An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.     

On optimal diet in a patchy environment

Optimal foraging theory has dealt with the following questions independently: (1) On what prey types should an individual predator feed (optimal diet)? (2) How long should a predator stay in each patch if prey is patchily distributed (optimal allocation of time to patches) ? This paper explores optimal foraging in patches containing several different kinds of prey. Results obtained by simulation show that deviations from recent predictions are to be expected, particularly for long interpatch travel times and rapid depletion of profitable prey types. In these situations the tactics of feeding as either specialist or as a generalist can be inferior to a tactic which starts as a specialist and then expands the diet after some time in the patch. Furthermore, predators should not necessarily stay longer in a patch if interpatch travel time increases. Some experimental tests of these new predictions are proposed.     

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.     

The Ross-Macdonald model in a patchy environment

We generalize to n patches the Ross-Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio R(0). We prove that if R(0)1, then the disease-free equilibrium is globally asymptotically stable. When R(0)>1, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain minus the disease-free equilibrium.     

Persistence and stability of seed-dispersed species in a patchy environment

A mathematical model is developed for the dynamics of seed-dispersed tree species in a region of forest patches or islands. The relevant variables are the population numbers of individual species populations on each patch. A model that considers only one tree species (of the many present) in a region of N distinct patches is formally analogous to a model of an N-species mutualistic community. Using the theory of M-matrices, which has been successfully applied to mutualistic communities, criteria for persistence and stability of the species are derived. The model is then extended to a community of L species that interact competitively and mutualistically in an N-island region. The possibilities of obtaining numerical parameters for the models are discussed.     

Predator-prey system structure in patchy and ephemeral phytotelmata: Aquatic communities in small aroid axils

Mechanisms allowing the persistence of an aquatic predator-prey system in tiny pools (the mean ca. 0.5 ml) held by taro axils were analyzed from the viewpoint of temporal and spatial patterns of habitat use. Predatory larvae of a mosquitoTopomyia tipuliformis concentrated in young axils, while 9 other taxa utilized both young and old axils or concentrated more in older axils. The total prey density was lower in axils with the predator but the density of each prey taxon (except for a few cases) and the number of prey taxa did not differ between axils with and without predators. This indicates thatT. tipuliformis is a general predator and does not influence prey community organization through selective removal of particular prey taxa. Inter-axil distribution ofT. tipuliformis was aggregated in the first instar but uniform in the third and fourth instars due to intraspecific cannibalism, which assures the survival of a single individual under short food supply. Distributions of prey taxa were mostly aggregated, fitting the negative binomial distribution. Thirty seven out of 45 associations of 10 taxa were independent with 3 negative (between the predator and some late-colonizing prey) and 5 positive (between some prey taxa) associations. Probabilistic refuges (produced by independent aggregated distributions) reduced interspecific encounters which may result in competition and predation, and thus probably facilitated prey coexistence. There was no evidence for the importance of predation in structuring the prey community. This system may be prey-dominated in that predator persistence depends on prey community existence but prey community structure depends less on predation.     

Cell communities and robustness in development

The robustness of patterning events in development is a key feature that must be accounted for in proposed models of these events. When considering explicitly cellular systems, robustness can be exhibited at different levels of organization. Consideration of two widespread patterning mechanisms suggests that robustness at the level of cell communities can result from variable development at the level of individual cells; models of these mechanisms show how interactions between participating cells guarantee community-level robustness. Cooperative interactions enhance homogeneity within communities of like cells and the sharpness of boundaries between communities of distinct cells, while competitive interactions amplify small inhomogeneities within communities of initially equivalent cells, resulting in fine-grained patterns of cell specialization.     

Agricultural intensification drives changes in hybrid network robustness by modifying network structure

Within ecological communities, species engage in myriad interaction types, yet empirical examples of hybrid species interaction networks composed of multiple types of interactions are still scarce. A key knowledge gap is understanding how the structure and stability of such hybrid networks are affected by anthropogenic disturbance. Using 15,169 interaction observations, we constructed 16 hybrid herbivore‐plant‐pollinator networks along an agricultural intensification gradient to explore changes in network structure and robustness to local extinctions. We found that agricultural intensification led to declines in modularity but increases in nestedness and connectance. Notably, network connectance, a structural feature typically thought to increase robustness, caused declines in hybrid network robustness, but the directionality of changes in robustness along the gradient depended on the order of local species extinctions. Our results not only demonstrate the impacts of anthropogenic disturbance on hybrid network structure, but they also provide unexpected insights into the structure‐stability relationship of hybrid networks.     

Transmission dynamics for vector-borne diseases in a patchy environment

In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross–MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number \(\mathcal{R }_0\) is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of \(\mathcal{R }_0\) . It is shown that the disease free equilibrium is asymptotically stable if \(\mathcal{R }_0<1\) , and it is unstable if \(\mathcal{R }_0>1\) ; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for \(\mathcal{R }_0\) are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for \(\mathcal{R }_0\) in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, \(\mathcal{R }_0\) is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned.     

Scaling the effects of predation and disturbance in a patchy environment

The effects of hydraulic disturbances on the impact of two predatory benthic invertebrates on their prey were examined in a stream at two distinct spatial scales. At the scale of small habitat patches (0.0625 m²), hydraulic patch type was an important determinant of the microdistribution of prey and predators. Prey abundances were similar across all patch types at baseflow, but local densities were higher in patches identified as low-flow refugia after periods of high and fluctuating flow. The microdistribution pattern of predatory larvae of a caddisfly, Plectrocnemia conspersa, was similar to that of its prey, whereas predatory larvae of an alderfly, Sialis fuliginosa, did not shift their microdistribution significantly with discharge and were always most abundant in lowflow refugia. There was little evidence of an aggregative response of predators with prey, even though both predators and prey are mobile. Both predator species showed similar patch-specific patterns of per capita consumption rates: uniform consumption rates across hydraulic patch types at low and moderate flows, but highest in flow refugia during high flows. Species-specific patterns, however, were apparent in the magnitude and direction of differences between consumption rates during disturbance events, and in comparable patches at base flow: At high flow, consumption rates for P. conspersa were exaggerated (3.9 times higher) in flow refugia but at par in other patches; for S. fuliginosa they were at par in flow refugia but reduced in other patches (up to 3.3. times lower). These differences may be related to species-specific foraging behaviours (search vs ambush predators) and the influence of prey movements on feeding success. Using the patch-scale results only, it is difficult to predict the effects of physical disturbance on predation intensity at the larger scales of whole habitats, populations or communities. At the large scale (>200 m²), net predator impacts were estimated over the stream reach, using a spatially explicit model that accounts, in an additive way, for habitat heterogeneity and patch-specific responses of predators and prey. The relationship between predator impact over the whole reach and hydraulic disturbance differed for the two predators. The predator impact of S. fuliginosa decreased with increasing hydraulic disturbance, as predicted by the harsh-benign hypothesis. There was no directional trend for P. conspersa, however, and maximum predator impact may occur at intermediate disturbance levels. For the prey community in this stream, predation pressure from S. fuliginosa appears to fluctuate directly with the discharge hydrograph, whereas predation from P. conspersa may be more persistent. Flow refugia may play a dual role in the sructure of stream communities by preventing catastrophic mortality of animals (predators and prey) from physical forces during disturbances, and by maintaining (or perhaps increasing) predation pressure. Summing the effects of species interactions in small habitat patches to the larger scale of a whole stream reach indicates that the scale of approach influences the observed patterns and their implied underlying process.     

A comparison of foraging strategies in a patchy environment.

In this paper we compare foraging strategies that might be used by predators seeking prey in a patchy environment. The strategies differ in the extent to which predators aggregate in response to prey density. The approach to the comparison is suggested by the idea of evolutionarily stable strategies. A strategy is said to be evolutionarily stable if it cannot be invaded by another strategy. Thus we examine scenarios where a small number of individuals using one strategy are introduced into a situation where a large number of individuals using the other strategy are already present. However, our foraging models do not explicitly incorporate predator population dynamics, so we use net energy uptake as a surrogate for reproductive fitness. In cases where all of the patches visited by predators sustain prey populations, we find that for any pair of strategies one of them will have a higher net energy uptake than the other whether it is the resident or the introduced strain. However, which one is higher will typically depend on the total predator population, which is determined by the resident strain. If the predators leave prey densities high, the more aggregative strain will have the advantage. If the predators reduce prey densities to low levels the less aggregative strain will have the advantage. In cases where one strain of predators aggregates in response to prey density and the other does not, then there might be patches which do not contain prey but do contain (non-aggregating) predators. In those cases, there is the possibility that whichever strategy is used by the introduced strain will yield a higher energy uptake than that used by the resident strain. This suggests that if some patches are empty of prey then aggregative and non-aggregative strategies may be able to coexist.     

Basic reproduction ratio for a fishery model in a patchy environment

We present a dynamical model of a multi-site fishery. The fish stock is located on a discrete set of fish habitats where it is catched by the fishing fleet. We assume that fishes remain on fishing habitats while the fishing vessels can move at a fast time scale to visit the different fishing sites. We use the existence of two time scales to reduce the dimension of the model : we build an aggregated model considering the habitat fish densities and the total fishing effort. We explore a regulation procedure, which imposes an average residence time in patches. Several equilibria exist, a Fishery Free Equilibria (FFEs) as well as a Sustainable Fishery Equilibria (SFEs). We show that the dynamics depends on a threshold which is similar to a basic reproduction ratio for the fishery. When the basic reproduction ratio is less or equal to 1, one of the FFEs is globally asymptotically stable (GAS), otherwise one of the SFEs is GAS.     

Spatial structure in ecological communities – a quantitative analysis

The spatial structure of communities has recently gained much attention in ecology. Spatial structure comprises an important element in communities, but the literature lacks a thorough investigation about possible among‐organism or among‐ecosystem differences in the degree of spatial structure. Here, I conducted a quantitative review to determine if the degree of spatial structure varied predictably between the major organism types and ecosystems. Spatial structure was quantified as the relative fraction of community variation explained purely by spatial variables (fraction S/E). I integrated data from 322 variation partition analyses both in a generalized linear model (GLM) and using a boosted regression tree method, and showed that a mean of 11.0% of the variation in community composition was explained purely by spatial variables. Across all taxa, a body size–S/E relationship was positive. In GLM, fraction S/E increased highly significantly with study extent, it was highest among terrestrial taxa and higher in ectotherms than in homoiotherms. Spatial structure was also higher in omnivores than in autotrophs. These results suggest that the degree of spatial structure is jointly driven by extrinsic factors such as study extent and ecosystem type, and intrinsic factors such as body size, thermoregulation and interactions between body size and dispersal mode. These results should be important not only for basic research, but also conservation and bioassessment programs would benefit from the information about the magnitude of spatial variation in nature. Synthesis Spatial processes comprise an important element in most ecological communities, but the degree to which spatial structure varies across organisms or ecosystems is poorly known. Here, a quantitative review of 322 variation partition analyses indicated that spatial component varied predictably across ecological communities – it was driven by study extent and ecosystem type as well as by species traits such as body size and thermoregulation. These results give deep insights into the magnitude of spatial variation in nature and should be highly beneficial for conservation and bioassessment programs that are built on the information about how communities vary in space.     

Sex allocation in a patchy environment: a marginal value theorem

The ESS sex allocation when male/female fitnesses vary with patch type is a set of values which either equalizes the marginal values of the male/female fitness tradeoffs, or are pure sexes. This is shown for a hermaphrodite; the result is then generalized to other sex allocation cases.     

Spatial network structure and metapopulation persistence

We explore the relationship between network structure and dynamics by relating the topology of spatial networks with its underlying metapopulation abundance. Metapopulation abundance is largely affected by the architecture of the spatial network, although this effect depends on demographic parameters here represented by the extinction-to-colonization ratio (e/c). Thus, for moderate to large e/c-values, regional abundance grows with the heterogeneity of the network, with uniform or random networks having the lowest regional abundances, and scale-free networks having the largest abundance. However, the ranking is reversed for low extinction probabilities, with heterogeneous networks showing the lowest relative abundance. We further explore the mechanisms underlying such results by relating a node's incidence (average number of time steps the node is occupied) with its degree, and with the average degree of the nodes it interacts with. These results demonstrate the importance of spatial network structure to understanding metapopulation abundance, and serve to determine under what circumstances information on network structure should be complemented with information on the species life-history traits to understand persistence in heterogeneous environments.     

Seed dispersal in a patchy environment with global age dependence

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.This work was supported in part by National Science Foundation Grants MCS-7903497 and MCS-790349701     

Population structure of a monophagous moth in a patchy landscape

1. The population structure of a monophagous noctuid moth, Abrostola asclepiadis , living on a patchily distributed perennial herb, Vincetoxicum hirundinaria is described. The study took place over 5 years at a landscape scale (about 12 km²).
2. Patch occupancy rates and population densities were studied in relation to patch size, degree of patch isolation, level of sun exposure and distance from the coast. In addition, flight tests in the laboratory were performed to estimate the potential dispersal capacity of the moth.
3. Occupancy rates were high and the likelihood of extinction depended on patch size. Small patches were less likely to be occupied than were large patches (> 10 m²). Sun-exposed patches were occupied for a lower proportion of years than were shaded patches. No distance effects could be discerned at the spatial scale of study, presumably because the insect is a strong flier.
4. Population densities in occupied patches decreased with increasing patch size. Furthermore, insect densities tended to increase with distance from the coast. Density changes in patches were synchronized.
5. The studied insect population can be described as a 'patchy population' sensu Harrison (1991) with spatially correlated population dynamics. These dynamics are superimposed on a landscape gradient.