Modeling selective local interactions with memory: Motion on a 2D lattice
Introduction
In a series of recent works [1], [2], [3], [4], [5], [6] we developed mathematical models for describing various aspects of the motion of the cyanobacteria Synechocystis sp., which are coccoidal bacteria that move towards light, a motion known as phototaxis. As a result of this motion, finger-like appendages form on a large scale [7], [8]. In contrast, in regions of low and medium density, cells follow a quasi-random pattern of motion in which small aggregates form, yet bacteria may still move in various directions without any observable bias in the direction of the light source.
This quasi-random motion in regions of low-density was the focus of our works in [2], [3] in which we developed mathematical models to describe the emerging patterns of motion. Our approach was to construct stochastic particle models in which we considered individual particles that move according to a prescribed set of rules at discrete time steps. The rules of motion allowed the particles to persist in their previous direction of motion, become stationary or start moving if already stationary, and change the direction of their motion. When a particle changes its direction of motion, it can only choose to move towards one of its neighbors. Particles can detect their neighbors within a given detection range. These models generated patterns of motion that qualitatively agree with the experimental data.
In order to gain a better understanding of the mathematical model, we developed a one-dimensional version of our stochastic model from [2], [3], in which particles were constrained to move on a one-dimensional lattice [1]. In this context, it became possible to develop a system of ODEs that quantify the expected number of particles at each position, following the method outlined in [9]. The results of the stochastic model agreed in many cases with the results of the deterministic model, depending on the choice of parameters. In addition, randomly chosen initial conditions in the deterministic model led to the formation of aggregates in most cases.
In this paper, we generalize the one-dimensional model from [1] to a motion on a two-dimensional lattice and use numerical simulations to study the emerging patterns. Similarly to [1], our study starts with a stochastic particle system and proceeds with a system of ODEs that capture the averaged behavior of the discrete system.
It is important to note that this study is an example of a flocking model. Mathematical models of flocking phenomena have became very popular in recent years, most of which intend to describe a process in which self-propelled individual organisms act collectively. Examples for such models include flocking models for fish [10], [11], [12], [13], birds [14], [15], and insects [16], [17], among many others. Various mechanisms have been proposed in the literature for changing the direction of motion. In [15], Reynolds models a flock of birds using the rules of collision avoidance, velocity matching, and attraction within a certain radius. Vicsek et al. propose a simple model where the only rule is for each individual to assume the average direction of its neighbors, with some random perturbation [18]. In the model of Couzin et al., particles have a zone of repulsion, a zone of orientation in which they match their neighbors’ directions, and a zone of attraction [19]. The Cucker–Smale model proposes that a bird changes its velocity at each time step by adding a weighted average of the differences between its velocity and those of other birds [14]. In contrast, our approach requires a particle to move towards one of its neighbors.
The structure of this paper is as follows. After reviewing the one-dimensional models in Section 2, we introduce the two-dimensional stochastic particle model in Section 3.1. Multiple simulations of the stochastic particle model are conducted in Section 3.2. We observe the formation of horizontal and vertical aggregates whose lengths depend upon the choice of parameters.
In Section 4.1, we derive a system of ODEs that captures the averaged behavior of the stochastic particle model. The correspondence between the stochastic particle model and the ODEs model is demonstrated in Section 4.2. The ODEs system also results in the formation of aggregates, at least when the model parameters are confined to a certain range. Concluding remarks are provided in Section 5.
Section snippets
Review of the one-dimensional models
We start by reviewing the one-dimensional model from [1]. Consider a set of particles that occupy the vertices of a one-dimensional lattice. There are no restrictions on the number of particles that can occupy each bin. We fix a detection radius which determines how far away a particle can detect neighboring particles. At every discrete time-step, each particle can either
- (i)
persist in its last direction with probability ,
- (ii)
become stationary with probability ,
- (iii)
choose to move towards
Model formulation
Assume that particles are located on the vertices of a periodic lattice. As in the one-dimensional model, we assume that particles remember their previous direction of movement and can either continue in that direction, choose a new direction, or remain stationary. The detection radius can be generalized to 2-D by counting the particles within a Euclidean distance of . To simplify the calculations, we fix the detection radius to be 1 so that particles can only detect only adjacent
Model derivation
Since simulating a large number of particles on a large grid is computationally intensive, we derive a system of ODEs to capture the mean number of particles in each bin. Let denote the probability of the system being in a given state . Here, is a matrix that denotes the number of right-moving particles at every node. Similarly, are matrices that correspond to the number of left, up, and down-moving particles in every node. The variables with a
Conclusions
In this paper we generalized the one-dimensional model of Galante and Levy [1] to two dimensions. At every time step particles may persist their motion in their current direction with probability , remain stationary with probability , or move toward one of their neighboring particles with equal probabilities. Since there are no exclusion principles in place, multiple particles are allowed to occupy every spot on the lattice, and hence when a particle changes its direction of motion, the new
Acknowledgments
This work was supported in part by the joint NSF/NIGMS program under Grant Number DMS-0758374 and in part by Grant Number R01CA130817 from the National Cancer Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Cancer Institute or the National Institutes of Health.
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