A transport equation model and diffusion approximation for the walk of whirligig beetles

Abstract

The circular walk of whirligig beetles, in the absence of swarming behaviour, is described by a coupled system of transport equations. For this system, an approximate description is derived in the form of coupled damped wave equations and ultimately in the form of a scalar diffusion equation. The resulting diffusion coefficient is quite different from what one would get for a walk on straight lines with tumbling (as in bacterial motion).

Keywords:

• animal locomotion
• transport equation
• moment approximation
• parabolic scaling
• damped wave equation
• diffusion approximation

Mathematics Subject Classification :

• 35Q92
• 60J70
• 92C17
• 92D50

1. A model for whirligig motion

The movements of whirligig beetles have been observed on various scales and also modelled by computer simulation. The population distribution in isolated ponds and lakes and population exchange between such habitats have been studied by genetic methods in 9 10. The daily changes in multi-species swarms on large lakes have been observed in 6. The motion of isolated individuals and of individuals in small swarms have been studied in 11, on calm surfaces and on running water, and the behaviour has been modelled by computer simulation 12. Finally, on a finer scale, the swimming mechanics of an individual beetle has been investigated in 3, see also 8. In this paper, we model the motion patterns of individual beetles in a homogeneous environment and derive differential equations for the (probability) density in space.

There are various types of paths of whirligig beetles, and the body axis of the moving beetle may not be parallel to the velocity vector (in particular on running water) 11. But most paths can be described as if the beetle is swimming in circles [11, in particular Figures 11 and 13]. However, pure circular movement prevails for at most two or three full turns after which the beetle will make an abrupt change of direction. To this extent the movements resemble the run and tumble behaviour of many bacteria which run on straight lines, then stop and choose a new direction. But bacteria tend to run on straight lines while whirligigs run on circles. However, the bacterium Escherichia coli swims in ‘right-hand’ circles when close to a (glass) surface, while it swims straight in a three-dimensional medium 2. The effect can be explained in terms of the asymmetric action of the flagella and the rotation of the body of the bacterium 13.

For bacteria there are various models in the form of transport equations and diffusion equations 1 7. We can expect that a similar approach leads to a model for whirligig motion. Here, we derive such a model from some basic idealizing assumptions. In deriving a diffusion approximation, we follow the general approach via moment approximations as in 4 5 (in the latter the peculiar random walk of the bacterium Azospirillum has been treated).

We describe the motion of the beetle in terms of three variables, its position x=(x ₁, x ₂) in the plane, its direction given by a point on the unit circle , and a discrete variable which assumes the values ± and tells whether the beetle runs on a circle counterclockwise or clockwise. Hence the probability distribution for the state of the beetle or the density of a group of independent beetles (neglecting swarming behaviour), at time t, is a function . Our model depends on three parameters, the speed γ, the turning rate μ, and the (constant) radius r of the assumed circles (1/r is the absolute value of the curvature of the path except at stopping positions). We show that, with the assumptions made above, the function satisfies the following system of two coupled transport equations,

The first three terms of each equation just state that the beetle moves in the direction with speed γ. If it were not for the next term, the beetle would stay on a straight line. The last term on the left-hand side changes the direction in such a way that the beetle stays on a circle. The right-hand side is the turning operator. This is based on the assumption of an exponentially distributed holding time with parameter μ. We assume that at a stop the beetle chooses a new direction and orientation (sign of curvature) according to the uniform distribution on .

In 13 a general transport equation model for bacterial motion has been designed that allows for circular and more general movements depending on the internal state of the bacterium and external forces. The model (1) is not a special case but perhaps can be seen, with some effort, as a limiting case. If so, it will be necessary to link the curvature variable ± to the concepts of velocity and internal state.

Field observations 11 show that the distribution of the path velocity (speed) is about log normal and the angular velocity is about normal, centred at zero. The typical characteristic sharp turns and changes of the sign of curvature show up in videotaped sections of paths [11, Figure 13]. On a larger time and space scale [11, Figure 8] the paths look more like Brownian motion, suggesting a diffusion approximation. If we assume that the angular velocity distribution in [11, Figure 7c] also represents the turn angle distribution, then our choice of the uniform distribution seems crude. It would be easy to implement another distribution into Equation (1) but difficult to derive a better diffusion approximation. On the other hand, it does not make sense to replace the turning operator by a diffusion operator (i.e. by u _φφ) because diffusion of direction does not really describe what whirligigs do (in contrast to Azospirillum).

For the system (1), we derive approximations which ultimately lead to diffusion equations. But there are intermediate steps in the form of coupled damped wave equations that are of independent interest. In Equation (1) we introduce the particle density

and the mean transversal gradient of the net flow

The unit vector ν is defined as follows. At a given space point, with a given direction, there are two osculating circles. The vector ν is transversal to the direction and points into the circle on which the beetle moves counterclockwise.

For the two functions ū(t, x) and p(t, x), we derive the approximate system of equations

It appears that the function p cannot be easily measured experimentally. Hence, we do not suggest the system (4) as a biological model. It is an intermediate step between the full model (1) and a diffusion approximation.

The main result is the diffusion approximation of the walk in (1)

which is obtained by appropriate parabolic scaling. We keep the quantities

fixed and let the turning rate μ go to infinity; i.e. we assume the standard diffusion approximation where is a constant and, in addition, we let the radius r become small. The assumptions say that we look at the walk in a large domain and for a long time. The diffusion coefficient

is the harmonic mean of the two quantities and r ²μ. The inequality between the harmonic mean and the geometric mean shows

which is independent of μ.

The coefficient is an increasing function of r ²μ and

for .

In Section 2, model (1) will be derived, and in Section 3 the diffusion approximation. Section 4 contains a discussion.

2. Derivation of the model

Assume a point (the beetle) walks on a circle of radius r at constant speed γ counterclockwise. If the circle has centre then the initial position of the point can be described as

Once the point moves counterclockwise, we get

We can eliminate the position of the centre of the circle (using (8)),

From here, we arrive at differential equations

These equations can be seen as the characteristic equations of a first-order partial differential equation (some physicists call this a ‘master equation’)

The angle θ describes the position of the point on the actual circle (with reference to the centre of that circle). The angle gives the direction of the actual tangent vector to the path. In Equation (12) (counterclockwise motion), we replace θ by φ and get

If the point runs clockwise on the circle then in Equation (11), we have and such that eventually we get the equation (only one sign is changed)

Now we must describe how whirligigs change direction. To keep things simple, we assume they run with exponential holding time and choose a new direction and a new osculating circle (i.e. make a choice between clockwise and counterclockwise motion) according to the uniform distribution. Hence, we couple Equations (13) and (14) with a turning operator which yields the system (1).

3. Deriving diffusion approximations

In Equation (1), we introduce new variables as

The function is the density of particles at position x that move in the direction φ. These variables satisfy the equations (which are equivalent with Equation (1))

For this system, we define six moments:

We multiply the equations by cosφ and sinφ and integrate. We get a system of moment equations that is not closed,

In the eight integrals containing space derivatives we expand and discard higher order Fourier coefficients, i.e., we replace cos² φ = sin² φ ≈ 1/2 and sin φ cos φ ≈ 0, then we exchange integration and differentiation. In the other four integrals we use integration by parts. Thus, we discard high frequency oscillations and arrive at a closed moment system

We differentiate the first equation by t, the third by x ₁, the fourth by x ₂. We multiply by appropriate factors and add,

We replace the last term from the first equation and get

We differentiate the fifth equation by x ₂ and the sixth equation by x ₁ and take the difference,

We define

and get, with m ₀=ū, the system (4). The form of this system, with a time derivative on the right-hand side, may appear unusual in animal locomotion, but it is not unusual in fluid dynamics and combustion. The system is equivalent with the standard form of a degenerate parabolic system with one time derivative in each equation,

We write the system (4) as

We assume that μ is large and hence we omit the highest time derivatives,

We replace p in the first equation from the second equation, and find

which is Equations (5) and (7).

4. Discussion

Whirligig beetles run rapidly in circles, either clockwise or counterclockwise, and perform rapid changes of direction. We design a basic model in the form of coupled transport equations with the independent variables space, direction, and orientation (clockwise/counterclockwise) and three parameters speed γ, radius (of the supposed circle) r, rate of change μ (between clockwise/counterclockwise).

We derive a diffusion approximation (5) with diffusion coefficient given by the formula (7). Essentially this coefficient depends not on three parameters separately but only on two parameters and μ r ². The diffusion coefficient depends in a monotone way on the parameter μ r ². It increases from zero to the ‘standard’ diffusion coefficient .

If γ and μ are large and r is small, then is considerably smaller than D _∞; it goes to zero for r→0. In this limiting case, the beetles run very fast on small circles but the overall displacement is small.

Notice that a correlated random walk on a straight line produces the diffusion coefficient . The factor 1/2 results from the fact that here the space dimension is 2.

The result says that when the movements of whirligig beetles are observed over large space and time ranges and hence appear as diffusion, then the curvature of the paths, i.e. the (average) radius r of the (approximate) circles, plays an essential role. It is only for a very large radius that the ‘standard’ diffusion coefficient is obtained.