The Stochastic Asymptotic Stability Analysis in Two Species Lotka-Volterra Model

The asymptotic stability of two species stochastic Lotka-Volterra model is explored in this paper. Firstly, the Lotka-Volterra model with random parameter is built and reduced into the equivalent deterministic system by orthogonal polynomial approximation. Then, the linear stability theory and Routh-Hurwitz criterion for nonlinear deterministic systems are applied to the equivalent one. At last, at the aid of Lyapunov second method, we obtain that as the random intensity or statistical parameter of random variable is changed, the stability about stochastic Lotka-Volterra model is different from the deterministic system.

GLOBAL STABILITY OF EXTENDED MULTI-GROUP SIR EPIDEMIC MODELS WITH PATCHES THROUGH MIGRATION AND CROSS PATCH INFECTION

In this article, we establish the global stability of an endemic equilibrium of multi-group SIR epidemic models, which have not only an exchange of individuals between patches through migration but also cross patch infection between different groups. As a result, we partially generalize the recent result in the article [16].

A modified multi-group model of angular and momentum distribution of cosmic ray muons for thickness measurement and material discrimination of slabs

Muon tomography is a capable imaging technique to measure the geometry of high-Z objects. However,most existed algorithms used in muon tomography have obscured the effects of angular distribution and momentum spectra of cosmic ray muons and reduced the spatial resolution. We present a modified multi-group model that takes into account these effects and calibrates the model by the material of lead. Performance tests establish that the model is capable of measuring the thickness of a Pb slab and identifying the material of an unknown slab on a reasonable exposure timescale, in both cases of complete and incomplete angular data. Results show that the modified multi-group model is helpful for improvements in image resolution in real applications.

Global dynamics of a novel multi-group model for computer worms

In this paper, we study worm dynamics in computer networks composed of many autonomous systems. A novel multigroup SIQR （susceptible-infected-quarantined-removed） model is proposed for computer worms by explicitly considering anti-virus measures and the network infrastructure. Then, the basic reproduction number of worm R0 is derived and the global dynamics of the model are established. It is shown that if R0 is less than or equal to 1, the disease-free equilibrium is globally asymptotically stable and the worm dies out eventually, whereas, if R0 is greater than 1, one unique endemic equilibrium exists and it is globally asymptotically stable, thus the worm persists in the network. Finally, numerical simulations are given to illustrate the theoretical results.

DYNAMIC FOR A STOCHASTIC MULTI-GROUP AIDS MODEL WITH SATURATED INCIDENCE RATE

In this paper,a stochastic multi-group AIDS model with saturated incidence rate is studied.We prove that the system is persistent in the mean under some parametric restrictions.We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function.Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system,which greatly improves upon previous results.

EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation.We obtain a critical value λ1^D(Ω0),and demonstrate that the existence of the predator inΩ0 only depends on the relationship of the growth rateμof the predator and λ1^D(Ω0),not on the prey.Furthermore,whenμ<λ1^D(Ω0),we obtain the existence and uniqueness of its positive steady state solution,while whenμ≥λ1^D(Ω0),the predator and the prey cannot coexist inΩ0.Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding regionΩ0,which is different from that of the classical Lotka-Volterra predator-prey model.

Traveling wave for a time-periodic Lotka-Volterra model with bistable nonlinearity

This paper studies bistable wavefronts of a diffusive time-periodic Lotka-Volterra system.We obtain a new condition for the existence,uniqueness and stability of bistable timeperiodic traveling waves.This condition is sharp and greatly improves the result established in the reference(X.Bao and Z.Wang,Journal of Differential Equations,255(2013)2402-2435).An example is given to demonstrate our consequence.

Extinction and Persistent of a Stochastic Multi-group SIR Epidemic Model

We establish a stochastic differential equation epidemic model of multi-group SIR type based on the deterministic multi-group SIR mode. Then, we define the basic reproduction number R0^S and show that it is a sharp threshold for the dynamic of the stochastic multi-group SIR model. More specially, if R0^S 〈 1, then the disease-free equilibrium will be asymptotically stable which means the disease will die out, if R0^S 〉 1, the disease-free equilibrium will unstable, and our model will positively recurrence to a positive domain which implies the persistence of our model. Numerical simulation examples are carried out to substantiate the analytical results.

Dynamics of the Austrian Food Market: Application of Lotka-Volterra Differential Equations

The organic food market has become an important part of food industry. We analyze sales data from Austria for 2014 to 2020 of 124 products from 25 product groups in six categories, each in conventional and organic form. We fitted their market shares by means of a modified Lotka-Volterra model with constant coefficients. When only organic and conventional products were compared, a significant increase in market shares was observed for 15 of 25 organic product groups, indicating a continuing growth of the organic food market. The typical Lotka-Volterra dynamics was a predator-prey dynamics with an organic product (group) predating on conventional products that were in symbiosis.

Study on coopetition relationship simulation among M-commerce information service subjects based on Lotka-Volterra model

In order to explore the evolution law of coopetition relationship among Mcommerce information service(MIS)subjects and further reveal the allocation mechanism of information resources,the research extends Lotka-Volterra model,builds the coopetition relationship model of MIS subjects and simulates their coopetition relationship by Python and MATLAB to obtain the evolution trend of information resources possession(IRP)of MIS subjects in the coopetition process.The results show that the mutualism cooperation pattern dominated by promoting effect can maximize the value of information resources.This paper has significant reference value for how to optimize the allocation of information resources among MIS subjects.

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.

Dynamics of dominant tree species in a forest ecotone on the northern slop of Changbai Mountain

The competition and dynamics of dominant trees species in the forest ecotone between the broad-leaved/Korean pine （Pinus koraiensis） mixed forest and the spruce-fir forest （also known as dark conifer forest） in Changbai Mountain, Jilin Province in Northeast China were studied by using Lotka-Volterra model, based on the data from twenty-eight sample plots with area of 20 mx90 m for each one. Results showed that under natural condition, differentiation of communities followed two directions： one would be Spruce （Picea jezoensis and few P. koraiensis） and fir （Abies nephrolepis） co-dominant conifer forest, and at the equilibrium fir was absolutely preponderant （77.1% of relative dominance （RD））; the other would be the conifer and broad-leaved mixed forest, and at equilibrium, the broad-leaved tree species was 50% of RD in the broad-leaved/Korean pine mixed forest and 66% of RD in the broad-leaved and spruce-fir mixed forest. The study demonstrated that both broad-leaved/Korean pine mixed forest and dark conifer forest were climax community, the ecotone had transitional characteristics, and the diversification of the forest communities suggested that the direction of succession was affected by local habitat.

Spatial pattern and driving factors of biomass carbon density for natural and planted coniferous forests in mountainous terrain,eastern Loess Plateau of China

Background:Understanding the spatial pattern and driving factors of forest carbon density in mountainous terrain is of great importance for monitoring forest carbon in support of sustainable forest management for mitigating climate change.Methods:We collected the forest inventory data in 2015 in Shanxi Province,eastern Loess Plateau of China,to explore the spatial pattern and driving factors of biomass carbon density(BCD)for natural and planted coniferous forests using Anselin Local Moran’s I,Local Getis-Ord G*and semivariogram analyses,and multi-group structural equation modeling,respectively.Results:The result of spatial pattern of BCDs for natural forests showed that the BCD was generally higher in the north but lower in the south of Shanxi.The spatial pattern for planted forests was substantially different from that for natural forests.The results of multi-group SEM suggested that elevation(or temperature as the alternative factor of elevation)and stand age were important driving factors of BCD for these two forest types.Compared with other factors,the effects of latitude and elevation on BCD showed much greater difference between these two forest types.The difference in indirect effect of latitude(mainly through affecting elevation and stand age)between natural and planted forests was to some extent a reflection of the difference between the spatial patterns of BCDs for natural and planted forests in Shanxi.Conclusions:The natural coniferous forests had a higher biomass carbon density,a stronger spatial dependency of biomass carbon density relative to planted coniferous forests in Shanxi.Elevation was the most important driving factor,and the effect on biomass carbon density was stronger for natural than planted coniferous forests.Besides,latitude presented only indirect effect on it for the two forest types.

Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age

Consider the heterogeneity(e.g.,heterogeneous social behaviour,heterogeneity due to different geography,contrasting contact patterns and different numbers of sexual partners etc.)of host population,in this paper,the authors propose an infection age multigroup SEIR epidemic model.The model system also incorporates the feedback variables,where the infectivity of infected individuals may depend on the infection age.In the direction of mathematical analysis of model,the basic reproduction number R_0 has been computed.The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R_(0).More precisely,for R_(0)≤1,the disease-free equilibrium is globally asymptotically stable and for R_(0)>1,they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method.By considering a numerical example,they investigate the effects of infection age and feedback on the prevalence of the disease.Their result shows that feedback parameters have different and even opposite effects on different groups.However,by choosing an appropriate value of feedback parameters,the disease could be eradicated or maintained at endemic level.Besides,the infection age of infected individuals may also change the behaviour of the disease,global stable to damped oscillations or damped oscillations to global stable.

STABILITY ANALYSIS OF A LOTKA-VOLTERRA COMMENSAL SYMBIOSIS MODEL INVOLVING ALLEE EFFECT

In this paper, we present a stability analysis of a Lotka-Volterra commensal symbiosis model subject to Allee effect on the unaffected population which occurs at low population density. By analyzing the Jacobian matrix about the positive equilibrium, we show that the positive equilibrium is locally asymptotically stable. By applying the differential inequality theory, we show that the system is permanent, consequently, the boundary equilibria of the system is unstable. Finally, by using the Dulac criterion, we show that the positive equilibrium is globally stable. Although Allee effect has no influence on the final densities of the predator and prey species, numeric simulations show that the system subject to an Allee effect takes much longer time to reach its stable steady-state solution, in this sense that Allee effect has unstable effect on the system, however, such an effect is controllable. Such an finding is greatly different to that of the predator-prey model.

A MULTI-GROUP SVEIR EPIDEMIC MODEL WITH DISTRIBUTED DELAY AND VACCINATION

In this paper, based on a class of multi-group epidemic models of SEIR type with bilinear incidences, we introduce a vaccination compartment, leading to multi-group SVEIR model. We establish that the global dynamics are completely determined by the basic reproduction number R0V which is defined by the spectral radius of the next generation matrix. Our proofs of global stability of the equilibria utilize a graph-theoretical approach to the method of Lyapunov functionals. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccines to obtain immunity or the possibility for them to be infected before acquiring immunity is neglected in each group, this condition will be satisfied and the disease can always be eradicated by suitable vaccination strategies. This may lead to over evaluation for the effect of vaccination.

Global stability for a delayed multi-group SIRS epidemic model with cure rate and incomplete recovery rate

In this paper, by applying Lyapunov functional approach, we establish a sufficient condition on the global stability of a ＂delayed＂ multi-group SIRS epidemic model with cure rate and incomplete recovery rate which does not depend on the delays and is an extension of the ＂light drug model＂ studied in the recent paper [Muroya, Li and Kuniya, Complete global analysis of an SIRS epidemic model with graded cure rate and incomplete recovery rate, J. Math. Anal. Appl. 410 （2014） 719-732] to a multi-group model. Applying a Lyapunov functional on total population of each compartment, we offer new techniques for the delayed system, how to prove the permanence, the existence of the endemic equilibrium and the global stability of disease-free equilibrium for the reproduction number R0 ≤ 1 and endemic equilibrium forR0 ≥ 1.

Threshold dynamics of a multi-group SEAR alcoholism model with public health education

In this paper, we investigate the threshold dynamics of a multi-group SEAR alcoholism epidemic model with public health education. The multi-group model allows us to describe interactions both within-group and inter-group separately. We prove that the basic reproduction number R0 plays the role of a threshold for the long-time behavior of the model. The alcohol-free equilibrium P0 of the model is globally asymptotically stable if R0≤1,while the alcohol-present equilibrium P* exists uniquely and is globally asymptotically stable if R0>1.For the proofs of main results, we use the classical method of Lyapunov functions and apply subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory. Our results expand the previous results which have been obtained in single-group models. Sensitivity analysis and numerical simulations are also performed to illustrate our results.

Variability of Prorocentrum donghaiense response to allelopathic action from Alexandrium pacificum in laboratory culture

The dinoflagellates Alexandrium pacificum and Prorocentrum donghaiense are two well-known harmful algal blooms(HABs)-forming species,both were usually found in the same sea areas in form of HABs in China.To date,there is no mechanistic model that can fully explain the occurrence of P.donghaiense blooms with A.pacificum.We found that different strains of P.donghaiense had different intrinsic growth rates of 0.107-0.215/d,and these strains exhibited different responses to the allelopathic action from A.pacificum.Some strains of P.donghaiense could grow well despite some degrees of inhibition in a short period,suggesting the two algal species P.donghaiense and A.pacificum could coexist,even if A.pacificum was allelopathic.Our findings may advance the understanding of phenotypes in P.donghaiense and provide a potential mechanism involved in the coexistence of P.donghaiense and A.pacificum in the same area.

Interspecific Competition Between Levanderina fissa(Dinophyceae) and the Other Three Microalgal Species in Co-Cultures

Growth interactions between the dinoflagellate Levanderina fissa (=Gyrodinium instriatum) and the other three algal bloom species Skeletonema costatum (Bacillariophyceae), Prorocentrum micans (Dinophyceae), and Chattonella marina (Raphido- phyceae) were studied using bi- or tri-algal co-cultures under laboratory conditions. The Lotka-Volterra model was applied to esti- mate the interspecific competition between them. Results showed that the growth of L. fissa was significantly suppressed in both bi-algal and tri-algal cultures, and the maximum cell densities in co-cultures were only between 21% and 69% of that in mono-cul- ture. L. fissa outcompeted P. micans;however, its effects on S. costatum and C. marina were generally less than the effects of their intraspecific competitions based on the competition coefficients. Contrarily, the inhibitory effects of S. costatum and C. marina on L. fissa were much greater. S. costatum outcompeted the other three species in all co-cultures, and its competition coefficients on the other species were more than 10000 times higher than the opposite. The growth of C. marina was suppressed by S. costatum;how- ever, it outcompeted L. fissa and P. micans. The growth of P. micans was most strongly inhibited in co-cultures, and the maximum cell densities in co-cultures were less than 15% of that in mono-species culture. The results suggested that L. fissa had few advan- tages in the interspecific competition. The adaptation to low salinity and the utilization of organic phosphorus might be the important reasons for L. fissa to take the competitive predominance in brackish estuarine environments.