A stochastic differential game approach toward animal migration

  • Hidekazu YoshiokaEmail author
Original Article


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.


Animal population Differential game Stochastic impulse control Social interaction Finite difference scheme 

List of symbols

Ai, j

Diagonal coefficients of the discretized HJBQVI at the (ij)-th vertex vi,j


Independent variable that represents the total population


The bivariate map that sums the cost and benefit of migration


1-D standard Brownian motion


A nonnegative, bounded, and smooth univariate function


Positive constants


Parameter shaping the regularity of the awareness ψ




Error norm in the policy iteration algorithm


Generic sufficiently smooth function


A continuous function


Free boundary of the solution to the HJBQVI


Performance index


Weighting constants appearing in the migration cost B


Maximum body weight


Intervals in the n-direction


Intervals in the x-direction


The total population in the habitat at the time t


The maximum value of the total population Nt

pi, j

Integer to define the optimal control η* at each vertex





\(u = \left({\tau_{1};\eta_{1},\tau_{2};\eta_{2}, \ldots} \right)\)

Migration strategy


Optimized u

vi, j

(ij)-th vertex


The uncertainty of the mortality rate at the time t


Optimized w


Maximum value of wt

\(W = \left\{{\left. w \right|0 \le w \le w_{\max}} \right\}\)

Admissible range of wt


The representative body weight at the time t


Independent variable that represents the body weight


Model parameters shaping the functional form of the migration cost B


Parameter shaping the regularity of the awareness ψ


Discount rate


Error tolerance in the policy iteration algorithm


Cell length in the n-direction


Cell length in the x-direction


Test functions


Value function


Piece-wise linear interpolation of discretized \(\Phi\)


Total number of migrants of the ith migration


Optimized η


Parameter shaping the accumulated profit in the current habitat


Deterministic body growth rate


Coefficient that represents the state-dependent awareness


Parameter shaping ψ


Dummy functions for defining viscosity solutions


Base mortality rate


Weighting content appearing in the accumulated profit in the current habitat


Penalty parameter


Amplitude of the stochastic growth rate


The smallest time s such that Ns = 0


Timing of the ith migration

χi, j

Characteristic function to penalize the non-local term of the HJBQVI in numerical computation


The domain to define the HJBQVI


Closure of \(\Omega\)


The migration sub-domain. A sub-domain of \(\Omega\)


The residency sub-domain. A sub-domain of \(\Omega\)


Natural filtration generated by \(B_{t}\). Its subscript is sometimes dropped when there is no confusion


Partial differential operator defining the HJBQVI


Partial differential operator defining a linear counterpart of the HJBQVI


Non-local operator defining the HJBQVI


The admissible set of u


The admissible set of w


1-D real space



JSPS Research Grant No. 17K15345 and a grant for ecological survey of a life history of the landlocked ayu Plecoglossus altivelis from MLIT of Japan supports this research. The author thanks the two reviewers for their critical comments and suggestions on biological and mathematical descriptions in this paper. The author is grateful to a reviewer for helpful comments, suggestions, and discussions on the performance index from a viewpoint of evolutionary biology.


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Authors and Affiliations

  1. 1.Faculty of Life and Environmental ScienceShimane UniversityMatsueJapan

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