An Optimal Switching Approach for Bird Migration

TL;DR

The paper develops a stochastic optimal switching framework for bird migration with three diffusion regimes, characterizing the value functions $V_i(t,x)$ as the unique viscosity solution to a system of Hamilton-Jacobi-Bellman variational inequalities and deriving optimal switching regions $\,\mathcal{S}_{ij}$. It applies this framework to study stopover-site deterioration and information quality (perfect vs partial), using finite-difference methods and stochastic simulations to compute $V_i$, switching regions, and migratory payoffs. Key findings show that deterioration contracts or eliminates switching to certain stopovers and reduces overall payoff, while stopover-based information can reduce mismatches between perceived and actual terminal conditions $g(t)$ vs $G(t)$, especially under climate-driven timing shifts. These results provide quantitative insights into how environmental change and information access shape migratory strategies and have implications for conserving critical stopover habitats to support population resilience.

Abstract

Bird migration is an adaptive behavior ultimately aiming at optimizing survival and reproductive success. We propose an optimal switching model to study bird migration, where birds' migration behaviors can be efficiently modeled as switching between different stochastic differential equations. For individuals with perfect information regarding the environment, we implement numeric methods to see the expected payoff and corresponding optimal control. For individual with only partial information of the environment, we combine the finite difference method and stochastic simulations to investigate the change of the bird's optimal strategy. Based on biological backgrounds, we characterizing the optimal strategies of birds under different scenarios and these behaviors depend on the specific assumptions of the model.

Figures (16)

• Figure 1: Geographical setting of the migration route.
• Figure 2: The three states of an individual (waiting/detour/direct) with switching control
• Figure 3: (An illustration of switching regions) Consider three different realizations of the diffusion process (representing three distinct individuals). In path 1, the individual arrives at the green region, so it is optimal for the individual to switch to the waiting state $i=3$ and stay there until it is time to switch to "direct flight" (which is the diffusion region $i=2$). In the second path, the individual arrives at the red region at which point it is optimal to adopt direct flight immediately. (Note that once the individual adopts direct flight, no further switching is possible so it does not matter that it enters the green region. The above diagram only concerns individuals adopting switching modes $i=1$ or $i=3$.) Finally, in path 3, the individual does not enter the green or red region, so it is optimal for the individual to stay at diffusion mode of "detour flight" $i=1$.
• Figure 4: stopover sites
• Figure 5: Optimal control at $\lambda = 0$

Theorems & Definitions (5)

• Lemma 2.1: Stochastic dynamic programming principle (SDPP)
• proof
• Proposition 3.1: Existence and uniqueness
• proof
• Proposition 3.2: Optimal switching strategy