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1.
J. M. Cushing 《Journal of mathematical biology》1984,20(3):259-276
The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population
is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the
trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional
of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in
the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of
the author.
The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability.
It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical.
Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states
as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield
formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates
as functionals of population density.
Analytical and numerical results for some examples are given which illustrate these results. 相似文献
2.
《Journal of biological dynamics》2013,7(1):278-287
ABSTRACTA stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium. 相似文献
3.
We study two- and three-compartment models of a hierarchical cell production system with cell division regulated by the level of mature cells. We investigate the structure of equilibria with respect to parameters as well as local stability properties for the equilibria. To interpret the results we adapt the concept of reproduction numbers, which is well known in ecology, to stem cell population dynamics. In the two-compartment model, the positive equilibrium is stable wherever it exists. In the three-compartment model, we find that the intermediate stage of differentiation is responsible for the emergence of an instability region in the parameter plane. Moreover, we prove that this region shrinks as the mortality rate for mature cells increases and discuss this result. 相似文献
4.
Summary The paper formulates a one locus two allele diploid model influenced by temporal random selection intensities. The concept of stochastic local stability for equilibria states is formulated. Necessary and sufficient conditions are derived for the stochastic local stability of the fixation states for non-dominant and dominant traits. A general model, where a fixed polymorphic equilibrium is maintained is investigated. The complete local evolutionary picture is derived and some cases of global convergence are established.Supported in part by N. I. H. Grant GM 10452-09 and N. S. F. Grant 7507129. 相似文献
5.
6.
A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable. 相似文献
7.
In this paper, we formulate a nonlinear system of difference equations that models the three-stage life cycle of the deer
tick over four seasons. We study the effect of seasonality on the stability and oscillatory behavior of the tick population
by comparing analytically the seasonal model with a non-seasonal one. The analysis of the models reveals the existence of
two equilibrium points. We discuss the necessary and sufficient conditions for local asymptotic stability of the equilibria
and analyze the boundedness and oscillatory behavior of the solutions. A main result of the mathematical analysis is that
seasonality in the life cycle of the deer tick can have a positive effect, in the sense that it increases the stability of
the system. It is also shown that for some combination of parameters within the stability region, perturbations will result
in a return to the equilibrium through transient oscillations. The models are used to explore the biological consequences
of parameter variations reflecting expected environmental changes. 相似文献
8.
Stability in chemostat equations with delayed nutrient recycling 总被引:2,自引:0,他引:2
The growth of a species feeding on a limiting nutrient supplied at a constant rate is modelled by chemostat-type equations with a general nutrient uptake function and delayed nutrient recycling. Conditions for boundedness of the solutions and the existence of non-negative equilibria are given for the integrodifferential equations with distributed time lags. When the time lags are neglected conditions for the global stability of the positive equilibrium and for the extinction of the species are provided. The positive equilibrium continues to be locally stable when the time lag in recycling is considered and this is proved for a wide class of memory functions. Computer simulations suggest that even in this case the region of stability is very large, but the solutions tend to the equilibrium through oscillations.Work performed within the activity of the research group Equazioni di evoluzione e applicazioni Fisico-Matematiche, MPI (Italy), and under the auspices of GNFM, CNR (Italy) 相似文献
9.
Mean-square stability of a stochastic model for bacteriophage infection with time delays 总被引:1,自引:0,他引:1
Carletti M 《Mathematical biosciences》2007,210(2):395-414
We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time. 相似文献
10.
A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to
be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models
raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we
use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria
(called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize
the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation
of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets. 相似文献
11.
D. Xie S. Gulnik E. Gustchina B. Yu W. Shao W. Qoronfleh A. Nathan J. W. Erickson 《Protein science : a publication of the Protein Society》1999,8(8):1702-1707
The monomer-dimer equilibrium for the human immunodeficiency virus type 1 (HIV-1) protease has been investigated under physiological conditions. Dimer dissociation at pH 7.0 was correlated with a loss in beta-sheet structure and a lower degree of ANS binding. An autolysis-resistant mutant, Q7K/L33I/L63I, was used to facilitate sedimentation equilibrium studies at neutral pH where the wild-type enzyme is typically unstable in the absence of bound inhibitor. The dimer dissociation constant (KD) of the triple mutant was 5.8 microM at pH 7.0 and was below the limit of measurement (approximately 100 nM) at pH 4.5. Similar studies using the catalytically inactive D25N mutant yielded a KD value of 1.0 microM at pH 7.0. These values differ significantly from a previously reported value of 23 nM obtained indirectly from inhibitor binding measurements (Darke et al., 1994). We show that the discrepancy may result from the thermodynamic linkage between the monomer-dimer and inhibitor binding equilibria. Under conditions where a significant degree of monomer is present, both substrates and competitive inhibitors will shift the equilibrium toward the dimer, resulting in apparent increases in dimer stability and decreases in ligand binding affinity. Sedimentation equilibrium studies were also carried out on several drug-resistant HIV-1 protease mutants: V82F, V82F/I84V, V82T/I84V, and L90M. All four mutants exhibited reduced dimer stability relative to the autolysis-resistant mutant at pH 7.0. Our results indicate that reductions in drug affinity may be due to the combined effects of mutations on both dimer stability and inhibitor binding. 相似文献
12.
Xin-You Meng 《Journal of biological dynamics》2018,12(1):342-374
In this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results. 相似文献
13.
Gang Huang Yasuhiro Takeuchi Wanbiao Ma Daijun Wei 《Bulletin of mathematical biology》2010,72(5):1192-1207
In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into
the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays
are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing
suitable Lyapunov functionals and using the Lyapunov–LaSalle invariance principle, we prove the global stability of the endemic
equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global
properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not
affect the stability, but the latent period in an infected host plays a positive role to control disease development. 相似文献
14.
Understanding how complexity persists in nature is a long-standing goal of ecologists. In theoretical ecology, local stability is a widely used measure of ecosystem persistence and has made a major contribution to the ecosystem stability-complexity debate over the last few decades. However, permanence is coming to be regarded as a more satisfactory definition of ecosystem persistence and has relatively recently become available as a tool for assessing the global stability of Lotka-Volterra communities. Here we document positive relationships between permanence and Lotka-Volterra food web complexity and report a positive correlation between the probability of local stability and permanence. We investigate further the frequency of discrepancy (attributed to fragile systems that are locally stable but not permanent or locally unstable systems that are permanent and have cyclic or chaotic dynamics), associate non-permanence with the local stability or instability of equilibria on the boundary of the state-space, and investigate how these vary with aspects of ecosystem complexity. We find that locally stable interior equilibria tend to have all locally unstable boundary equilibria. Since a locally stable boundary is inconsistent with permanent dynamics, this can explain the observed positive correlation between local interior stability and permanence. Our key finding is that, at least in Lotka-Volterra model ecosystems, local stability may be a better measure of persistence than previously thought. 相似文献
15.
We study a system of two integrodifierential equations which models the evolution of a biotic species feeding on an abiotic resource. We also consider nutrient recycling with time delay. By Hopf bifurcation theory we prove the existence of stable oscillations for a range of values of the input of nutrients.Work performed within the activity of the research group Evolution Equations and Physico-Mathematical Applications, M.P.I. (Italy), and under the auspices of G.N.F.M., C.N.R. (Italy) 相似文献
16.
HBV(HePatitis B virus)是一种具有严重传染性的肝炎病毒,迄今为止,人们对它的免疫和慢性化的机制等方面还不甚了解。本文基于相关的病理知识,对应的建立了具有时滞的微分方程数学模型,系统地探讨了肝炎B病毒与宿主细胞之间的关系,利用Lyapunov函数方法研究了病毒动力学模型感染平衡点的局部稳定性和未感染平衡点全局稳定性,并利用数学模拟验证了理论分析。结果表明时滞的存在不会影响到感染平衡点的局部稳定性,但能影响平衡点到达的时间跨度,对于药物治疗的疗程和治疗时机的确定有参考意义。 相似文献
17.
捕食者有病的生态-流行病模型的分析 总被引:11,自引:1,他引:10
建立并分析了捕食者具有疾病且有功能反应的生态-流行病(SI)模型,讨论了解的有界性.应用特征根法得到了平衡点局部渐近稳定的充分条件,进一步分析了平衡点的全局稳定性,得到了边界平衡点和正平衡点全局稳定的充分条件。 相似文献
18.
In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster. 相似文献
19.
We present a mathematical bioeconomic model of a fishery with a variable price. The model describes the time evolution of the resource, the fishing effort and the price which is assumed to vary with respect to supply and demand. The supply is the instantaneous catch while the demand function is assumed to be a monotone decreasing function of price. We show that a generic market price equation (MPE) can be derived and has to be solved to calculate non trivial equilibria of the model. This MPE can have 1, 2 or 3 equilibria. We perform the analysis of local and global stability of equilibria. The MPE is extended to two cases: an age-structured fish population and a fishery with storage of the resource. 相似文献