The mathematics of propagation phenomena

Anything that propagates – infectious diseases, cancerous cells, bushfires – must behave in a certain way which can be encapsulated in a mathematical model.

Dr Ting-Ying Chang, Postdoctoral Research Fellow, School of Science and Technology

In 1905, five muskrats introduced into the modern-day Czech Republic became the unwitting epicentre of a muskrat invasion in central Europe. Using information gleaned from the subsequent widespread eradication efforts, John Skellam in 1951, a statistician and ecologist, took the square root of the area of the muskrat range – from the spreading radius – and plotting it against the years. He observed that the resulting data followed a straight line, a phenomenon that was predicted by the Fisher-KPP model back in 1937!

“In fact, it’s not just the spread of animal species that can be predicted,” enthused Dr Ting-Ying Chang, a postdoctoral research fellow at the University of New England. Dr Chang is currently working with Professor Yihong Du, looking intocthe mathematics of propagation phenomena. “Anything that propagates – infectious diseases, cancerous cells, bushfires – must behave in a certain way which can be encapsulated in a mathematical model. We can then use these models to predict how things can spread and how we might go about solving these problems by encouraging or containing the spread. This of course has many practical applications that can benefit society.”

For a new species in a new environment, there are only two possibilitiescestablish or vanish. How we think about which of these might occur has remained a central concern for mathematical models that sought to encapsulate propagation phenomena: under what threshold criteria will a new species successfully spread or vanish in a new environment?

One of the limitations of the aforementioned Fisher-KPP model is that it predicts successful establishment of a new species regardless of the initial population size, even though numerous experiments have pointed to initial population size as a determinant for a successful establishment. In order to more precisely capture what is happening in real life, many mathematical models have since been proposed with successive generalisations which better reflect the irregularities and environmental constraints of the physical world.

Dr Chang’s current work with Professor Du involves investigating the propagation phenomenon using the framework of models with nonlocal diffusion, an extension from local diffusion in the classical model. Despite being more complicated and requiring the development of new techniques, the nonlocal diffusion model has been observed to be more precise in its predictions and insights in many physical phenomena related to the dispersal of biological or chemical species.

“The goal is to establish a more general mathematical model which can predict propagation phenomena more accurately. These models can answer not only the question of when can spreading happen, but also where and how. It can provide information on the precise location of spreading fronts, the speed of the spread, how the environment can impact the spread, and many more. Real-life experiments take considerable time and money to conduct, and mathematical models can help guide scientists in planning their experiments and to have better expectations and predictions of the outcome of their experiments. So it’s not just something we tell first-year students,” she chuckles, “maths really can be very useful in the real world.”