On the Spatial Modelling of Biological Invasions
Abstract
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.