一个精确可解的带重置的捕食者-食饵模型
An exactly solvable predator prey model with resetting
摘要：
我们研究了一个简单的扩散粒子（猎物）的模型，在遇到一群扩散捕食者之一，可以死亡或重置到其原始位置的起源。我们发现，猎物的生存概率$t$的时间代数衰减为$\sim t^{-θ(p, γ)}$的指数$θ$连续依赖于模型的两个参数，与$p$表示的概率，猎物生存后遇到捕食者和$γ= D_A/(D_A+D_B)$，其中$D_A$和$D_B$分别是扩散常数的猎物和捕食者。我们还精确计算的概率分布$P(N|t_c)$的遭遇总数，直到捕获时间$t_c$，并表明，它表现出一个异常的大偏差形式$P(N|t_c)\sim t_c^{- Φ\left(\frac{N}{\ln t_c}=z\right)}$大$t_c$。速率函数$Φ(z)$是显式计算的。数值模拟与我们的分析结果非常一致。
英文摘要：
We study a simple model of a diffusing particle (the prey) that on encounter with one of a swarm of diffusing predators can either perish or be reset to its original position at the origin. We show that the survival probability of the prey up to time $t$ decays algebraically as $\sim t^{-θ(p, γ)}$ where the exponent $θ$ depends continuously on two parameters of the model, with $p$ denoting the probability that a prey survives upon encounter with a predator and $γ= D_A/(D_A+D_B)$ where $D_A$ and $D_B$ are the diffusion constants of the prey and the predator respectively. We also compute exactly the probability distribution $P(N|t_c)$ of the total number of encounters till the capture time $t_c$ and show that it exhibits an anomalous large deviation form $P(N|t_c)\sim t_c^{- Φ\left(\frac{N}{\ln t_c}=z\right)}$ for large $t_c$. The rate function $Φ(z)$ is computed explicitly. Numerical simulations are in excellent agreement with our analytical results.
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