The paper is focused on the Predator-Prey model modified in the case of harvesting one or both populations. Firstly there is given a short description of the basic model and the sensitivity analysis. The first essential modification is percentage harvesting. This model could be easily converted to the basic one using a substitution. The next modification is constant harvesting. Solving this system requires linearization, which was properly done and brought valuable results applicable even for the basic or the percentage harvesting model. The next chapter describes regulation models, which could be used especially in applying environmental policies. All reasonable regulation models are shown after distinguishing between discrete and continuous harvesting. The last chapter contains an algorithm for maximizing the profit of a harvester using econometrical modelling tools.

In this thesis, Optimum Decoding-based Smoothing Algorithm (ODSA) is applied to well-known Discrete Lotka-Volterra Model. The performance of the algorithm is investigated for various parameters by simulations. Moreover, ODSA is compared with the SIR Particle Filter Algorithm. The advantages and disadvantages of the both algorithms are presented.

Dans ce travail de thèse, nous étudions des modèles mathématiques de la dynamique des populations en environnements déterministe et stochastique. Pour les environnements déterministes, nous considérons trois modèles. Le premier est un modèle intra-guilde prenant en compte des effets d'un environnement spatial hétérogène avec une migration rapide des individus entre les différents sites. Le deuxième est un modèle de pêche dans une zone constituée d’une aire marine protégée où la pêche est interdite et d’une zone où la population de poissons est pêchée. Enfin le troisième est un modèle prédateur-proie considérant une proie et deux prédateurs avec des réponses fonctionnelles de Beddington-DeAngelis. Pour les environnements stochastiques, nous étudions un modèle épidémique SIRS et un modèle prédateur-proie en prenant en compte un bruit télégraphique. Nous étudions le comportement dynamique de ces modèles et nous recherchons les conditions de maintien ou de disparition des espèces modélisées. / In this thesis, we consider mathematical population dynamics models in deterministic and stochastic environments. For deterministic environments, we study three models: an intraguild model with the effects of spatial heterogeneous environment and fast migration of individuals; a fishery model with Marine Protected Area where fishing is prohibited and an area where the fish population is harvested; a predator-prey model which has one prey and two predators with Beddington-DeAngelis functional responses. For stochastic environments, we study SIRS epidemic model and predator-prey models under telegraph noise. We try to present the dynamical behavior of these models and show out the existence or vanishing of species in the models.

Non-native species and climate change pose serious threats to global biodiversity. However, the roles of climate, dispersal, and competition are difficult to disentangle in heterogeneous landscapes. We combine empirical data and theory to examine how these forces influence the spread of non-native species in Lake Baikal. We analyze the potential for Daphnia longispina to establish in Lake Baikal, potentially threatening an endemic, cryophillic copepod Epischurella baikalensis. We collected field samples to establish current community composition and compared them to model predictions informed by flow rates, present-day temperatures, and temperature projections. Our data and model agree that expansion is currently limited by dispersal. However, projected increases in temperature reverse this effect, allowing D. longispina to establish in Lake Baikal’s main basin. A strong negative impact emerges from the interaction between climate change and dispersal, outweighing their independent effects. Climate, dispersal, and competition have complex, interactive effects on expansion with important implications for global biodiversity.

We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).

<p>We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).</p>

This thesis explores the methodology of state, and in particular, parameter estimation for time series datasets. Various approaches are investigated that are suitable for nonlinear models and non-Gaussian observations using state space models. The methodologies are applied to a dataset consisting of the historical lynx and hare populations, typically modeled by the Lotka- Volterra equations. With this model and the observed dataset, particle filtering and parameter estimation methods are implemented as a way to better predict the state of the system. Methods for parameter estimation considered include: maximum likelihood estimation, state augmented particle filtering, multiple iterative filtering and particle Markov chain Monte Carlo (PMCMC) methods. The specific advantages and disadvantages for each technique are discussed. However, in most cases, PMCMC is the preferred parameter estimation solution. It has the advantage over other approaches in that it can well approximate any posterior distribution from which inference can be made. / Master's thesis

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.

Nesta dissertação, tratamos do estudo das bifurcações de um modelo bi-dimensional de presa-predador, que estende e aperfeiçoa o sistema de Lotka-Volterra. Tal modelo apresenta cinco parâmetros e uma função resposta não monotônica do tipo Holling IV: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ Estudamos as bifurcações do tipo sela-nó, Hopf, transcrítica, Bogdanov-Takens e Bogdanov-Takens degenerada. O método dos centros organizadores é usado para estudar o comportamento qualitativo do diagrama de bifurcação. / In this work are studied the bifurcations of a bi-dimensional predator-prey model, which extends and improves the Volterra-Lotka system. This model has five parameters and a non-monotonic response function of Holling IV type: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ They studied the sadle-node, Hopf, transcritic, Bogdanov-Takens and degenerate Bogdanov-Takens bifurcations. The method of organising centers is used to study the qualitative behavior of the bifurcation diagram.

I detta tvärvetenskapliga arbete studeras från den matematiska sidan tre klassiska populationsmodeller: Malthus tillväxtmodell, Verhulsts logistiska modell och Lotka-Volterras jägarebytesmodell. De klassiska modellerna jämförs med stokastiska. De stokastiska modeller som studeras är födelsedödsprocesser och deras diffusionsapproximation. Jämförelse görs med medelvärdesbildade simuleringar. Det krävs många simuleringar för att kunna genomföra jämförelserna. Dessa simuleringar måste utföras i datormiljö och det är här den datalogiska aspekten av arbetet kommer in. Modellerna och deras resultathantering har implementerats i både MatLab och i C, för att kunna möjliggöra en undersökning om skillnaderna i tidsåtgången mellan de båda språken, under genomförandet av ovan nämnda jämförelser. Försök till tidsoptimering utförs och även användarvänligheten under implementeringen av de matematiska problemen i de båda språken behandlas. Följande matematiska slutsatser har dragits, att de medelvärdesbildade lösningarna inte alltid sammanfaller med de klassiska modellerna när de simuleras på stora tidsintervall. I den logistiska modellen samt i Lotka-Volterras modell dör förr eller senare de stokastiska simuleringarna ut när tiden går mot oändligheten, medan deras deterministiska representation lever vidare. I den exponentiella modellen sammanfaller medelvärdet av de stokastiska simuleringarna med den deterministiska lösningen, dock blir spridningen stor för de stokastiska simuleringarna när de utförs på stora tidsintervall. Datalogiska slutsatser som har dragits är att när det kommer till att implementera få modeller, samt resultatbearbetning av dessa, som ska användas upprepade gånger, är C det bäst lämpade språket då det visat sig vara betydligt snabbare under exekvering än vad MatLab är. Dock måste hänsyn tas till alla de svårigheter som implementeringen i C drar med sig. Dessa svårigheter kan till stor del undvikas om implementeringen istället sker i MatLab, då det därmed finns tillgång till en uppsjö av väl lämpade funktioner och färdiga matematiska lösningar. / In this interdisciplinary study, three classic population models will be studied from a mathematical view: Malthus’ growth, Verhulst’s logistic model and Lotka-Volterra’s model for hunter and prey. The classic models are being compared to the stochastic ones. The stochastic models studied are the birthdeath processes and their diffusion approximation. Comparisons are made by averaging simulations. It requires numerous simulations to carry out the comparisons. The simulations must be carried out on a computer and this is where the computer science emerges to the project. The models, along with the handling of the results, have been implemented in both MatLab and in C in order to allow a comparison between the two languages whilst executing the above mentioned study. Attempts to time optimization and an evaluation concerning the user-friendliness regarding the implementation of mathematical problems will be performed. Mathematic conclusions, which have been drawn, are that the averaging solutions do not always coincide with the traditional models when they are being simulated over large time. In the logistic model and in Lotka-Volterra’s model the stochastic simulations will sooner or later die when the time is moving towards infinity, whilst their deterministic representation keeps on living. In the exponential model, the mean values of the stochastic simulations and of the deterministic solution coincide. There is, however, a large spread for the stochastic simulations when they are carried out over a large time. Computer scientific conclusions drawn from the study includes that when it comes to implementing a few models, along with the handling of the results, to be used repeatedly, C is the most appropriate language as it proved to be significantly faster during execution. However, all of the difficulties during the implementation of mathematical problems in C must be kept in mind. These difficulties can be avoided if the implementation instead takes place in MatLab, where a numerous of mathematical functions and solutions will be available.