A prey-predator model with harvesting for fishery resource with reserve area

Considering that over exploitation would result in the extinction of the population, we propose and investigate a Holling II functional response prey-predator model with harvesting for fishery resource in a two-patch environment: a free fishing zone (patch 1) and a reserve zone (patch 2) where fishing is strictly prohibited. First, the presence of harvesting can impact the existence of equilibria. Further, stability criteria of the model is analyzed both from local and global point of view. Our results indicate that so long as the prey population in the reserved zone does not extinct, the both prey always exist, that is marine reserves should ensure the sustainability of system. Thus, marine reserves not only protect species inside the reserve area but they can also increase fish abundance in adjacent areas. Next, the existence of bionomic equilibrium and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle. It is established that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

A modified epidemiological model for computer viruses

Since the computer viruses pose a serious problem to individual and corporative computer systems, a lot of effort has been dedicated to study how to avoid their deleterious actions, trying to create anti-virus programs acting as vaccines in personal computers or in strategic network nodes. Another way to combat viruses propagation is to establish preventive policies based on the whole operation of a system that can be modeled with population models, similar to those that are used in epidemiological studies. Here, a modified version of the SIR (Susceptible-Infected-Removed) model is presented and how its parameters are related to network characteristics is explained. Then, disease-free and endemic equilibrium points are calculated, stability and bifurcation conditions are derived and some numerical simulations are shown. The relations among the model parameters in the several bifurcation conditions allow a network design minimizing viruses risks.     

The analysis of an epidemic model on networks

The paper consider an epidemic model with birth and death on networks. We derive the epidemic threshold R₀ dependent on birth rate b, death rate d (natural death) and μ from the infectious disease and natural death, and cure rate γ. And the stability of the equilibriums (the disease-free equilibrium and endemic equilibrium) are analysed. Finally, the effects of various immunization schemes are studied and compared. We show that both targeted, and acquaintance immunization strategies compare favorably to a proportional scheme in terms of effectiveness. For active immunization, the threshold is easier to apply practically. To illustrate our theoretical analysis, some numerical simulations are also included.     

A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.     

Optimal harvesting in a predator-prey model with Allee effect and sigmoid functional response

This work deals with the determination of the optimal harvest policy in an open access fishery in which both prey and predator species are subjected to non-selective harvesting.The model is described by autonomous ordinary differential equation systems, the functional response of the predators is Holling type III and the prey growth is affected by the Allee effect. The catch-rate functions are based on the catch per unit effort (CPUE) or Schaefer’s hypothesis.The problem of determining the optimal harvest policy is solved by using Pontryagin’s maximal principle. The problem here studied is to maximize a cost function representing the present value of a continuous time-stream of revenue of the fishery.     

Dynamic model of worm propagation in computer network

In this paper, an attempt has been made to mathematically formulate a compartmental susceptible – exposed – infectious – susceptible with vaccination (that is, anti-virus treatment) (SEIS-V) epidemic transmission model of worms in a computer network with natural death rate (which depends on the total number of nodes). The stability of the result is stated in terms of modified reproductive number R_v. We have derived an explicit formula for the modified reproductive number R_v, and have shown that the worm-free equilibrium, whose component of infective is zero, is globally asymptotically stable if R_v < 1, and unstable if R_v > 1. The contribution of vertical transmission to the modified reproductive number is also analyzed. Numerical methods are employed to solve and simulate the system of equations developed and interpretation of the model yields interesting revelations. Analysis of efficient antivirus software is also performed.     

An EOQ model for salesmen’s initiatives, stock and price sensitive demand of similar products - A dynamical system

The paper deals with an inventory model to determine the retailer’s optimal order quantity for similar products. It is assumed that the amount of display space is limited and the demand of the products depends on the display stock level where more stock of one product makes a negative impression of the another product. Besides it, the demand rate is also dependent on selling price and salesmen’s initiatives. Also, the replenishment rate depends on the level of stock of the items. The objective of the model is to maximize the profit function, including the effect of inflation and time value of money by Pontryagin’s Maximal Principles. The stability analysis of the concerned dynamical system has been done analytically.     

一类平面系统稳定区域的最佳估计及其在电力系统分析中的应用

本文利用现代动力系统几何理论研究一类平面系统的平衡点的吸引区域估计问题 ,并将其应用于一类具体的电力系统 .     

An epidemic model of a vector-borne disease with direct transmission and time delay

This paper considers an epidemic model of a vector-borne disease which has direct mode of transmission in addition to the vector-mediated transmission. The incidence term is assumed to be of the bilinear mass-action form. We include both a baseline ODE version of the model, and, a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction number R₀. If R₀?1, the disease-free equilibrium is globally stable and the disease dies out. If R₀>1, a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region. The delay in the differential-delay model accounts for the incubation time the vectors need to become infectious. We study the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation.     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.     

Global exponential stability and global attractivity of impulsive Hopfield neural networks with time delays

In this paper, by means of constructing the extended impulsive delayed Halanay inequality and by Lyapunov functional methods, we analyze the global exponential stability and global attractivity of impulsive Hopfield neural networks with time delays. Some new sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive Hopfield neural networks with time delays are obtained. Those conditions are more feasible than that given in the earlier references to some extent. Some numerical examples are also discussed in this work to illustrate the advantage of the results we obtained.     

A novel computer virus model and its dynamics

In this paper, we propose a novel computer virus propagation model and study its dynamic behaviors; to our knowledge, this is the first time the effect of anti-virus ability has been taken into account in this way. In this context, we give the threshold for determining whether the virus dies out completely. Then, we study the existence of equilibria, and analyze their local and global asymptotic stability. Next, we find that, depending on the anti-virus ability, a backward bifurcation or a Hopf bifurcation may occur. Finally, we show that under appropriate conditions, bistable states may be around. Numerical results illustrate some typical phenomena that may occur in the virus propagation over computer network.     

A mathematical model for assessing the impact of poverty on yaws eradication

A neglected disease with a nearly forgotten name is making a comeback following a global control programme that almost eradicated it more than forty years ago. Until the 1970s the prevalence of non-venereal treponematosis, including yaws, was greatly reduced after worldwide mass treatment. In 2005, cases were again reported in the Democratic Republic of the Congo. A deterministic model is formulated to investigate the impact of poverty on yaws eradication. Threshold parameters are determined and stabilities analysed. The reproductive number was also used to assess the impact of birth rate in resource-constrained families on the dynamics of yaws. The model was shown to be globally stable whenever the associated reproductive number is less than a unity. Using the Lyapunov function it was proved that whenever the associated reproductive number is greater than a unity an endemic equilibrium exists and is globally asymptotically stable. Results from this theoretical study suggests that if the population of children in the community is dominated by those from resource-constrained families, then yaws eradication will remain difficulty to attain. Thus, more needs to be done in addressing issues such as high fertility rate, overcrowding, poor sanitation, etc. and poverty in general so that yaws epidemic which was successfully controlled several decades ago will cease to reemerge and can easily be eradicated.     

A polycycle and limit cycles in a non-differentiable predator-prey model

For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.     

A discrete hierarchical model of intra-specific competition

A discrete hierarchical model with either age, size, or stage structure is derived. The resulting scalar equation for total population level is then used to study contest and scramble intra-specific competition. It is shown how equilibrium levels and resilience are related for the two different competition situations. In particular, scramble competition yields a higher population level while contest competition is more resilient if the uptake rate as a function of resource density is concave down. The conclusions are reversed if the uptake rate is concave up.     

On the existence and stability of approximate solutions of perturbed vector equilibrium problems

In this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of Painlevé-Kuratowski to the initial set. Then, we introduce the concept of approximate solution for equilibrium problem governed by set-valued maps and we study the stability of these solutions. The particular case of linear continuous operators is considered as well.     

A propagation model of computer virus with nonlinear vaccination probability

This paper is intended to examine the effect of vaccination on the spread of computer viruses. For that purpose, a novel computer virus propagation model, which incorporates a nonlinear vaccination probability, is proposed. A qualitative analysis of this model reveals that, depending on the value of the basic reproduction number, either the virus-free equilibrium or the viral equilibrium is globally asymptotically stable. The results of simulation experiments not only demonstrate the validity of our model, but also show the effectiveness of nonlinear vaccination strategies. Through parameter analysis, some effective strategies for eradicating viruses are suggested.     

Dynamic model and simulation analysis of the genetic impact of population harvesting

In this paper the impact of exploitation of a sexually reproducing population is investigated by means of a selection model. Our aim is to find genetic systems for which the fishing results in selection effect, provided that the growth and reproduction rates of individuals are genetically determined. To this end a complex dynamic model is presented, providing long-term predictions both on the size structure and on the genetic composition of the population. For a minimal nontrivial model, the two-locus two-allele case is considered, where the survival, transition and reproduction rates depend on size and genotype. For each size class and genotype the corresponding density is a state variable. The mating system is supposed to be totally panmictic and the gamete production is described in terms of the meiosis matrix.Based on the above model, an in silico analysis is carried out. The simulation results show that the long-term behaviour of the genetic structure can be characterized by a cyclic convergence, which means that the state sequences corresponding to different phases of the reproduction cycle tend to an asymptotic genotype distribution. For an illustration of the effect of exploitation on the genetic composition the “fishing effort” model is considered. If the totally homozygous genotype possesses the best phenotype, fishing does not seem to influence the genetic distribution in the long term. The same is true in case of heterozygote advantage. In some situations, however, fishing modifies the genetic distribution of the population. Meanwhile there is a significant change in the size of the harvested individuals. This result points out to the importance of the genotype-phenotype correspondence while building up fishing strategies.