This thesis is concerned with the effects of fractional derivatives in predator-prey like systems, including models of plant water interaction. In Chapter 3, a fractional order predator-prey model is introduced, and we show how fractional derivative order can change the system from monostable to bistable. The observable domains of attraction of the two stable points will also be considered, in particular how they change as the fractional order is changed. In Chapter 4, we will generalise the predator-prey model studied in Chapter 3 by considering different fractional orders for each species. This system is referred to as an incommensurate system. We will explain how the different fractional orders affect the stability of this model. Then, in order to see if this change in stability is a more general result, we will consider a plant-herbivore incommensurate system and study the stability of this system. We will also find an approximate analytical solution for the characteristic equation of the incommensurate system when the two fractional orders α and β are similar and both close to the critical value of the fractional order of the commensurate system.In this case, we are able to map out the stable and unstable boundary as a functionof both parameters. We will compare the analytical and numerical solutions in these two incommensurate systems. In Chapter 5, we consider two different modelsof the interaction between surface water, soil water and plants. The first is similar to the model of Dagbovie and Sherratt, without spatial derivatives. We study the steady states of this model and observe the effect of adding the fractional order on the system. In the second model the soil water equation is replaced with the more realistic the Richards equation. In this model, we will also study the steady state and dynamic behaviour in the integer model and then consider the incommensurate fractional system. In this case, we see that a fractional order can affect the transient behaviour of the system.
|Date of Award||31 Jul 2020|
- University Of Strathclyde
|Supervisor||Alison Ramage (Supervisor) & Nigel Mottram (Supervisor)|