Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

The thesis describes various modifications of the predator-prey model. The modifications are considering several harvesting methods. At the beginning a solution and a sensitivity analysis of the basic model are provided. The first modification is the percentage harvesting model, which could be easily converted to the basic model. Secondly a constant harvesting including a linearization is derived. A significant part is devoted to regulation models with special a focus on environmental applications and the stability of the system. Optimization algorithms for one and both species harvesting are derived and back-tested. One species harvesting is based on econometrical tools; the core of two species harvesting is the modified Newton's method. The economic applications of the model in macroeconomics and oligopoly theory are expanded using the methods derived in the thesis.

Dans cette thèse nous nous sommes intéressés à la formation des bancs de poissons et à l'étude des interactions proies-prédateurs en présence de comportements collectifs par une approche de modélisation. Ce phénomène, bien qu'il soit le résultat d'interactions qui ont lieu à l'échelle individuelle, il engendre des conséquences à plus grande échelle, spatiale ou temporelle. L'objectif principal de cette thèse a été celui de comprendre l'influence du processus d'agrégation sur la réponse fonctionnelle. Nous avons élaboré un modèle centré sur l'individu (IBM) qui décrit les interactions intra-spécifiques ainsi que celle entre proies et prédateurs. Les agents peuvent ou pas avoir du comportement collectif.Dans un premier modèle les prédateurs sont attirés par les proies qu'ils voient. Ensuite, nous avons ajouté une composante à la vitesse du prédateur dépendante du bruit émis par les proies. Pour les deux cas, nous avons comparé les réponses fonctionnelles dans quatre configurations différentes. Nos résultats suggèrent que la prédation est plus efficace lorsque les proies s'agrègent. De plus, deux différents types de réponse fonctionnelle émergent : celle de Holling type II si les proies ne forment pas des bancs, celle de Holling type III autrement.Ensuite, nous nous sommes focalisés sur les conséquences à l'échelle de la population d'un phénomène (l'agrégation) se déroulant à l'échelle individuelle. Nous avons cherché un indice capable de détecter les bancs dans l'espace. Ensuite, en suivant une démarche mathématique, nous avons écrit une équation aux dérivées partielles représentant l'évolution spatio-temporelle de la densité des proies. / This thesis deals with the modelling of schooling phenomena and its consequences on predator-prey dynamics. Many marine species exhibit collective behaviour. While this phenomenon depends on individual interactions, it can have important effects at larger spatial and temporal scales.The main goal of this work is to understand the influence of aggregation on functional response, which represents predator feeding behaviour. We elaborated an individual-based model (IBM) describing schooling behaviour as well as predator-prey interactions. Predators can either be attracted towards visible prey, or hear, and so move towards, the noise produced by prey at a larger distance and then attack once they are close enough to see them. We analysed four different configurations, in which prey and/or predators can school or just move randomly. Our results shown an increased predation efficiency when prey school, and also different functional response shapes: Holling type II emerges if prey do not aggregate, while Holling type III emerges when prey school. Then, we focused on schooling consequences at higher scale. In particular, we analysed how to fit classical models to our emergent functional responses. Moreover, we found out a possible way to detect aggregates. Finally, we wrote a model representing the spatio-temporal evolution of prey density.

Ce travail est consacré à l’étude de la dynamique d’un système proie-prédateur de type Leslie-Gower défini par un système d’équations différentielles ordinaires (EDO) ou d’équations différentielles stochastiques (EDS), ou par des systèmes couplés d’EDO ou d’EDS. L’objectif principal est de faire l’analyse mathématique et la simulation numérique des modèles construits. Cette thèse est divisée en deux parties : La première partie est consacrée à un système proie-prédateur où les proies utilisent un refuge, le modèle est donné par un système d’équations différentielles ordinaires ou d’équations différentielles stochastiques. Le but de cette partie est d’étudier l’impact du refuge ainsi que la perturbation stochastique sur le comportement des solutions du système. Dans la deuxième partie, nous considérons un système proie-prédateur couplé en réseau. Il s’agit d’étudier comment des couplages plus ou moins forts entre plusieurs systèmes affectent l’existence et la position des points d’équilibre, et la stabilité de ces systèmes. / This work is devoted to the study of the dynamics of a predator-prey system of Leslie-Gower type defined by a system of ordinary differential equations (EDO) or stochastic differential equations (EDS), or by coupled systems of EDO or EDS. The main objective is to do mathematical analysis and numerical simulation of the models built. This thesis is divided into two parts : The first part is dedicated to a predator-prey system where the prey uses a refuge, the model is given by a system of ordinary differential equations or stochastic differential equations. The purpose of this part is to study the impact of the refuge as well as the stochastic perturbation on the behavior of the solutions of the system. In the second part, we consider a networked predator-prey system. We show that symmetric couplings speed up the convergence to a stationary distribution.

In introductory biology, college students are taught to predict how populations will grow and change over time by using population growth models. These models are commonly represented as mathematical equations. However, students consistently struggle when math and biology concepts intersect in the classroom, and these struggles lead to suboptimal understanding of how mathematical population models are designed and used. Education literature suggests that students may struggle with population modeling because of math anxiety, the high cognitive load of the task, and the lack of scaffolding for abstract concepts. In our study, we sought to improve student mastery modeling exponential growth, logistic growth, and Lotka-Volterra predator-prey interactions through using pictorial diagrams in modeling pedagogy. We predicted that these diagrams would reduce the amount of triggered math anxiety, lower the cognitive load of the task through reducing element interactivity, and allow for a more scaffolding for abstract symbols through a pictorial representation bridge. To test the effectiveness of population diagrams, we created two versions of a population modeling lesson plan: one version taught using diagrams then equations, while the other taught using purely equations. We also designed practice and assessment questions that tested calculation and model-building ability. We assessed math anxiety, scientific reasoning ability, and math ability at the beginning of the semester and state anxiety, effort of tasks, and difficulty of tasks during each lesson. Over 200 students from a non-major biology course were randomly assigned to each group, and all were given a pre-assessment, four lessons, a practice test, and a unit test on population modeling. Our findings show that while the addition of pictorial models to the traditional pedagogy did not have a significant effect on exponential and logistic growth model mastery, students that were exposed to predator-prey diagrams were more able to create a new model for a three-level predator-prey interaction than students that were only given traditional pedagogy. In addition, students who were exposed to predator-prey interaction diagrams before they derived equations reported a lower cognitive load than students who were only exposed to equations. Although diagrams were not a more helpful calculation tool for students than traditional equations, using population diagrams before to equation derivation may help improve student mastery of growth model creation.

Predators often have strong top-down effects on ecosystems and are considered a priority for conservation and management. Predator activity can influence prey distribution, abundance, and foraging behaviors and are likely to influence habitat by impacting ecological and environmental characteristics as well as presence of competitor species. There are knowledge gaps of the functional diversity of fish assemblages, non-consumptive predator effects, and environmental effects on fish assemblages. With this study, effects of top marine predators, such as sharks and great barracuda, on diversity and abundance of prey communities were examined in putative low (north side of South Bimini = lagoon) and high-risk (south side of South Bimini = flat) areas around South Bimini, The Bahamas. Baited remote underwater video surveys (BRUVs) deployed in the nearshore habitat captured abundance and potential predator-prey interactions. Predator and prey abundances at each site were compared to determine potential risk affect within high and low risk environments. A general baseline of predator and prey species was established throughout six months of observation (January- June 2018). Results showed a difference in prey communities between high and low risk habitats. Teleost abundance was highest on the south side of South Bimini. There were no differences in flight behavior of prey from predator (sharks vs barracuda). Longitude, depth, temperature, salinity, and dissolved oxygen were significantly linked to biotic assemblages. The identification of significant factors influencing predator-prey interaction is important in understanding community composition and for future implementation of conservation and management practices pertaining to nearby mangrove and seagrass habitats.

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