# An ecological invasion mimics a drunken walk

January 09, 2017A theory that uses the mathematics of a drunken walk describes ecological invasions better than waves, according to Tim Reluga, associate professor of mathematics and biology, Penn State.

The ability to predict the movement of an ecological invasion is important because it determines how resources should be spent to stop an invasion in its tracks. The spread of disease such as the black plague in Europe or the spread of an invasive species such as the gypsy moth from Asia are examples of ecological invasions.

Two camps of scientists work on this problem -- mathematicians and ecologists. Mathematicians focus on creating models to describe invasion waves, while ecologists go to the field to measure observations of invasions, building computer simulations to predict the phenomenon they observe. Ideally both camps should agree on the underlying theory to explain their model results. But an ongoing argument continues among these scientists due to one seemingly simple detail -- how randomness affects an ecological invasion. Reluga hopes his approach will settle the argument, reconciling mathematical models with ecological observations.

"I hope this paper makes things clear that different kinds of randomness have different effects on invasions," Reluga said.

Previously, ecologists made inconsistent theories about how randomness influences an invasion. Some said it sped up while others said it slowed down an invasion. This is in contrast to mathematicians who said randomness had no affect on invasions, but randomness affects an ecological invasion in a number of different ways.

Reluga's work categorizes this randomness into three factors -- spatial, demographic, and temporal. The invasion of a forest population, such as the spread of acorn trees in England and Scotland at the end of the last ice age, can show how all three random factors affect this ecological invasion. The presence of squirrels in the forest can increase spatial randomness as squirrels disperse acorns further away from trees. Demographic randomness describes the variation in the average number of acorns trees produce. Finally, temporal randomness refers to how regularly the trees disperse seeds through time.

For his research Reluga constructed a mathematical model of an ecological invasion that behaves like a random walk, or movement that resembles the way someone who has had too much to drink tries to walk. He then showed the model replicates four key properties observed in computer simulations -- increasing spatial and temporal randomness sped up an invasion, and increasing demographic randomness and population density slowed down an invasion. By mathematically proving his model results replicated these properties, he concluded his take on spatial, demographic, and temporal random factors resembles the real world. Reluga's results, published in

*Theoretical Population Biology*, agree with what ecologists observe in the field and mathematicians predict with models, covering a wide class of invasion phenomenon.

"This is the way we should be thinking about the problem of randomness in ecological invasions," Reluga said. "If we think about it in this different frame, all the results make natural sense."

Reluga's different frame describes the movement of individuals in a population, stepping away from the conventional wave model used by mathematicians. This approach gives his model the fourth key property -- invasions slow down when the density of the population increases. This means that survival is affected by how many and how close neighbors are. His results replicate real-world observations that many nearby neighbors decrease the chance of survival as resources become scarcer.

-end-

The National Science Foundation provided funding for this research.Penn State

**Related Mathematics Articles:**

More democracy through mathematics

For democratic elections to be fair, voting districts must have similar sizes.

For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Mathematics supports a new way to classify viruses based on structure

New research supports a structure-based classification system for viruses which could help in the identification and treatment of emerging viruses.

New research supports a structure-based classification system for viruses which could help in the identification and treatment of emerging viruses.

US educators awarded for exemplary teaching in mathematics

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

The mathematics of coffee extraction: Searching for the ideal brew

Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world.

Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world.

Even physicists are 'afraid' of mathematics

Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests.

Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests.

Mathematics and music: New perspectives on the connections between these ancient arts

World-leading experts on music and mathematics present insights on the connections between these two ancient arts, especially as they relate to composition and performance, as well as creativity, education, and geometry.

World-leading experts on music and mathematics present insights on the connections between these two ancient arts, especially as they relate to composition and performance, as well as creativity, education, and geometry.

Kindergarteners' mathematics success hinges on preschool skills

In a study funded by the National Science Foundation, researchers at the University of Missouri discovered that preschoolers who better process words associated with numbers and understand the quantities associated with these words are more likely to have success with math when they enter kindergarten.

In a study funded by the National Science Foundation, researchers at the University of Missouri discovered that preschoolers who better process words associated with numbers and understand the quantities associated with these words are more likely to have success with math when they enter kindergarten.

First international mathematics research institute launched in Australia

World leaders in the mathematical sciences are visiting Melbourne for a series of research programs at Australia's first international research institute for mathematics and statistics.

World leaders in the mathematical sciences are visiting Melbourne for a series of research programs at Australia's first international research institute for mathematics and statistics.

**Related Mathematics Reading:**

## Best Science Podcasts 2019

We have hand picked the**best science podcasts**for 2019. Sit back and enjoy new science podcasts updated daily from your favorite science news services and scientists.

**Now Playing: TED Radio Hour**

**Bias And Perception**

How does bias distort our thinking, our listening, our beliefs... and even our search results? How can we fight it? This hour, TED speakers explore ideas about the unconscious biases that shape us. Guests include writer and broadcaster Yassmin Abdel-Magied, climatologist J. Marshall Shepherd, journalist Andreas EkstrÃ¶m, and experimental psychologist Tony Salvador.

**Now Playing: Science for the People**

**#514 Arctic Energy (Rebroadcast)**

This week we're looking at how alternative energy works in the arctic. We speak to Louie Azzolini and Linda Todd from the Arctic Energy Alliance, a non-profit helping communities reduce their energy usage and transition to more affordable and sustainable forms of energy. And the lessons they're learning along the way can help those of us further south.