Search

Enclosure theorems are derived for homogeneous bounded order-preserving operators and illustrated for operators involving pair-formation functions introduced by Karl-Peter Hadeler in the late 1980s. They are applied to a basic discrete-time two-sex population model and to the relation between the basic turnover number and the basic reproduction number.

Feller kernels are a concise means to formalize individual structural transitions in a structured discrete-time population model. An iteroparous populations (in which generations overlap) is considered where different kernels model the structural transitions for neonates and for older individuals. Other Feller kernels are used to model competition between individuals. The spectral radius of a suitable Feller kernel is established as basic turnover number that acts as threshold between population extinction and population persistence. If the basic turnover number exceeds one, the population shows various degrees of persistence that depend on the irreducibility and other properties of the transition kernels. [ABSTRACT FROM AUTHOR]

Ixodid ticks are acknowledged as one of the most important hematophagous arthropods because of their ability in transmitting a variety of tick-borne diseases. Mathematical models have been developed, based on emerging knowledge about tick ecology, pathogen epidemiology and their interface, to understand tick population dynamics and tick-borne diseases spread patterns. However, no serious effort has been made to model and assess the impact of host immunity triggered by tick feeding on the distribution of the tick population according to tick stages and on tick population extinction and persistence. Here, we construct a novel mathematical model taking into account the effect of host immunity status on tick population dynamics, and analyze the long-term behaviours of the model solutions. Two threshold values, R 11 and R 22 , are introduced to measure the reproduction ratios for the tick-host interaction in the absence and presence of host immunity. We then show that these two thresholds (sometimes under additional conditions) can be used to predict whether the tick population goes extinct ( R 11 < 1 ) and the tick population grows without bound ( R 22 > 1 ). We also prove tick permanence (persistence and boundedness of the tick population) and the existence of a tick persistence equilibrium if R 22 < 1 < R 11 . As the host species adjust their immunity to tick infestation levels, they form for the tick population an environment with a carrying capacity very much like that in logistic growth. Numerical results show that the host immune reactions decrease the size of the tick population at equilibrium and apparently reduce the tick-borne infection risk. [ABSTRACT FROM AUTHOR]

Ticks play a critical role as vectors in the transmission and spread of Lyme disease, an emerging infectious disease which can cause severe illness in humans or animals. To understand the transmission dynamics of Lyme disease and other tick-borne diseases, it is necessary to investigate the population dynamics of ticks. Here, we formulate a system of delay differential equations which models the stage structure of the tick population. Temperature can alter the length of time delays in each developmental stage, and so the time delays can vary geographically (and seasonally which we do not consider). We define the basic reproduction number $${\mathcal R}_0$$ of stage structured tick populations. The tick population is uniformly persistent if $${\mathcal R}_0>1$$ and dies out if $${\mathcal R}_0<1$$ . We present sufficient conditions under which the unique positive equilibrium point is globally asymptotically stable. In general, the positive equilibrium can be unstable and the system show oscillatory behavior. These oscillations are primarily due to negative feedback within the tick system, but can be enhanced by the time delays of the different developmental stages. [ABSTRACT FROM AUTHOR]

A model is considered for a spatially distributed population of male and female individuals that mate and reproduce only once in their life during a very short reproductive season. Between birth and mating, females and males move by diffusion on a bounded domain Ω. Mating and reproduction is described by a (positively) homogeneous function (of degree one). We identify a basic reproduction number R0 that acts as a threshold between extinction and persistence. If R0 < 1, the population dies out while it persists (uniformly weakly) if R0 > 1. R0 is the cone spectral radius of a bounded homogeneous map. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of solutions to a measure-valued selection–mutation model that we formulated in [14] . We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy. We also show that when the strategy space is discrete, the selection–mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy. [ABSTRACT FROM AUTHOR]