# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

August 2004 , Volume 4 , Issue 3

Guest Editors: Lansun Chen, Yang Kuang, Shigui Ruan and Glenn Webb

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2004, 4(3): 501-515 doi: 10.3934/dcdsb.2004.4.501 +[Abstract](2554) +[PDF](215.8KB)
Abstract:
Local stability and instability of the disease-free equilibriums of an age-structured predator-prey model with disease in the prey is examined. The basic idea is to apply the linearized stability principle and the theory of semigroups.
2004, 4(3): 517-525 doi: 10.3934/dcdsb.2004.4.517 +[Abstract](2219) +[PDF](212.9KB)
Abstract:
A stochastic model for the dynamics of a single species of a stage structured population is presented. The model (in Lagrangian or Monte Carlo formulation) describes the life history of an individual assumed completely determined by the biological processes of development, mortality and reproduction. The dynamics of the overall population is obtained by the time evolution of the number of the individuals and of their physiological age. No other assumption is requested on the structure of the biological cycle and on the initial conditions of the population. Both a linear and a nonlinear models have been implemented. The nonlinearity takes into account the feedback of the population size on the mortality rate of the offsprings. For the linear case, i.e. when the population growths without any feedback dependent on the population size, the balance equations for the overall population density are written in the Eulerian formalism (equations of Von Foerster type in the deterministic case and of Fokker-Planck type in the stochastic case). The asymptotic solutions to these equations, for sufficiently large time, are in good agreement with the results of the numerical simulations of the Lagrangian model. As a case study the model is applied to simulate the dynamics of the greenhouse whitefly, Trialeurodes vaporarioum (Westwood), a highly polyphagous pest insect, on tomato host plants.
2004, 4(3): 527-546 doi: 10.3934/dcdsb.2004.4.527 +[Abstract](2531) +[PDF](308.2KB)
Abstract:
In this paper we first consider a two consumer-one resource model with one of the consumer species exhibits intraspecific feeding interference but there is no interspecific competition between the two consumer species. We assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington-DeAngelis functional response. Using dynamical systems theory, it is shown that the two consumer species can coexist upon the single limiting resource in the sense of uniform persistence. Moreover, by constructing a Liapunov function it is shown that the system has a globally stable positive equilibrium. Second, we consider a model with an arbitrary number of consumers and one single limiting resource. By employing practical persistence techniques, it is shown that multiple consumer species can coexist upon a single resource as long as all consumers exhibit sufficiently strong conspecific interference, that is, each of them exhibits Beddington-DeAngelis functional response.
2004, 4(3): 547-554 doi: 10.3934/dcdsb.2004.4.547 +[Abstract](2657) +[PDF](168.6KB)
Abstract:
We consider a periodic predator-prey system where the prey has a history that takes them through two stages, immature and mature. We provide a sufficient and necessary condition to guarantee the permanence of the system.
2004, 4(3): 555-562 doi: 10.3934/dcdsb.2004.4.555 +[Abstract](1952) +[PDF](175.6KB)
Abstract:
In this paper, a monotone-iterative scheme is established for finding positive periodic solutions of a competition model of tumor-normal cell interaction. The model describes the evolution of a population with normal and tumor cells in a periodically changing environment. This population is under periodical chemotherapeutic treatment. Competition among the two kinds of cells is considered. The mathematical problem involves a coupled system of Lotka-Volterra together with periodically pulsed conditions. The existence of positive periodic solutions is proved by the monotone iterative technique and in a special case, the uniqueness of a periodic solution is obtained by proving that any two periodic solutions have the same average. Moreover, we also show that the system is permanent under the conditions which guarantee the existence of the periodic solution. Some computer simulations are carried out to demonstrate the main results.
2004, 4(3): 563-574 doi: 10.3934/dcdsb.2004.4.563 +[Abstract](2447) +[PDF](202.3KB)
Abstract:
In this paper, we establish sufficient criteria for the existence of positive periodic solutions for a class of discrete time semi-ratio-dependent predator-prey interaction models based on systems of nonautonomous difference equations. The approach involves the coincidence degree and its related continuation theorem as well as some priori estimates.
2004, 4(3): 575-587 doi: 10.3934/dcdsb.2004.4.575 +[Abstract](2151) +[PDF](306.0KB)
Abstract:
We analyze the replicator equation for two games closely related with the social dilemma occurring in public goods situations. In one case, players can punish defectors in their group. In the other case, they can choose not to take part in the game. In both cases, interactions are not pairwise and payoffs non-linear. Nevertheless, the qualitative dynamics can be fully analyzed. The games offer potential solutions for the problem of the emergence of cooperation in sizeable groups of non-related individuals -- a basic question in evolutionary biology and economics.
2004, 4(3): 589-594 doi: 10.3934/dcdsb.2004.4.589 +[Abstract](2228) +[PDF](164.8KB)
Abstract:
We study the least cost-size problem and the least cost-deviation problem for a nonlinear population model with age-dependence, which takes fertility rate as the control variable. The existence of a unique optimal control and the optimality conditions of first order are investigated by means of Ekeland's variational principle and normal cone technique. Our conclusion extends a known result in the literature.
2004, 4(3): 595-605 doi: 10.3934/dcdsb.2004.4.595 +[Abstract](2865) +[PDF](192.9KB)
Abstract:
The impulsive vaccination strategies of the epidemic SIR models with nonlinear incidence rates $\beta I^{p}S^{q}$ are considered in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic infection-free solution of the impulsive epidemic system and prove that the periodic infection-free solution is globally asymptotically stable. In order to apply vaccination pulses frequently enough so as to eradicate the disease, the threshold for the period of pulsing, i.e. $\tau _{max}$ is shown, further, by bifurcation theory, we obtain a supercritical bifurcation at this threshold, i.e. when $\tau>\tau_{max}$ and is closing to $\tau_{max}$, there is a stable positive periodic solution. Throughout the paper, we find impulsive epidemiological models with nonlinear incidence rates $\beta I^{p}S^{q}$ show a much wider range of dynamical behaviors than do those with bilinear incidence rate $\beta SI$ and our paper extends the previous results, at the same time, theoretical results show that pulse vaccination strategy is distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination, therefore impulsive vaccination strategy provides a more natural, more effective vaccination strategy.
2004, 4(3): 607-614 doi: 10.3934/dcdsb.2004.4.607 +[Abstract](2505) +[PDF](181.6KB)
Abstract:
In this paper, we study the impact of feedback control on a predator-prey model with functional response. It is proven that the position and number of positive equilibria and limit cycles, parameter domain of stability and bifurcations of such model can be changed by some feedback control which has the form $u=kx+h.$ The main results of this paper show that a constant control has a stronger impact on the properties of this model than a proportional state feedback.
2004, 4(3): 615-622 doi: 10.3934/dcdsb.2004.4.615 +[Abstract](2351) +[PDF](144.6KB)
Abstract:
The stability analysis of the interior equilibria, whose components are all positive, of non linear ordinary differential equation models describing in vivo dynamics of infectious diseases are complicated in general. Liu, "Nonlinear oscillation in models of immune responses to persistent viruses, Theor. Popul. Biol. 52(1997), 224-230" and Murase, Sasaki and Kajiwara, "Stability analysis of pathogen-immune interaction dynamics (submitted)" proved the stability of the interior equilibria of such models using symbolic calculation software on computers. In this paper, proofs without using symbolic calculation software of the stability theorems given by Liu and Murase et al. are presented. Simple algebraic manipulations, properties of determinants, and their derivatives are used. The details of the calculation given by symbolic calculation software can be seen clearly.
2004, 4(3): 623-628 doi: 10.3934/dcdsb.2004.4.623 +[Abstract](2115) +[PDF](149.8KB)
Abstract:
Detailed studies of single species population dynamics are important for understanding population behaviour and the analysis of large complex ecosystems. Here we present two general models for simulating insect population dynamics: The distributed delay processes and Poisson Process models. In the distributed delay processes model, the simulated population has the characteristic property that the time required for maturation from one stage of growth (instar) to another is directly related to ambient temperature. In this model the parameters DEL and K are significant to the simulated process. The discrete Poisson model deals with the individual development of a group of free entities with random forward movement. These two general component models can be used to simulate the population growth of many insects currently the subject of research interest. The application of distributed delay processes to dynamics of cotton bollworm helicoverpa armigera is presented. The results show the simulation data quite "fit" the observed data.
2004, 4(3): 629-634 doi: 10.3934/dcdsb.2004.4.629 +[Abstract](2884) +[PDF](142.2KB)
Abstract:
Some assumptions of Logistic Equation are frequently violated. We applied the Allee effect to the Logistic Equation so as to avoid these unrealistic assumptions. Following basic principles of Catastrophe theory, this new model is identical to a Fold catastrophe type model. An ecological interpretation of the results is provided.
2004, 4(3): 635-642 doi: 10.3934/dcdsb.2004.4.635 +[Abstract](2842) +[PDF](156.8KB)
Abstract:
This paper investigates two types of SIS epidemic model with vaccination and constant population size to determine to the thresholds, equilibria, and stabilities. One of SIS models is a delay differential equations, in which the period of immunity due to vaccination is a constant. Another is an ordinary differential equations, in which the loss of immunity due to vaccination is in the exponent form. We find all of their thresholds respectively, and compare them. The disease-free equilibrium is globally asymptotically stable if the threshold is not greater than one; the endemic equilibrium is globally asymptotically stable if the threshold is greater than one.
2004, 4(3): 643-652 doi: 10.3934/dcdsb.2004.4.643 +[Abstract](2786) +[PDF](183.2KB)
Abstract:
This paper focuses on the study of an age-structured SIRS epidemic model with a vaccination program. We first give the explicit expression of the reproductive number $\mathcal{R}(\psi)$ in the presence of vaccine, and show that the infection-free steady state is locally asymptotically stable if $\mathcal{R}(\psi)<1$ and unstable if $\mathcal{R}(\psi)>1$. Second, we prove that the infection-free state is globally stable if the basic reproductive number $\mathcal{R}_0 <1$, and that an endemic equilibrium exists when the reproductive number $\mathcal{R}(\psi)>1$.
2004, 4(3): 653-662 doi: 10.3934/dcdsb.2004.4.653 +[Abstract](2412) +[PDF](183.0KB)
Abstract:
A nonautonomous diffusion model with Holling III functional response and continuous time delay is considered in this paper, where all parameters are time dependent and the prey can diffuse between two patches of a heterogeneous environment with barriers between patches, but for the predator the diffusion does not involve a barrier between patches. It is shown that the system is persistent under any diffusion rate effect. Moreover, sufficient conditions that guarantee the existence of a positive periodic solution which is globally asymptotic stable are obtained.
2004, 4(3): 663-670 doi: 10.3934/dcdsb.2004.4.663 +[Abstract](2018) +[PDF](158.2KB)
Abstract:
In this paper, we consider the question of global stability of the positive equilibrium in a chemostat-type system with delayed nutrient recycling. By constructing Liapunov function, we obtain a sufficient condition for the global stability of the positive equilibrium.
2004, 4(3): 671-678 doi: 10.3934/dcdsb.2004.4.671 +[Abstract](2125) +[PDF](162.2KB)
Abstract:
In this paper, we consider a delayed $SIR$ epidemic model with density dependent birth process. For the model with larger birth rate, we discuss the asymptotic property of its solutions. Furthermore, we also study the existence of Hopf bifurcation from the endemic equilibrium of the model and local asymptotic stability of the endemic equilibrium.
2004, 4(3): 679-686 doi: 10.3934/dcdsb.2004.4.679 +[Abstract](1778) +[PDF](147.8KB)
Abstract:
It is the major task of the researches of conservation biology to explore species existing necessary conditions and endanger mechanism [1]. Presently, population viability analysis models mainly focus on a single species and few of them take into account the influence of inter-species effect to aimed species [2][3]. It is more difficult to apply traditional population viability analysis to insects, as compared to birds or mammals. First, insects have complex life histories, small body and various species. For animals that have body length between 10m and 1cm, the number of the species increases by 100 times with the body length shorten by 1/10 [4]. Biologists' knowledge is far from completely understanding insect species, or even the number of insect, because it is very difficult to obtain the life parameters of wild insect populations. Second, biologists are accustomed to study the key species of the community, which are often the topmost taxa in biology chain or the dominant species in communities. These insect species are rare to be found playing a key role independently in ecosystem maintenance or community succession. Last, many insect species have become extinct before people know them well. The efficient and comprehensive approach is required to detect why the population of some insect specify is descending and what kind of protective strategies should be applied. In this paper, we have proposed the competition index of Parmassius nomion species by combining the aimed species population dynamics with the diversity index. The results have shown that the alteration of competition index is able to detect the danger of shrinking population.
2004, 4(3): 687-694 doi: 10.3934/dcdsb.2004.4.687 +[Abstract](1887) +[PDF](129.3KB)
Abstract:
Gause's experiments of Paramecium caudatum have been thought as one of the most accurate experiments in ecology. Although it has been hypothesized by ecologists that the population dynamics can be approximated by the classical sigmoid curve, there are still some questions as to whether the analytical method is accurate enough in relation to experimental data. Therefore analytical results are frequently encountered with doubt. In this study, we estimated some growing parameters based strictly on the life history of Paramecium caudatum and with a more flexible logistic model. Since the intrinsic growth rate values fell in different regions, the population dynamics were considered to follow a complex pattern.
2004, 4(3): 695-704 doi: 10.3934/dcdsb.2004.4.695 +[Abstract](2001) +[PDF](223.0KB)
Abstract:
A continuous probability density model for the spatial distribution and migration pattern for a pelagic fish stock is described. The model is derived as the continuum limit of a random walk in the plane which leads to an advection-diffusion equation. The direction of the velocity vector is given by the gradient of a "comfort function" which incorporates factors such as temperature, food density, distance to spawning grounds, etc., which are believed to affect the behaviour of the capelin. An application to Barents Sea capelin is presented.
2004, 4(3): 705-711 doi: 10.3934/dcdsb.2004.4.705 +[Abstract](2287) +[PDF](300.8KB)
Abstract:
The spatiotemporal pattern formation in a prey-predator dynamics is studied numerically. External noise as well as the productivity of the prey population control emergence, symmetry and stability of as well as transitions between structures. Diffusive Turing structures and invasion waves are presented as example.
2004, 4(3): 713-720 doi: 10.3934/dcdsb.2004.4.713 +[Abstract](1844) +[PDF](158.9KB)
Abstract:
This paper provides a minimally simple theory that accounts for the foraging behaviour of animals. It presents three separate systems of differential equations that predict the selection of diets from various types of food, and also the time-budgets of the occupancy of patches of food without, and with regeneration of food. The theory subsumes the whole of optimal foraging theory as one special case of foraging behaviour defined by the physiological requirements of animals. The theory explains foraging in terms of both the acquisition of food and the utilization of food in the maintenance of life.
2004, 4(3): 721-727 doi: 10.3934/dcdsb.2004.4.721 +[Abstract](2302) +[PDF](167.9KB)
Abstract:
In this paper, the Chemostat model with stage-structure and the Beddington-DeAngelies functional responses is studied. Sufficient conditions for uniform persistence of this model with delay are obtained via uniform persistence of infinite dimensional dynamical systems; and for the model without delay, sufficient conditions for the global asymptotic stability of the positive equilibrium are presented.
2004, 4(3): 729-738 doi: 10.3934/dcdsb.2004.4.729 +[Abstract](2161) +[PDF](194.0KB)
Abstract:
In [1] a simplified model for the control of testosterone secretion is given by

$\frac{dR}{dt}=f(T)-b_1R,\qquad\qquad\qquad\qquad$(*)

$\frac{dT}{dt}=b_2R(t-\tau)-b_3T,$

where $R$ denotes the luteinizing hormone releasing hormone, $T$ denotes the hormone testosterone and the negative feedback function $f(T)$ is a positive monotonic decreasing differentiable function of $T$. The delay $\tau$ is associated with the blood circulation time in the body, and $b_1$, $b_2$ and $b_3$ are positive parameters. In this paper, developing the method given in [2], we establish necessary and sufficient conditions for the steady state of (*) to be asymptotic stable or linearly unstable.

2004, 4(3): 739-746 doi: 10.3934/dcdsb.2004.4.739 +[Abstract](1797) +[PDF](164.0KB)
Abstract:
The effect of spatially partial prevention of infectious disease is considered as an application of population models in inhomogeneous environments. The area is divided into two ractangles, and the local contact rate between infectives and susceptibles is sufficiently reduced in one rectangle. The dynamics of the infection considered here is that described by an SIS model with diffusion. Then the problem can be reduced to a Fisher type equation, which has been fully studied by many authors, under some conditions. The steady states of the linearized equation are considered, and a Nagylaki type result for predicting whether the infection will become extinct over time or not is obtained. This result leads to some necessary conditions for the extinction of the infection.
2004, 4(3): 747-758 doi: 10.3934/dcdsb.2004.4.747 +[Abstract](3014) +[PDF](188.2KB)
Abstract:
A ratio-dependent predator-prey model with stage structure for the prey is proposed and analyzed, which improves the assumption that each individual prey has the same ability to be captured by predator. In this paper, mathematical analysis of the model equations with regard to boundedness of solutions, nature of equilibria, permanence are analyzed. We obtain conditions that determine the permanence of the populations. Furthermore, we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model. By the application of comparing argument and exploiting the monotonicity of one equation of the model, we obtain sufficient conditions for the global attractivity of positive equilibrium.
2004, 4(3): 759-768 doi: 10.3934/dcdsb.2004.4.759 +[Abstract](4989) +[PDF](161.3KB)
Abstract:
Two impulsive models concerning integrated pest management(IPM) are proposed according to impulsive effect with fixed moments and unfixed moments, respectively. The first model has the potential to protect the natural enemies from extinction, but under some conditions may also serve to extinction of the pest. The second model is constructed according to the practices of IPM, that is, when the pest population reaching the economic injury level, a combination of biological, cultural, and chemical tactics that reduce pests to tolerable levels is used. By using analytical method, we show that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given economic threshold. Further, the complete expression of period of the periodic solution is given. Thus, IPM strategy proved firstly by mathematical models is more effective than the classical method.
2004, 4(3): 769-776 doi: 10.3934/dcdsb.2004.4.769 +[Abstract](1908) +[PDF](152.6KB)
Abstract:
In this paper, we introduce a method of stepwise correspondence analysis. The mathematical model, criterion of selecting variable and computational procedure of this method are given in the paper. Using this method, we study the relationship among 26 Chinese ethnic groups based on body form characteristics data.
2004, 4(3): 777-788 doi: 10.3934/dcdsb.2004.4.777 +[Abstract](1999) +[PDF](202.3KB)
Abstract:
This article discusses the structure of weed reproduction incorporating the application of a mathematical model. This mathematical methodology enables the construction, testing and application of distribution models for the analysis of the structure of weed reproduction and weed ecology. The mathematical model was applied, at the individual level, to the weed species, Bromus sterilis. The application of this method, to the weed under competition, resulted in an analysis of the overall reproduction structure of the weed which follows approximately Gaussian distribution patterns and an analysis of the shoots in the weed plant which follow approximately Sigmoid distribution patterns. It was also discovered that the application of the mathematical distribution models, when applied under specific conditions could, effectively estimate the seed production and total number of shoots in a weed plant. On the average, a weed plant has 3 shoots, with each shoot measuring 90cm in height and being composed of 21 spikelets. Besides the estimations of the total shoots and seed production within the experimental field, one may also apply these mathematical distribution models to estimate the germination rate of the species within the experimental field in following years.
2004, 4(3): 789-795 doi: 10.3934/dcdsb.2004.4.789 +[Abstract](2361) +[PDF](174.8KB)
Abstract:
A system of functional differential equations is used to model the single microorganism in the chemostat environment with a periodic nutrient and antibiotic input. Based on the technique of Razumikhin, we obtain the sufficient condition for uniform persistence of the microbial population. For general periodic functional differential equations, we obtain a sufficient condition for the existence of periodic solution, therefore, the existence of positive periodic solution to the chemostat-type model is verified.
2004, 4(3): 797-804 doi: 10.3934/dcdsb.2004.4.797 +[Abstract](2492) +[PDF](165.2KB)
Abstract:
An epidemic model is studied to understand the effect of a population dispersal on the spread of a disease in two patches. Under the assumption that the dispersal of infectious individuals is barred, it is found that susceptive dispersal may cause the spread of the disease in one patch even though the disease dies out in each isolated patch. For the case where the disease spreads in each isolated patch, it is shown that suitable susceptive dispersal can lead to the extinction of the disease in one patch.
2004, 4(3): 805-811 doi: 10.3934/dcdsb.2004.4.805 +[Abstract](1676) +[PDF](167.7KB)
Abstract:
In this article we focus on clinical trials in which the compliance is measured with random errors, and develop an error-in-variables model for the analysis of the clinical trials. With this model, we separate the efficacy of prescribed treatment from that of the compliance. With additional information correlated with compliance, we prove that the model is identifiable, and get estimators for the parameters of interest, including the parameter reflecting the efficacy of the treatment. Furthermore, we extend the model to stratified populations, and discuss the asymptotic properties of the estimators.
2004, 4(3): 813-821 doi: 10.3934/dcdsb.2004.4.813 +[Abstract](2004) +[PDF](173.5KB)
Abstract:
In the longitudinal studies of certain diseases, subjects are assessed periodically. In fact, many Alzheimer's Disease Research Centers (ADRC) in the United States typically assess their subjects annually, resulting in grouped or interval censored data for the progression time from one stage of dementia to a more severe stage of dementia. This paper studies the likelihood ratio test for increasing hazard rate associated with the progression time of dementia based on grouped progression time data. We first give the maximum likelihood estimators (MLEs) for model parameters under the assumption that the hazard rate of the progression time is nondecreasing. We then present the likelihood ratio test for testing the null hypothesis that the hazard rate is constant against the alternative that it is increasing. Finally, the methodology is applied to the dementia progression time from the Consortium to Establish a Registry for Alzheimer's Disease (CERAD). The statistical methodology developed here, although specifically referred to the study of dementia in the paper, can be easily applied to other longitudinal medical studies in which the disease status is categorized according to the severity and the hazard rate associated with the transition time among disease stages is to be tested.
2004, 4(3): 823-831 doi: 10.3934/dcdsb.2004.4.823 +[Abstract](2305) +[PDF](176.9KB)
Abstract:
A discrete periodic two-species Lotka-Volterra predator-prey model with time delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the model.
2004, 4(3): 833-840 doi: 10.3934/dcdsb.2004.4.833 +[Abstract](2113) +[PDF](936.5KB)
Abstract:
This paper considers population dynamics of sea bass and young sea bass which are modeled by stage-structured delay-differential equations. It is shown that time delay can stabilize the dynamics. That is, as time delay increases, system becomes periodic and stable even if system without time delay is chaotic.
2004, 4(3): 841-850 doi: 10.3934/dcdsb.2004.4.841 +[Abstract](2558) +[PDF](193.8KB)
Abstract:
Age is an important factor in the dynamics of epidemic process. Great attention has been paid to continuous age-structured epidemic models. The discrete epidemic models are in their infancy. In this paper a discrete age-structured epidemic SIS model is formulated. The dynamical behavior of this model is studied. The basic reproductive number is defined and threshold for the persistence or extinction of disease is found.
2004, 4(3): 851-863 doi: 10.3934/dcdsb.2004.4.851 +[Abstract](2210) +[PDF](210.3KB)
Abstract:
We consider periodic solutions of a system of difference equations with delay arising from a discrete neural network. We show that such a small network possesses a huge amount of stable periodic orbits with large domains of attraction if the delay is large, and thus the network has the potential large capacity for associative memory and for temporally periodic pattern recognition.

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