Heterogeneous Metapopulations Research Articles

Self-segregation in heterogeneous metapopulation landscapes

Complex interactions are at the root of the population dynamics of many natural systems, particularly for being responsible for the allocation of species and individuals across apposite niches of the ecological landscapes. On the other side, the randomness that unavoidably characterises complex systems has increasingly challenged the niche paradigm providing alternative neutral theoretical models. We introduce a network-inspired metapopulation individual-based model (IBM), hereby named self-segregation, where the density of individuals in the hosting patches (local habitats) drives the individuals spatial assembling while still constrained by nodes’ saturation. In particular, we prove that the core–periphery structure of the networked landscape triggers the spontaneous emergence of vacant habitat patches, which segregate the population in multistable patterns of isolated (sub)communities separated by empty patches. Furthermore, a quantisation effect in the number of vacant patches is observed once the total system mass varies continuously, emphasising thus a striking feature of the robustness of population stationary distributions. Notably, our model reproduces the patch vacancy found in the fragmented habitat of the Glanville fritillary butterfly Melitaea cinxia, an endemic species of the Åland islands. We argue that such spontaneous breaking of the natural habitat supports the concept of the highly contentious (Grinnellian) niche vacancy and also suggests a new mechanism for the endogeneous habitat fragmentation and consequently the peripatric speciation.

Global Dynamics of an SIS Model on Metapopulation Networks with Demographics

In this paper, we propose a susceptible-infected-susceptible (SIS) epidemic model with demographics on heterogeneous metapopulation networks. We analytically derive the basic reproduction number, which determines not only the existence of endemic equilibrium but also the global dynamics of the model. The model always has the disease-free equilibrium, which is globally asymptotically stable when the basic reproduction number is less than unity and otherwise unstable. We also provide sufficient conditions on the global stability of the unique endemic equilibrium. Numerical simulations are performed to illustrate the theoretical results and the effects of the connectivity and diffusion. Furthermore, we find that diffusion rates play an active role in controlling the spread of infectious diseases.

Resource heterogeneity and the evolution of public goods cooperation.

Heterogeneity in resources is a ubiquitous feature of natural landscapes affecting many aspects of biology. However, the effect of environmental heterogeneity on the evolution of cooperation has been less well studied. Here, using a mixture of theory and experiments measuring siderophore production by the bacterium Pseudomonas aeruginosa as a model for public goods based cooperation, we explore the effect of heterogeneity in resource availability. We show that cooperation in metapopulations that were spatially heterogeneous in terms of resources can be maintained at a higher level than in homogeneous metapopulations of the same average resource value. The results can be explained by a positive covariance between fitness of cooperators, population size, and local resource availability, which allowed cooperators to have a disproportionate advantage within the heterogeneous metapopulations. These results suggest that natural environmental variation may help to maintain cooperation.

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.

Moment closure of infectious diseases model on heterogeneous metapopulation network

The global transmission of infectious diseases poses huge threats to human. Traditional heterogeneous mean-field models on metapopulation networks ignore the heterogeneity of individuals who are in different disease states in subpopulations with the same degree, resulting in inaccuracy in predicting the spread of disease. In this paper, we take heterogeneity of susceptible and infectious individuals in subpopulations with the same degree into account, and propose a deterministic unclosed general model according to Markov process on metapopulation networks to curve the global transmission of diseases precisely. Then we make the general model closed by putting forward two common assumptions: a two-dimensional constant distribution and a two-dimensional log-normal distribution, where the former is equivalent to the heterogeneous mean-field model, and the latter is a system of weighted ordinary differential equations. Further we make a stability analysis for two closed models and illustrate the results by numerical simulations. Next, we conduct a series of numerical simulations and stochastic simulations. Results indicate that our general model extends and optimizes the mean-field model. Finally, we investigate the impacts of total mobility rate on disease transmission and find that timely and comprehensive travel restriction in the early stage is an effective prevention and control of infectious diseases.

A spatially heterogeneous network-based metapopulation software model applied to the simulation of a pulmonary tuberculosis infection

Tuberculosis (TB) is an ancient disease that, although curable, still accounts for over 1 million deaths worldwide. Shortening treatment time is an important area of research but is hampered by the lack of models that mimic the full range of human pathology. TB shows distinct localisations during different stages of infection, the reasons for which are poorly understood. Greater understanding of how heterogeneity within the human lung influences disease progression may hold the key to improving treatment efficiency and reducing treatment times.In this work, we present a novel in silico software model which uses a networked metapopulation incorporating both spatial heterogeneity and dissemination possibilities to simulate a TB infection over the whole lung and associated lymphatics. The entire population of bacteria and immune cells is split into a network of patches: members interact within patches and are able to move between them. Patches and edges of the lung network include their own environmental attributes which influence the dynamics of interactions between the members of the subpopulations of the patches and the translocation of members along edges.In this work, we detail the initial findings of a whole-organ model that incorporates distinct spatial heterogeneity features which are not present in standard differential equation approaches to tuberculosis modelling. We show that the inclusion of heterogeneity within the lung landscape when modelling TB disease progression has significant outcomes on the bacterial load present: a greater differential of oxygen, perfusion and ventilation between the apices and the basal regions of the lungs creates micro-environments at the apex that are more preferential for bacteria, due to increased oxygen availability and reduced immune activity, leading to a greater overall bacterial load present once latency is established.These findings suggest that further whole-organ modelling incorporating more sophisticated heterogeneities within the environment and complex lung topologies will provide more insight into the environments in which TB bacteria persist and thus help develop new treatments which are factored towards these environmental conditions.

Asymmetric migration decreases stability but increases resilience in a heterogeneous metapopulation

Many natural populations are spatially distributed, forming a network of subpopulations linked by migration. Migration patterns are often asymmetric and heterogeneous, with important consequences on the ecology and evolution of the species. Here we investigate experimentally how asymmetric migration and heterogeneous structure affect a simple metapopulation of budding yeast, formed by one strain that produces a public good and a non-producer strain that benefits from it. We study metapopulations with star topology and asymmetric migration, finding that all their subpopulations have a higher fraction of producers than isolated populations. Furthermore, the metapopulations have lower tolerance to challenging environments but higher resilience to transient perturbations. This apparent paradox occurs because tolerance to a constant challenge depends on the weakest subpopulations of the network, while resilience to a transient perturbation depends on the strongest ones.

A Landscape Approach to Invasive Species Management.

Biological invasions are not only a major threat to biodiversity, they also have major impacts on local economies and agricultural production systems. Once established, the connection of local populations into metapopulation networks facilitates dispersal at landscape scales, generating spatial dynamics that can impact the outcome of pest-management actions. Much planning goes into landscape-scale invasive species management. However, effective management requires knowledge on the interplay between metapopulation network topology and management actions. We address this knowledge gap using simulation models to explore the effectiveness of two common management strategies, applied across different extents and according to different rules for selecting target localities in metapopulations with different network topologies. These management actions are: (i) general population reduction, and (ii) reduction of an obligate resource. The reduction of an obligate resource was generally more efficient than population reduction for depleting populations at landscape scales. However, the way in which local populations are selected for management is important when the topology of the metapopulation is heterogeneous in terms of the distribution of connections among local populations. We tested these broad findings using real-world scenarios of European rabbits (Oryctolagus cuniculus) infesting agricultural landscapes in Western Australia. Although management strategies targeting central populations were more effective in simulated heterogeneous metapopulation structures, no difference was observed in real-world metapopulation structures that are highly homogeneous. In large metapopulations with high proximity and connectivity of neighbouring populations, different spatial management strategies yield similar outcomes. Directly considering spatial attributes in pest-management actions will be most important for metapopulation networks with heterogeneously distributed links. Our modelling framework provides a simple approach for identifying the best possible management strategy for invasive species based on metapopulation structure and control capacity. This information can be used by managers trying to devise efficient landscape-oriented management strategies for invasive species and can also generate insights for conservation purposes.

Modeling vaccination in a heterogeneous metapopulation system

We present here a multicity SIS epidemic model with vaccination. The model describes the dynamics of heterogeneous metapopulations that contain imperfectly vaccinated individuals. The effect of vaccination on heterogeneous multicity models has not been previously studied. We show that under very generic conditions, the epidemic threshold does not depend on the diffusion coefficient of the vaccinated individuals, but it does depend on the diffusion coefficient of the infected population. We then show, using a novel methodology, that the reproduction number is determined by the homogeneous model parameters and by the maximal number of neighbors a city can have, when the diffusion coefficient of the infected population is low. Finally, we present numerical simulations to support the analytical results.

Impact of dispersal on the stability of metapopulations

Dispersal is a key ecological process that enables local populations to form spatially extended systems called metapopulations. In the present study, we investigate how dispersal affects the linear stability of a general single-species metapopulation model. We discuss both the influence of local within-patch dynamics and the effects of various dispersal behaviours on stability. We find that positive density-dependent dispersal and positive density-dependent settlement are destabilizing dispersal behaviours while negative density-dependent dispersal and negative density-dependent settlement are stabilizing. It is also shown that dispersal has a stabilizing impact on heterogeneous metapopulations that correlates positively with the number of patches and the connectance of metapopulation networks.

The Scaling of Human Contacts and Epidemic Processes in Metapopulation Networks.

We study the dynamics of reaction-diffusion processes on heterogeneous metapopulation networks where interaction rates scale with subpopulation sizes. We first present new empirical evidence, based on the analysis of the interactions of 13 million users on Twitter, that supports the scaling of human interactions with population size with an exponent γ ranging between 1.11 and 1.21, as observed in recent studies based on mobile phone data. We then integrate such observations into a reaction- diffusion metapopulation framework. We provide an explicit analytical expression for the global invasion threshold which sets a critical value of the diffusion rate below which a contagion process is not able to spread to a macroscopic fraction of the system. In particular, we consider the Susceptible-Infectious-Recovered epidemic model. Interestingly, the scaling of human contacts is found to facilitate the spreading dynamics. This behavior is enhanced by increasing heterogeneities in the mobility flows coupling the subpopulations. Our results show that the scaling properties of human interactions can significantly affect dynamical processes mediated by human contacts such as the spread of diseases, ideas and behaviors.

Cluster formation in a heterogeneous metapopulation model.

A spatially explicit heterogeneous metapopulation model with two different patch types is analyzed. Some network topologies support a partially synchronized dynamics, a state where two different clusters of patches are formed. Within each cluster the dynamics of all patches are synchronized. The linearized asymptotic stability of the partially synchronized attractor is studied. The transversal stability is analyzed and a simple expression for the transversal Lyapunov number of partially synchronized attractors is obtained.

Impact of density-dependent migration flows on epidemic outbreaks in heterogeneous metapopulations.

We investigate the role of migration patterns on the spread of epidemics in complex networks. We enhance the SIS-diffusion model on metapopulations to a nonlinear diffusion. Specifically, individuals move randomly over the network but at a rate depending on the population of the departure patch. In the absence of epidemics, the migration-driven equilibrium is described by quantifying the total number of individuals living in heavily or lightly populated areas. Our analytical approach reveals that strengthening the migration from populous areas contains the infection at the early stage of the epidemic. Moreover, depending on the exponent of the nonlinear diffusion rate, epidemic outbreaks do not always occur in the most populated areas as one might expect.

Stochastic dynamics of interacting haematopoietic stem cell niche lineages.

Since we still know very little about stem cells in their natural environment, it is useful to explore their dynamics through modelling and simulation, as well as experimentally. Most models of stem cell systems are based on deterministic differential equations that ignore the natural heterogeneity of stem cell populations. This is not appropriate at the level of individual cells and niches, when randomness is more likely to affect dynamics. In this paper, we introduce a fast stochastic method for simulating a metapopulation of stem cell niche lineages, that is, many sub-populations that together form a heterogeneous metapopulation, over time. By selecting the common limiting timestep, our method ensures that the entire metapopulation is simulated synchronously. This is important, as it allows us to introduce interactions between separate niche lineages, which would otherwise be impossible. We expand our method to enable the coupling of many lineages into niche groups, where differentiated cells are pooled within each niche group. Using this method, we explore the dynamics of the haematopoietic system from a demand control system perspective. We find that coupling together niche lineages allows the organism to regulate blood cell numbers as closely as possible to the homeostatic optimum. Furthermore, coupled lineages respond better than uncoupled ones to random perturbations, here the loss of some myeloid cells. This could imply that it is advantageous for an organism to connect together its niche lineages into groups. Our results suggest that a potential fruitful empirical direction will be to understand how stem cell descendants communicate with the niche and how cancer may arise as a result of a failure of such communication.

Impact of commuting on disease persistence in heterogeneous metapopulations

We use a stochastic metapopulation model to study the combined effects of seasonality and spatial heterogeneity on disease persistence. We find a pronounced effect of enhanced persistence associated with strong heterogeneity, intermediate coupling strength and moderate seasonal forcing. Analytic calculations show that this effect is not related to the phase lag between epidemic bursts in different patches, but rather with the linear stability properties of the attractor that describes the steady state of the system in the large population limit.

Spectral Properties of the Connectivity Matrix and the SIS-epidemic Threshold for Mid-size Metapopulations

We consider the spread of an infectious disease on a heterogeneous metapopulation defined by any (correlated or uncorrelated) network. The infection evolves under transmission, recovery and migration mechanisms. We study some spectral properties of a connectivity matrix arising from the continuous-time equations of the model. In particular we show that the classical sufficient condition of instability for the disease-free equilibrium, well known for the particular case of uncorrelated networks, works also for the general case. We give also an alternative condition that yields a more accurate estimation of the epidemic threshold for correlated (either assortative or dissortative) networks.

Effects of symmetric and asymmetric dispersal on the dynamics of heterogeneous metapopulations: Two-patch systems revisited

Although the effects of dispersal on the dynamics of two-patch metapopulations are well studied, potential interactions between local dynamics and asymmetric dispersal remain unexplored. We examined the dynamics of two Ricker models coupled by symmetric or asymmetric constant-fraction dispersal at different rates. Unlike previous studies, we extensively sampled the r1−r2 space and found that stability of the coupled system was markedly affected by interactions between dispersal (in terms of strength and asymmetry) and local dynamics. When both subpopulations were intrinsically chaotic, increased symmetry in the exchange of individuals had a greater stabilizing impact on the dynamics of the system. When one subpopulation showed considerably more unstable dynamics than the other, higher asymmetry in the exchange of individuals had a stabilizing or destabilizing effect on the dynamics depending on whether the net dispersal bias was from the relatively stable to the relatively unstable subpopulation, or vice versa. The sensitivity of chaotic dynamics to stabilization due to dispersal varied with r-value in the chaotic subpopulation. Under unidirectional or bidirectional symmetric dispersal, when one subpopulation was intrinsically chaotic and the other had stable dynamics, the stabilization of chaotic subpopulations with r~3.3–4.0 occurred at the lowest dispersal rates, followed by chaotic subpopulations with r~2.7–2.95 and, finally, chaotic subpopulations with r~2.95–3.3. The mechanism for this pattern is not known but might be related to the range and number of different attainable population sizes possible in different r-value zones.

Social and landscape effects on food webs: a multi-level network simulation model

One source of complexity in ecological systems is the hierarchical organization of parallel biological processes. Our ‘horizontal’ knowledge describing different levels is quite massive, but the understanding of their vertical interactions is very poor. We present a toy model linking social networks, food webs and a landscape graph. Horizontal processes refer to population, community and metacommunity dynamics, while vertical processes connect the three organizational levels. The model is stochastic and individual-based. We parametrized it by using reasonable empirical values found in the literature. Sensitivity analysis shows how the parameters describing the dynamics of a particular species (e.g., probability of social tie formation with conspecific individuals, or migration rate) can affect metapopulation size and spatial heterogeneity of all food web species. Changing the values of various parameters at any of the three levels have commensurable effects on the population size of all species. In contrast to the general intuition, community dynamics do not dominate population biology; social and landscape processes can trigger greater effects than food web interactions. More rapidly changing social relationships lead to a decrease in social network cohesion, thus impairing the feeding efficiency of consumers. In food webs, trophic specialization provides an advantage when it contributes to avoid competition, being detrimental otherwise. Highest migration rates result in a more heterogeneous metapopulation distribution of the generalist consumer, indirectly supporting its specialist competitor. We discuss conceptual and methodological aspects of the model, demonstrating the importance of an integrative view. We also emphasize the relevance of vertical connections, suggesting how such a modelling framework could support conservation biology. Further studies should focus on methods to approximate external pressures with changes in model parameters, thus allowing to characterize possible impacts on ecological systems.

Time-varying human mobility patterns with metapopulation epidemic dynamics

In this paper, explicitly considering the influences of an epidemic outbreak on human travel, a time-varying human mobility pattern is introduced to model the time variation of global human travel. The impacts of the pattern on epidemic dynamics in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals, are investigated by using a mean-field approach. The results show that the pattern does not alter the epidemic threshold, but can slightly lower the final average density of infected individuals as a whole. More importantly, we also find that the pattern produces different impacts on nodes with different degree, and that there exists a critical degree kc. For nodes with degree smaller than kc, the pattern produces a positive impact on epidemic mitigation; conversely, for nodes with degree larger than kc, the pattern produces a negative impact on epidemic mitigation.

Dynamics of Adaptation in Spatially Heterogeneous Metapopulations

The selection pressure experienced by organisms often varies across the species range. It is hence crucial to characterise the link between environmental spatial heterogeneity and the adaptive dynamics of species or populations. We address this issue by studying the phenotypic evolution of a spatial metapopulation using an adaptive dynamics approach. The singular strategy is found to be the mean of the optimal phenotypes in each habitat with larger weights for habitats present in large and well connected patches. The presence of spatial clusters of habitats in the metapopulation is found to facilitate specialisation and to increase both the level of adaptation and the evolutionary speed of the population when dispersal is limited. By showing that spatial structures are crucial in determining the specialisation level and the evolutionary speed of a population, our results give insight into the influence of spatial heterogeneity on the niche breadth of species.