A simple game-theoretic model for upstream fish migration

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.

A stochastic differential game approach toward animal migration

A stochastic differential game model for animal migration between two habitats under uncertain environment, a new population dynamics model, is formulated. Its novelty is the use of an impulse control formalism to naturally describe migrations with different timings and magnitudes that the conventional models could not handle. Uncertainty of the environment that the population faces with is formulated in the context of the multiplier robust control. The optimal migration strategy to give the maximized minimal profit is found through a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). A key message from HJBQVI is that its free boundary determines the optimal migration strategy. Solving the HJBQVI is carried out with a specialized stable and convergent finite difference scheme. This paper theoretically suggests that the sub-additivity of the performance index, the index to be optimized through the migration, critically affects the resulting strategy. The computational results with the established scheme are consistent with the theoretical predictions and support importance of the sub-additivity property. Social interaction to reduce the net mortality rate is also quantified, suggesting a linkage between the present and existing population dynamics models.

An optimal stopping approach for onset of fish migration

Comprehending life history of migratory fish, onset of migration in particular, is a key biological and ecological research topic that still has not been clarified. In this paper, we propose a simple mathematical model for the onset of fish migration in the context of a stochastic optimal stopping theory, which is a new attempt to our knowledge. Finding the criteria of the onset of migration reduces to solving a variational inequality of a degenerate elliptic type. As a first step of the new mathematical modeling, mathematical and numerical analyses with particular emphasis on whether the model is consistent with the past observation results of fish migration are examined, demonstrating reasonable agreement between the theory and observation results. The present mathematical model thus potentially serves as a simple basis for analyzing onset of fish migration.