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In this paper we develop an encounter-based model of a run-and-tumbleparticle (RTP) confined to a finite interval $[0,L]$ with partially absorbing,sticky boundaries at both ends. We assume that the particle switches betweentwo constant velocity states $\pm v$ at a rate $\alpha$. Whenever the particlehits a boundary, it becomes stuck by pushing on the boundary until either atumble event reverses the swimming direction or it is permanently absorbed. Weformulate the absorption process by identifying the first passage time (FPT)for absorption with the event that the time $A(t)$ spent attached to eitherwall up to time $t$ (the occupation time) crosses some random threshold$\hat{A}$. Taking $\Psi({a})\equiv\mathbb{P}[\hat{A}\gt{a}]$ to be anexponential distribution, $\Psi[{a}]=e^{-\kappa_0a}$, we show that the jointprobability density for particle position $X(t)$ and velocity state$\sigma(t)=\pm $ satisfies a well-defined boundary value problem (BVP) with$\kappa_0$ representing a constant absorption rate. The solution of this BVPdetermines the so-called occupation time propagator, which is the jointprobability density for the triplet $(X(t),A(t),\sigma(t))$. The propagator isthen used to incorporate more general models of absorption based onnon-exponential (non-Markovian) distributions $\Psi(a)$. We illustrate thetheory by calculating the mean FPT (MFPT) and splitting probabilities forabsorption. We also show how our previous results for partially absorbing,non-sticky boundaries can be recovered in an appropriate limit. Absorption nowdepends on the number of collisions $\ell(t)$ of the RTP with the boundary.Finally, we extend the theory by taking absorption to depend on the individualoccupation times at the two ends.

Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries