-protein interaction using countercurrent distribution in aqueous two-phase systems: partition behavior of five glycolytic enzymes from crude baker's yeast extract

The partition behavior of five glycolytic enzymes, in extracts from baker's yeast (Saccharomyces cerevisiae), between two aqueous phases has been studied by countercurrent distribution. All enzymes showed distribution patterns which indicated homogeneity and a similar partition behavior. In purified form, three of the enzymes (glyceraldehyde-phosphate dehydrogenase, 3-phosphoglycerate kinase, and enolase) showed the same partition behavior as in the extracts. Pure 6-phosphofructokinase, on the other hand, changed its partition distinctively relative to what was found in the extracts. These results indicate interactions between this enzyme and macromolecular compounds in the extracts and support a model suggested by Kurganov et al. (1985, J. Theor. Biol. 116, 509-526) describing a "glycolytic particle."     

A user’s guide to PDE models for chemotaxis

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.     

Modelling Cell Migration and Adhesion During Development

Cell?Ccell adhesion is essential for biological development: cells migrate to their target sites, where cell?Ccell adhesion enables them to aggregate and form tissues. Here, we extend analysis of the model of cell migration proposed by Anguige and Schmeiser (J. Math. Biol. 58(3):395?C427, 2009) that incorporates both cell?Ccell adhesion and volume filling. The stochastic space-jump model is compared to two deterministic counterparts (a system of stochastic mean equations and a non-linear partial differential equation), and it is shown that the results of the deterministic systems are, in general, qualitatively similar to the mean behaviour of multiple stochastic simulations. However, individual stochastic simulations can give rise to behaviour that varies significantly from that of the mean. In particular, individual simulations might admit cell clustering when the mean behaviour does not. We also investigate the potential of this model to display behaviour predicted by the differential adhesion hypothesis by incorporating a second cell species, and present a novel approach for implementing models of cell migration on a growing domain.     

The effects of time delays in a phosphorylation-dephosphorylation pathway

Complex signaling cascades involve many interlocked positive and negative feedback loops which have inherent delays. Modeling these complex cascades often requires a large number of variables and parameters. Delay differential equation models have been helpful in describing inherent time lags and also in reducing the number of governing equations. However the consequences of model reduction via delay differential equations have not been fully explored. In this paper we systematically examine the effect of delays in a complex network of phosphorylation-dephosphorylation cycles (described by Gonze and Goldbeter, J. Theor. Biol., 210, (2001) 167-186), which commonly occur in many biochemical pathways. By introducing delays in the positive and negative regulatory interactions, we show that a delay differential model can indeed reduce the number of cycles actually required to describe the phosphorylation-dephosphorylation pathway. In addition, we find some of the unique properties of the network and a quantitative measure of the minimum number of delay variables required to model the network. These results can be extended for modeling complex signalling cascades.     

Stochastic models of kleptoparasitism

In this paper, we consider a model of kleptoparasitism amongst a small group of individuals, where the state of the population is described by the distribution of its individuals over three specific types of behaviour (handling, searching for or fighting over, food). The model used is based upon earlier work which considered an equivalent deterministic model relating to large, effectively infinite, populations. We find explicit equations for the probability of the population being in each state. For any reasonably sized population, the number of possible states, and hence the number of equations, is large. These equations are used to find a set of equations for the means, variances, covariances and higher moments for the number of individuals performing each type of behaviour. Given the fixed population size, there are five moments of order one or two (two means, two variances and a covariance). A normal approximation is used to find a set of equations for these five principal moments. The results of our model are then analysed numerically, with the exact solutions, the normal approximation and the deterministic infinite population model compared. It is found that the original deterministic models approximate the stochastic model well in most situations, but that the normal approximations are better, proving to be good approximations to the exact distribution, which can greatly reduce computing time.     

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this “paradox,” a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.     

Testing fundamental evolutionary hypotheses

Sober and Steel (J. Theor. Biol. 218, 395-408) give important limits on the use of current models with sequence data for studying ancient aspects of evolution; but they go too far in suggesting that several fundamental aspects of evolutionary theory cannot be tested in a normal scientific manner. To the contrary, we show examples of how some alternatives to the theory of descent can be formulated in such a way that they lead to predictions that can be evaluated (and rejected). The critical factor is a logical formulation of the alternatives, even though not all possible alternatives can be tested simultaneously. Similarly, some of the limits using DNA sequence data can be overcome by other types of sequence derived characters. The uniqueness (or not) of the origin of life, though still difficult, is similarly amenable to the testing of alternative hypotheses.     

Tumour suppression by immune system through stochastic oscillations

The well-known Kirschner-Panetta model for tumour-immune System interplay [Kirschner, D., Panetta, J.C., 1998. Modelling immunotherapy of the tumour-immune interaction. J. Math. Biol. 37 (3), 235-252] reproduces a number of features of this essential interaction, but it excludes the possibility of tumour suppression by the immune system in the absence of therapy. Here we present a hybrid-stochastic version of that model. In this new framework, we show that in reality the model is also able to reproduce the suppression, through stochastic extinction after the first spike of an oscillation.