Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article


Doctor of Philosophy


Applied Mathematics


Zou, Xingfu


We investigate problems of biological spatial invasion through the use of spatial modelling. We begin by examining the spread of an invasive weed plant species through a forest by developing a system of partial differential equations (PDEs) involving an invasive weed and a competing native plant species. We find that extinction of the native plant species may be achieved by increasing the carrying capacity of the forest as well as the competition coefficient between the species. We also find that the boundary conditions exert long-term control on the biomass of the invasive weed and hence should be considered when implementing control measures. We then consider biological invasion on a smaller scale – the spread of melanoma, an invasive cancer. We investigate oncolytic virotherapy using adenoviruses as a treatment modality by using a system of ordinary differential equations (ODEs). Our model incorporates the oxygen concentration of the tumour microenvironment, as it is well known that hypoxic conditions reduce the efficacy of adenoviruses. As in the case of invasive weed spreading, our modelling highlights the importance of a favourable environment. In particular, our investigation into the infection rate of the virus and the oncolysis rate supports the notion of bounding the oncolysis rate for optimal clinical outcomes. Furthermore, our modelling suggests that the virus’ oncolytic potency should be increased under hypoxic conditions, but should not be too large, so as to avoid inhibiting the replication of the virus. We find that these results are consistent after extending the model to a regional model which accounts for spreading of the melanoma via the lymphatic system. We then continue our investigation of oncolytic virotherapy by analyzing a PDE model of melanoma spreading through the skin. We find results which are consistent with our ODE model: Placing infection rate-dependent bounds on the oncolysis rate leads to more favourable clinical outcomes, We provide some quantitative estimates on how to determine these bounds. Our theoretical modelling provides further evidence to suggest that auxiliary topical (regenerative) treatment of the skin can be a useful complement to virotherapy.

Summary for Lay Audience

Using mathematical and computational tools can offer new insights on existing problems in biology. We consider some problems of biological invasion from a mathematical perspective: (i) The spread of an invasive weed plant species through a forest, taking up the resources of the pre-existing native plants and (ii) the spread of melanoma, an invasive skin cancer with high a mortality rate when diagnosed at an advanced stage. Even though these problems may seem very disconnected, very similar mathematical tools can be used to analyze them and to determine the similarities which exist between them. When investigating invasive plants with this approach, we find some potentially useful control measures to reduce the harm caused by invasive weeds. In particular, we find that taking control measures at the boundaries of where the weeds grow rather than throughout the entire forest can help control the spreading. When investigating the spread of melanoma, we consider the possible outcomes of treating the cancer with an oncolytic virus. This is a genetically engineered virus that attacks cancer cells, uses them to replicate, then destroys them while leaving healthy tissue unharmed. We build models which let us make suggestions on how to engineer these viruses by determining what features the viruses need. In particular, we find that a useful oncolytic virus should be effective at infecting the cancer cells, but not too potent at destroying these cancer cells. Why is that? Because if the virus kills the cancer cells faster than it may infect them, then it won’t have any hosts through which to replicate. We find that there is a very delicate balance between how infectious the virus is and how efficient it is at killing the cancer cells. Our models give guidance on how to build these viruses under various conditions, such as the available oxygen at the tumour site. By using math to investigate these biological problems and using computers to run simulations, we can make predictions on how to mitigate the negative impacts of biological invasions without having to wait for months or years. Of course this approach cannot replace ecological field work or clinical trials, but it can help guide them.