Interactions between different predator-prey states: a method for the derivation of the functional and numerical response

Inferring stochastic group interactions within structured populations via coupled autoregression

The internal behaviour of a population is an important feature to take account of when modelling its dynamics. In line with kin selection theory, many social species tend to cluster into distinct groups in order to enhance their overall population fitness. Temporal interactions between populations are often modelled using classical mathematical models, but these sometimes fail to delve deeper into the, often uncertain, relationships within populations. Here, we introduce a stochastic framework that aims to capture the interactions of animal groups and an auxiliary population over time. We demonstrate the model's capabilities, from a Bayesian perspective, through simulation studies and by fitting it to predator-prey count time series data. We then derive an approximation to the group correlation structure within such a population, while also taking account of the effect of the auxiliary population. We finally discuss how this approximation can lead to ecologically realistic interpretations in a predator-prey context. This approximation also serves as verification to whether the population in question satisfies our various assumptions. Our modelling approach will be useful for empiricists for monitoring groups within a conservation framework and also theoreticians wanting to quantify interactions, to study cooperation and other phenomena within social populations.

Coevolution of the reckless prey and the patient predator

The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator-prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.