Two fish species competition model with nonlinear interactions and equilibrium catches

Two species competition model is built up by assuming the hypothetical second order interactions in order to consider effects of exploitation on two competing fish species with non-linear interactions. Most important characteristic of this model, compared withLotka-Volterra type linear competition model, is that this model can possess multiple stable equilibrium points. Therefore there is a possibility that two species keeping the equilibrium state at one stable equilibrium point will be attracted to the other stable equilibrium point after a heavy perturbation. In this model reversible change of the fishing pressure does not always results in that of the equilibrium catch. In this sence MSY concept for single species can not be extended to this model. If there are multiple stable equilibrium points, the change of the dominant fish species, catastrophic and irreversible change of each equilibrium catch may be observed when the perturbation by the exploitation is added. This phenomenon immediately reminds us of the change of the dominant fish species between Japanese common mackerel and Pacific saury in the northwest Pacific Ocean. In case of the management of two competing fish species with nonlinear interactions, the consideration on the balance between the fishing pressure for each species may be as important as the decision on the catch limit for each species. MSY level for each species based on the single-species theory could be quite erroneous.     

Qualitative analysis of an HIV transmission model.

In 1988, a multiple-group model for HIV transmission with preferred mixing was proposed by Jacquez and coworkers. In the present paper, the work done by Jacquez et al. is extended. It is shown that the stability modulus of the Jacobian matrix at the no-disease equilibrium is a threshold for this model. Furthermore, if the no-disease equilibrium is unstable, the number of infected individuals will remain above a certain positive level regardless of initial levels; that is, the disease will persist uniformly. The stability of the endemic equilibrium in the case of restricted mixing is also studied. A series of sufficient conditions for local and global asymptotic stability of the endemic equilibrium are stated.     

Rational behavioral response and the transmission of STDs

The susceptible-infected (SI) model is extended by allowing for individual optimal choices of self-protective actions against infection, where agents differ with respect to preferences and costs of self-protection. It is shown that a unique endemic equilibrium prevalence exists when the basic reproductive number of a STD is strictly greater than unity, and that the disease-free equilibrium is the unique steady state equilibrium when the basic reproductive number is less than or equal to one. Unlike in models that take individual behavior as given and fixed, the endemic equilibrium prevalence need not vary monotonically with respect to the basic reproductive number. Specifically, with endogenously determined self-protective behavior, a reduction in the basic reproductive number may in fact increase the endemic equilibrium prevalence. The global stability of the endemic steady state is established for the case of a homogeneous population by showing that, for any non-zero initial disease prevalence, there exists an equilibrium path which converges to the endemic steady state.     

Equilibrium modeling of extractive enzymatic hydrolysis of penicillin G with concomitant 6-aminopenicillanic acid crystallization

In the present downstream processing of penicillin G, penicillin G is extracted from the fermentation broth with an organic solvent and purified as a potassium salt via a number of back-extraction and crystallization steps. After purification, penicillin G is hydrolyzed to 6-aminopenicillanic acid, a precursor for many semisynthetic beta-lactam antibiotics. We are studying a reduction in the number of pH shifts involved and hence a large reduction in the waste salt production. To this end, the organic penicillin G extract is directly to be added to an aqueous immobilized enzyme suspension reactor and hydrolyzed by extractive catalysis. We found that this conversion can exceed 90% because crystallization of 6-aminopenicillanic acid shifts the equilibrium to the product side. A model was developed for predicting the equilibrium conversion in batch systems containing both a water and a butyl acetate phase, with either potassium or D-p-hydroxyphenylglycine methyl ester as counter-ion of penicillin G. The model incorporates the partitioning equilibrium of the reactants, the enzymatic reaction equilibrium, and the crystallization equilibrium of 6-aminopenicillanic acid. The model predicted the equilibrium conversion of Pen G quite reasonably for different values of pH, initial penicillin G concentration and phase volume ratio. The model can be used as a tool for optimizing the enzymatic hydrolysis.     

Prey dispersal and predator persistence

To understand how patchiness influences population dynamics of a tri-trophic interaction, a tractable model is formulated in terms of differential equations. Motivated by the structure of systems such as plants, phytophagous mites and predatory mites, the model takes dispersal into account at the middle trophic level. The effect of dispersal for the middle level in a tri-trophic system could be either stabilising or destabilising since the middle level acts both as prey and as predator. First a simple model with logistic growth for the lowest level is formulated. A model with logistic growth for the lowest level and instantaneous dispersal has a globally stable three-species equilibrium, if this equilibrium exists. Addition of a middle level dispersal phase of non-negligible duration influences equilibrium stability. In the absence of the top trophic level a limit cycle can occur, caused by the delay that exists in the reaction of the middle level to the changes in the lowest level. With low predator efficiency, it is also possible to have an unstable three-species equilibrium. So addition of a middle level dispersal phase of non-negligible duration can work destabilising. Next the persistence of the third trophic level is studied. Even when the three-species equilibrium exists, the third trophic level need not be persistent. A two-species limit cycle can keep its stability when a three-species equilibrium exists; the system is then bistable. It is argued that such a bistability can offer an alternative explanation for pesticide-induced outbreaks of spider mites and failure of predator introduction.     

Determination of dependence of binding parameters on receptor occupancy

Measurements of the binding of ligand to receptors that are macromolecules, either free or components of biomembranes, often show deviation from what is expected of a simple reaction described by an association and a dissociation rate constant. A more versatile model and more discriminating experiments are required for a satisfactory explanation. This paper is based on a general model of the binding reaction in which the rate constants and equilibrium constant are dependent upon occupancy of receptors. The analysis of the model leads to three kinds of experiments: (1) equilibrium measurements which permit quantitative determination of a dissociation equilibrium parameter as a function of receptor occupancy; (2) measurements prior to equilibrium which yield the same information; and (3) measurements prior to equilibrium which reveal quantitatively the dependence of both association and dissociation rate parameters separately, on occupancy.     

A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant

The classical Michaelis-Menten model is widely used as the basis for modeling of a number of biological systems. As the model does not consider the inhibitory effect of endproducts that accumulate in virtually all bioprocesses, it is often modified to prevent the overestimation of reaction rates when products have accumulated. Traditional approaches of model modification use the inclusion of irreversible, competitive, and noncompetitive inhibition factors. This article demonstrates that these inhibition factors are insufficient to predict product inhibition of reactions that are close the dynamic equilibrium. All models investigated were found to violate thermodynamic laws as they predicted positive reaction rates for reactions that were endergonic due to high endproduct concentrations. For modeling of biological processes that operate close to the dynamic equilibrium (e.g., anaerobic processes), it is critical to prevent the prediction of positive reaction rates when the reaction has already reached the dynamic equilibrium. This can be achieved by using a reversible kinetic model. However, the major drawback of the reversible kinetic model is the large number of empirical parameters it requires. These parameters are difficult to determine and prone to experimental error. For this reason, the reversible model is not practical in the modeling of biological processes.This article uses the fundamentals of steady-state kinetics and thermodynamics to establish an equation for the reversible kinetic model that is of practical use in bio-process modeling. The behavior of this equilibrium-based model is compared with Michaelis-Menten-based models that use traditional inhibition factors. The equilibrium-based model did not require any empirical inhibition factor to correctly predict when reaction rates must be zero due to the free energy change being zero. For highly exergonic reactions, the equilibrium-based model did not deviate significantly from the Michaelis-Menten model, whereas, for reactions close to equilibrium, the reaction rate was mainly controlled by the quotient of mass action ratio (concentration of all products over concentration of all substrates) over the equilibrium constant K. This quotient is a measure of the displacement of the reaction from its equilibrium. As the new equation takes into account all of the substrates and products, it was able to predict the inhibitor effect of multiple endproducts. The model described is designed to be a useful basis for a number of different model applications where reaction conditions are close to equilibrium.     

Modelling the activation,opening, inactivation and reopening of the voltage-gated sodium channel.

A model of the voltage-gated sodium channel is put forward suggesting that the four S4 voltage-sensors behave as screw-helices making a series of discrete transitions that carry one elementary charge for each notch of the screw helix. After the channel has been activated by the first two steps R in equilibrium with P in equilibrium with A in all four domains, followed by a voltage-independent rearrangement, it is opened by a third cooperative step A in equilibrium with B in domains I, II and III in conjunction with hydration. Inactivation is a voltage-dependent process controlled by the third step A in equilibrium with I in sensor IVS4, and the closing of the channel is brought about its dehydration. From the inactivated steady state the channel may be reopened by a fourth step, I in equilibrium with C in sensor IVS4 and rehydration. The computed kinetics of the model are shown to conform closely with those observed experimentally.     

Eigenvalues and structure of compartmental models

The “spread” of the nonzero eigenvalues of a compartmental matrix is studied by reference to the associated directed graph. It is related to the eigenvalues of the matrices of the individual cycles for certain strongly connected directed graphs. The equilibrium solution to the entire model is also an equilibrium solution to the model consisting of the individual cycles.     

Theoretical analysis of mixed Plasmodium malariae and Plasmodium falciparum infections with partial cross-immunity

A deterministic model for assessing the dynamics of mixed species malaria infections in a human population is presented to investigate the effects of dual infection with Plasmodium malariae and Plasmodium falciparum. Qualitative analysis of the model including positivity and boundedness is performed. In addition to the disease free equilibrium, we show that there exists a boundary equilibrium corresponding to each species. The isolation reproductive number of each species is computed as well as the reproductive number of the full model. Conditions for global stability of the disease free equilibrium as well as local stability of the boundary equilibria are derived. The model has an interior equilibrium which exists if at least one of the isolation reproductive numbers is greater than unity. Among the interesting dynamical behaviours of the model, the phenomenon of backward bifurcation where a stable boundary equilibrium coexists with a stable interior equilibrium, for a certain range of the associated invasion reproductive number less than unity is observed. Results from analysis of the model show that, when cross-immunity between the two species is weak, there is a high probability of coexistence of the two species and when cross-immunity is strong, competitive exclusion is high. Further, an increase in the reproductive number of species i increases the stability of its boundary equilibrium and its ability to invade an equilibrium of species j. Numerical simulations support our analytical conclusions and illustrate possible behaviour scenarios of the model.     

Resource limited and equilibrium models of populations

Summary The logistic equation has been used as the basis of two distinct models of population dynamics, a resource limited model and an equilibrium model. It is argued that although the two models are very different, the distinction between them has been obscured by the use of the same terminology for the parameters of both. It is further argued that the simple equilibrium model developed from the logistic equation has inconsistencies which make it untenable as a population model.     

Coexistence in MacArthur-style consumer-resource models

The nature of and conditions for permanent coexistence of consumers and resources are characterized in a family of models that generalize MacArthur's consumer-resource model. The generalization is of the resource dynamics, which need not be of Lotka-Volterra form but are subject only to certain restrictions loose enough to admit many resource dynamics of biological interest. For any such model, (1) if there is an interior equilibrium, then it is globally attracting, else some boundary equilibrium is globally attracting-thus permanent coexistence is coexistence at a globally attracting equilibrium; (2) there is an interior equilibrium if and only if for any species, the equilibrium approached in the absence of that species and the presence of the others is invasible by that species--thus permanent coexistence is equivalent to mutual invasibility; (3) for resources without direct interactions, the conditions for permanent coexistence of the consumers admit an instructive formulation in terms of regression statistics. The significance and limitations of the models and results are discussed.     

Epistasis of quantitative trait loci under different gene action models

Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham's model, F2-metric, and F(infinity)-metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham's model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F2-metric and F(infinity)-metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F2 population with two unlinked loci, Cockerham's model is reduced to the F2-metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F(infinity)-metric model are not orthogonal but they can be easily translated into those under the F2-metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies.     

Population and prehistory II: space-limited human populations in constant environments

We present a population model to examine the forces that determined the quality and quantity of human life in early agricultural societies where cultivable area is limited. The model is driven by the non-linear and interdependent relationships between the age distribution of a population, its behavior and technology, and the nature of its environment. The common currency in the model is the production of food, on which age-specific rates of birth and death depend. There is a single non-trivial equilibrium population at which productivity balances caloric needs. One of the most powerful controls on equilibrium hunger level is fertility control. Gains against hunger are accompanied by decreases in population size. Increasing worker productivity does increase equilibrium population size but does not improve welfare at equilibrium. As a case study we apply the model to the population of a Polynesian valley before European contact.     

Heterozygosity in pin-thrum plants or with partial sex linkage

A two locus model is constructed for selection of a gene closely linked to the S locus in pin-thrum plants or to the sex determining part of the Y chromosome. Using this model, conditions for stability at the equilibrium point which is predicted by one-locus theory when there is heterozygotic superiority are derived. If the recombination value is small, it is found that this equilibrium point is unstable and that the gene frequencies go to a new stable equilibrium point at which the population has a higher average fitness. A few simple cases of selection and the implication of these to the theory of the evolution of the Y chromosome are discussed.     

Modelling of phytoplankton allelopathy with Monod–Haldane-type functional response—A mathematical study

In this paper, a three-tier model of phytoplankton, zooplankton and nutrient is considered and stability of different equilibrium points is analyzed along with Hopf-bifurcation around coexisting equilibrium point. Here, we have assumed toxication process as the guiding factor for bloom formation as well as its termination and this process is incorporated into our model by choosing the zooplankton grazing function as a Monod–Haldane function due to the phytoplankton toxicity. Extensive numerical simulations have been performed to validate the analytical findings and these simulation work reveal the chaotic oscillation exhibited by the model system for certain choice of the parameter values.     

Differential susceptibility epidemic models

We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either constant or proportional to the total population. We derive an explicit formula for the reproductive number of infection for each model by investigating the local stability of the infection-free equilibrium. We further prove that the infection-free equilibrium of each model is globally asymptotically stable by qualitative analysis of the dynamics of the model system and by utilizing an appropriately chosen Liapunov function. We show that if the reproductive number is greater than one, then there exists a unique endemic equilibrium for all of the DS models studied in this paper. We prove that the endemic equilibrium is locally asymptotically stable for the models with no disease-induced mortality and the models with contact numbers proportional to the total population. We also provide sufficient conditions for the stability of the endemic equilibrium for other situations. We briefly discuss applications of the DS models to optimal vaccine strategies and the connections between the DS models and predator-prey models with multiple prey populations or host-parasitic interaction models with multiple hosts are also given.This research was partially supported by the Department of Energy under contracts W-7405-ENG-36 and the Applied Mathematical Sciences Program KC-07-01-01.     

Statistical mechanical model of the energetics of coated vesicle formation.

Adsorptive endocytosis is mediated by a structural transformation of a two-dimensional hexagonal lattice to a polyhedron or coated vesicle. Clathrin is the structural protein involved in this process. A theoretical model that focuses on the energetics of clathrin in coated vesicle formation is presented. Fisher's cluster model of phase transitions is applied to this problem. The equilibrium constant for the process of converting a large two-dimensional patch to a coated vesicle and a smaller patch is calculated. There are three energetic contributions to be considered. They are the surface energy, the interior or lattice energy, and a loop entropy. Features of the Ising model are introduced into this model by scaling the critical exponents. In determining the configurational partition functions required for equilibrium constant calculations, the polyhedron is represented as a planar graph with no surfaces. The equilibrium constants are extremely sensitive to changes in the lattice and surface energies. The loop entropy contributions favor bimodal distributions in vesicle size under certain conditions. Conditions can also be established where the energy required to form the vesicle is small.     

Aquaculture,pollution and fishery - dynamics of marine industrial interactions

We model bioeconomic interrelations between a commercial fishery and an aquaculture industry by using a dynamical systems theory approach. The biomass follows a logistic growth where the pollution emerging from aquaculture is accounted for by means of a retardation term. We investigate the existence and stability of the equilibrium states of this model as a function of the growth-retardation parameter and find that a necessary (but not sufficient) condition for stability is low and moderate values of the emission-remediation ratio. Three intervals of the growth-retardation parameter are identified in this regime of the emission-remediation ratio. The regime of low and negligible influence of the pollution on the biomass evolution gives rise to the existence of an asymptotically stable equilibrium state characterized by a finite biomass and a finite effort in the fishery. In the same regime we identify two unstable equilibrium states of which the former one is characterized by no effort in the fishery, whereas the latter one is characterized by no biomass and no effort. When the growth retardation parameter exceeds a certain threshold, the fishery becomes unprofitable and the equilibrium state characterized by no effort in the fishery becomes asymptotically stable. By a further increase in this parameter above a higher threshold value, also the biomass is wiped out and the equilibrium state characterized by no biomass and no effort becomes asymptotically stable.     

Frequency-Dependent Selection: The High Potential for Permanent Genetic Variation in the Diallelic, Pairwise Interaction Model

A detailed analytic and numerical study is made of the potential for permanent genetic variation in frequency-dependent models based on pairwise interactions among genotypes at a single diallelic locus. The full equilibrium structure and qualitative gene-frequency dynamics are derived analytically for a symmetric model, in which pairwise fitnesses are chiefly determined by the genetic similarity of the individuals involved. This is supplemented by an extensive numerical investigation of the general model, the symmetric model, and nine other special cases. Together the results show that there is a high potential for permanent genetic diversity in the pairwise interaction model, and provide insight into the extent to which various forms of genotypic interactions enhance or reduce this potential. Technically, although two stable polymorphic equilibria are possible, the increased likelihood of maintaining both alleles, and the poor performance of protected polymorphism conditions as a measure of this likelihood, are primarily due to a greater variety and frequency of equilibrium patterns with one stable polymorphic equilibrium, in conjunction with a disproportionately large domain of attraction for stable internal equilibria.