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An introduction to structured population dynamics

01 Jan 1987-
TL;DR: In this paper, the authors present a case study of multispecies interactions with continuous models of age-structured models and show that these models can be used in a variety of applications.
Abstract: Preface 1 Discrete Models Matrix Models Autonomous Single Species Models Some Applications A Case Study Multispecies Interactions 2 Continuous Models Age-Structured Models Autonomous Age-Structured Models Some Applications Multispecies Interactions Other Structured Models 3 Population Level Dynamics Ergodicity and Nonlinear Models The Linear Chain Trick Hierarchical Models Total Population Size in Age-Structured Models Appendix A Stability Theory for Maps Linear Maps Linearization of Maps Appendix B Bifurcation Theorems A Global Bifurcation Theorem Local Parameterization Appendix C Miscellaneous Proofs Bibliography Index
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Journal ArticleDOI
TL;DR: The main objective of this paper is to provide practical guidelines that combine graphical information with analytical work to effectively study the local stability of some models involving delay dependent parameters.
Abstract: In most applications of delay differential equations in population dynamics,the need of incorporation of time delays is often the result of the existence of some stage structure. Since the through-stage survival rate is often a function of time delays,it is easy to conceive that these models may involve some delay dependent parameters. The presence of such parameters often greatly complicates the task of an analytical study of such models. The main objective of this paper is to provide practical guidelines that combine graphical information with analytical work to effectively study the local stability of some models involving delay dependent parameters. Specifically,we shall show that the stability of a given steady state is simply determined by the graphs of some functions of τ which can be expressed explicitly and thus can be easily depicted by Maple and other popular software. In fact,for most application problems,we need only look at one such function and locate its zeros. This function often has only two zeros,providing thresholds for stability switches. The common scenario is that as time delay increases,stability changes from stable to unstable to stable, implying that a large delay can be stabilizing. This scenario often contradicts the one provided by similar models with only delay independent parameters.

544 citations


Cites background from "An introduction to structured popul..."

  • ...In most applications of delay differential equations in population dynamics, the need of incorporation of a time delay is often the result of the existence of some stage structure [1], [3], [10], [11], [12], [15]....

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Journal ArticleDOI
TL;DR: A critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field.
Abstract: This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.

424 citations

Journal ArticleDOI
TL;DR: An ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations is presented and it is found that the disease‐free equilibrium is locally asymptotically stable when $R_0 1$.
Abstract: We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, and recovered classes, before reentering the susceptible class. Susceptible mosquitoes can become infected when they bite infectious or recovered humans, and once infected they move through the exposed and infectious classes. Both species follow a logistic population model, with humans having immigration and disease‐induced death. We define a reproductive number, $R_0$, for the number of secondary cases that one infected individual will cause through the duration of the infectious period. We find that the disease‐free equilibrium is locally asymptotically stable when $R_0 1$. We prove the existence of at least one endemic equilibrium point for all $R_0 > 1$. In the absence of disease‐induced death, we prove that the tran...

404 citations


Cites methods from "An introduction to structured popul..."

  • ...8), in the absence of disease-induced death (δh = 0), we prove, using the Lyapunov–Schmidt expansion as described by Cushing [9], that the bifurcation is supercritical (forward)....

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  • ...In this model (2.8), in the absence of disease-induced death (δh = 0), we prove, using the Lyapunov–Schmidt expansion as described by Cushing [9], that the bifurcation is supercritical (forward)....

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  • ...We can determine the initial direction of the continua of solution-pairs, C1 and C2, using the Lyapunov–Schmidt expansion, as described by Cushing [9]....

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Book
15 Dec 2010
TL;DR: The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term as mentioned in this paper, and applies to infinite-dimensional as well as to finite-dimensional dynamical systems.
Abstract: The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term. It applies to infinite-dimensional as well as to finite-dimensional dynamical systems, and to discrete-time as well as to continuous-time semiflows. This monograph provides a self-contained treatment of persistence theory that is accessible to graduate students. The key results for deterministic autonomous systems are proved in full detail such as the acyclicity theorem and the tripartition of a global compact attractor. Suitable conditions are given for persistence to imply strong persistence even for nonautonomous semiflows, and time-heterogeneous persistence results are developed using so-called ""average Lyapunov functions"". Applications play a large role in the monograph from the beginning. These include ODE models such as an SEIRS infectious disease in a meta-population and discrete-time nonlinear matrix models of demographic dynamics. Entire chapters are devoted to infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat.

352 citations


Cites methods from "An introduction to structured popul..."

  • ...Nonlinear matrix models, such as those introduced in Chapter 3, are increasingly being used in population modeling as indicated by the recent monographs [29, 44, 39]....

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Journal ArticleDOI
Sanyi Tang1, Lansun Chen1
TL;DR: Using the discrete dynamical system determined by the stroboscopic map, an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions are obtained, and the threshold conditions for their stability are obtained.
Abstract: In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos.

292 citations