Chapter 1 Introduction and Background
Pest control is essential to agricultural production since pest outbreaks can cause significant economic damage to crops ([xiang2014discrete]). Integrated Pest Management (IPM) strategies that combine biological and chemical control methods have been widely used to prevent pest population outbreaks ([liang2018discrete]). Biological control refers to the action of natural enemies such as parasites, predators, or pathogens in maintaining pest density at a lower average ([bellows1999handbook]). Natural enemies may be released periodically in a few numbers at a critical time of the season, referred to as classical or inoculative release ([mills1996modelling]) or in large quantities at a single time, known as augmentation or inundative release ([bale2008biological]). For example, the periodic introduction of the parasitoid Encarsia formosa has been found to be effective at controlling the greenhouse whitefly, Trialeurodes vaporariorum, a serious pest of tomato and cucumber crops ([bellows1999handbook]). In host-parasitoid dynamics, parasitoid species are considered as biological control agents. The larvae of parasitoids grow inside or on the host, which they eventually kill ([mckenzie1977ecology]). For example, Diaeretiella rapae is the common parasitoid of the cabbage aphid Brevicoryne brassicae ([mckenzie1977ecology]).
Chemical control is one of the most effective components of IPM in reducing pest population density ([jang2012discrete]). However, long-term use of pesticides can cause environmental pollution and damage to the land by killing natural enemies, resulting in pest population outbreaks ([heinrichs1984secondary]). Long-term use of chemical pesticides in the crop fields can be harmful to the environment, and pests can develop strong resistance to some pesticides. To defeat pest resistance, pesticide rotation or switching, avoiding unnecessary pesticide applications, and using non-chemical control techniques are suggested ([liang2018discrete]). As a result, a combination of both chemical and biological control is often used to suppress pest populations. A lot of research have studied in host-parasitoid dynamics with Integrated Pest Management (IPM), which employs biological control along with pesticide spraying, as a sustainable strategy for controlling agricultural pests ([tang2008models, tang2010optimum]). For example, IPM host-parasitoid models in which controls are applied impulsively ([tang2004modelling, tang2008multiple]), based on an economic threshold thresholds (ET) ([saphores2000economic, higley1986economic, alston2011pest, stejskal2003economic]), switching discrete model where the switches is guided by economic threshold (ET) ([xiang2014dynamic, he2020holling, xiang2014discrete]) have been studied. However, the compatibility of these control methods is critical to the success of an integrated pest management strategy. Aside from the fact that different natural enemies may respond to toxicants differently, variations in population structure can also influence outcomes. For instance, D rapae (M’Intosh), a common parasitoid of B brassicae (L.), a cabbage aphid, exhibits different stage susceptibility after exposure to the insecticide imidacloprid, with mortality in the adult stage being double that of the pupae stage for various concentrations of the insecticide ([stark2020population]). Motivated by this, in the first part of the dissertation, we develop a stage-structured host-parasitoid model with an integrated pest management approach to reduce pesticide use and prevent the pest outbreaks.
The host-parasitoid models and the compartmental epidemiological models share the same matrix model structure. In particular, in host–parasitoid systems, new parasitoids are produced in proportion to the host population, since hosts are converted into parasitoids through parasitism. Similarly, in epidemiological models, new infections arise in proportion to the susceptible population, as susceptible individuals become infectious. This contrasts with predator–prey and competition models, where population growth is typically proportional to the existing number of individuals within the same population, reflecting self-replication rather than conversion between classes. The mathematical compartmental epidemic model is an essential tool for forecasting disease transmission dynamics. One area of interest in epidemic models is both epidemics, sudden outbreaks of a disease, and endemic situations, in which a disease is always present, and the control of the disease, with vaccination and treatment, has been the subject of theoretical analysis ([kribs2000simple]). Continuous-time epidemic models with partially effective vaccination have been shown to exhibit backward bifurcation ([brauer2004backward]). This naturally raises the question of whether similar backward bifurcation phenomena arise in discrete-time epidemic models.
In models with backward bifurcation, the model often exhibits bistability between the disease-free and endemic steady states for . Numerous studies have been done on backward bifurcation in continuous time models ([brauer2004backward, kribs2000simple, hadeler1997backward, wang2006backward, gumel2012causes, wang2009backward]). In contrast, much of the work on discrete-time epidemic models has focused primarily on local stability of the disease-free equilibrium or cycles by computing the basic reproduction number (), ([allen2008basic, cushing2016many, diekmann2010construction, van2017reproduction]), computing the basic reproduction number () for the demographic population cycles ([van2019demographic]), analyzing the local stability of the disease-free dynamics when the basic reproduction number is less than one ([van2019disease, van2019demographic]). Moreover, in [van2019disease], the authors establish the global stability of the disease-free equilibrium under certain conditions when and determine the persistence of disease when using a Lyapunov function. However, compared with continuous-time models, relatively few studies in discrete-time epidemic models address the dynamics of endemic equilibria. Based on this, in the second part of my dissertation, we develop a discrete-time SIS model with vaccination where the infection term is modeled by a negative exponential term, and the model may exhibit backward bifurcation.
The host-parasitoid and epidemic models presented in this dissertation are connected the bifurcation theorem presented in Chapter LABEL:chapter1.5. This theorem, which is an extension of the Fundamental Bifurcation Theorem, provides an analytical tool to study model dynamics when an invader invades a resident-only cycle. In Chapter LABEL:chapter1.5, we extended the Bifurcation theorem presented in [Meissen2017Auth] for prey-predator type interaction to the general resident-invader type population interaction. In this proof, we allow the off-diagonal elements () of the projection matrix to be non-zero, which represent the creation of new residents from the invaders, and vise versa. In particular, we provide explicit expressions that determine the direction of bifurcation and stability. As an application, we present a discrete-time SIR model with demographic population cycles and show that, when the disease-free equilibrium or the two-cycle has a steady state, a branch of stable endemic equilibria or a stable endemic two-cycle bifurcates, respectively, if . In other words, we show that for , the endemic steady state inherits the structure of the disease-free steady state when this steady state is an equilibrium or 2-cycle.
In Chapter LABEL:chapter2, we develop a discrete-time host-parasitoid model with structured in both species to investigate how population structure impacts the interaction of both species. We assume a general density-dependent growth rate in host reproduction, and only adult parasitoids can attack both hosts. We establish conditions regarding the extinction of both species, the existence of hosts while parasitoids are extinct, and the persistence of both species. To study the coexistence dynamics of the host-parasitoid species, we use the bifurcation theorem presented in Chapter LABEL:chapter1.5. We also study, through numerical simulations, how pesticide spraying may interact with natural enemies (parasitoids) to control a pest (host) species, considering two spraying strategies: spraying at the beginning or end of the time interval.
In Chapter LABEL:chapter3, we extended the host-parasitoid model to an impulsive difference system by incorporating periodic pesticide spraying along with additional parasitoid releases. This modelling approach is more realistic than the prior one. We investigate the impact of periodic control strategies on host-parasitoid interactions. We first show that all solutions of the parasitoid-only model converge to a periodic solution. We determine when parasitoids can eradicate the host when control measures are applied periodically. In other words, we establish the conditions under which host eradication cycles are globally attracting. Finally, through numerical simulations, we show the impact of periodic combined control strategies applied at the same and different time units. We also show the impact of the control strategies on the host eradication.
In Chapter LABEL:chapter4, we develop a discrete-time susceptible, infected, susceptible, and vaccinated (SISV) population model in which vaccination reduces the risk of infection but does not eliminate it. We assume vaccination occurs prior to disease transmission in this model. We show that under a condition, the disease-free state is globally asymptotically stable. Then, using the Bifurcation Theorem, we establish conditions under which disease-free equilibria undergo a backward bifurcation at . We found that imperfect vaccination and disease recovery are necessary for backward bifurcation. Finally, we compare our findings with an alternative discrete-time SISV model where disease transmission occurs before vaccination. The latter model may show backward bifurcation even when the vaccination is completely effective.
Finally, in Chapter LABEL:chapter5, we conclude with a discussion and potential future extensions.