{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/28921"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/28921","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"Two-patch herbivore/vegetation models with density-dependent migration","abstract":"In this thesis we constructed two mathematical models for herbivore/vegetation interactions in environment of two patches, using the metaphysiological approach and a density-dependent migrations. In the first model we considered the case when the environment is constant, and we constructed a system of four perturbed ordinary differential equations describing the dynamics when only herbivores allowed to move between the two patches searching for food. The model contain two different timescales, fast for migrations and slow for the other demographic changes in the system. We used the geometric singular perturbation theory in order to reduce the dimension of the system. Using the continuation software AUTO we provided bifurcation diagrams for the reduced systems and we also provided some numerical illustrations to show the dynamics of the system for different migrations propensities. We analyzed the bifurcation diagrams using Morse decompositions and Conley index theory, to confirm their correctness. We constructed a second mathematical model, by considering that the vegetation growth depends on seasonal rainfall and the soil moisture. We provided some numerical simulations to illustrate several variates of dynamics for different migration speed and, when the migration propensities and the vegetation quality are change.","abstract_html":"In this thesis we constructed two mathematical models for herbivore/vegetation interactions in environment of two patches, using the metaphysiological approach and a density-dependent migrations. In the first model we considered the case when the environment is constant, and we constructed a system of four perturbed ordinary differential equations describing the dynamics when only herbivores allowed to move between the two patches searching for food. The model contain two different timescales, fast for migrations and slow for the other demographic changes in the system. We used the geometric singular perturbation theory in order to reduce the dimension of the system. Using the continuation software AUTO we provided bifurcation diagrams for the reduced systems and we also provided some numerical illustrations to show the dynamics of the system for different migrations propensities. We analyzed the bifurcation diagrams using Morse decompositions and Conley index theory, to confirm their correctness. We constructed a second mathematical model, by considering that the vegetation growth depends on seasonal rainfall and the soil moisture. We provided some numerical simulations to illustrate several variates of dynamics for different migration speed and, when the migration propensities and the vegetation quality are change.","abstract_has_math":false,"creators":["Bakheet, Mohamed Abdalaziz Abdalla"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Laurie, Henri"],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013","date_published":"2013","updated_at":"2026-07-22T22:22:47Z","subjects":[],"languages":["eng"],"rights":["Creative Commons Attribution 4.0 International (CC BY 4.0)"],"rights_urls":["http://creativecommons.org/licenses/by/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/28921","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Laurie, Henri"]},{"key":"dc:creator","label":"Author","values":["Bakheet, Mohamed Abdalaziz Abdalla"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-10-10T08:52:16Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-10-10T08:52:16Z"]},{"key":"dc:date.issued","label":"Date","values":["2013"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cape Town"]},{"key":"dc:type","label":"Dc Type","values":["Master Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Masters"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Creative Commons Attribution 4.0 International (CC BY 4.0)"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://creativecommons.org/licenses/by/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/28921"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis we constructed two mathematical models for herbivore/vegetation interactions in environment of two patches, using the metaphysiological approach and a density-dependent migrations. In the first model we considered the case when the environment is constant, and we constructed a system of four perturbed ordinary differential equations describing the dynamics when only herbivores allowed to move between the two patches searching for food. The model contain two different timescales, fast for migrations and slow for the other demographic changes in the system. We used the geometric singular perturbation theory in order to reduce the dimension of the system. Using the continuation software AUTO we provided bifurcation diagrams for the reduced systems and we also provided some numerical illustrations to show the dynamics of the system for different migrations propensities. We analyzed the bifurcation diagrams using Morse decompositions and Conley index theory, to confirm their correctness. We constructed a second mathematical model, by considering that the vegetation growth depends on seasonal rainfall and the soil moisture. We provided some numerical simulations to illustrate several variates of dynamics for different migration speed and, when the migration propensities and the vegetation quality are change."]},{"key":"dc:title","label":"Title","values":["Two-patch herbivore/vegetation models with density-dependent migration"]}]}],"canonical_facts":{"dc:contributor.advisor":["Laurie, Henri"],"dc:creator":["Bakheet, Mohamed Abdalaziz Abdalla"],"dc:date.accessioned":["2018-10-10T08:52:16Z"],"dc:date.available":["2018-10-10T08:52:16Z"],"dc:date.issued":["2013"],"dc:description.abstract":["In this thesis we constructed two mathematical models for herbivore/vegetation interactions in environment of two patches, using the metaphysiological approach and a density-dependent migrations. In the first model we considered the case when the environment is constant, and we constructed a system of four perturbed ordinary differential equations describing the dynamics when only herbivores allowed to move between the two patches searching for food. The model contain two different timescales, fast for migrations and slow for the other demographic changes in the system. We used the geometric singular perturbation theory in order to reduce the dimension of the system. Using the continuation software AUTO we provided bifurcation diagrams for the reduced systems and we also provided some numerical illustrations to show the dynamics of the system for different migrations propensities. We analyzed the bifurcation diagrams using Morse decompositions and Conley index theory, to confirm their correctness. We constructed a second mathematical model, by considering that the vegetation growth depends on seasonal rainfall and the soil moisture. We provided some numerical simulations to illustrate several variates of dynamics for different migration speed and, when the migration propensities and the vegetation quality are change."],"dc:identifier.uri":["http://hdl.handle.net/11427/28921"],"dc:language":["eng"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:publisher.institution":["University of Cape Town"],"dc:rights":["Creative Commons Attribution 4.0 International (CC BY 4.0)"],"dc:rights.uri":["http://creativecommons.org/licenses/by/4.0/"],"dc:title":["Two-patch herbivore/vegetation models with density-dependent migration"],"dc:type":["Master Thesis"],"dc:type.qualificationlevel":["Masters"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-22T22:22:47Z"}