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.

Dynamical Systems and Population Persistence

Stochastic population dynamics under regime switching II

On competitive Lotka-Volterra model in random environments

We study the global behavior of a non-linear susceptible-infectious-removed (SIR)-like epidemic model with a non-bilinear feedback mechanism, which describes the influence of information, and of information-related delays, on a vaccination campaign. We upgrade the stability analysis performed in d’Onofrio et al. [A. d’Onofrio, P. Manfredi, E. Salinelli, Vaccinating behavior, information, and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol. 71 (2007) 301] and, at same time, give a special example of application of the geometric method for global stability, due to Li and Muldowney. Numerical investigations are provided to show how the stability properties depend on the interplay between some relevant parameters of the model.

Global stability of an SIR epidemic model with information dependent vaccination

Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching

Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

In this paper, we consider population survival by using single-species stage-structured models. As a criterion of population survival, we employ the mathematical notation of permanence. Permanence of stage-structured models has already been studied by Cushing (1998). We generalize his result of permanence, and obtain a condition which guarantees that population survives. The condition is applicable to a wide class of stage-structured models. In particular, we apply our results to the Neubert-Caswell model, which is a typical stage-structured model, and obtain a condition for population survival of the model.

Permanence of single-species stage-structured models.

In this short note, we consider attenuant cycles of population models. This study concerns the second conjecture of Cushing and Henson [A periodically forced Beverton-Holt equation, J. Diff. Eq. Appl., 8 (2002), pp. 1119–1120], which was recently resolved affirmatively by Elaydi and Sacker [Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures, Proc. 8th Inter. Conf. Diff. Eq., Brno, (in press)]. They showed that the periodic fluctuations in the carrying capacity always reduce the average of population densities in the Beverton-Holt equation. We extend this result and give a class of population models in which the periodic fluctuations in the carrying capacity always reduce the average of population densities.

A Note on Attenuant Cycles of Population Models with Periodic Carrying Capacity

This paper considers attenuation of cycles generated by periodic difference equations for population dynamics. This study concerns the second conjecture of Cushing and Henson [A periodically forced Beverton-Holt equation, Journal of Difference Equations and Applications, 8, 2002, pp. 1119–1120], which was recently resolved affirmatively by Elaydi and Sacker [Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures, Proceedings of the 8th International Conference on Difference Equations and Applications, Brno, Czech Republic (in press)]. We extend their result and obtain a sufficient condition for attenuation of cycles in population models. This sufficient condition is applicable to a wide class of periodic difference equations with arbitrary period. For an illustration, the result is applied to the Beverton-Holt equation and other specific population models.

Attenuant cycles of population models with periodic carrying capacity

This paper considers the dynamics of a discrete-time Kolmogorov system for two-species populations. In particular, permanence of the system is considered. Permanence is one of the concepts to describe the species' coexistence. By using the method of an average Liapunov function, we have found a simple sufficient condition for permanence of the system. That is, nonexistence of saturated boundary fixed points is enough for permanence of the system under some appropriate convexity or concavity properties for the population growth rate functions. Numerical investigations show that for the system with population growth rate functions without such properties, the nonexistence of saturated boundary fixed points is not sufficient for permanence, actually a boundary periodic orbit or a chaotic orbit can be attractive despite the existence of a stable coexistence fixed point. This result implies, in particular, that existence of a stable coexistence fixed point is not sufficient for permanence.

Ryusuke Kon

Papers

Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

Permanence of single-species stage-structured models.

A Note on Attenuant Cycles of Population Models with Periodic Carrying Capacity

Attenuant cycles of population models with periodic carrying capacity

Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points.