Synchronous dynamics and rates of extinction in spatially structured populations

We explore extinction rates using a spatially arranged set of subpopulations obeying Ricker dynamics. The population system is subjected to dispersal of individuals among the subpopulations as well as to local and global disturbances. We observe a tight positive correlation between global extinction rate and the level of synchrony in dynamics among thesubpopulations. Global disturbances and to a lesser extent, migration, are capable of synchronizing the temporal dynamics of the subpopulations over a rather wide span of the population growth rate r. Local noise decreases synchrony, as does increasing distance among the subpopulations. Synchrony also levels off with increasing r: in the chaotic region, subpopulations almost invariably behave asynchronously. We conclude that it is asynchrony that reduces the probability of global extinctions, not chaos as such: chaos is a special case only. The relationship between global extinction rate, synchronous dynamics and population growth rate is robust to changes in dispersal rates and ranges.     

A spatially structured metapopulation model with patch dynamics

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 It? type based on our deterministic model. Our simulations reveal good agreement between the deterministic model and the SDE model.     

Reconstructing the dynamics of unobserved variables in spatially extended systems

Attractor reconstruction using embedding techniques is a widely used tool when analysing data from real systems. It allows reconstruction of the system dynamics from only one observable and is thus extremely powerful. We show here that this reconstruction is also possible from spatially coupled systems. We use a common host–parasitoid model as an example as ecological systems are virtually always spatially extended. Additionally, data from ecological systems has often only one observable, e.g. population density, from a potentially much higher-dimensional system. Singular value decomposition is used to show the existence of a functional relationship mapping the time delayed coordinates of one variable to the full spatially coupled system. We investigate the effects of noise and indicate two important spatial scales. Finally, we illustrate that a reconstruction can be obtained from a system that is only partially sampled.     

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.     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

Biodiversity and reduced extinction risks in spatially isolated rodent populations

Ecologists have rarely explored the potential influence of local (alpha) biodiversity on the stability and local extinction of spatially isolated populations. Twenty years of annual counts of a small, grazing rodent (Utah prairie dogs, Cynomys parvidens ) from 20 different isolated local populations (colonies) in southern Utah, U.S.A. were analysed. These prairie dogs exhibited large fluctuations and repeated extinctions at individual colonies during the census period. Frequency of extinction at a colony declined dramatically as the number of locally occurring plant species increased. This pattern was not explained by differences among colonies in plant productivity, plant species composition, colony size, or variability in annual counts. Thus, lower extinction risk of consumer populations may be associated with greater resource diversity, and maintaining high local plant diversity may help sustain spatially isolated herbivore populations in fragmented habitats.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

A management-oriented framework for selecting metrics used to assess habitat- and path-specific quality in spatially structured populations

Mobile species with complex spatial dynamics can be difficult to manage because their population distributions vary across space and time, and because the consequences of managing particular habitats are uncertain when evaluated at the level of the entire population. Metrics to assess the importance of habitats and pathways connecting habitats in a network are necessary to guide a variety of management decisions. Given the many metrics developed for spatially structured models, it can be challenging to select the most appropriate one for a particular decision. To guide the management of spatially structured populations, we define three classes of metrics describing habitat and pathway quality based on their data requirements (graph-based, occupancy-based, and demographic-based metrics) and synopsize the ecological literature relating to these classes. Applying the first steps of a formal decision-making approach (problem framing, objectives, and management actions), we assess the utility of metrics for particular types of management decisions. Our framework can help managers with problem framing, choosing metrics of habitat and pathway quality, and to elucidate the data needs for a particular metric. Our goal is to help managers to narrow the range of suitable metrics for a management project, and aid in decision-making to make the best use of limited resources.     

Scale-free dynamics of somatic adaptability in immune system

The long-time dynamics of somatic adaptability in immune system is simulated by a simple physical model defined in the shape space. The immune system described by the model exhibits a scale free behavior as is observed in living systems. The balance between the positive and negative feedbacks of the model leads to a robust immune system where the positive one corresponds to the formation of memory cells and the negative one to immunosuppression. Also the immunosenescence of the system is discussed based on the time-dependence of the epigenetic landscape of the adaptive immune cells in the shape space.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.     

Self-organization of mobile populations in cyclic competition

The formation of out-of-equilibrium patterns is a characteristic feature of spatially extended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the ‘rock-paper-scissors’ game. Both experimentally and theoretically, such non-transitive interactions have been found to induce self-organization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals’ mobility on the spatial structures emerging in rock-paper-scissors games. We devise a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system's behavior—in the proper continuum limit—is aptly captured by a set of stochastic partial differential equations. The system's stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals’ wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex Ginzburg-Landau equation, their entanglement results from the inherent stochastic nature of the system. These findings and our methods have important applications for understanding the formation of noisy patterns, e.g. in ecological and evolutionary contexts, and are also of relevance for the kinetics of (bio)-chemical reactions.     

Patterns of vegetative growth and reproduction in relation to branch orders: the plant as a spatially structured population

The patterns of vegetative growth and reproduction in relation to orders of terminal branches were examined in the evergreen woody plant, Eurya japonica. The branch order number was determined centrifugally. The trunk was given order number 1; branches issuing directly from the trunk were order 2; branches growing on order 2 branches were order 3, and so on. The results of this study show the differential patterns of vegetative growth and reproduction in relation to the branch orders. Lower-order shoots of terminal branches grew more, but produced few flowers. On the other hand, for the higher-order terminal branches, shoot growth was very limited but flowering was more intense. The results show that a tree can be interpreted not as a mere population of equivalent modules but as a spatially structured population. Thus, it is essential to consider the position of modules within the branch system when patterns of vegetative growth and reproduction are examined. It is hypothesized that the difference in the opportunity cost in relation to the branch orders is the main cause of the spatial structure for patterns of vegetative growth and reproduction. Furthermore, for same-order terminal branches, current-year shoot elongation was independent of flowering intensity. These results suggest that this species only invests resources in reproduction that are surplus to its requirements for the functions associated with survival, such as growth and/or storage. Received: 22 July 1999 / Accepted: 12 January 2000     

Spatial models of virus-immune dynamics

To date, the majority of theoretical models describing the dynamics of infectious diseases in vivo are based on the assumption of well-mixed virus and cell populations. Because many infections take place in solid tissues, spatially structured models represent an important step forward in understanding what happens when the assumption of well-mixed populations is relaxed. Here, we explore models of virus and virus-immune dynamics where dispersal of virus and immune effector cells was constrained to occur locally. The stability properties of our spatial virus-immune dynamics models remained robust under almost all biologically plausible dispersal schemes, regardless of their complexity. The various spatial dynamics were compared to the basic non-spatial dynamics and important differences were identified: When space was assumed to be homogeneous, the dynamics generated by non-spatial and spatially structured models differed substantially at the peak of the infection. Thus, non-spatial models may lead to systematic errors in the estimates of parameters underlying acute infection dynamics. When space was assumed to be heterogeneous, spatial coupling not only changed the equilibrium properties of the uncoupled populations but also equalized the dynamics and thereby reduced the likelihood of dynamic elimination of the infection. In line with experimental and clinical observations, long-lasting oscillation periods were virtually absent. When source-sink dynamics were considered, the long-term outcome of the infection depended critically on the degree of spatial coupling. The infection collapsed when emigration from source sites became too large. Finally, we discuss the implications of spatially structured models on medical treatment of infectious diseases, and note that a huge gap exists in data accurately describing infection dynamics in solid tissues.     

Eco-evolutionary dynamics of dispersal in spatially heterogeneous environments

Ecology Letters (2011) 14: 1025-1034 ABSTRACT: Evolutionary changes in natural populations are often so fast that the evolutionary dynamics may influence ecological population dynamics and vice versa. Here we construct an eco-evolutionary model for dispersal by combining a stochastic patch occupancy metapopulation model with a model for changes in the frequency of fast-dispersing individuals in local populations. We test the model using data on allelic variation in the gene phosphoglucose isomerase (Pgi), which is strongly associated with dispersal rate in the Glanville fritillary butterfly. Population-specific measures of immigration and extinction rates and the frequency of fast-dispersing individuals among the immigrants explained 40% of spatial variation in Pgi allele frequency among 97 local populations. The model clarifies the roles of founder events and gene flow in dispersal evolution and resolves a controversy in the literature about the consequences of habitat loss and fragmentation on the evolution of dispersal.     

Diversity and extinction in the Cenozoic history of Caribbean reefs

Occurrences of reef corals are examined at Caribbean fossil localities to determine how biodiversity has changed within the region over the past 50 million years. Analyses of 294 species (66 genera) at 58 fossil localities show that Caribbean generic diversity rose to 44 between 50–22 Ma, ranged from 32–39 between 22–2 Ma, and dropped to 25 afterwards. Regional species diversity was high at 40–36 Ma, 28–22 Ma, and 5–2 Ma. Origination rates were elevated throughout each high diversity interval, but extinction was concentrated near the end of each interval. Regional highs of origination and extinction, therefore, differed in timing and duration, causing the observed regional diversity increases during the three remarkably long intervals of turnover. Highs of generic origination decreased in magnitude as immigration from the Mediterranean ceased, but speciation highs increased in association with emergence of the Central American isthmus. Peaks of extinction coincided with regional changes in climate and oceanic circulation. Maximum species diversities within assemblages increased to 40–60 between 50–36 Ma, and have remained relatively constant ever since. Assemblage compositions differed among localities having similar ages and environments, suggesting that the timing and pattern of turnover varied across the region. Stable diversities but variable compositions within assemblages suggest that dispersal and recruitment influenced the pattern of faunal change during turnover. Accepted: 22 August 1999     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Population and evolutionary dynamics in spatially structured seasonally varying environments

Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life‐history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco‐evolutionary theory and models have not yet fully encompassed within‐individual and among‐individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life‐history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal‐tracking technologies are increasingly demonstrating substantial within‐population variation in the occurrence and form of migration versus year‐round residence, generating diverse forms of ‘partial migration’ spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year‐round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio‐temporal population dynamics, we define a ‘partially migratory meta‐population’ system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within‐individual and among‐individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco‐evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta‐populations given diverse forms of seasonal environmental variation and change, and to forecast system‐specific dynamics. To demonstrate one such approach, we use an evolutionary individual‐based model to illustrate that multiple forms of partial migration can readily co‐exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco‐evolutionary responses to spatio‐temporal seasonal environmental variation and change.     

Multi-species spatially-explicit indicators reveal spatially structured trends in bird communities

Multi-species indicators are often used to assess biodiversity trends. By combining population trends across several species they summarise trends across a community. Composite indicators such as these are useful for examining general temporal patterns and may suggest important drivers of biodiversity change. However, they may also mask substantial spatial variation in population trends, particularly when they are calculated over large spatial regions. We produced spatially-explicit indicators for farmland and woodland bird communities in the UK and further separate these into trends for generalist and specialist species within each group. We found considerable spatial variation in the indicators, which is masked by indicators calculated at the national level. The farmland community indicator showed mostly positive trends in western areas and extensive declines in south-east England. The woodland community indicator showed a north–south divide, with increases in Scotland and northern England and stability in the southern regions. For both communities, indicator trends for specialist species were more negative than those for generalists. We found no significant difference in farmland community indicators between arable land and improved grassland. Woodland specialists had significantly more negative trends in broadleaf compared to coniferous woodlands, suggesting habitat-type is one of the drivers of changes in the woodland community. These spatial patterns in bird population trends may be used to highlight regional conservation priorities and identify where those may differ from the national scale. In combination with information about other environmental changes, they may also be used to develop hypotheses about potential drivers of change. We advocate that this approach is adopted for other taxa and geographical areas.     

Asynchronization of local population dynamics and persistence of a metapopulation: a lesson from an endangered composite plant, Aster kantoensis

Fragmentation of a large habitat makes local populations less linked to others, and a whole population structure changes to a metapopulation. The smaller a local population is, the more strengthened extinction factors become. Then, frequent extinctions of local populations threaten persistence of the metapopulation unless recolonizations occur rapidly enough after local extinctions. Spatially structured models have been more widely used for predicting future population dynamics and for assessing the extinction risk of a metapopulation. In this article, we first review such spatially structured models that have been applied to conservation biology, focusing on effects of asynchronization among local population dynamics on persistence of the whole metapopulation. Second, we introduce our ongoing project on extinction risk assessment of an endangered composite biennial plant, Aster kantoensis, in the riverside habitat, based on a lattice model for describing its spatiotemporal population dynamics. The model predicted that the extinction risk of A. kantoensis depends on both the frequency of flood occurrence and the time to coverage of a local habitat by other competitively stronger perennials. Finally, we present a measure (Hassell and Pacala's CV ²) for quantifying the effect of asynchronization among local population dynamics on the persistence of a whole metapopulation in conservation ecology. Received: January 12, 2000 / Accepted: February 8, 2000