Predator-prey dynamics: Advancing ecological sustainability through mathematical modelling
By Clara Wong
It’s been happening since 1970 – a drastic, half-century-long drop in global wildlife populations by 60%. In a 2018 Living Planet Report, the WWF called the figures an “urgent sign that nature needs life support.”
Few today could successfully deny that human activity is pushing earth’s ecological balance to near tipping points. Among the many causes are hunting and fishing. Is there a sweet-spot of coexistence inside which humans and animals can interact in this context without driving each other to extinction?
Under the supervision of Dr. Kunquan Lan at Ryerson’s Biomathematics and Fluids Group, PhD candidate Marvin Hoti and two fellow researchers proposed a mathematical answer. Their solution: an upgrade to one of biomathematics’ most popular tools for analyzing predator-prey dynamics, the Lotka Volterra model. Their findings were published in the journal, Mathematical Methods in the Applied Sciences.
Improving on an over-simplified model
The Lotka-Volterra is a classic in biomathematics. Italian and US mathematicians Volterra and Lotka developed the pair of ordinary differential equations in the 1920s. It was a basic model reflecting the oscillating cycles of growth and decline in predator-prey populations. Workable, but built on simple assumptions that failed to account for other dynamic factors. Two such: prey refuge and prey harvesting.
Prey refuge occurs when a prey species employs strategies – such as migration – to avoid its predator. Prey harvesting is the unnatural elimination of prey due to human interference – typically through hunting or fishing.
Both processes remove a predator’s food source, with potential threat to either species. Both have also been happening for centuries. In a modern-day context where humans and animals increasingly converge, ecological sustainability demands the question: what is an acceptable range that allows for humans to harvest prey, and for prey to migrate normally, but ensures coexistence for both species?
Hoti and his colleagues reviewed previous papers on the Lotka-Volterra model. “To our knowledge, no one has ever incorporated prey harvesting and prey migration terms into this particular model,” he says.
After a year of research, the team developed a more realistic version of the Lotka-Volterra model, with improved utility as a predictive tool for long-term behaviour in a given ecosystem.
“A very reasonable model”
By using qualitative theory for planar dynamical systems, they found that an ecosystem’s stability is heavily affected by prey refuge and prey harvesting. “Our paper shows how sensitive these two terms are in maintaining balance,” Hoti explains.
Based on the new model, predator and prey cannot coexist if the prey harvesting term exceeds 1/4 and the prey refuge term exceeds 1. Hoti interprets: “That’s a very small margin. Any higher and the system would not be stable and either species could die out.”
Hoti reflects: “Even without the math, it’s natural to assume that if you hunt and your harvesting rate is too high, the species will be eliminated. But through our work, we can now give specific ranges that we need to stay within to prevent extinction.”
Managing the impact of human activity
Since prey migration occurs naturally in many ecosystems, the onus falls on humans to moderate their impact on wildlife. Hoti’s work is one of many that push biomathematics forward and could ultimately play a part in helping policymakers regulate human interactions with ecology.
“Prey migration already reduces a predator’s food supply. If you add harvesting by humans, it could be very damaging to the system’s stability,” explains Hoti. “The theory is here. It’s developed. We need to be cautious about managing it.”
After completing the study using ordinary differential equations of the first order, Hoti is now experimenting with fractional differential equations and models that can capture additional phenomena. “We could always add new equations,” says Hoti. “One possibility is a term for harvesting predator rather than prey. Another might be the impact of toxins on aquatic populations.”
Looking into the future, Hoti foresees more work ahead. “You can take even the best model and always improve on it,” he concludes. “The biomathematics field is strong and will likely continue growing for years. So, the work is never done.”