Mathematical Tri-State Model for Bee Shimmering Propagation Dynamics

TL;DR

The first analytical tri-state Inactive-Active-Relapse (IAR) model is introduced to formulate the intrinsic process of bee shimmering, providing a foundation for further theoretical understanding of bee shimmering wave dynamics and could serve as inspiration for modelling other self-organised phenomena across scientific applications.

Abstract

Bees undergo a self-organised process known as shimmering, where they form emergent patterns when they interact with each other on the nest surface as a defence mechanism in response to predator attacks. Many experimental studies have empirically investigated how the transfer of information to neighbouring bees propagates in various shimmering processes by measuring shimmering wave strength. However, there is no analytical modelling of the collective defence mechanism in nature. Here we introduce the first analytical tri-state Inactive-Active-Relapse (IAR) model to formulate the intrinsic process of bee shimmering. The major shimmering behaviour is shown to emerge under theoretical conditions which is demonstrated numerically and visually by simulating 1,000,000 bee agents, while the number of agents is scalable. Furthermore, we elaborate on these mathematical results to construct a wave strength function to demonstrate the accuracy of shimmering dynamics. The constructed wave strength function can be adapted to peak between 50-150ms which supports the experimental studies. Our results provide a foundation for further theoretical understanding of bee shimmering wave dynamics and could serve as inspiration for modelling other self-organised phenomena across scientific applications.

Figures (7)

• Figure 1: Bee shimmering agents information transfer Bee shimmering takes place on the surface of the bee nest and is initiated when a wasp (a) is in close vicinity. There are three possible types of agents during shimmering known as generator (b), bucket bridger agents (c), and chain tail (d). The red arrows show the origin of the transfer of information, the green arrows determine where the information is being propagated to. The bucket bridger agents (c) acts as both an emitter and a receiver of information. The chain tail agents (d) do not respond to information in any way, whereas the generator agents (b) act as the transmitters to start the wave. Each agent has its own respective neighbourhood angle of where the wave would be received as indicated in the reference axis. The green arrow indicates the direction from which the wave is originating, pointing towards the nearest bee transmitting the signal.
• Figure 2: State transitions of the various agents Three states have been classified, inactive, active and relapse states. These states classify the motion of each of the bees. Inactive states indicate the state at which bees are not expanding their abdomens, the active state classifies the process at which the bee goes from an inactive state to its maximum expansion. Thirdly the relapse state is the process during which the bee returns to the inactive state from its maximum flexion. Reaction (d) indicates the stages at which active state Bees return to the inactive state. The reactions showcase the interaction dynamics of each individual type of agent, which corresponds to generator agent (a), bucket bridger agents (b), and chain tail agents (c), and the relapsing process (d). Diagrams (a)-(d) show the state transitions for the state interaction model. Diagram (e) shows the complete tri-state model interactions showcasing the full interaction dynamics of bee shimmering showing a redundancy of the chain tail agent where the rate $r_{5}$ is encapsulated when the rates $r_{1}$ from (b) and $r_{4}$ (a) has a probability for the reactions not being undergone.
• Figure 3: Proportion of Inactive Active and Relapsing agents for Minor Shimmering Figure (a) displays a saltatoric process plot of the proportion of Inactive, Active and Relapsing agents over time by choosing the initial conditions $a(0)=0.3$ and $i(0)=1-a(0)$, $r_{2}=0.2$, $\lambda=0.3$, $r_{3}=0.9$, $\left\langle k_{B}\right\rangle=8$, $\left\langle k_{G}\right\rangle=0.1$, $\alpha=0.9$ and $\beta=0.9$. Figure (b) displays a bucket bridging plot of the proportion of Inactive, Active and Relapsing agents over time by choosing the initial conditions $a(0)=0.1$ and $i(0)=1-a(0)$, $r_{2}=0.01$, $\lambda=0.001$, $r_{3}=0.1$, $\left\langle k_{B}\right\rangle=8$, $\left\langle k_{G}\right\rangle=1$, $\alpha=0.9$ and $\beta=0.9$
• Figure 4: Proportion of Inactive Active and Relapsing agents for Major Shimmering Part (a) displays the proportion of inactive, active, and relapsing agents over time, calculated using the following initial conditions: $a(0)=0.00001$, $i(0)=1-a(0)$, $r_{2}=0.9$, $\lambda=0.8$, $r_{3}=0.1$, $\left\langle k_{B}\right\rangle=10$, $\left\langle k_{G}\right\rangle=1$, $\alpha=0.9$, and $\beta=0.9$ without further subsequent waves. Part (b) shows the same information, but with initial conditions of $a(0)=0.000001$, $i(0)=1-a(0)$, $r_{2}=0.5$, $\lambda=0.8$, $r_{3}=0.006$, $\left\langle k_{B}\right\rangle=8$, $\left\langle k_{G}\right\rangle=1$, $\alpha=0.9$, and $\beta=0.6$, including daughter waves.
• Figure 5: Wave Strength Function results Plot (a) shows the wave strength function over time with varying $\lambda$ in intervals of 0.1 and choosing $a(0)=1\times 10^{-13}$, $i(0)=1-a(0)$, $r_{2}=0.05$, $r_{3}=0.001$, $\left\langle k_{B}\right\rangle=10$, $\left\langle k_{G}\right\rangle=4$, $\alpha=0.9$, $\beta=0.03$. Plot (b) shows the strength of the wave moving from right to left using the same parameters as (a) in a given direction $\theta$ with an angle tolerance of $\epsilon=0.01$ radians

Theorems & Definitions (3)

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