Local and global stability for population models

In general, local stability does not imply global stability. We show that this is true even if one only considers population models.We show that a population model is globally stable if and only if it has no cycle of period 2. We also derive easy to test sufficient conditions for global stability. We demonstrate that these sufficient conditions are useful by showing that for a number of population models from the literature, local and global stability coincide.We suggest that the models from the literature are in some sense simple, and that this simplicity causes local and global stability to coincide.     

Discrete three-stage population model: persistence and global stability results

A general three-stage discrete-time population model is studied. The inherent net reproductive number for this model is derived. Global stability of the origin is established provided that the inherent net reproductive number is less than one. If it is larger than one the existence of a unique positive fixed point is proved and the persistence of the system is established. Finally, for certain parameter ranges global stability of the positive fixed point is proved.     

Discrete three-stage population model: persistence and global stability results

A general three-stage discrete-time population model is studied. The inherent net reproductive number for this model is derived. Global stability of the origin is established provided that the inherent net reproductive number is less than one. If it is larger than one the existence of a unique positive fixed point is proved and the persistence of the system is established. Finally, for certain parameter ranges global stability of the positive fixed point is proved.     

Local and global principles of striate cortical organization: an advanced model

A model of the functional architecture in monkey striate cortex is proposed, based on recent data, in which the global structure is a result of the local structure. The local structure itself is organized around the cytochrome oxidase blobs. It is characterized by different anisotropic distributions of preferred orientations in upper and lower layers. In particular, the upper layers show a bias for radial orientations and the lower layers show a bias for concentric orientations (Bauer and Dow 1989). This organization may derive from the symmetry of the visual field and visual system which tends to be radial or concentric rather than cartesian. Functionally, the increased sensitivities of the different radial and concentric maps of orientation and movement detectors seem to be an adaptive fit to optic flow fields on the retina during complex movements of the subject in its environment.     

Effective competition determines the global stability of model ecosystems

We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.     

Trophic interactions and the relationship between species diversity and ecosystem stability

Several theoretical studies propose that biodiversity buffers ecosystem functioning against environmental fluctuations, but virtually all of these studies concern a single trophic level, the primary producers. Changes in biodiversity also affect ecosystem processes through trophic interactions. Therefore, it is important to understand how trophic interactions affect the relationship between biodiversity and the stability of ecosystem processes. Here we present two models to investigate this issue in ecosystems with two trophic levels. The first is an analytically tractable symmetrical plant-herbivore model under random environmental fluctuations, while the second is a mechanistic ecosystem model under periodic environmental fluctuations. Our analysis shows that when diversity affects net species interaction strength, species interactions--both competition among plants and plant-herbivore interactions--have a strong impact on the relationships between diversity and the temporal variability of total biomass of the various trophic levels. More intense plant competition leads to a stronger decrease or a lower increase in variability of total plant biomass, but plant-herbivore interactions always have a destabilizing effect on total plant biomass. Despite the complexity generated by trophic interactions, biodiversity should still act as biological insurance for ecosystem processes, except when mean trophic interaction strength increases strongly with diversity.     

Some results on global stability of a predator-prey system

In this paper we derive some results to ensure the global stability of a predator-prey system. The results cover most of the models which have been proposed in the ecological literature for predator-prey systems. The first result is very geometric and it is very easy to check from the graph of prey and predator isoclines. The second one is purely algebraic, however, it covers the defects of the first one especially in dealing with Holling's type-3 functional response in some sense. We also discuss the global stability of Kolmogorov's model. Some examples are presented in the discussion section.Works partially supported by the National Science Council of the Republic of China     

The volterra model for three species predator-prey systems: Boundedness and stability

Summary The classical Volterra equations modelling three species predatorprey interactions are considered with the intention of developing global properties of their solutions. Thirty-four different cases are exhibited. Global asymptotic stability is established for all non-loop cases. Global boundedness is established for all but one kind of loop case, and a condition is exhibited which insures boundedness for that case also. Finally a particular case is shown to have unbounded solutions.     

The relationship between global and regional distribution diminishes among phylogenetically basal species

Phylogenetic legacy and phylogenetic trends affect the ecology of species-except, apparently, for the width of their distribution. As a result, "macroecological" patterns of species distributions emerge constantly in phylogenetically very distinct species assemblages. The width of the global distribution of species, for instance, constantly correlates positively to the width of their regional distribution. However, such patterns primarily reflect the phylogenetically derived species that dominate most assemblages. Basal species, in contrast, might show different macroecological patterns. We tested the hypothesis that the correlation between global and regional distributions of species diminishes among the phylogenetically basal species. We considered central European higher plants and defined global distribution as the occupancy of global floristic zones, regional distribution as the grid occupancy in Eastern Germany, and phylogenetic position as the rank distance to tree base. We also took into account a number of confounding variables. We found that, across all lineages, the global/regional correlation diminished among basal species. We then reanalyzed 19 lineages separately and always found the same pattern. The pattern reflected both increases in global distributions and decreases in regional distributions among basal species. The results indicate that many basal species face a risk of global or at least regional extinction, but have escaped the downward spiral of mutually reinforcing extinction risks at multiple scales. We suggest that many basal species had much time to expand their global ranges but are presently displaced locally by more derived species. Overall, the study shows that macroecological patterns may not be static and universal, but may undergo macroevolutionary trends. Analyses of macroecological patterns across a phylogeny may thus provide insights into macroevolutionary processes.     

Characterization of topological keystone species: Local,global and “meso-scale” centralities in food webs

An important question in the network representation of ecological systems is to determine how direct and indirect interactions between species determine the positional importance of species in the ecosystem. Here we present a quantitative analysis of the similarities and differences of six different topological centrality measures as indicators of keystone species in 17 food webs. These indicators account for local, global and “meso-scale” – intermediate between local and global – topological information about species in the food webs. Using factor analysis we shown that most of these centrality indices share a great deal of topological information, which range from 75% to 96%. A generalized keystone indicator is then proposed by considering the factor loadings of the six-centrality measures, which contains most of the information encoded by these indices. However, the individual ordering of species according to these criteria display significant differences in most food webs. We simulate the effects of species extinction by removing species ranked according to a local and a “meso-scale” centrality indicator. The differences observed on three network characteristics – size, average distance and clustering coefficient of the largest component – after the removal of the most central nodes indicate that the consideration of these indices have different impacts for the ranking of species with conservational biology purposes. The “meso-scale” indicator appears to play an important role in determining the relative importance of species in epidemic spread and parasitism rates.     

Local and global cycles in an acarine predator-prey system: a frequency domain analysis

Techniques of time series analysis are applied to the dynamics of the phytophagous miteTetranychus urticae Koch and its predatorTyphlodromus occidentalis Nesbitt in six experimental mini-orchards, sampled weekly during two years (van de Klashorst et al., 1992). Autocovariance and crosscovariance functions characterize local and global behaviour in the time domain. Spectral density function and cross amplitude spectrum provide important information of the system's behaviour in the frequency domain. Because of a strikingly different behaviour, the total record was cut into two distinct periods. During the first period, all local systems oscillate with a frequency of four cycles per year and all in phase, resulting in a strongly periodical global behaviour at a high mean level. Since the mean local power spectra coincide with the global power spectrum over a wide range of frequencies, it is concluded that the total system of six orchards is homogeneous. During the second period, all local power spectra are mutually different, more or less smoothed with no apparent peak pointing to a periodical component. The resulting global power spectrum is almost flat at a low mean level. The system is heterogeneous with a quasi-stable global behaviour.From the cross-amplitude spectrum it became clear that the dynamics of the two species within the orchards remained strongly coupled over a wide range of frequencies.Since the experimental circumstances had not been intentionally changed, the origin of the drastic change in behaviour could not be identified. If high rates of dispersal have been the synchronizer during the first period, it is not clear why asynchrony suddenly occured in the second period.A comparison of the overall record with the results of Nachman's (1987) stochastic simulation model suggests that the change could possibly be of stochastic origin.     

On the global stability of a generalized cholera epidemiological model

In this paper, we conduct a careful global stability analysis for a generalized cholera epidemiological model originally proposed in [J. Wang and S. Liao, A generalized cholera model and epidemic/endemic analysis, J. Biol. Dyn. 6 (2012), pp. 568–589]. Cholera is a water- and food-borne infectious disease whose dynamics are complicated by the multiple interactions between the human host, the pathogen, and the environment. Using the geometric approach, we rigorously prove the endemic global stability for the cholera model in three-dimensional (when the pathogen component is a scalar) and four-dimensional (when the pathogen component is a vector) systems. This work unifies the study of global dynamics for several existing deterministic cholera models. The analytical predictions are verified by numerical simulation results.     

A global organism detection and monitoring system for non-native species

Harmful invasive non-native species are a significant threat to native species and ecosystems, and the costs associated with non-native species in the United States is estimated at over $120 Billion/year. While some local or regional databases exist for some taxonomic groups, there are no effective geographic databases designed to detect and monitor all species of non-native plants, animals, and pathogens. We developed a web-based solution called the Global Organism Detection and Monitoring (GODM) system to provide real-time data from a broad spectrum of users on the distribution and abundance of non-native species, including attributes of their habitats for predictive spatial modeling of current and potential distributions. The four major subsystems of GODM provide dynamic links between the organism data, web pages, spatial data, and modeling capabilities. The core survey database tables for recording invasive species survey data are organized into three categories: “Where, Who & When, and What.” Organisms are identified with Taxonomic Serial Numbers from the Integrated Taxonomic Information System. To allow users to immediately see a map of their data combined with other user's data, a custom geographic information system (GIS) Internet solution was required. The GIS solution provides an unprecedented level of flexibility in database access, allowing users to display maps of invasive species distributions or abundances based on various criteria including taxonomic classification (i.e., phylum or division, order, class, family, genus, species, subspecies, and variety), a specific project, a range of dates, and a range of attributes (percent cover, age, height, sex, weight). This is a significant paradigm shift from “map servers” to true Internet-based GIS solutions. The remainder of the system was created with a mix of commercial products, open source software, and custom software. Custom GIS libraries were created where required for processing large datasets, accessing the operating system, and to use existing libraries in C++, R, and other languages to develop the tools to track harmful species in space and time. The GODM database and system are crucial for early detection and rapid containment of invasive species.     

On the global stability of an epidemic model of computer viruses

In this paper, we study the global properties of a computer virus propagation model. It is, interesting to note that the classical method of Lyapunov functions combined with the Volterra–Lyapunov matrix properties, can lead to the proof of the endemic global stability of the dynamical model characterizing the spread of computer viruses over the Internet. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in computer virus propagation model. A numerical study of the model is also carried out to investigate the analytical results.     

A simple model of the relationship between asymmetry and developmental stability

The relationship between developmental stability and morphological asymmetry is derived under the standard view that structures on each side of an individual develop independently and are normally distributed. I use developmental variance of sizes of parts, V_D, as the converse of developmental stability, and assume that V_D follows a gamma distribution. Repeatability of asymmetry, a measure of how informative asymmetry is about V_D, is quite insensitive to the variance in V_D, for example only reaching 20% when the coefficient of variation of V_D is 100%. The coefficient of variation of asymmetry, CV_FA, also increases very slowly with increasing population variation in V_D. CV_FA values from empirical data are sometimes over 100%, implying that developmental stability is sometimes more variable than any previously studied type of trait. This result suggests that alternatives to this model may be needed.     

Resource partitioning within a single species population and population stability: A theoretical model

Discrete population models which assume unequal resource partitioning among population members bring about population stability. These models also assume that individual resource share is independent of population density. The model presented here is an attempt to answer the question What does bring about population stability—the inequality of resource partitioning itself or the independence of resource share of population density? By developing a theoretical model with varying dependence of the resource share on the population size, it is shown that the inequality itself is not sufficient for population stability; rather it is the independence of the resource share from population size which brings about this property.     

Classification and stability of global inhomogeneous solutions of a macroscopic model of cell motion

Many micro-organisms use chemotaxis for aggregation, resulting in stable patterns. In this paper, the amoeba Dictyostelium discoideum serves as a model organism for understanding the conditions for aggregation and classification of resulting patterns. To accomplish this, a 1D nonlinear diffusion equation with chemotaxis that models amoeba behavior is analyzed. A classification of the steady state solutions is presented, and a Lyapunov functional is used to determine conditions for stability of inhomogenous solutions. Changing the chemical sensitivity, production rate of the chemical attractant, or domain length can cause the system to transition from having an asymptotic steady state, to having asymptotically stable single-step solution and multi-stepped stable plateau solutions.     

Local coexistence of tree species and the dynamics of global distribution pattern along an environmental gradient: a simulation study

A spatially explicit tree-based model was used to demonstrate the effects of a mechanism promoting multiple-species coexistence on the development of vegetation zonation and its response to climate change. Temporal fluctuation in reproduction was incorporated as the mechanism, which facilitates the persistence of less competitive species. Four hypothetical tree species with different temperature dependencies of seed production were randomly located over a landscape represented by 2,000×40 cells. Each cell can sustain a single tree at most. A zonal distribution pattern emerged corresponding to the temperature gradient along the long axis of the landscape. When there was a temporal variation in seed production, species became distributed over a wider range than that when seed production was constant. When the whole landscape was warmed, the distribution range of each species shifted towards the cool end of the landscape. However, the migration was retarded due to competition for vacant spaces with the remnant species which had dominated the location before the warming. Temporal fluctuation in reproduction facilitated the migration because it enhanced the persistence of minority species and, thus, the invasion and establishment of new species in the area dominated by other species.