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Mite Versus Mite in Deadly Numbers Game

By Gio Braidotti

Tuesday, September 1, 2009

It is the defining image of the African savannah: a feline predator making the final leap after an electrifying chase to drag down its prey. But that basic predator/prey relationship – in which the survival of one species is dependent on the ‘supply’ of another – has a mathematical underside with applications far beyond the ecology of a savannah.

The stock exchange, biological pest control, energy policy and cancer – these are all instances where equations that capture predator/prey dynamics are finding applications. Solving these equations is the task of a little-known field of academia, mathematical biology.

The classic – and the simplest – solution is described by the Lotka–Volterra system of differential equations that were first proposed in the mid-1920s. Represented graphically, the solution appears as two wavy lines that predict oscillating numbers of predators and prey. As the prey population increases, predator numbers follow suit until the prey declines and drives down the ability of predators to survive.

While there is a satisfying supply/demand simplicity to these dynamics, mathematical biology since the 1970s has been on a quest to produce mathematical models that are more life-like and the dynamics more realistically complex.

For a young Indian mathematician, Dr Aspriha Chakraborty, the quest for human-world relevance resonated deeply. It saw her abandon the study of pure mathematics at a large Indian university in favour of moving to Swinburne University of Technology to work with Dr Manmohan Singh at the Faculty of Engineering and Industrial Sciences.

In just five years at Swinburne, she has managed the extraordinary feat of completing both a master’s degree and a doctorate. Her master’s thesis, Numerical Study of Biological Problems in a Predator and Prey System, has been published in book form by Germany’s VDM Verlag, while her doctorate produced four peer-reviewed articles. Her supervisor, Dr Singh, simply calls her “brilliant”.

One of Dr Chakraborty’s widely acclaimed achievements relates to the concept of ‘prey taxis’ – the natural tendency of predators to follow the greatest concentration of prey. This tendency affects predator and prey movement, their spatial distribution in an environment, and consequentially, the possible outcomes of predation.

To capture the impact of prey taxis, Dr Chakraborty succeeded in adding a third equation to the standard model (in addition to the two wavy predator/prey lines mentioned above).

She says prey taxis helps shift the models from simplistic assumptions (that ignore everything but the predator and prey) to more biologically realistic, complex and dynamic capabilities – which provides useful modelling tools beyond basic biology and ecology.

Crucial to her work was the opportunity provided by Dr Singh to apply ‘numerical methods’ (that require real data) to mathematical biology rather than restrict the discipline to theory. It is this dual approach that distinguishes the Swinburne mathematicians from more traditional approaches.

Dr Singh explains that the theoretical approach involves creating theorems and then examining stability – whether predator and prey populations stabilise. “In numerical methods what we do is use field data generated by experimental biologists and show graphically what a stable equilibrium involves in a particular environment, what is unstable, and why there are changes in the system,” he says.

The requirement for field data anchors the mathematicians to a real situation where the predator/prey equations can find pragmatic applications. In Dr Chakraborty’s case, the application involves biological control of agricultural pests.

For her doctorate at Swinburne, the field data was provided by Dr Peter Ridland, formerly of the Victorian Department of Primary Industries (DPI), and involves the two-spotted spider mite and its predator, a mite called Phytoseiulus persimilis.

Dr Ridland says this predator is routinely introduced into European glasshouses and fruit orchards to protect a broad range of fruits, vegetables and flower crops. Uses are increasingly being found for the predator in Australia, especially in Queensland in the cultivation of strawberries in glasshouses.

However, the two-spotted spider mite is a so-called ‘secondary pest’ – a normally harmless species that becomes a pest for farmers only subsequent to the intensive use of insecticides – because, over time, insecticides kill off the two-spotted spider mite’s natural predators while the pest acquires pesticide-resistance and flourishes.

“The ideal situation is to maintain stable predator and prey population at low numbers for long periods of time,” Dr Ridland says. “That makes ‘balance’ the overwhelming idea for biological pest management and the goal is to avoid a boom in pest numbers because then it is extremely difficult to control the pest and prevent damage to crops.”

Balance among natural enemies – or a ‘stable equilibrium’, as Dr Chakraborty would say – is precisely what her innovation to the predator/prey equations can help to model and predict.

“What we have proven theoretically is that if we can introduce prey taxis, we can keep the equilibrium between predator and prey going for far longer,” Dr Singh says. “However, to prove it numerically requires a new round of glasshouse experiments and we would definitely like to pursue the experimental work with the DPI if funds are available.”

For the ecologist, Dr Ridland, the beauty of mathematical biology lies in the gain in understanding of what affects predator/prey stability. “The interaction with natural enemies is an important part of the ecology of insects, so anything that can help us better understand how to achieve balance is valuable.”

As to Dr Chakraborty herself, she would like to see the power of the predator/prey equations applied in a new direction. “I want to apply it either to cancer modelling or evolutionary biology. In both fields we use the same equations – it is still about the movement of populations as they interact. So in cancer research you are looking at cancer cells (the predator) invading normal tissue (the prey).”

Both her chosen areas are even more biologically complex, providing the perfect stepping-stone in Dr Chakraborty’s quest to infuse the mathematical models with some of the vital dynamics typical of living systems.

Photo

Caption: Dr Aspriha Chakraborty moved to Swinburne to explore applying numerical methods to mathematical biology.

Photo: Paul Jones