Populations and Evolution (q-bio.PE)

Abstract: We consider a moving boundary mathematical model of biological invasion. The model describes the spatiotemporal evolution of two populations: each population undergoes linear diffusion and logistic growth, and the boundary between the two populations evolves according to a two--phase Stefan condition. This mathematical model describes situations where one population invades into regions occupied by the other population, such as the spreading of a malignant tumour into surrounding tissues. Full time--dependent numerical solutions are obtained using a level--set numerical method. We use these numerical solutions to explore several properties of the model including: (i) survival and extinction of one population initially surrounded by the other; and (ii) linear stability of the moving front boundary in the context of a travelling wave solution subjected to transverse perturbations. Overall, we show that many features of the well--studied one--phase single population analogue of this model can be very different in the more realistic two--phase setting. These results are important because realistic examples of biological invasion involve interactions between multiple populations and so great care should be taken when extrapolating predictions from a one--phase single population model to cases for which multiple populations are present. Open source Julia--based software is available on GitHub to replicate all results in this study.

Abstract: Ecological networks describe the interactions between different species, informing us of how they rely on one another for food, pollination and survival. If a species in an ecosystem is under threat of extinction, it can affect other species in the system and possibly result in their secondary extinction as well. Understanding how (primary) extinctions cause secondary extinctions on ecological networks has been considered previously using computational methods. However, these methods do not provide an explanation for the properties which make ecological networks robust, and can be computationally expensive. We develop a new analytic model for predicting secondary extinctions which requires no non-deterministic computational simulation. Our model can predict secondary extinctions when primary extinctions occur at random or due to some targeting based on the number of links per species or risk of extinction, and can be applied to an ecological network of any number of layers. Using our model, we consider how false positives and negatives in network data affect predictions for network robustness. We have also extended the model to predict scenarios in which secondary extinctions occur once species lose a certain percentage of interaction strength, and to model the loss of interactions as opposed to just species extinction. From our model, it is possible to derive new analytic results such as how ecological networks are most robust when secondary species degree variance is minimised. Additionally, we show that both specialisation and generalisation in distribution of interaction strength can be advantageous for network robustness, depending upon the extinction scenario being considered.