Plant virus epidemiology: applications and prospects for mathematical modeling and analysis to improve understanding and disease control

In recent years, mathematical modeling has increasingly been used to complement experimental and observational studies of biological phenomena across different levels of organization. In this article, we consider the contribution of mathematical models developed using a wide range of techniques and uses to the study of plant virus disease epidemics. Our emphasis is on the extent to which models have contributed to answering biological questions and indeed raised questions related to the epidemiology and ecology of plant viruses and the diseases caused. In some cases, models have led to direct applications in disease control, but arguably their impact is better judged through their influence in guiding research direction and improving understanding across the characteristic spatiotemporal scales of plant virus epidemics. We restrict this article to plant virus diseases for reasons of length and to maintain focus even though we recognize that modeling has played a major and perhaps greater part in the epidemiology of other plant pathogen taxa, including vector-borne bacteria and phytoplasmas

Assessing recovery potential of aquatic macroinvertebrate populations using ecological models

Doordat de groeiende wereldbevolking een steeds grotere druk op natuurlijke ecosystemen legt, wordt de vraag naar een goede methode voor het beoordelen van de mogelijkheid tot herstel van een systeem steeds groter. Dit is vooral relevant voor agrarische ecosystemen welke traditioneel als functie hebben om voedsel voor de menselijke populatie te produceren. Agrarische ecosystemen leveren echter ook andere ecosysteemdiensten zoals omzettingen van nutriënten, bestuiving, het op peil houden van een bepaalde bodemkwaliteit en structuur, maar ook esthetische en recreatieve diensten, waarvan de duurzaamheid moet worden gewaarborgd

Mathematical modeling of fall armyworm spodoptera frugiperda infestations in maize crops and its impact on final maize biomass

A Dissertation Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in Mathematical and Computer Sciences and Engineering of the Nelson Mandela African Institution of Science and TechnologyFall armyworm (FAW-Spodoptera frugiperda), a highly destructive and fast spreading agricul tural pest native to North and South America, poses a real threat to global food security. It is estimated that intermittent FAW outbreaks could cause up to $US 13 billion per annum in crop losses throughout sub-Saharan Africa. Considering this projected loss it is imperative that various tools and techniques be utilized to infer on the various factors that affect FAW maize in teraction and in-turn affect the final maize biomass. Mathematical modeling has proved to be an important tool that is capable of shedding light on the FAW-maize interaction dynamics. In this study, three mathematical models were proposed to evaluate the impact of memory effects and controls, seasonality and Integrated Pest Management strategy (farming awareness and larvae predation) on FAW infestations in maize crops and on final maize biomass. Firstly, to evaluate the impact of memory effects and control, a new dynamical system for FAW-maize biomass interaction via Caputo fractional-order operator was proposed and analyzed. In the proposed model, four equilibrium points which revealed the existence of a threshold parameter defined by R0 were computed and analyzed. Further, it was observed that, R0, the average number of newborns produced by one individual female moth during its life span was an integral compo nent for stability of the aforementioned model equilibria. Secondly, to evaluate the implications of seasonality on FAW maize interaction and on the final maize biomass, a non-autonomous mathematical model was proposed and analyzed. The analysis revealed that the model solution was non-negative, unique, permanent and bounded admitting global asymptotic and continuous periodic function. Further, the model was extended into an optimal control problem with the aim of determining optimal pesticides and traditional methods that are capable of minimizing FAW egg and larvae populations at minimum cost. Results from the study demonstrated that a combination of pesticides use at low intensity with traditional methods at higher intensity could eradicate FAW in a maize field in a period less than half the life span of the crop in the field. Thirdly, to evaluate the impact of farming awareness campaigns and larvae predation, a fractional-order model that incorporated farming awareness campaigns and larvae predation was proposed and analysed. Overall, the study highlighted that, non-time dependent farming awareness campaigns should be close to 100% all the time to eradicate the FAW. However, when time-dependent farming awareness was implemented, it was observed that even less than 50% intensity level could lead to eradication of FAW. In all the proposed models, comprehen sive numerical simulations were carried out in MATLAB programming language to support the analytical findings. In a nutshell, the results of this study showed that mathematical models can be important tools to evaluate FAW and maize interaction dynamics

Prey-predator “Host-parasite” Models with Adaptive Dispersal: Application to Social Animals

abstract: Foraging strategies in social animals are often shaped by change in an organism's natural surrounding. Foraging behavior can hence be highly plastic, time, and condition dependent. The motivation of my research is to explore the effects of dispersal behavior in predators or parasites on population dynamics in heterogeneous environments by developing varied models in different contexts through closely working with ecologists. My models include Ordinary Differential Equation (ODE)-type meta population models and Delay Differential Equation (DDE) models with validation through data. I applied dynamical theory and bifurcation theory with carefully designed numerical simulations to have a better understanding on the profitability and cost of an adaptive dispersal in organisms. My work on the prey-predator models provide important insights on how different dispersal strategies may have different impacts on the spatial patterns and also shows that the change of dispersal strategy in organisms may have stabilizing or destabilizing effects leading to extinction or coexistence of species. I also develop models for honeybee population dynamics and its interaction with the parasitic Varroa mite. At first, I investigate the effect of dispersal on honeybee colonies under infestation by the Varroa mites. I then provide another single patch model by considering a stage structure time delay system from brood to adult honeybee. Through a close collaboration with a biologist, a honeybee and mite population data was first used to validate my model and I estimated certain unknown parameters by utilizing least square Monte Carlo method. My analytical, bifurcations, sensitivity analysis, and numerical studies first reveal the dynamical outcomes of migration. In addition, the results point us in the direction of the most sensitive life history parameters affecting the population size of a colony. These results provide novel insights on the effects of foraging and Varroa mites on colony survival.Dissertation/ThesisDoctoral Dissertation Applied Mathematics for the Life and Social Sciences 201

Spreading speeds and traveling waves in some population models.

Virtually every ecosystem has been invaded by exotic organisms with potentially drastic consequences for the native fauna or flora. Studying the forms and rates of invading species has been an important topic in spatial ecology. We investigate two two-species competition models with Allee effects in the forms of reaction-diffusion equations and integro-difference equations. We discuss the spatial transitions from a mono-culture equilibrium to a coexistence equilibrium or a different mono-culture equilibrium in these models. We provide formulas for the spreading speeds based on the linear determinacy and show the results on the existence of traveling waves. We also study a two-sex stage-structured model. We carry out initial analysis for the spreading speed and conduct numerical simulations on the traveling waves and spreading speeds in the two-sex model

The price of defence: Maternal effects in an aposematic ladybird

Offspring phenotype can be adaptively altered via maternal non-genetic inheritance. Such ‘maternal effects’ enable females to adjust their per offspring investment in response to variation in the offspring environment, and thus maximise their reproductive success. Consequently they play a pivotal role in population dynamics and the response of species to environmental change. Despite this, little is known about how maternal effects mediate reproductive investment in response to multiple or novel environmental changes, such as those driven by anthropogenic activity. I use the 2-spot ladybird intraguild predation system, where resources and predation risk are highly variable, to explore the role of maternal effects in the response of a native species to an invasive predator, as well as answering outstanding questions about how maternal effects function under complex and antagonistic sets of variables. The results indicate that it is unlikely that maternally mediated changes in egg phenotype will improve the survival of 2-spot ladybird offspring in the face of predation from larvae of the invasive harlequin ladybird. They do, however, demonstrate the importance of studying maternal effects in the context of the multiple environmental factors, which more accurately represent the complex environments in which organisms live and evolve, corroborating recent theoretical predictions. Finally I provide evidence of the multifaceted nature of parental effects in aposematic species and reveal the role that they may play in shaping the variation in defence and warning coloration observed in adult populations.NER

Repulsive-attractive models for the impact of two predators on prey species varying in anti-predator response

This study considers the dynamical interaction of two predatory carnivores (Lions (Panthera leo) and Spotted Hyaenas (Crocuta crocuta)) and three of their common prey (Buffalo (Syncerus caffer), Warthog (Phacochoerus africanus) and Kudu (Tragelaphus strepsiceros)). The dependence on spatial structure of species’ interaction stimulated the author to formulate reaction-diffusion models to explain the dynamics of predator-prey relationships in ecology. These models were used to predict and explain the effect of threshold populations, predator additional food and prey refuge on the general species’ dynamics. Vital parameters that model additional food to predators, prey refuge and population thresholds were given due attention in the analyses. The stability of a predator-prey model for an ecosystem faced with a prey out-flux which is analogous to and modelled as an Allee effect was investigated. The results highlight the bounds for the conversion efficiency of prey biomass to predator biomass (fertility gain) for which stability of the three species ecosystem model can be attained. Global stability analysis results showed that the prey (warthog) population density should exceed the sum of its carrying capacity and threshold value minus its equilibrium value i.e., W >(Kw + $) −W . This result shows that the warthog’s equilibrium population density is bounded above by population thresholds, i.e., W < (Kw+$). Besides showing the occurrence under parameter space of the so-called paradox of enrichment, early indicators of chaos can also be deduced. In addition, numerical results revealed stable oscillatory behaviour and stable spirals of the species as predator fertility rate, mortality rate and prey threshold were varied. The stabilising effect of prey refuge due to variations in predator fertility and proportion of prey in the refuge was studied. Formulation and analysis of a robust mathematical model for two predators having an overlapping dietary niche were also done. The Beddington-DeAngelis functional and numerical responses which are relevant in addressing the Principle of Competitive Exclusion as species interact were incorporated in the model. The stabilizing effect of additional food in relation to the relative diffusivity D, and wave number k, was investigated. Stability, dissipativity, permanence, persistence and periodicity of the model were studied using the routine and limit cycle perturbation methods. The periodic solutions (b 1 and b 3), which influence the dispersal rate (') of the interacting species, have been shown to be controlled by the wave number. For stability, and in order to overcome predator natural mortality, the nutritional value of predator additional food has been shown to be of high quality that can enhance predator fertility gain. The threshold relationships between various ecosystem parameters and the carrying capacity of the game park for the prey species were also deduced to ensure ecosystem persistence. Besides revealing irregular periodic travelling wave behaviour due to predator interference, numerical results also show oscillatory temporal dynamics resulting from additional food supplements combined with high predation rates