# A mathematical model for the control of Eldana saccharina Walker using the sterile insect technique

Thesis (PhD)--Stellenbosch University, 2013.

Thesis

ENGLISH ABSTRACT: Two mathematical models are formulated in this dissertation for the population growth of an Eldana saccharina Walker infestation of sugarcane under the influence of partially sterile released insects. The first model describes the population growth of and interaction between normal and sterile E. saccharina moths in a temporally variable, but spatially homogeneous environment. The model consists of a deterministic system of difference equations subject to strictly positive initial data. The primary objective of this model is to determine suitable parameters in terms of which the above population growth and interaction may be quantified and according to which E. saccharina infestation levels and the associated sugarcane damage may be measured. The second model describes this growth and interaction under the influence of partially sterile insects which are released in a temporally variable and spatially heterogeneous environment. The model consists of a discretized reaction-diffusion system with variable diffusion coefficients, subject to strictly positive initial data and zero-flux Neumann boundary conditions on a bounded spatial domain. The primary objectives in this case are to establish a model which may be used within an area-wide integrated pest management programme for E. saccharina in order to investigate the efficiency of different sterile moth release strategies in various scenarios without having to conduct formal field experiments, and to present guidelines by which release ratios, frequencies and distributions may be estimated that are expected to lead to suppression of the pest. In addition to the mathematical models formulated, two practical applications of the models are described. The first application is the development of a user-friendly simulation tool for simulating E. saccharina infestation under the influence of sterile insect releases over differently shaped spatial domains. This tool provides the reader with a deeper understanding as to what is involved in applying mathematical models, such as the two described in this dissertation, to real-life scenarios. In the second application, an optimal diversification of sugarcane habitats is considered as an option for minimising average E. saccharina infestation levels, and as a further consequence, improving the cost-efficiency of sterile insect releases. Although many special cases of the above model classes have been used to model the sterile insect technique in the past, few of these models describe the technique for Lepidopteran species with more than one life stage and where F1-sterility is relevant. In addition, none of these models consider the technique when fully sterile females and partially sterile males are being released. The models formulated in this dissertation are also the first to describe the technique applied specifically to E. saccharina, and to consider the economic viability of applying the technique to this species. Furthermore, very few examples exist of such models which go beyond a theoretical description and analysis towards practical, real-life applications as illustrated in this dissertation.

AFRIKAANSE OPSOMMING: Twee wiskundige modelle word in hierdie proefskrif vir die populasiegroei van ’n Eldana saccha- rina Walker infestasie van suikerriet onder die invloed van gedeeltelik steriele, vrygelate insekte daargestel. Die eerste model beskryf die populasiegroei van en -interaksie tussen normale en steriele E. saccharina motte in ’n dinamiese, maar ruimtelik-homogene omgewing. Die model is ’n stelsel deterministiese verskilvergelykings onderhewig aan streng positiewe aanvangswaardes. Die primˆere doelstelling met hierdie model is om geskikte parameters te bepaal in terme waarvan die bogenoemde groei en interaksie gekwantifiseer kan word, en waarvolgens E. saccharina infestasievlakke en die gepaardgaande suikerrietskade gemeet kan word. Die tweede model beskryf hierdie groei en interaksie onder die invloed van gedeeltelik steriele insekte wat in ’n dinamiese en ruimtelik-heterogene omgewing vrygelaat word. Die model is ’n gediskretiseerde stelsel reaksie-diffusievergelykings met veranderlike diffusieko¨effisi¨ente onderhewig aan streng positiewe aanvangswaardes en zero-vloei Neumann-randwaardes op ’n begrensde ruimtelike gebied. Die primˆere doelstellings in hierdie geval is om ’n model tot stand te bring wat in ’n area-wye, ge¨ıntegreerde pesbestrydingsprogram vir E. saccharina gebruik kan word om die doeltreffendheid van verskillende steriele motvrylatingstrategie¨e te bepaal sonder om daadwerklik veldeksperimente uit te voer, en om riglyne daar te stel waarvolgens vrylatingsverhoudings, -frekwensies en -verspreidings bepaal kan word wat na verwagting na ’n onderdrukking van die pes sal lei. Bykomend tot die wiskundige modelle in hierdie proefskrif, word twee praktiese toepassings van die modelle ook beskryf. In die eerste toepassing word ’n gebruikersvriendelike simulasie hulpmiddel ontwikkel om E. saccharina infestasie onder die invloed van steriele insekvrylatings in verskillende ruimtelike gebiede te simuleer. Hierdie toepassing fasiliteer ’n dieper begrip van wat ter sprake is in die toepassing van wiskundige modelle, soos die twee modelle in hierdie tesis, tot werklike scenario’s. In die tweede toepassing word ’n optimale diversifisering van suikerriet habitats as ’n opsie vir die vermindering van die gemiddelde E. saccharina infestasie vlakke beskou, en gevolglik word die verbetering van die koste-doeltreffendheid van steriele insekvrylatings afgeskat. Alhoewel verskeie spesiale gevalle van die bogenoemde twee klasse van modelle reeds in die verlede gebruik is om die doeltreffendheid van die steriele-insektegniek te modelleer, beskryf weinig van hierdie modelle die tegniek vir Lepidopteriese spesies met meer as een lewensfase en waar F1-steriliteit ter sprake is. Verder beskryf geen van hierdie modelle die tegniek waar algeheel steriele wyfies en gedeeltelik steriele mannetjies vrygelaat word nie. Die modelle in hierdie tesis is ook die eerste waar die tegniek spesifiek op E.saccharina toegepas word, en waar die ekonomiese lewensvatbaarheid van die tegniek vir hierdie spesie oorweeg word. Verder bestaan daar min voorbeelde van soortgelyke modelle wat verder gaan as ’n teoretiese beskrywing en wiskundige ontleding na praktiese, werklike toepassings, soos in hierdie proefskrif ge¨ıllustreer.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/85865

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