Probing the effects of the well-mixed assumption on viral infection dynamics

Viral kinetics have been extensively studied in the past through the use of spatially well-mixed ordinary differential equations describing the time evolution of the diseased state. However, emerging spatial structures such as localized populations of dead cells might adversely affect the spread of infection, similar to the manner in which a counter-fire can stop a forest fire from spreading. In a previous publication [Beauchemin, C., Samuel, J., Tuszynski, J., 2005. A simple cellular automaton model for influenza A viral infections. J. Theor. Biol. 232(2), 223-234], a simple two-dimensional cellular automaton model was introduced and shown to be accurate enough to model an uncomplicated infection with influenza A. Here, this model is used to investigate the effects of relaxing the well-mixed assumption. Particularly, the effects of the initial distribution of infected cells, the regeneration rule for dead epithelial cells, and the proliferation rule for immune cells are explored and shown to have an important impact on the development and outcome of the viral infection in our model.     

Viability analyses with habitat-based metapopulation models

Population viability analysis (PVA) models incorporate spatial dynamics in different ways. At one extreme are the occupancy models that are based on the number of occupied populations. The simplest occupancy models ignore the location of populations. At the other extreme are individual-based models, which describe the spatial structure with the location of each individual in the population, or the location of territories or home ranges. In between these are spatially structured metapopulation models that describe the dynamics of each population with structured demographic models and incorporate spatial dynamics by modeling dispersal and temporal correlation among populations. Both dispersal and correlation between each pair of populations depend on the location of the populations, making these models spatially structured. In this article, I describe a method that expands spatially structured metapopulation models by incorporating information about habitat relationships of the species and the characteristics of the landscape in which the metapopulation exists. This method uses a habitat suitability map to determine the spatial structure of the metapopulation, including the number, size, and location of habitat patches in which subpopulations of the metapopulation live. The habitat suitability map can be calculated in a number of different ways, including statistical analyses (such as logistic regression) that find the relationship between the occurrence (or, density) of the species and independent variables which describe its habitat requirements. The habitat suitability map is then used to calculate the spatial structure of the metapopulation, based on species-specific characteristics such as the home range size, dispersal distance, and minimum habitat suitability for reproduction. Received: April 1, 1999 / Accepted: October 29, 1999     

Metapopulation dynamics with quasi-local competition

Stepping-stone models for the ecological dynamics of metapopulations are often used to address general questions about the effects of spatial structure on the nature and complexity of population fluctuations. Such models describe an ensemble of local and spatially isolated habitat patches that are connected through dispersal. Reproduction and hence the dynamics in a given local population depend on the density of that local population, and a fraction of every local population disperses to neighboring patches. In such models, interesting dynamic phenomena, e.g. the persistence of locally unstable predator-prey interactions, are only observed if the local dynamics in an isolated patch exhibit non-equilibrium behavior. Therefore, the scope of these models is limited. Here we extend these models by making the biologically plausible assumption that reproductive success in a given local habitat not only depends on the density of the local population living in that habitat, but also on the densities of neighboring local populations. This would occur if competition for resources occurs between neighboring populations, e.g. due to foraging in neighboring habitats. With this assumption of quasi-local competition the dynamics of the model change completely. The main difference is that even if the dynamics of the local populations have a stable equilibrium in isolation, the spatially uniform equilibrium in which all local populations are at their carrying capacity becomes unstable if the strength of quasi-local competition reaches a critical level, which can be calculated analytically. In this case the metapopulation reaches a new stable state, which is, however, not spatially uniform anymore and instead results in an irregular spatial pattern of local population abundance. For large metapopulations, a huge number of different, spatially non-uniform equilibrium states coexist as attractors of the metapopulation dynamics, so that the final state of the system depends critically on the initial conditions. The existence of a large number of attractors has important consequences when environmental noise is introduced into the model. Then the metapopulation performs a random walk in the space of all attractors. This leads to large and complicated population fluctuations whose power spectrum obeys a red-shifted power law. Our theory reiterates the potential importance of spatial structure for ecological processes and proposes new mechanisms for the emergence of non-uniform spatial patterns of abundance and for the persistence of complicated temporal population fluctuations.     

Dispersal: a central and independent trait in life history

The study of tradeoffs among major life history components (age at maturity, lifespan and reproduction) allowed the development of a quantitative framework to understand how environmental variation shapes patterns of biodiversity among and within species. Because every environment is inherently spatially structured, and in most cases temporally variable, individuals need to move within and among habitats to maximize fitness. Dispersal is often assumed to be tightly integrated into life histories through genetic correlations with other vital traits. This assumption is particularly strong within the context of a fast‐slow continuum of life‐history variation. Such a framework is to date used to explain many aspects of population and community dynamics. Evidence for a consistent and context‐independent integration of dispersal in life histories is, however, weak. We therefore advocate the explicit integration of dispersal into life history theory as a principal axis of variation influencing fitness, that is free to evolve, independently of other life history traits. We synthesize theoretical and empirical evidence on the central role of dispersal and its evolutionary dynamics on the spatial distribution of ecological strategies and its impact on population spread, invasions and coexistence. By applying an optimality framework we show that the inclusion of dispersal as an independent dimension of life histories might substantially change our view on evolutionary trajectories in spatially structured environments. Because changes in the spatial configuration of habitats affect the costs of movement and dispersal, adaptations to reduce these costs will increase phenotypic divergence among and within populations. We outline how this phenotypic heterogeneity is anticipated to further impact population and community dynamics.     

Forecasting spatially structured populations: the role of dispersal and scale

We forecasted spatially structured population models with complex dynamics, focusing on the effect of dispersal and spatial scale on the predictive capability of nonlinear forecasting (NLF). Dispersal influences NLF ability by its influence on population dynamics. For simple 2-cell models, when dispersal is small, our ability to predict abundance in subpopulations decreased and then increased with increasing dispersal. Spatial heterogeneity, dispersal manner, and environmental noise did not qualitatively change this result. But results are not clear for complex spatial configurations because of complicated dispersal interactions across subpopulations. Populations undergoing periodic fluctuations could be forecasted perfectly for all deterministic cases that we studied, but less reliably when environmental noise was incorporated. More importantly, for all models that we have examined, NLF was much worse at larger spatial scales as a consequence of the asynchronous dynamics of subpopulations when the dispersal rate was below some critical value. The only difference among models was the critical value of dispersal rate, which varied with growth rate, carrying capacity, mode of dispersal, and spatial configuration. These results were robust even when environmental noise was incorporated. Intermittency, common in the dynamics of spatially structured populations, lowered the predictive capability of NLF. Forecasting population behaviour is of obvious value in resource exploitation and conservation. We suggest that forecasting at local scales holds promise, whereas forecasting abundance at regional scales may yield poor results. Improved understanding of dispersal can enhance the management and conservation of natural resources, and may help us to understand resource-exploitation strategies employed by local indigenous humans.     

Intraguild predation with spatially structured interactions

Consumer–resource interactions with intraguild predation (IGP) were studied in a spatial setting (i.e., predators catch prey and individuals reproduce within local neighborhoods only). Pair approximation (a method for deriving ordinary differential equations that approximate the dynamics of a community that interacts in a lattice environment) was used to study the effect of spatially structured species interactions. An individual-based computer simulation was used to extend the study to a case with spatially variable resource densities. The qualitative results of the pair approximation model were similar to those of the corresponding non-spatial model. However, the spatial model predicted coex((istence over a wider range of parameters than the non-spatial model when intraguild prey are nutritionally valuable to intraguild predators. Spatially heterogeneous resource distributions and spatially structured interaction could overturn the qualitative predictions of non-spatial models.     

A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration

The drive to understand the invasion, spread and fade out of infectious disease in structured populations has produced a variety of mathematical models for pathogen dynamics in metapopulations. Very rarely are these models fully coupled, by which we mean that the spread of an infection within a subpopulation affects the transmission between subpopulations and vice versa. It is also rare that these models are accessible to biologists, in the sense that all parameters have a clear biological meaning and the biological assumptions are explained. Here we present an accessible model that is fully coupled without being an individual-based model. We use the model to show that the duration of an epidemic has a highly non-linear relationship with the movement rate between subpopulations, with a peak in epidemic duration appearing at small movement rates and a global maximum at large movement rates. Intuitively, the first peak is due to asynchrony in the dynamics of infection between subpopulations; we confirm this intuition and also show the peak coincides with successful invasion of the infection into most subpopulations. The global maximum at relatively large movement rates occurs because then the infectious agent perceives the metapopulation as if it is a single well-mixed population wherein the effective population size is greater than the critical community size.     

The effect of dispersal and neighbourhood in games of cooperation

The prisoner's dilemma (PD) and the snowdrift (SD) games are paradigmatic tools to investigate the origin of cooperation. Whereas spatial structure (e.g. nonrandom spatial distribution of strategies) present in the spatially explicit models facilitates the emergence of cooperation in the PD game, recent investigations have suggested that spatial structure can be unfavourable for cooperation in the SD game. The frequency of cooperators in a spatially explicit SD game can be lower than it would be in an infinitely large well-mixed population. However, the source of this effect cannot be identified with certainty as spatially explicit games differ from well-mixed games in two aspects: (i) they introduce spatial correlations, (ii) and limited neighbourhood. Here we extend earlier investigations to identify the source of this effect, and thus accordingly we study a spatially explicit version of the PD and SD games with varying degrees of dispersal and neighbourhood size. It was found that dispersal favours selfish individuals in both games. We calculated the frequency of cooperators at strong dispersal limit, which in concordance with the numerical results shows that it is the short range of interactions (i.e. limited neighbourhood) and not spatial correlations that decreases the frequency of cooperators in spatially explicit models of populations. Our results demonstrate that spatial correlations are always beneficial to cooperators in both the PD and SD games. We explain the opposite effect of dispersal and neighbourhood structure, and discuss the relevance of distinguishing the two effects in general.     

异质种群动态模型:破碎化景观动态模拟的新途径

景观破碎化导致物种以异质种群方式存活,使得基于异质种群动态模拟破碎化景观动态成为可能。异质种群动态模型的发展为景观动态模拟奠定了良好基础。根据空间处理方式的不同,异质种群模型可分为三大类,可不同程度地用于描述破碎化景观动态。(1)空间不确定异质种群模型,假定所有局域种群间均等互联,模型中不包含空间信息,仅能用于景观斑块动态描述;(2)空间确定异质种群模型,假设局域种群在二维空间上以规则格子形式排列,是一种准现实的空间处理方式,可用于景观动态的简单描述;(3)空间现实异质种群模型,包含了破碎化景观中局域种群的几何特征,可直接用于真实景观动态的模拟研究。空间现实的和基于个体的异质种群模型不但是未来异质种群模型发展的主流,也将成为未来破碎化景观动态研究的重要工具。为了更加准确完整地描述破碎化景观动态,不但应该综合运用已有的各种异质种群模型方法,更要引进新模型来刎画多物种、多变量、高维度、复杂连接的破碎化景观格局与过程。     

Effect of diffusion and spatially varying predation risk on the dynamics and equilibrium density of a predator-prey system

Starting from natural planktonic systems, we present a new mechanism involving spatial heterogeneity, and develop a new spatial structure model of planktonic predation systems. Firstly, the effect of diffusion on the dynamics of the system is investigated. We find that diffusion of only prey or both prey and predator between different patches with different predation risk may stabilize the dynamics, depending on the flow rate. Only a medium flow rate can lead to the stability of the system. Too large a rate can cause the system to approach the non-spatial limit case of a well-mixed system. Too large a rate can cause the system to approach the non-spatial limit case as a well-mixed system, which is characterized by its strongly oscillatory dynamics. When only prey diffuse, the smaller the parameter f (the proportion of the patchy volume with larger predation risk to the total volume), the more stable the system. If both populations can diffuse, however, only medium and very small f values may stabilize the system. Also, the response of the spatially averaged equilibrium densities of the system to the increasing of the flow rate is examined. With increasing flow rate, the spatial-averaged equilibrium density of prey decreases, while that of predator depends on which species can diffuse. For the case of prey diffusion only, it first remains unchanged and then slightly decreases, while it increases for the case of combinations as the flow rate increases. Our results are, qualitatively, determined by the spatially heterogeneous mechanism that we propose, and further regulated by top-down forces. Of practical importance, the results reported here indicate that which species diffuse plays a key role in the ways in which diffusion influences the dynamics and the spatial-average equilibrium densities of the system responses to the flow rate's increasing.     

A two-phase epidemic driven by diffusion

In this paper, I present and analyse a model for the spatial dynamics of an epidemic following the point release of an infectious agent. Under conditions where the infectious agent disperses rapidly, relative to the dispersal rate of individuals, the resulting epidemic exhibits two distinct phases: a primary phase in which an epidemic wavefront propagates at constant speed and a secondary phase with a decelerating wavefront. The behavior of the primary phase is similar to standard results for diffusive epidemic models. The secondary phase may be attributed to the environmental persistence of the infectious agent near the release point. Analytic formulas are given for the invasion speeds and asymptotic infection levels. Qualitatively similar results appear to hold in an extended version of the model that incorporates virus shedding and dispersal of individuals.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

The influence of habitat heterogeneity on host-pathogen population dynamics

The influence of spatial heterogeneity on the population dynamics of a naturally occurring invertebrate host-pathogen system was experimentally investigated. At ten week intervals over a two year period, I quantified the spatial distribution of natural populations of the terrestrial isopod crustacean Porcellio scaber infected with the isopod iridescent virus (IIV). During the seasonally dry periods of summer and early fall in central California, isopod populations were highly aggregated and the degree of patchiness and distance between inhabited patches was greatest. Coincident with increased patchiness and patch spacing was an increase in isopod density within patches. During the wet seasons of winter and spring, isopod population patchiness, inter-patch spacing, and within-patch density was low. Seasonal changes in virus prevalence were negatively correlated with within-patch density, patchiness, and inter-patch spacing. The influence of the spatial distribution of isopods on virus prevalence was also tested in field experiments. The virus was introduced into arrays of artificial habitat patches colonized by isopods in which interpatch distance was varied. The prevalence of resulting infections was monitored at weekly intervals. In addition, dispersal rates between artificial patches and natural patches were quantified and compared. The results showed that isopods in treatments with the smallest inter-patch spacing had the highest virus prevalence, with generally lower prevalence among isopods in more widely spaced patches. The spacing of experimental patches significantly affected virus prevalence, although the experiments did not resolve a clear relationship between patch spacing and virus prevalence. Rates of dispersal between patches decreased with increased patch spacing, and these rates did not differ significantly from dispersal between natural patches. The results suggest that rates of dispersal between isopod subpopulations may be an important component of the infection dynamics in this system. I discuss the consequences of these findings for host-pathogen dynamics in fragmented habitats, and for other ecological interactions in spatially heterogeneous habitats.     

Spatial heterogeneity enhance robustness of large multi-species ecosystems

Understanding ecosystem stability and functioning is a long-standing goal in theoretical ecology, with one of the main tools being dynamical modelling of species abundances. With the help of spatially unresolved (well-mixed) population models and equilibrium dynamics, limits to stability and regions of various ecosystem robustness have been extensively mapped in terms of diversity (number of species), types of interactions, interaction strengths, varying interaction networks (for example plant-pollinator, food-web) and varying structures of these networks. Although many insights have been gained, the impact of spatial extension is not included in this body of knowledge. Recent studies of spatially explicit modelling on the other hand have shown that stability limits can be crossed and diversity increased for systems with spatial heterogeneity in species interactions and/or chaotic dynamics. Here we show that such crossing and diversity increase can appear under less strict conditions. We find that the mere possibility of varying species abundances at different spatial locations make possible the preservation or increase in diversity across previous boundaries thought to mark catastrophic transitions. In addition, we introduce and make explicit a multitude of different dynamics a spatially extended complex system can use to stabilise. This expanded stabilising repertoire of dynamics is largest at intermediate levels of dispersal. Thus we find that spatially extended systems with intermediate dispersal are more robust, in general have higher diversity and can stabilise beyond previous stability boundaries, in contrast to well-mixed systems.     

Spatially extended host-parasite interactions: the role of recovery and immunity

Techniques for determining the long-term dynamics of host-parasite systems are well established for mixed populations. The field of spatial modelling in ecology is more recent but a number of key advances have been made. In this paper, we use state-of-the-art approximation techniques, supported by simulations, in order to investigate the role of recovery and immunity in spatially structured populations. Our approach is to use correlation models, namely pair-wise models, to capture the spatial relationships of contacts and interactions between individuals. We use the pair-wise framework to address a number of key ecological questions; including, the persistence of endemic limit cycles and regions of parasite-driven extinction--features which differentiate spatial from non-spatial models--and the effects on invasion fitness. We demonstrate a loss of limit cycle behaviour, in addition to an increase in the critical transmissibility and extinction thresholds, when recovery is included. This approach allows for a better analytical understanding of the dynamics of host-parasite interactions and demonstrates the importance of recovery and immunity in local interactions.     

Spatial dynamics of feedback and feedforward regulation in cell lineages

Feedback mechanisms within cell lineages are thought to be important for maintaining tissue homeostasis. Mathematical models that assume well-mixed cell populations, together with experimental data, have suggested that negative feedback from differentiated cells on the stem cell self-renewal probability can maintain a stable equilibrium and hence homeostasis. Cell lineage dynamics, however, are characterized by spatial structure, which can lead to different properties. Here, we investigate these dynamics using spatially explicit computational models, including cell division, differentiation, death, and migration / diffusion processes. According to these models, the negative feedback loop on stem cell self-renewal fails to maintain homeostasis, both under the assumption of strong spatial restrictions and fast migration / diffusion. Although homeostasis cannot be maintained, this feedback can regulate cell density and promote the formation of spatial structures in the model. Tissue homeostasis, however, can be achieved if spatially restricted negative feedback on self-renewal is combined with an experimentally documented spatial feedforward loop, in which stem cells regulate the fate of transit amplifying cells. This indicates that the dynamics of feedback regulation in tissue cell lineages are more complex than previously thought, and that combinations of spatially explicit control mechanisms are likely instrumental.     

Spatial processes that maintain biodiversity in plant communities

The composition of communities of sessile organisms, and the change in species diversity with time, is a spatially explicit phenomenon. Three spatial factors clearly affect diversity: (1) the structure and heterogeneity of the landscape that limits species immigration and ultimate community size; (2) neighborhood interactions that determine colonization and extinction rates and influence residence times of local populations; and (3) disturbances that open spatially contiguous areas for recolonization by less abundant species. The importance of these three factors was first reviewed and then examined with a spatially explicit, multi-species model of plant dispersal, competition and establishment, with an assumption of neutrality (all species had equivalent life histories) that reduced the initial dimensionality of the problem. The simulations assumed that the probability of immigration was a linear function of mainland abundance and distance to islands, similar to the equilibrium theory of island biogeography and the unified neutral theory of biodiversity. The rate of increase in species richness was not constant across island sizes, declining as island area became very large. This pattern was explained by the spatial dynamics of colonization and establishment, a non-random process that cannot be explained by passive sampling alone. Simulations showed that population establishment depended critically on rare long-distance dispersal events while population persistence was achieved by the formation of aggregated species distributions that developed through restricted dispersal and local competitive interactions. Nevertheless, species richness always declined to a single species in the absence of disturbances, while up to 40 species could persist to 10,000 years when spatially dependent mortality was added. Further explorations with spatially explicit models will be required to fully appreciate the consequence of land use change and altered disturbance regimes on patterns of species distribution and the maintenance of diversity.     

Habitat choice stabilizes metapopulation dynamics by enabling ecological specialization

Dispersal is a key trait responsible for the spread of individuals and genes among local populations, thereby generating eco‐evolutionary interactions. Especially in heterogeneous metapopulations, a tight coupling between dispersal, population dynamics and the evolution of local adaptation is expected. In this respect, dispersal should counteract ecological specialization by redistributing locally selected phenotypes (i.e. migration load). Habitat choice following an informed dispersal decision, however, can facilitate the evolution of ecological specialization. How such informed decisions influence metapopulation size and variability is yet to be determined. By means of individual‐based modelling, we demonstrate that informed decisions about both departure and settlement decouple the evolution of dispersal and that of generalism, selecting for highly dispersive specialists. Choice at settlement is based on information from the entire dispersal range, and therefore decouples dispersal from ecological specialization more effectively than choice at departure, which is only based on local information. Additionally, habitat choice at departure and settlement reduces local and metapopulation variability because of the maintenance of ecological specialization at all levels of dispersal propensity. Our study illustrates the important role of habitat choice for dynamics of spatially structured populations and thus emphasizes the importance of considering that dispersal is often informed.     

The effect of static and dynamic spatially structured disturbances on a locally dispersing population

Previous models of locally dispersing populations have shown that in the presence of spatially structured fixed habitat heterogeneity, increasing local spatial autocorrelation in habitat generally has a beneficial effect on such populations, increasing equilibrium population density. It has also been shown that with large-scale disturbance events which simultaneously affect contiguous blocks of sites, increasing spatial autocorrelation in the disturbances has a harmful effect, decreasing equilibrium population density. Here, spatial population models are developed which include both of these spatially structured exogenous influences, to determine how they interact with each other and with the endogenously generated spatial structure produced by the population dynamics. The models show that when habitat is fragmented and disturbance occurs at large spatial scales, the population cannot persist no matter how large its birth rate, an effect not seen in previous simpler models of this type. The behavior of the model is also explored when the local autocorrelation of habitat heterogeneity and disturbance events are equal, i.e. the two effects occur at the same spatial scale. When this scale parameter is very small, habitat fragmentation prevents the population from persisting because sites attempting to reproduce will drop most of their offspring on unsuitable sites; when the parameter is very large, large-scale disturbance events drive the population to extinction. Population levels reach their maximum at intermediate values of the scale parameter, and the critical values in the model show that the population will persist most easily at these intermediate scales of spatial influences. The models are investigated via spatially explicit stochastic simulations, traditional (infinite-dispersal) and improved (local-dispersal) mean-field approximations, and pair approximations.