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A stochastic differential game approach toward animal migration

  • Hidekazu YoshiokaEmail author
Original Article

Abstract

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.

Keywords

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

n

Independent variable that represents the total population

B

The bivariate map that sums the cost and benefit of migration

Bt

1-D standard Brownian motion

c

A nonnegative, bounded, and smooth univariate function

C1C2

Positive constants

Cψ

Parameter shaping the regularity of the awareness ψ

E

Expectation

E

Error norm in the policy iteration algorithm

f

Generic sufficiently smooth function

G

A continuous function

h

Free boundary of the solution to the HJBQVI

J

Performance index

k0k1k2

Weighting constants appearing in the migration cost B

K

Maximum body weight

mn

Intervals in the n-direction

mx

Intervals in the x-direction

Nt

The total population in the habitat at the time t

Nmax

The maximum value of the total population Nt

pi, j

Integer to define the optimal control η* at each vertex

s

Time

t

Time

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

Migration strategy

u*

Optimized u

vi, j

(ij)-th vertex

wt

The uncertainty of the mortality rate at the time t

w*

Optimized w

wmax

Maximum value of wt

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

Admissible range of wt

Xt

The representative body weight at the time t

x

Independent variable that represents the body weight

αβ

Model parameters shaping the functional form of the migration cost B

γ

Parameter shaping the regularity of the awareness ψ

δ

Discount rate

ɛ

Error tolerance in the policy iteration algorithm

Δn

Cell length in the n-direction

Δx

Cell length in the x-direction

φ

Test functions

\(\Phi\)

Value function

\(\hat{\Phi}\)

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

ηi

Total number of migrants of the ith migration

η*

Optimized η

κ

Parameter shaping the accumulated profit in the current habitat

μ

Deterministic body growth rate

ψ

Coefficient that represents the state-dependent awareness

ψ0

Parameter shaping ψ

Ψ

Dummy functions for defining viscosity solutions

λ

Base mortality rate

ϑ

Weighting content appearing in the accumulated profit in the current habitat

ρ

Penalty parameter

σ

Amplitude of the stochastic growth rate

τ

The smallest time s such that Ns = 0

τi

Timing of the ith migration

χi, j

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

\(\Omega\)

The domain to define the HJBQVI

\(\bar{\Omega}\)

Closure of \(\Omega\)

\(\Omega_{\text{m}}\)

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

\(\Omega_{\text{r}}\)

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

\({\mathcal{F}}_{t}\)

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

\({\mathcal{L}}\)

Partial differential operator defining the HJBQVI

\({\mathcal{L}}_{w}\)

Partial differential operator defining a linear counterpart of the HJBQVI

\({\mathcal{M}}\)

Non-local operator defining the HJBQVI

\({\mathcal{U}}\)

The admissible set of u

\({\mathcal{W}}\)

The admissible set of w

\({\mathbf{\mathbb{R}}}\)

1-D real space

Notes

Acknowledgements

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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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Life and Environmental ScienceShimane UniversityMatsueJapan

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