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A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone
Thesis (PhD)--Stellenbosch University, 2021.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2021
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| _version_ | 1867614089335799808 |
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| access_status_str | Open Access |
| author | Kit Daniel, Searle |
| author2 | Van Vuuren, Jan Harm |
| author_browse | Kit Daniel, Searle Van Vuuren, Jan Harm |
| author_facet | Van Vuuren, Jan Harm Kit Daniel, Searle |
| author_sort | Kit Daniel, Searle |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2021. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/123689 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:46:29.473Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/123689 A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone Kit Daniel, Searle Van Vuuren, Jan Harm Stellenbosch University. Faculty of Engineering. Dept. of Industrial Engineering. Population dynamics Reaction-diffusion equations Harvesting machinery Asymptotic analysis UCTD Sustainable agriculture Thesis (PhD)--Stellenbosch University, 2021. ENGLISH ABSTRACT: As a result of the rapid increase in the global population there is a continued increase in the demand for natural resources in the form of food. Apart from the contributions of modern agriculture, a number of foods derive from harvesting living biological food species residing in their natural habitats. This increase in demand for food, and the subsequent increase in the harvesting of biological species, may cause the populations of these species to become extinct, if the resources are over-exploited. The following natural questions therefore arise: What kind of harvesting strategy should be adopted so that biological food species can be harvested sustain- ably? Moreover, for a given harvesting strategy, at what rate should the species be harvested so as to maximise the total harvest per unit time? Answers to these questions are complex because many considerations are involved in defining the notion of a harvesting strategy. This notion may be abstracted by acknowledging that harvesting a biological species is a spatio- temporal activity. A harvesting strategy may, for example, entail harvesting the species every- where in its habitat at a maximally sustainable constant harvest rate. A harvesting strategy may alternatively involve harvesting the species at varying rates in different regions of its habi- tat, perhaps implementing protection zones (demarcated areas in which the species may not be harvested at all). Another alternative is that the species may remain unharvested in the interior of its habitat, while only being harvested along the boundary of the habitat. In this dissertation, the growth and movement of a hypothetical biological species population is modelled as an initial-boundary value problem involving a reaction-diffusion equation in two cases. In the first case, where the population’s reproduction rate does not depend on time, the objective of the study is to determine analytically a maximally sustainable species density pro-rata harvesting rate along the boundary of species’ habitat. This is achieved by studying the long-term asymptotic behaviour of solutions of the aforementioned model and establishing a maximal density pro-rata harvest rate for which a globally stable equilibrium attractor of solutions to the model exists, attracting all other solutions. Important necessary properties of this equilibrium attractor are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest per unit time at equilibrium. In the second case, where the population’s reproduction rate depends seasionally on time, the objective of the study is again to determine analytically a maximally sustainable species density pro-rata harvesting rate along the boundary of species’ habitat. This is again achieved by studying the long-term asymptotic behaviour of solutions of the aforementioned model and establishing a maximal density pro-rata harvest rate for which a globally stable periodic attractor to solutions for the model exists. Important necessary properties of this periodic attractor are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest over a single interval of seasionality. AFRIKAANSE OPSOMMING: As gevolg van die snelle toename in die wˆereldbevolking, is daar voortdurend ’n toename in die vraag na natuurlike hulpbronne in die vorm van voedsel. Afgesien van bydraes uit die moderne landbou, is ’n aantal voedselsoorte afhanklik van die oesting van lewende biologiese spesies in hul natuurlike habitatte. Hierdie toename in die vraag na voedsel, en die gevolglike toename in die oesting van biologiese spesies, kan veroorsaak dat die populasies van sulke spesies uitsterf as die hulpbronne oorbenut word. Die volgende natuurlike vrae ontstaan dus: Watter soort oesstrategie behoort gevolg te word sodat biologiese voedselspesies volhoubaar ge-oes kan word? Boonop, teen watter tempo behoort sulke spesies ge-oes te word om die totale oes per tydseenheid te maksimeer? Antwoorde op hierdie vrae is ingewikkeld omdat baie oorwegings betrokke is by die definisie van die konsep van ’n oesstrategie. Hierdie konsep kan geabstraeer word deur op te let dat die oesting van biologiese spesies ’n tyd- ruimtelike aktiwiteit is. ’n Oesstrategie kan byvoorbeeld daaruit bestaan dat die spesies regoor sy habitat teen ’n maksimaal-volhoubare konstante tempo ge-oes word. Alternatiewelik kan die oesting van die spesie teen verskillende tempo’s in verskillende areas van sy habitat plaasvind, moontlik met die implementering van beskerm-sones (afgebakende gebiede waarin die spesie glad nie ge-oes mag word nie). ’n Ander alternatief is dat die spesie intern tot sy habitat onge-oes bly, terwyl dit net langs die rand van die habitat ge-oes word. In hierdie proefskrif word die groei en beweging van ’n hipotetiese biologiese spesiepopulasie as ’n begin- en randwaarde-probleem wat ’n reaksie-diffusievergelyking insluit, in twee gevalle gemodelleer. In die eerste geval, waar die voortplantingskoers nie van tyd afhang nie, is die doel van die studie om ’n maksimaal-volhoubare spesiedigtheidspro-rata oestempo langs die rand van die spesie se habitat analities te bepaal. Hierdie doel word bereik deur die langtermyn asimptotiese gedrag van oplossings tot die bovermelde model te bestudeer en ’n maksimale spesiedigtheidspro-rata oestempo te bepaal waarvoor ’n globaal-stabiele ewewigsattraktor vir die model bestaan, wat alle oplossings daarvan aantrek. Belangrike nodige eienskappe van hierdie ewewigsattraktor word bepaal waarvoor die bestaan van ’n spesiedigtheidspro-rata oestempo gewaarborg word wat die totale oes per tydseenheid in ewewig maksimeer. In die tweede geval, waar die voortplantingskoers seisonaal van tyd afhang, is die doel van die studie ook om ’n maksimaal-volhoubare spesiedigtheidspro-rata oestempo langs die rand van die spesie se habi- tat analities te bepaal. Hierdie doel word weereens bereik deur die langtermyn asimptotiese gedrag van oplossings tot die bovermelde model te bestudeer en ’n maksimale spesiedigtheidspro- rata oestempo te bepaal waarvoor ’n globaal-stabiele, periodiese attraktor van modeloplossings bestaan. Belangrike nodige eienskappe van hierdie periodiese attraktor word bepaal waarvoor die bestaan van ’n spesiedigtheidspro-rata oestempo gewaarborg word wat die totale oes per seisoen in ewewig maksimeer. Doctoral 2021-10-12T08:22:36Z 2021-12-22T14:16:02Z 2021-10-12T08:22:36Z 2021-12-22T14:16:02Z 2021-12 Thesis http://hdl.handle.net/10019.1/123689 en_ZA Stellenbosch University 150 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Population dynamics Reaction-diffusion equations Harvesting machinery Asymptotic analysis UCTD Sustainable agriculture Kit Daniel, Searle A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| title | A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| title_full | A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| title_fullStr | A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| title_full_unstemmed | A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| title_short | A mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| title_sort | mathematical model for sustainable harvesting of a biological species on the boundary of a protected zone |
| topic | Population dynamics Reaction-diffusion equations Harvesting machinery Asymptotic analysis UCTD Sustainable agriculture |
| url | http://hdl.handle.net/10019.1/123689 |
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