Dinamika grabežljivac-plijen populacijskih modela

master's thesis

• Title (english): Predator-prey dynamics of population models
• Author: Matea Vila
• Mentor: Maja Resman (mentor)
• Committee member: Maja Resman (predsjednik povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Matko Ljulj (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatramo modele ponašanja populacija u vremenu. Prvo promatramo ponašanje jedne populacije kroz Malthusov i logistički model rasta. Zatim promatramo interakcije dvije populacije pomoću grabežljivac-plijen modela. Radimo kvalitativnu analizu ponašanja trajektorija u vremenu bez eksplicitnog rješavanja jednadžbi. Kroz rad proučavamo ekvilibrije linearnih i nelinearnih sustava te pripadajuće fazne portrete. Analizu faznih portreta linearnih sustava radimo s obzirom na svojstvene vrijednosti matrice sustava. Koristimo Hartman-Grobmanov teorem koji povezuje izgled trajektorija nelinearnih sustava i njegove linearizacije u slučaju hiperboličkog ekvilibrija. Objasnili smo da za grabežljivac-plijen (Lotka-Volterra) model postoje dva ekvilibrija, jedan je sedlo, a drugi centar. Na drugi ekvilibrij \(\left(\frac{c}{d},\frac{a}{b}\right) \) grabežljivac-plijen modela ne možemo primijeniti Hartman-Grobmanov teorem, nego dokazujemo da su trajektorije rješenja oko ekvilibrija \(\left(\frac{c}{d},\frac{a}{b}\right) \) zatvorene. To objašnjava periodično ponašanje populacije grabežljivaca i populacije plijena u vremenu.

Abstract (english)

In this thesis, we consider population behavior models over time. First, we examine the behavior of a single population through the Malthusian and logistic growth models. Next, we consider the interactions of two populations using the predator-prey model. We conduct a qualitative analysis of the system over time without explicitly solving the equations. Throughout the work, we study the equilibria of linear and nonlinear systems, along with their corresponding phase portraits. The analysis of phase portraits of linear systems is based on the eigenvalues of the matrix of the system. We use the Hartman-Grobman theorem, which connects the trajectories of nonlinear systems and their linearizations in the case of hyperbolic equilibrium. We explain that for the predator-prey model, two equilibria exist: one is a saddle, and the other is a center. For the second equilibrium \(\left(\frac{c}{d},\frac{a}{b}\right) \) of the predator-prey model, the Hartman-Grobman theorem cannot be applied. We show that around second equlibrium solution trajectories are closed, which explains the periodic behavior of the predator and prey populations over time.

• Keywords: ponašanja populacija; Malthusov model rasta; logistički model rasta; Hartman-Grobmanov teorem
• Keywords (english): population behavior; Malthusian model; logistic growth model; Hartman-Grobman theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:144887
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-06 16:11:56