A stochastic differential game approach toward animal migration

Gaps in modelling animal migration with evolutionary game theory: infection can favour the loss of migration

Ongoing environmental changes alter how natural selection shapes animal migration. Understanding how these changes play out theoretically can be done using evolutionary game theoretic (EGT) approaches, such as looking for evolutionarily stable strategies. Here, we first describe historical patterns of how EGT models have explored different drivers of migration. We find that there are substantial gaps in both the taxa (mammals, amphibians, reptiles, insects) and mechanisms (mutualism, interspecific competition) included in past EGT models of migration. Although enemy interactions, including parasites, are increasingly considered in models of animal migration, they remain the least studied of factors for migration considered to date. Furthermore, few papers look at changes in migration in response to perturbations (e.g. climate change, new species interactions). To address this gap, we present a new EGT model to understand how infection with a novel parasite changes host migration. We find three possible outcomes when migrants encounter novel parasites: maintenance of migration (despite the added infection cost), loss of migration (evolutionary shift to residency) or population collapse, depending on the risk and cost of getting infected, and the cost currency. Our work demonstrates how emerging infection can alter animal behaviour such as migration. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Effects of water currents on fish migration through a Feynman-type path integral approach under [Formula: see text] Liouville-like quantum gravity surfaces

A stochastic differential game theoretic model has been proposed to determine optimal behavior of a fish while migrating against water currents both in rivers and oceans. Then, a dynamic objective function is maximized subject to two stochastic dynamics, one represents its location and another its relative velocity against water currents. In relative velocity stochastic dynamics, a Cucker-Smale type stochastic differential equation is introduced under white noise. As the information regarding hydrodynamic environment is incomplete and imperfect, a Feynman type path integral under [Formula: see text] Liouville-like quantum gravity surface has been introduced to obtain a Wick-rotated Schrödinger type equation to determine an optimal strategy of a fish during its migration. The advantage of having Feynman type path integral is that, it can be used in more generalized nonlinear stochastic differential equations where constructing a Hamiltonian-Jacobi-Bellman (HJB) equation is impossible. The mathematical analytic results show exact expression of an optimal strategy of a fish under imperfect information and uncertainty.

Stochastic optimal switching model for migrating population dynamics

An optimal switching control formalism combined with the stochastic dynamic programming is, for the first time, applied to modelling life cycle of migrating population dynamics with non-overlapping generations. The migration behaviour between habitats is efficiently described as impulsive switching based on stochastic differential equations, which is a new standpoint for modelling the biological phenomenon. The population dynamics is assumed to occur so that the reproductive success is maximized under an expectation. Finding the optimal migration strategy ultimately reduces to solving an optimality equation of the quasi-variational type. We show an effective linkage between our optimality equation and the basic reproduction number. Our model is applied to numerical computation of optimal migration strategy and basic reproduction number of an amphidromous fish Plecoglossus altivelis altivelis in Japan as a target species.