The Lotka-Volterra predator-prey game

The Lotka-Volterra predator-prey model is one of the earliest and, perhaps, the best known example used to explain why predators can indefinitely coexist with their prey. The population cycles resulting from this model are well known. In this article I show how adaptive behavior of prey and predators can destroy these cycles and stabilize population dynamics at an equilibrium. The classical predator-prey model assumes that interaction strength between prey and predators is fixed, which means that coefficients describing interactions between prey and predators do not change in time. However, there is increasing evidence that individuals adjust their activity levels in response to predation risk and availability of resources. For example, a high predation risk due to large predator numbers leads to prey behaviors that make them less vulnerable. They can either move to a refuge or become vigilant. However, such avoidance behaviors usually also decrease animal opportunities to forage which leads to foraging-predation risk trade-off. The present article shows that such a trade-off can have a strong bearing on population dynamics. In fact, while the classical Lotka-Volterra model has isoclines that are straight lines, the foraging-predation risk trade-off leads to prey (predator) isoclines with vertical (horizontal) segments. Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey models showed that such isoclines have stabilizing effect on population dynamics because they limit maximum possible fluctuations in prey and predator populations. The present article shows that not only population fluctuations are limited, but they can even be completely eliminated.

Krivan, V. 2013. Behavioral refuges and predator-prey coexistence. Journal of Theoretical Biology 339:112-121.

Krivan, V. 2011. On the Gause predator-prey model with a refuge: A fresh look at the history. Journal of Theoretical Biology 274:67-73.

Krivan, V., Pryiadarshi, A. 2015. L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge. Journal of theoretical biology 370:21-26

Krivan, V. 2007. The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs. American Naturalist 170: 771-782.

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