ECOP Subgroup Contributed Talks

Abdennasser Chekroun

University of Tlemcen

"Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase"

We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

Laura Wadkin

Newcastle University

"Mathematical modelling of the spread of tree disease through forests"

The past decades have seen a dramatic rise in the number of emerging diseases of plants and trees across the world. These diseases threaten the survival of native trees and have huge social, economic, and environmental impacts. The Department for Environmental, Food and Rural Affairs have highlighted the importance of mathematical modelling in developing robust management policies to minimise the impacts of these threats. We are working to mathematically model the spread of tree diseases, using a combination of agent-based models, partial differential equations, and statistical inference techniques. The aim is to combine local lattice modelling approaches with global continuum models to perform systemic modelling and parameter inference of past and present tree epidemics in the mainland UK. The results can be used to deepen our understanding of the process of tree disease spread and crucially, explore intervention and management strategies to find the best methods of stopping the disease spread.

Laura Jimenez

University of Hawaii

"Statistical models to estimate the fundamental niche of a species using occurrence data"

The fundamental niche of a species is the set of environmental conditions that allow the species to survive in the absence of biotic interactions and dispersal limitations. Estimating the center (i.e., the optimal environmental conditions for the species) and extent of the fundamental niche is of great importance when the fitted models are used to predict the effects of climate change on the geographic distribution of the species. However, most of the existing approaches to estimate niches use occurrence samples that are biased, and often fit complex models that are not a biologically realistic representation of the fundamental niche' border. Occurrence samples come from the realized niche (a subset of the fundamental niche that includes biotic interactions and dispersal limitations) and may not represent the full environmental potentiality of a species; samples may be biased towards well-represented regions of niche space. I will present two new models to estimate the fundamental niche of a species that use occurrence data and assume a simple, biologically realistic shape for the fundamental niche. I will show how to incorporate known tolerance ranges for the species into the models and how to account for environmental biases in the samples.

Lee Altenberg

University of Hawai`i at Mānoa

"Going Against the Flow: Selection for Counter-Current Dispersal in Gyres"

Much attention has been given to the 'drift paradox' for river organisms: how populations in streams can maintain themselves despite being constantly swept downstream. Here we shall consider a different situation: where circular currents produce time-irreversible dispersal dynamics. We will see that when there is environmentally produced cyclical dispersal among habitats with spatial variability in quality, organisms that disperse against the cyclical flows will have an aggregate population growth advantage. These results are obtained by applying some classical results from spectral theory, including theorems by Karlin and Levinger. Temporal variation in habitat quality or dispersal is not addressed. Open problems for further work include the degree to which these result extend to dispersal that is only partially or approximately counter-current. The widespread occurrence of positive rheotaxis among ocean organisms may conceivably be a manifestation of these selection dynamics.