University Sétif 1 FERHAT ABBAS Faculty of Sciences
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Auteur Aya MELAAB |
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Titre : On the predator–prey system with strong Allee effect in prey Type de document : texte imprimé Auteurs : Aya MELAAB, Auteur ; Nabil Beroual, Directeur de thèse Editeur : Sétif:UFA1 Année de publication : 2025 Importance : 1 vol (64 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Predator–prey
System strong Allee effect
Bifurcation
Limit cycles
Coexistence
Heteroclinic loopIndex. décimale : 510-Mathématique Résumé : Abstract
In this thesis, we conduct a global bifurcation analysis of a general class of predator–prey models incorporating a strong Allee effect in the prey population. We highlight the existence of a heteroclinic loop connecting two saddle points and investigate the occurrence of Hopf bifurcation. Depending on parameter ranges, we also prove the existence, uniqueness, or nonexistence of a limit cycle. For a unique critical parameter value, a threshold curve separates regions of overexploitation (prey extinction) and coexistence (successful predator invasion) in the space of initial conditions. To illustrate our theoretical results, we provide several numerical examples. This work is primarily based on the article by Jinfeng Wang, Junping Shi, and Junjie Wei entitled *Predator–prey system with strong Allee effect in prey*.Note de contenu : Contents
Introduction 3
1 Basicconceptsofecologyanddynamicalsystems 5
1.1 Basicconceptsofecology . .............................. 5
1.1.1 Glossary . .................................. 5
1.1.2 Interactionbetweenpopulations . ...................... 6
1.1.3 DefinitionoftheAlleeEffect . ........................ 7
1.2 Basicconceptsofmathematics . ........................... 8
1.2.1 SomeDefinitions . .............................. 8
1.2.2 ConceptofLimitCycle . ........................... 16
1.3 BasicConceptsintheEcologicalModel . ...................... 20
1.3.1 DynamicModelofaSinglePopulation . .................. 20
1.3.2 Two-SpeciesModels . ............................. 23
2 Onthepredator–preysystemwithstrongAlleeeffectinprey 31
2.1 Phaseplaneanalysis . ................................ 32
2.1.1 EquilibriumPoints . ............................. 33
2.2 StabilityAnalysis . .................................. 34
2.3 Hopfbifurcation . ................................... 42
2.4 Theuniquenessoflimitcycle . ........................... 45
3 Examplesanddiscussion 52
3.1 CubicModelwithHollingTypeIIFunctionalResponse . ............. 52
3.1.1 APrototypicalModel . ............................ 52
3.1.2 Equilibruimpoints . ............................. 53
3.1.3 HopfBifurcation . .............................. 53
3.1.4 UniquenessoftheLimitCycle . ....................... 53
3.1.5 NonexistenceofPeriodicOrbits . ...................... 54
3.2 CubicModelwithLinearFunctionalResponse . ................. 56
3.3 Boukal–Sabelis–Berecmodel . ............................ 58
Conclusion 61Côte titre : MAM/0768 On the predator–prey system with strong Allee effect in prey [texte imprimé] / Aya MELAAB, Auteur ; Nabil Beroual, Directeur de thèse . - [S.l.] : Sétif:UFA1, 2025 . - 1 vol (64 f.) ; 29 cm.
Langues : Anglais (eng)
Catégories : Thèses & Mémoires:Mathématique Mots-clés : Predator–prey
System strong Allee effect
Bifurcation
Limit cycles
Coexistence
Heteroclinic loopIndex. décimale : 510-Mathématique Résumé : Abstract
In this thesis, we conduct a global bifurcation analysis of a general class of predator–prey models incorporating a strong Allee effect in the prey population. We highlight the existence of a heteroclinic loop connecting two saddle points and investigate the occurrence of Hopf bifurcation. Depending on parameter ranges, we also prove the existence, uniqueness, or nonexistence of a limit cycle. For a unique critical parameter value, a threshold curve separates regions of overexploitation (prey extinction) and coexistence (successful predator invasion) in the space of initial conditions. To illustrate our theoretical results, we provide several numerical examples. This work is primarily based on the article by Jinfeng Wang, Junping Shi, and Junjie Wei entitled *Predator–prey system with strong Allee effect in prey*.Note de contenu : Contents
Introduction 3
1 Basicconceptsofecologyanddynamicalsystems 5
1.1 Basicconceptsofecology . .............................. 5
1.1.1 Glossary . .................................. 5
1.1.2 Interactionbetweenpopulations . ...................... 6
1.1.3 DefinitionoftheAlleeEffect . ........................ 7
1.2 Basicconceptsofmathematics . ........................... 8
1.2.1 SomeDefinitions . .............................. 8
1.2.2 ConceptofLimitCycle . ........................... 16
1.3 BasicConceptsintheEcologicalModel . ...................... 20
1.3.1 DynamicModelofaSinglePopulation . .................. 20
1.3.2 Two-SpeciesModels . ............................. 23
2 Onthepredator–preysystemwithstrongAlleeeffectinprey 31
2.1 Phaseplaneanalysis . ................................ 32
2.1.1 EquilibriumPoints . ............................. 33
2.2 StabilityAnalysis . .................................. 34
2.3 Hopfbifurcation . ................................... 42
2.4 Theuniquenessoflimitcycle . ........................... 45
3 Examplesanddiscussion 52
3.1 CubicModelwithHollingTypeIIFunctionalResponse . ............. 52
3.1.1 APrototypicalModel . ............................ 52
3.1.2 Equilibruimpoints . ............................. 53
3.1.3 HopfBifurcation . .............................. 53
3.1.4 UniquenessoftheLimitCycle . ....................... 53
3.1.5 NonexistenceofPeriodicOrbits . ...................... 54
3.2 CubicModelwithLinearFunctionalResponse . ................. 56
3.3 Boukal–Sabelis–Berecmodel . ............................ 58
Conclusion 61Côte titre : MAM/0768 Exemplaires (1)
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