Lattice population dynamics for plants with dispersing seeds and Vegetative propagation

Complex nonmonotonic responses of biodiversity to habitat destruction

It has typically been assumed that habitat destruction, characterized by habitat loss and fragmentation, has consistently negative effects on biodiversity. While numerous empirical studies have shown the detrimental effects of habitat loss, debate continues as to whether habitat fragmentation has universally negative effects. To explore the effects of habitat fragmentation, we developed a simple model for site-occupancy dynamics in fragmented landscapes. With the model, we demonstrate that a competition-colonization trade-off can result in nonlinear oscillatory responses in biodiversity to both habitat loss and fragmentation. However, the overall pattern of habitat loss reducing species richness is still established, in line with empirical observations. Interestingly, the existence of localized oscillations in biodiversity can explain the mixed responses of species richness to habitat fragmentation per se observed in nature, thereby reconciling the debate on the fragmentation-diversity relationship. Therefore, this study offers a parsimonious mechanistic explanation for empirically observed biodiversity patterns in response to habitat destruction.

Data-driven optimal closures for mean-cluster models: Beyond the classical pair approximation

This study concerns the mean-clustering approach to modeling the evolution of lattice dynamics. Instead of tracking the state of individual lattice sites, this approach describes the time evolution of the concentrations of different cluster types. It leads to an infinite hierarchy of ordinary differential equations which must be closed by truncation using a so-called closure condition. This condition approximates the concentrations of higher-order clusters in terms of the concentrations of lower-order ones. The pair approximation is the most common form of closure. Here, we consider its generalization, termed the "optimal approximation," which we calibrate using a robust data-driven strategy. To fix attention, we focus on a recently proposed structured lattice model for a nickel-based oxide, similar to that used as cathode material in modern commercial Li-ion batteries. The form of the obtained optimal approximation allows us to deduce a simple sparse closure model. In addition to being more accurate than the classical pair approximation, this "sparse approximation" is also physically interpretable which allows us to a posteriori refine the hypotheses underlying construction of this class of closure models. Moreover, the mean-cluster model closed with this sparse approximation is linear and hence analytically solvable such that its parametrization is straightforward, although it offers a good approximation of the actual time evolution of the cluster concentrations on short timescales only. On the other hand, parametrization of the mean-cluster model closed with the pair approximation is shown to lead to an ill-posed inverse problem.

Species exclusion and coexistence in a noisy voter model with a competition-colonization tradeoff

We introduce an asymmetric noisy voter model to study the joint effect of immigration and a competition-dispersal tradeoff in the dynamics of two species competing for space in regular lattices. Individuals of one species can invade a nearest-neighbor site in the lattice, while individuals of the other species are able to invade sites at any distance but are less competitive locally, i.e., they establish with a probability g≤1. The model also accounts for immigration, modeled as an external noise that may spontaneously replace an individual at a lattice site by another individual of the other species. This combination of mechanisms gives rise to a rich variety of outcomes for species competition, including exclusion of either species, monostable coexistence of both species at different population proportions, and bistable coexistence with proportions of populations that depend on the initial condition. Remarkably, in the bistable phase, the system undergoes a discontinuous transition as the intensity of immigration overcomes a threshold, leading to a half loop dynamics associated to a cusp catastrophe, which causes the irreversible loss of the species with the shortest dispersal range.

The role of omnivory in mediating metacommunity robustness to habitat destruction

Omnivores have long been known to play an important role in determining the stability of ecological communities. Recent theoretical studies have suggested that they may also increase the resilience of their communities to habitat destruction, one of the major drivers of species extinctions globally. However, these outcomes were obtained for minimal food webs consisting of only a single omnivore and its prey species, while much more complex communities can be anticipated in nature. In this study, we undertake a systematic comparative analysis of the robustness of metacommunities containing various omnivory structures to habitat loss and fragmentation using a mathematical model. We observe that, in general, omnivores are better able to survive facing habitat destruction than specialist predators of similar trophic level. However, the community as a whole does not always benefit from the presence of omnivores, as they may drive their intraguild prey to extinction. We also analyze the frequency with which these modules occur in a set of empirical food webs, and demonstrate that variation in their rate of occurrence is consistent with our model predictions. Our findings demonstrate the importance of considering the complete food web in which an omnivore is embedded, suggesting that future study should focus on more holistic community analysis.

A unifying framework for interpreting and predicting mutualistic systems

Coarse-grained rules are widely used in chemistry, physics and engineering. In biology, however, such rules are less common and under-appreciated. This gap can be attributed to the difficulty in establishing general rules to encompass the immense diversity and complexity of biological systems. Furthermore, even when a rule is established, it is often challenging to map it to mechanistic details and to quantify these details. Here we report a framework that addresses these challenges for mutualistic systems. We first deduce a general rule that predicts the various outcomes of mutualistic systems, including coexistence and productivity. We further develop a standardized machine-learning-based calibration procedure to use the rule without the need to fully elucidate or characterize their mechanistic underpinnings. Our approach consistently provides explanatory and predictive power with various simulated and experimental mutualistic systems. Our strategy can pave the way for establishing and implementing other simple rules for biological systems.

Evolution of Complex Asexual Reproductive Strategies in Jellyfish

Many living organisms in terrestrial and aquatic ecosystems rely on multiple reproductive strategies to reduce the risk of extinction in variable environments. Examples are provided by the polyp stage of several bloom-forming jellyfish species, which can reproduce asexually using different budding strategies. These strategies broadly fall into three categories: (1) fast localized reproduction, (2) dormant cysts, or (3) motile and dispersing buds. Similar functional strategies are also present in other groups of species. However, mechanisms leading to the evolution of this rich reproductive diversity are yet to be clarified. Here we model how risk of local population extinction and differential fitness of alternative modes of asexual reproduction could drive the evolution of multiple reproductive modes as seen in jellyfish polyps. Depending on environmental parameters, we find that evolution leads to a unique evolutionarily stable strategy, wherein multiple reproductive strategies generally coexist. As the extinction risk increases, this strategy shifts from a pure budding mode to a dual strategy and finally to one characterized by allocation into all three modes. We identify relative fitness-dependent thresholds in extinction risk where these transitions can occur and discuss our predictions in light of observations on polyp reproduction in laboratory and natural systems.

Food web persistence in fragmented landscapes

Habitat destruction, characterized by patch loss and fragmentation, is a key driver of biodiversity loss. There has been some progress in the theory of spatial food webs; however, to date, practically nothing is known about how patch configurational fragmentation influences multi-trophic food web dynamics. We develop a spatially extended patch-dynamic model for different food webs by linking patch connectivity with trophic-dependent dispersal (i.e. higher trophic levels displaying longer-range dispersal). Using this model, we find that species display different sensitivities to patch loss and fragmentation, depending on their trophic position and the overall food web structure. Relative to other food webs, omnivory structure significantly increases system robustness to habitat destruction, as feeding on different trophic levels increases the omnivore's persistence. Additionally, in food webs with a dispersal-competition trade-off between species, intermediate levels of habitat destruction can enhance biodiversity by creating refuges for the weaker competitor. This demonstrates that maximizing patch connectivity is not always effective for biodiversity maintenance, as in food webs containing indirect competition, doing so may lead to further species loss.

The analysis of an effect of seed propagation on defense strategy against pathogen transmission within clonal plant population using lattice model

Many clonal plants have two breeding systems, vegetative and seed propagation. In vegetative propagation, plants reproduce genetically identical offspring that have lower mortality rates. By contrast, the seed propagated offspring has higher mortality rate, however, the seed propagation acts an important role in maintaining the genetic diversity and reproduce widely. According to the experimental studies, the balance between the breeding systems, vegetative and seed propagation, is determined by several functions, such as resource allocation. The infection and spread of systemic pathogen also affect the optimal balance of the breeding systems. Thus, we examine the effect of invasion of systemic pathogen on the optimal balance of the breeding systems of clonal plant using lattice model in two cases, single population and mixed population. In the analysis, the equilibrium and its local stability were derived using approximation method and numerical simulation in single population. Additionally, two situations were assumed in mixed population, infected and uninfected populations, and the efficacy of seed propagation on the suppression of epidemic infections was examined by comparing the results in the two situations. As a result, seed propagation is an effective defensive behavior against systemic pathogens. In the single population, the plants increase their population by increasing the proportion of seed propagation when the epidemic pathogen has highly infective. In mixed population, the increasing proportion of seed propagation is the optimal breeding strategy to defend against the spread of a systemic pathogen.

Coexistence of species with different dispersal across landscapes: a critical role of spatial correlation in disturbance

Disturbance is key to maintaining species diversity in plant communities. Although the effects of disturbance frequency and extent on species diversity have been studied, we do not yet have a mechanistic understanding of how these aspects of disturbance interact with spatial structure of disturbance to influence species diversity. Here we derive a novel pair approximation model to explore competitive outcomes in a two-species system subject to spatially correlated disturbance. Generally, spatial correlation in disturbance favoured long-range dispersers, while distance-limited dispersers were greatly suppressed. Interestingly, high levels of spatial aggregation of disturbance promoted long-term species coexistence that is not possible in the absence of disturbance, but only when the local disperser was intrinsically competitively superior. However, spatial correlation in disturbance led to different competitive outcomes, depending on the disturbed area. Concerning ecological conservation and management, we theoretically demonstrate that introducing a spatially correlated disturbance to the system or altering an existing disturbance regime can be a useful strategy either to control species invasion or to promote species coexistence. Disturbance pattern analysis may therefore provide new insights into biodiversity conservation.

Robustness of metacommunities with omnivory to habitat destruction: disentangling patch fragmentation from patch loss

Habitat destruction, characterized by patch loss and fragmentation, is a major driving force of species extinction, and understanding its mechanisms has become a central issue in biodiversity conservation. Numerous studies have explored the effect of patch loss on food web dynamics, but ignored the critical role of patch fragmentation. Here we develop an extended patch-dynamic model for a tri-trophic omnivory system with trophic-dependent dispersal in fragmented landscapes. We found that species display different vulnerabilities to both patch loss and fragmentation, depending on their dispersal range and trophic position. The resulting trophic structure varies depending on the degree of habitat loss and fragmentation, due to a tradeoff between bottom-up control on omnivores (dominated by patch loss) and dispersal limitation on intermediate consumers (dominated by patch fragmentation). Overall, we find that omnivory increases system robustness to habitat destruction relative to a simple food chain.

An extended patch-dynamic framework for food chains in fragmented landscapes

Habitat destruction, a key determinant of species loss, can be characterized by two components, patch loss and patch fragmentation, where the former refers to the reduction in patch availability, and the latter to the division of the remaining patches. Classical metacommunity models have recently explored how food web dynamics respond to patch loss, but the effects of patch fragmentation have largely been overlooked. Here we develop an extended patch-dynamic model that tracks the patch occupancy of the various trophic links subject to colonization-extinction-predation dynamics by incorporating species dispersal with patch connectivity. We found that, in a simple food chain, species at higher trophic level become extinct sooner with increasing patch loss and fragmentation due to the constraint in resource availability, confirming the trophic rank hypothesis. Yet, effects of fragmentation on species occupancy are largely determined by patch loss, with maximal fragmentation effects occurring at intermediate patch loss. Compared to the spatially explicit simulations that we also performed, the current model with pair approximation generates similar community patterns especially in spatially clustered landscapes. Overall, our extended framework can be applied to model more complex food webs in fragmented landscapes, broadening the scope of existing metacommunity theory.

Spatial Competition: Roughening of an Experimental Interface

Limited dispersal distance generates spatial aggregation. Intraspecific interactions are then concentrated within clusters, and between-species interactions occur near cluster boundaries. Spread of a locally dispersing invader can become motion of an interface between the invading and resident species, and spatial competition will produce variation in the extent of invasive advance along the interface. Kinetic roughening theory offers a framework for quantifying the development of these fluctuations, which may structure the interface as a self-affine fractal, and so induce a series of temporal and spatial scaling relationships. For most clonal plants, advance should become spatially correlated along the interface, and width of the interface (where invader and resident compete directly) should increase as a power function of time. Once roughening equilibrates, interface width and the relative location of the most advanced invader should each scale with interface length. We tested these predictions by letting white clover (Trifolium repens) invade ryegrass (Lolium perenne). The spatial correlation of clover growth developed as anticipated by kinetic roughening theory, and both interface width and the most advanced invader’s lead scaled with front length. However, the scaling exponents differed from those predicted by recent simulation studies, likely due to clover’s growth morphology.

Species extinction thresholds in the face of spatially correlated periodic disturbance

The spatial correlation of disturbance is gaining attention in landscape ecology, but knowledge is still lacking on how species traits determine extinction thresholds under spatially correlated disturbance regimes. Here we develop a pair approximation model to explore species extinction risk in a lattice-structured landscape subject to aggregated periodic disturbance. Increasing disturbance extent and frequency accelerated population extinction irrespective of whether dispersal was local or global. Spatial correlation of disturbance likewise increased species extinction risk, but only for local dispersers. This indicates that models based on randomly simulated disturbances (e.g., mean-field or non-spatial models) may underestimate real extinction rates. Compared to local dispersal, species with global dispersal tolerated more severe disturbance, suggesting that the spatial correlation of disturbance favors long-range dispersal from an evolutionary perspective. Following disturbance, intraspecific competition greatly enhanced the extinction risk of distance-limited dispersers, while it surprisingly did not influence the extinction thresholds of global dispersers, apart from decreasing population density to some degree. As species respond differently to disturbance regimes with different spatiotemporal properties, different regimes may accommodate different species.

Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure

We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem for generating exact closures (here expressing joint probabilities in terms of their constituent marginal probabilities) for susceptible-infectious-removed (SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and removal processes, this enables us to obtain a systematic reduction in the number of differential equations needed for an exact 'moment closure' representation of the underlying stochastic model. We define 'transmission blocks' as a possible extension of the block concept in graph theory and show that the order at which the exact moment closure representation is curtailed is the size of the largest transmission block. More generally, approximate closures of the hierarchy of moment equations for these dynamics are typically defined for the first and second order yielding mean-field and pairwise models respectively. It is frequently implied that, in principle, closed models can be written down at arbitrary order if only we had the time and patience to do this. However, for epidemic dynamics on networks, these higher-order models have not been defined explicitly. Here we unambiguously define hierarchies of approximate closed models that can utilise subsystem states of any order, and show how well-known models are special cases of these hierarchies.

Exact Equations for SIR Epidemics on Tree Graphs

We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this “deterministic” representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

Extinction threshold for spatial forest dynamics with height structure

We present a pair-approximation model for spatial forest dynamics defined on a regular lattice. The model assumes three possible states for a lattice site: empty (gap site), occupied by an immature tree, and occupied by a mature tree, and considers three nonlinearities in the dynamics associated to the processes of light interference, gap expansion, and recruitment. We obtain an expression of the basic reproduction number R(0) which, in contrast to the one obtained under the mean-field approach, uses information about the spatial arrangement of individuals close to extinction. Moreover, we analyze the corresponding survival-extinction transition of the forest and the spatial correlations among gaps, immature and mature trees close to this critical point. Predictions of the pair-approximation model are compared with those of a cellular automaton.

The role of reproductive plant traits and biotic interactions in the dynamics of semi-arid plant communities

The dynamics of semi-arid plant communities are determined by the interplay between competition and facilitation among plants. The sign and strength of these biotic interactions depend on plant traits. However, the relationships between plant traits and biotic interactions, and the consequences for plant communities are still poorly understood. Our objective here was to investigate, with a modelling approach, the role of plant reproductive traits on biotic interactions, and the consequences for processes such as plant succession and invasion. The dynamics of two plant types were modelled with a spatially-explicit integrodifferential model: (1) a plant with seed dispersal (colonizer of bare soil) and (2) a plant with local vegetative propagation (local competitor). Both plant types were involved in facilitation due to a local positive feedback between vegetation biomass and soil water availability, which promoted establishment and growth. Plants in the system also competed for limited water. The efficiency in water acquisition (dependent on reproductive and growth plant traits) determined which plant type dominated the community at the steady state. Facilitative interactions between plant types also played an important role in the community dynamics, promoting establishment in the driest conditions and recovery from low biomass. Plants with vegetative propagation took advantage of the ability of seed dispersers to establish on bare soil from a low initial biomass. Seed dispersers were good invaders, maintained high biomass at intermediate and high rainfall and showed a high ability in taking profit from the positive feedback originated by plants with vegetative propagation under the driest conditions. However, seed dispersers lost competitiveness with an increasing investment in fecundity. All together, our results showed that reproductive plant traits can affect the balance between facilitative and competitive interactions. Understanding this effect of plant traits on biotic interactions provides insights in processes such as plant succession and shrub encroachment.

Dynamics of distribution and performance of ramets constructing genets: a demographic-genetic study in a clonal plant, Convallaria keiskei

BACKGROUND AND AIMS

In clonal plants producing vegetative offspring, performance at the genet level as well as at the ramet level should be investigated in order to understand the entire picture of the population dynamics and the life history characteristics. In this study, demography, including reproduction and survival, the growth patterns and the spatial distributions of ramets within genets of the clonal herb Convallaria keiskei were explored.

METHODS

Vegetative growth, flowering and survival of shoots whose genets were identified using microsatellite markers were monitored in four study plots for 3 years (2003-2005). The size structures of ramets in genets and their temporal shifts were then analysed. Their spatial distributions were also examined.

KEY RESULTS

During the census, 274 and 149 ramets were mapped in two 1 x 2 m plots, and 83 and 94 ramets in two 2 x 2 m quadrats. Thirty-eight genotypes were identified from 580 samples. Each plot included 5-18 genets, and most ramets belonged to the predominant genet(s) in each plot. Shoots foliated yearly for several years, but flowering ramets did not have an inflorescence the next year. A considerable number of new clonal offspring persistently appeared, forming a bell-shaped curve of the size structure of ramets in each genet. Comparing the structures modelled by the normal distributions suggested variation among ramets belonging to a single genet and variation among genets. Furthermore, spatial analyses revealed clumped and distant distributions of ramet pairs in a genet, in which the distant patterns corresponded to the linearly elongating clonal growth pattern of this species.

CONCLUSION

Characteristics of ramet performances such as flowering and recruitment of clonal offspring, in addition to growth, played a large part in the regulation of genet dynamics and distribution, which were different among the studied genets. These might be characteristics particularly relevant to clonal life histories.

Annual plants under cyclic disturbance regime: better understanding through model aggregation

In their application for conservation ecology, "classical" analytical models and individual-based simulation models (IBMs) both entail their specific strengths and weaknesses, either in providing a detailed and realistic representation of processes or in regard to a comprehensive model analysis. This well-known dilemma may be resolved by the combination of both approaches when tackling certain problems of conservation ecology. Following this idea, we present the complementary use of both an IBM and a matrix population model in a case study on grassland conservation management. First, we develop a spatially explicit IBM to simulate the long-term response of the annual plant Thlaspi perfoliatum (Brassicaceae), claspleaf pennycress, to different management schemes (annual mowing vs. infrequent rototilling) based on field experiments. In order to complement the simulation results by further analyses, we aggregate the IBM to a spatially nonexplicit deterministic matrix population model. Within the periodic environment created by management regimes, population dynamics are described by periodic products of annual transition matrices. Such periodic matrix products provide a very conclusive framework to study the responses of species to different management return intervals. Thus, using tools of matrix model analysis (e.g., loop analysis), we can both identify dormancy within the age-structured seed bank as the pivotal strategy for persistence under cyclic disturbance regimes and reveal crucial thresholds in some less certain parameters. Results of matrix model analyses are therefore successfully tested by comparing their results to the respective IBM simulations. Their implications for an enhanced scientific basis for management decisions are discussed as well as some general benefits and limitations of the use of aggregating modeling approaches in conservation.

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.

Host-parasite interactions between the local and the mean-field: how and when does spatial population structure matter?

The assumption that populations are completely mixed is reasonable for many populations, but there is likely to be some degree of local interaction whether spatially or socially in many systems. An important question is therefore how strong these local interactions need to be before there are significant effects on the dynamics of the system. Here, our approach is to use a multi-scale pair-approximation model to move between completely local and completely mixed host-parasite interactions. We show that systems dominated by near neighbour effects have less persistence of disease, and a greater possibility of parasite driven extinction and limit cycles. Furthermore this reduction in persistence occurs over a wide range of infection scales and is still significant in predominantly mixed host populations. Deterministic extinctions are only likely in highly spatial SI systems while oscillations also persist over a wide range of infection ranges, but only in hosts that reproduce mostly locally. In general the mean-field may well be a good approximation for many systems, even when there are a significant proportion of near neighbour events, but this depends crucially on the ecological context.

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.

Competing populations on fragmented landscapes with spatially structured heterogeneities: improved landscape generation and mixed dispersal strategies

Interactions between two species competing for space were studied using stochastic spatially explicit lattice-based simulations as well as pair approximations. The two species differed only in their dispersal strategies, which were characterized by the proportion of reproductive effort allocated to long-distance (far) dispersal versus short-distance (near) dispersal to adjacent sites. All population dynamics took place on landscapes with spatially clustered distributions of suitable habitat, described by two parameters specifying the amount and the local spatial autocorrelation of suitable habitat. Whereas previous results indicated that coexistence between pure near and far dispersers was very rare, taking place over only a very small region of the landscape parameter space, when mixed strategies are allowed, multiple strategies can coexist over a much wider variety of landscapes. On such spatially structured landscapes, the populations can partition the habitat according to local conditions, with one species using pure near dispersal to exploit large contiguous patches of suitable habitat, and another species using mixed dispersal to colonize isolated smaller patches (via far dispersal) and then rapidly exploit those patches (via near dispersal). An improved mean-field approximation which incorporates the spatially clustered habitat distribution is developed for modeling a single species on these landscapes, along with an improved Monte Carlo algorithm for generating spatially clustered habitat distributions.

Food encounter rates of simulated termite tunnels with variable food size/distribution pattern and tunnel branch length

Subterranean termites excavate tunnels in a search pattern to encounter food in soil. To investigate the effect of food size, food distribution and the branch length of tunnels on food encounter rate we used a lattice gas model to simulate tunnels of the Formosan subterranean termite, Coptotermes formosanus Shiraki. The model made use of minimized local rules derived from empirical data to simulate termite tunnel patterns in featureless soil. Food distributions with three types (uniform, random, and clumped) were defined by using an I-index proposed by Zimmer and Johnson (1985). The food encounter rate was higher in a clumped than in non-clumped (uniform and random) distribution of food particles. When food particle size was varied in random distributions of food particles a maximum encounter rate was found, with particles of larger or smaller size being encountered less frequently. We also discussed the relationship between the branch tunnel length and the tunnel search pattern in minimizing the redundancy of overlapping branches.

The role of trade-off shapes in the evolution of parasites in spatial host populations: an approximate analytical approach

Given the substantial changes in mixing in many populations, there is considerable interest in the role that spatial structure can play in the evolution of disease. Here we examine the role of different trade-off shapes in the evolution of parasites in a spatially structured host population where infection can occur locally or globally. We develop an approximate adaptive dynamic analytical approach, to examine how the evolutionarily stable (ES) virulence depends not only on the fraction of global infection/transmission but also on the shape of the trade-off between transmission and virulence. Our analysis can successfully predict the ES virulence found previously by simulation of the full system. The analysis confirms that when there is a linear trade-off between transmission and virulence spatial structure may lead to an ES virulence that increases as the proportion of global transmission increases. However, we also show that the ESS disappears above a threshold level of global infection, leading to maximization. In addition just below this threshold, there is the possibility of evolutionary bi-stabilities. When we assume the realistic trade-off between transmission and virulence that results in an ESS in the classical mixed model, we find that spatial structure can increase or decrease the ES virulence. A relatively high proportion of local infection reduces virulence but intermediate levels can select for higher virulence. Our work not only emphasizes the importance of spatial structure to the evolution of parasites, but also makes it clear that situations between the local and the global need to be considered. We also emphasize the key role that the shape of trade-offs plays in evolutionary outcomes.

Spatial Patterns of Prisoner’s Dilemma Game in Metapopulations

Because to defect is the evolutionary stable strategy in the prisoner's dilemma game (PDG), understanding the mechanism generating and maintaining cooperation in PDG, i.e. the paradox of cooperation, has intrinsic significance for understanding social altruism behaviors. Spatial structure serves as the key to this dilemma. Here, we build the model of spatial PDG under a metapopulation framework: the sub-populations of cooperators and defectors obey the rules in spatial PDG as well as the colonization-extinction process of metapopulations. Using the mean-field approximation and the pair approximation, we obtain the differential equations for the dynamics of occupancy and spatial correlation. Cellular automaton is also built to simulate the spatiotemporal dynamics of the spatial PDG in metapopulations. Join-count statistics are used to measure the spatial correlation as well as the spatial association of the metapopulation. Simulation results show that the distribution is self-organized and that it converges to a static boundary due to the boycotting of cooperators to defectors. Metapopulations can survive even when the colonization rate is lower than the extinction rate due to the compensation of cooperation rewards for extinction debt. With a change of parameters in the model, a metapopulation can consist of pure cooperators, pure defectors, or cooperator-defector coexistence. The necessary condition of cooperation evolution is the local colonization of a metapopulation. The spatial correlation between the cooperators tends to be weaker with the increase in the temptation to defect and the habitat connectivity; yet the spatial correlation between defectors becomes stronger. The relationship between spatial structure and the colonization rate is complicated, especially for cooperators. The metapopulation may undergo a temporary period of prosperity just before the extinction, even while the colonization rate is declining.

Moment Equations and Dynamics of a Household SIS Epidemiological Model

An SIS epidemiological model of individuals partitioned into households is studied, where infections take place either within or between households, the latter generally happening much less frequently. The model is explored using stochastic spatial simulations, as well as mathematical models which consist of an infinite system of ordinary differential equations for the moments of the distribution describing the proportions of individuals who are infectious among households. Various moment-closure approximations are used to truncate the system of ODEs to finite systems of equations. These approximations can sometimes lead to a system of ill-behaved ODEs which predict moments which become negative or unbounded. A reparametrization of the ODEs is then developed, which forces all moments to satisfy necessary constraints. Changing the proportion of contacts within and between households does not change the endemic equilibrium, but does affect the amount of time it takes to approach the fixed point; increasing the proportion of contacts within households slows the spread of the infection toward endemic equilibrium. The system of moment equations does describe this phenomenon, although less accurately in the limit as the proportion of between-household contacts approaches zero. The results indicate that although controlling the movement of individuals does not affect the long-term frequency of an infection with SIS dynamics, it can have a large effect on the time-scale of the dynamics, which may provide an opportunity for other controls such as immunizations to be applied.