Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.     

A reexamination of stability in randomly varying versus deterministic environments with comments on the stochastic theory of limiting similarity

The focus of this paper is a general relationship proposed by May (Amer. Natur. 107 (1973)) between the stability properties of stochastic models incorporating environmental variation and the stability properties of the deterministic models from which they are derived. The concepts of stochastic stability underlying this conjectured relationship are discussed and compared to the standard definitions of deterministic stability as well as alternative criteria for stability in stochastic models. It is shown by example that May's qualitative stability criterion does not ensure stability in any sense unless restrictive conditions on the form of the model are satisfied. Even under these conditions, the criterion, which is based on linearization, generally provides information only about the local dynamics of multispecies models. The applicability of such information to stochastic limiting similarity theory is discussed and alternative methods of analysis are proposed.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Cellular control models with linked positive and negative feedback and delays. II. Linear analysis and local stability

An analysis of local behavior is made of two nonlinear models which incorporate both an induction or positive feedback control mechanism and a repression or negative feedback control mechanism. The systems of differential equations with delays are linearized about their equilibria. The related characteristic equations which are exponential polynomials are studied to determine the local stability of the models. Computer studies are included to show the range of stability for different parameter values, and the biological significance is discussed briefly.     

How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions

We try to answer the question of to what extent details in nutrient uptake and phytoplankton physiology matter for population and community dynamics. To this end, we study how two nutrients interact in limiting phytoplankton growth. A popular formulation uses a product-rule for nutrient uptake, which we compare with that on the basis of synthesizing units. We first fit different nutrient uptake models to a dataset and conclude that the quantitative differences between the models are small. Then we study the sensitivity of phytoplankton growth and zooplankton-phytoplankton interactions (ZPi) models to uptake formulations. Two population models are compared; they are based on different assumptions on the relation between nutrient uptake and phytoplankton growth. We find that the population and community models are sensitive to uptake formulations. According to the uptake formulation used in the ZPi models, qualitative differences can be observed. Indeed, although two models based on functions with similar shapes have close equilibria, these can differ in stability properties. Since stability involves the derivatives of formulas, even if two formulas provide close values, large numerical differences in the stability criterion may occur after derivation. We conclude that mechanistic details can be of importance for community modelling.     

On the structure and stability of mutualistic systems: Analysis of predator-prey and competition models as modified by the action of a slow-growing mutualist

Two basic models of mutualism are presented in which interactions among three species lead to mutualism between two of them. The models represent 2-species predator-prey or competition systems in which a third species acts as a mutualist with either the predator, the prey, or one of the competitors. The models include the assumptions that there is a cost of associating with the mutualist and that the mutualist population grows much more slowly than the other two populations. Special cases of these two models correspond to six qualitatively different types of mutualistic benefit, all of which are known to occur in nature: deterring predation, increasing prey availability, feeding on (or competing with) a predator, increasing competitive interactions, decreasing competitive interactions, and feeding on (or competing with) a competitor. These models and their special cases are subjected to a local stability analysis. The results show that mutualism based upon deterring predation, competing with a predator, or decreasing competitive interactions enhances local stability, while mutualism based upon increasing prey availability or increasing competitive interactions reduces local stability. These results clearly reject the idea that mutualism is an inherently unstable process, and reinforces the idea that each different kind of mutualism will have to be considered separately. Compared to 2-species models of mutualism, the 3-species models provide a more realistic representation of the structure of many mutualistic systems, the mechanisms by which one species benefits another, and the regulation of the interaction.     

Stability properties of 2-species models of mutualism: Simulation studies

Summary Most previous analyses of the stability properties of models of mutualism have emphasized the destabilizing effects of mutualism. However, these analyses can be shown to be based upon inappropriate assumptions, or to be applicable only for special cases of mutualism. In this paper three basic 2-species models of mutualism are presented and their six combinations are analyzed by computer simulation for their return time stability and persistence stability. Four out of six models show greater return time stability than an appropriate model without mutualism, and all models show higher persistence stability than the model without mutualism. It is argued that real biological systems can be related to the qualitative structure of each of the basic models of mutualism, and that therefore none of the basic models or their stability properties can be eliminated a priori as being inappropriate. The conclusion follows that while some kinds of mutualistic interactions may be relatively unstable, other mutualisms, probably representing the majority of cases, can be considered to be relatively stable. The limitations of these models and analyses are considered.     

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 of static behavior of a class of unstructured models of continuous bioprocesses

The stability characteristics of a class of unstructured models of continuous bioreactors are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the non-biomass product formation rate is linearly proportional to the utilization rate of limiting substrate. The kinetics expressions of cell growth and product synthesis are allowed to assume general forms of substrate and product. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of non-trivial steady states with wash-out conditions, and multistability resulting from hysteresis. These general results are applied to specific examples of bioprocesses and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models.     

Equilibrium and Nonequilibrium Models in Ecological Anthropology: An Evaluation of "Stability" in Maring Ecosystems in New Guinea

Three models pertaining to the stability of Maring ecosystems have been proposed. The first is the local stability model, in which a population seeks its own equilibrium state; the second is the regional stability model, in which each population is ultimately unstable, but populations persist somewhere in space; and the third is the disequilibrium model, in which neither stability nor population regulation is attained. In the disequilibrium model, exogenous factors prevent a population, which is moving toward some equilibrium state, from reaching it. The large number of quantitative anthropological and ecological studies in Highlands New Guinea has not shown clearly which of these three models best describes reality. Simulation of shifting agriculture in New Guinea shows that the Highlands systems are equilibrium-seeking, but have such limited recovery rates from disturbance that even small perturbations are sufficient to keep them from reaching equilibrium. When the influences of technological innovation, environmental change, and social-cultural evolution are taken into account, the disequilibrium model is the model of choice. These systems remain away from their stable equilibrium points most of the time, if those exist at all. Thus, New Guinea agroecosystems can be stable or unstable depending upon how stability is defined.     

On the mechanical stability of porous coated press fit titanium implants: a finite element study of a pushout test

Pushout tests can be used to estimate the shear strength of the bone implant interface. Numerous such experimental studies have been published in the literature. Despite this researchers are still some way off with respect to the development of accurate numerical models to simulate implant stability. In the present work a specific experimental pushout study from the literature was simulated using two different bones implant interface models. The implant was a porous coated Ti-6Al-4V retrieved 4 weeks postoperatively from a dog model. The purpose was to find out which of the interface models could replicate the experimental results using physically meaningful input parameters. The results showed that a model based on partial bone ingrowth (ingrowth stability) is superior to an interface model based on friction and prestressing due to press fit (initial stability). Even though the present study is limited to a single experimental setup, the authors suggest that the presented methodology can be used to investigate implant stability from other experimental pushout models. This would eventually enhance the much needed understanding of the mechanical response of the bone implant interface and help to quantify how implant stability evolves with time.     

Stability in a class of cyclic epidemic models with delay

A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.     

The contribution of the glenoid labrum to glenohumeral stability under physiological joint loading using finite element analysis

The labrum contributes to passive glenohumeral joint stability. Cadaveric studies have demonstrated that this has position and load dependency, which has not been quantified under physiological loads. This study aims to validate subject-specific finite element (FE) models against in vitro measurements of joint stability and to utilise the FE models to predict joint stability under physiological loads. The predicted stability values were within ± one standard deviation of experimental data and the FE models showed a reduction in stability of 10–15% with high, physiological, loads. The developed regression equations provide the first representation of passive glenohumeral stability and will aid surgical decision-making.     

PLOIDY AND EVOLUTION BY SEXUAL SELECTION: A COMPARISON OF HAPLOID AND DIPLOID FEMALE CHOICE MODELS NEAR FIXATION EQUILIBRIA

We compare the stability properties of haploid and diploid models of Fisherian sexual selection (with male contribution limited to sperm) by examining both models at equilibria for which a male trait is fixed or absent. Haploid and diploid two locus diallelic models share the property that the stability of such fixation equilibria is determined by the relationship between the harmonic mean of relative preference values for the common male trait, weighted by the frequency of the preferences, and the relative viability associated with the common male trait. When diploid females with heterozygotic-based preferences express preference strengths intermediate between homozygote-based preferences, then boundary equilibria of haploid and diploid models share many stability properties. However, even with intermediate heterozygote preferences, haploid and diploid models do differ: (1) for a particular frequency of the preference allele, both fixation boundaries can be stable for the diploid model, and (2) with over- or underdominance at the preference locus (a possibility precluded in the haploid model), a fixation boundary in the diploid model may show two switches in its stability state for increasing frequencies of one of the preference alleles. These differences are due not just to the impossibility of dominance in haploid models, but also to the larger number of diploid genotypes.     

Strain energy descriptions of biological swelling. I: Single fluid compartment models

Strain energy functions are derived from biphasic soft tissue models in order to describe large-deformation, large-swelling, elastic behavior of nonlinear materials. The resulting analysis leads to calculations of stress-extension relations and tissue fluid pressure. Also explored are the elastic stability of the biphasic tissue models and the manner in which tissue pressure is altered by material deformation.     

Most networks in Wagner's model are cycling

In this paper we study a model of gene networks introduced by Andreas Wagner in the 1990s that has been used extensively to study the evolution of mutational robustness. We investigate a range of model features and parameters and evaluate the extent to which they influence the probability that a random gene network will produce a fixed point steady state expression pattern. There are many different types of models used in the literature, (discrete/continuous, sparse/dense, small/large network) and we attempt to put some order into this diversity, motivated by the fact that many properties are qualitatively the same in all the models. Our main result is that random networks in all models give rise to cyclic behavior more often than fixed points. And although periodic orbits seem to dominate network dynamics, they are usually considered unstable and not allowed to survive in previous evolutionary studies. Defining stability as the probability of fixed points, we show that the stability distribution of these networks is highly robust to changes in its parameters. We also find sparser networks to be more stable, which may help to explain why they seem to be favored by evolution. We have unified several disconnected previous studies of this class of models under the framework of stability, in a way that had not been systematically explored before.     

Bifurcation, stability, and cluster formation of multi-strain infection models

Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.     

Models of reproductive skew: A review and synthesis (Invited Article)

Animal societies vary markedly in reproductive skew, the extent to which breeding is monopolised by dominant individuals. In the last few years, a large number of different models have been developed to explain this variation. Here, I review existing models of reproductive skew, distinguishing between two basic types. Transactional models focus on group stability and the constraints this places on the division of reproduction. Compromise models, by contrast, ignore issues of group stability and view the division of reproduction as the outcome of a conflict in which each group member has a limited or partial ability to enforce its own optimum. I go on to show, however, that the division between transactional and compromise models is somewhat artificial, and that both approaches may be combined in a single, synthetic treatment. Different models of reproductive skew are thus better seen as special cases of a general underlying theory, rather than alternative paradigms. I conclude with a brief discussion of the possibilities and problems of empirically testing this unified theory of skew, and the prospects for future theoretical advances.