‘Mathematical exercise’ on a solvable stochastic control model for animal migration

Abstract

Animal migration is a mass biological phenomenon indispensable for comprehension and assessment of food-webs. So far, theoretical models to describe decision-making processes inherent in the animal migration have not been well established, which is the motivation of this research. It is natural to formulate the animal migration based on a stochastic control theory, which can describe system dynamics and its optimization in stochastic environment. To address this issue, a conceptual stochastic control model for the decision-making processes in animal migration is introduced and mathematically analysed. Its novelty is mathematical simplicity and the new theoretical, stochastic control viewpoint. Stochastic differential equations govern the animal population dynamics with gradual and radical migrations from the current habitat toward the next one. The population decides the occurrences, magnitudes, and timings of the migrations, so that a heuristic performance index is maximised. I derive a variational inequality that governs the maximised performance index and is exactly solvable. Its free boundaries govern the gradual and radical migrations. Despite the model simplicity, the exact solution is consistent with the empirical observation results of fish migration, implying its potential applicability to animal migration. The present model can be used for assessing fish migration. References S. Bauer and B. J. Hoye. Migratory animals couple biodiversity and ecosystem functioning worldwide. Science, 344, 2014. Article No. 1242552. N. E. Leonaerd. Multi-agent system dynamics: Bifurcation and behavior of animal groups. Annual Reviews in Control, 38(2):171–183, 2014. A. M. Oberman. Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–jacobi equations and free boundary problems. SIAM Journal on Numerical Analysis, 44(2):879–895, 2006. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. B. \T1\O ksendal and A. Sulem. Applied Stochastic Control of Jump Diffusions. Springer Berlin Heidelberg, 2007. Y. Yaegashi, H. Yoshioka, K. Unami, and M. Fujihara. An optimal management strategy for stochastic population dynamics of released \(plecoglossus\) \(altivelis\) in rivers. International Journal of Modeling, Simulation, and Scientific Computing, 8(2), 207. Article No. 1750039. H. Yoshioka, T. Shirai, and D. Tagami. Viscosity solutions of a mathematical model for upstream migration of potamodromous fish. In Proceedings of 12th SDEWES Conference, October 4-8, 2017, Dubrovnik, Croatia, 2017. In press.