Discrete three-stage population model: persistence and global stability results

The interplay between multiple control mechanisms in a host-parasitoid system: a discrete-time stage-structured modelling approach

We propose a discrete-time host-parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria as well as conditions for the existence and local stability of an interior equilibrium and system persistence. We study the model numerically to examine how pesticide spraying may interact with natural enemies (parasitoids) to control the pest (host) species. We then extend the model to an impulsive difference system that incorporates both periodic pesticide spraying and augmentation of the natural enemies to suppress the pest population. For this system, we determine when the pest-eradication periodic solution is globally attracting. We also examine how varying the control measures (pesticide concentration, natural enemy augmentation and the frequency of applications) may lead to different pest outbreak or persistence outcomes when eradication does not occur.

Discrete stage-structured tick population dynamical system with diapause and control

A discrete stage-structured tick population dynamical system with diapause is studied, and spraying acaricides as the control strategy is considered in detail. We stratify vector populations in terms of their maturity status as immature and mature subgroups. The immature subgroup is divided into two categories: normal immature and diapause immature. We compute the net reproduction number $ R_0 $ and perform a qualitative analysis. When $ R_0 < 1 $, the global asymptotic stability of tick-free fixed point is well proved by the inherent projection matrix; there exists a unique coexistence fixed point and the conditions for its asymptotic stability are obtained if and only if $ R_0 > 1; $ the model has transcritical bifurcation if $ R_0 = 1. $ Moreover, we calculate the net reproduction numbers of the model with constant spraying acaricides and periodic spraying acaricides, respectively, and compare the effects of the two methods on controlling tick populations.

Modeling the invasion and establishment of a tick-borne pathogen

We develop a discrete-time tick–host–pathogen model to describe the spread of a disease in a hard-bodied tick species. This model incorporates the developmental stages for a tick, the dependence of the tick life-cycle and disease transmission on host availability, and three sources of pathogen transmission. We first establish the global dynamics of the disease-free system. We then apply the model to two pathogens, Borellia burgdorferi and Anaplasma phagocytophila, using Ixodes ricinus as the tick species to study properties of the invasion and establishment of a disease numerically. In particular, we consider the basic reproduction number, which determines whether a disease can invade the tick-host system, as well as disease prevalence and time to establishment in the case of successful disease invasion. Using Monte Carlo simulations, we calculate the means of each of these disease metrics and their elasticities with respect to various model parameters. We find that increased tick survival may help enable disease invasion, decrease the time to disease establishment, and increase disease prevalence once established. In contrast, though disease invasion is sensitive to tick-to-host transmission and tick searching efficiencies, neither disease prevalence nor time to disease establishment is sensitive to these parameters. These differences emphasize the importance of developing approaches, such as the one highlighted here, that can be used to study disease dynamics beyond just pathogen invasion, including transitional and long-term dynamics.

A discrete tick population dynamics model with continuous and seasonal birth breeding

In this paper, we formulate a discrete two-stage model with two types of birth mechanisms, continuous and seasonal. We divide tick population into two subgroups, i.e. immature and mature. We also consider diapause as an important adaptive process for ticks in respond to climate change in both stages. We derive a formula for the inherent net reproductive numbers and explore their stability analysis. When breeding is continuous, there exists a unique globally asymptotically stable positive fixed point provided that the inherent net reproductive number is larger than one; the extinction fixed point is globally asymptotically stable if the inherent net reproductive number is less than one. When breeding is seasonal with 2-periodic birth function, there exists a unique globally asymptotically stable periodic solution provided that the inherent net reproductive number is larger than one, while the population goes to extinction if this value is less than one. For the cases with multi-periodic birth function, we use numerical simulation to compare their complex behaviors.

Temperature- and Turbidity-Dependent Competitive Interactions Between Invasive Freshwater Mussels

We develop a staged-structured population model that describes the competitive dynamics of two functionally similar, congeneric invasive species: zebra mussels and quagga mussels. The model assumes that the population survival rates are functions of temperature and turbidity, and that the two species compete for food. The stability analysis of the model yields conditions on net reproductive rates and intrinsic growth rates that lead to competitive exclusion. The model predicts quagga mussel dominance leading to potential exclusion of zebra mussels at mean water temperatures below [Formula: see text] and over a broad range of turbidities, and a much narrower set of conditions that favor zebra mussel dominance and potential exclusion of quagga mussels at temperatures above [Formula: see text] and turbidities below 35 NTU. We then construct a two-patch dispersal model to examine how the dispersal rates and the environmental factors affect competitive exclusion and coexistence.

A discrete stage-structured two-species competition model with sexual and clonal reproduction

In this paper, we analyse a discrete stage-structured model which is a generalization of the two-species competition model studied in [2]. Motivated by plant populations, each species is assumed to reproduce both sexually and clonally. We show that this model has a dynamical behaviour that is similar to that of the classical continuous two-dimensional Lotka-Volterra model under weak nonlinearities of the Beverton-Holt type. By allowing the species to have different competition efficiencies, we show that it is possible to obtain different dynamics including coexistence, bistability and competitive exclusion, in contrast with the model studied in [2], which exhibits only competitive exclusion behaviour.

The global dynamics of a discrete juvenile-adult model with continuous and seasonal reproduction

A general discrete juvenile-adult population model with time-dependent birth rate and nonlinear survivorship rates is studied. When breeding is continuous, it is shown that the model has a unique globally asymptotically stable positive equilibrium provided the net reproductive number is larger than one. If it is smaller than one, then the extinction equilibrium is globally asymptotically stable. When breeding is seasonal, it is shown that there exists a unique globally asymptotically stable periodic solution provided the net reproductive number is larger than one. When this value is less than one, the population goes to extinction. Conditions on the birth rate where the population with seasonal breeding survives while the population with continuous breeding becomes extinct are provided.