Abstract: Using a differential equation, the spiraling flight trajectory of an insect around a point light source can be modeled.

Introduction

    It is a common sight to see an insect approach a night-light. Nearly inevitably, an insect will spiral into the light source as if drawn by a recurring natural attraction. One might imagine that simple attraction to the light would result in a straight flight path to optimize the insect’s fulfillment in the least time. The spiraling phenomenon discounts the idea of a simple attractive force by the light on the insect.

    Some primitive animals orient themselves with respect to light sources. In the case of an insect, the animal’s compound eyes give it an ideal photographic positioning mechanism. An insect’s compound eyes are composed of many ommatidia, which are single detectors of light. Light beams from a single source stimulate a small group of ommatidia determining the insect’s angular position with the light. Insects use this positioning to adjust their path during flight. In a parallel light beam field, like that of the sun, an insect desiring to fly in a straight path would maintain constant angles, keeping the same group of ommatidia activated, between its path and the light beams in the field. See Figure 1.

In Figure 1 the parallel lines represent light beams from an infinite distance away. Due to man, today we have light sources that occur at much closer distances. As a result of light sources occurring at much closer distances the light beams occur in radial directions as shown in Figure 2.

Figure 2 represents the effect of a radial light source, which causes the flight of the insect to spiral towards the light source, while keeping a constant angle a with the radial light beam and the tangent vector, that is formed at each position on the spiral trajectory.

The Differential Equation

The spiraling flight trajectory of an insect can be modeled by the following Differential Equation, Equation 1:

In the differential equation, r is the radius, or the insect’s distance to the light source. The variable q is the insect’s radial angle around the point light source; v is the speed of the insect, which remains constant in our analysis. Lastly, a is the angle between the direction of flight and the direction towards the point light source.

Derivation

 With the light at the origin of a polar plot (See Figure 3) the insect flies by the law:

where t represents time and r and q are polar coordinates. We will allow q to go from negative infinity to positive infinity and take the derivative of the set of equations to find a solution to the functions r(t) and q(t).

The angle between the radial position vector and the velocity vector (tangent to the insect’s path) is the constant angle a. The dot product can be used to find the angle and relate the functions r(t) and q(t).

We assume that the insect flies with constant speed:

and substitute the resultant equation for velocity into the previous equations to arrive at the first polar differential equation. (Equation 2)

Using this equation we can find the second polar DE with a simple substitution.

Manipulation of the D.E.

Now we want to put the differential equation into a form that can be put in Matlab’s ODE45 solver. We can define r and q as shown in Equation 4.

 We can then take the derivative of the expressions in Equation 4 to get Equation 5.

Recall that r’ and q’ are equal to the differential equation, so the expressions can be substituted from our original differential equation to get Equation 6.

Now the differential equation is in a form that can be solved in Matlab using its ODE45 solver. In our analysis v=2ft./sec., q(0)=0 degrees , r(0)=5ft., and a varies depending on the orientation of the insect around the point light source.

Solving the D.E. using ODE45

The following M-file was used in part of the process of solving the differential equation using ODE45:

function xprime=spiral(t,x,flag,v,alpha);
xprime=zeros(2,1);
xprime(1)=-v*cos(alpha);
xprime(2)=v*sin(alpha)/x(1);

The following commands were used in Matlab to solve the differential equation, using Matlab:

>>t=linspace(0,100,2000);
>>[t,x]=ode45('spiral',[0,10],[5 0],[],2,pi/4);
>>r=x(:,1);
>>theta=x(:,2);
>>xx=r.*cos(theta);
>>yy=r.*sin(theta);
>>plot(xx,yy)

In Figure 4 the plot obtained from using Matlab’s ODE45 solver can be viewed. Later in this paper, one will see a very distinct similarity between Figure 4 and Figure 5. Both were for the case when a=45 degrees.

It can be seen that at the start of the solution t=0, and x=5. The x-axis can be defined to be rcos(q). Since q(0)=0, the expression is just equal to r, which was 5ft. If a tangent vector were drawn at the start of the solution it would have a slope of 2, which represents the speed, v.

Varying a angles

Depending on the constant angle a, the spiraling flight trajectory of an insect will have multiple spirals or very few. In Figure 5, an insect’s flight trajectory for a=45 degrees is shown.

In Figure 5 it can be seen that the insect does not have to make many spirals around the light source to reach it, due to the fact that the insect has oriented itself at a smaller angle than the larger angles that will be analyzed later in this paper. In Figure 6 it can be seen that the insect must make more spirals around the light source until the light source is reached.

For the plot in Figure 6 the insect orients itself at an angle a=82 degrees, and as a result the insect must make more spirals around the light source, while keeping the same ommatidia activated. In Figure 7 the angle a is now changed to a=89 degrees.

Lastly, in Figure 7 it can be seen that the insect must make many revolutions to reach the light source. The reason that the insect must make so many revolutions around the light source is because it is orienting itself around the light source at a much larger angle a.

Model in Three-Dimensions

Adding a new dimension to carry the analysis over to three space is easily done. As the insect approaches the light source from either above or below, it spirals inward toward the source just as it did in two space but a new constant angle, b, is introduced as in Figure 8. An angle between the vertical direction and the position vector toward the insect must remain constant to ensure that the same ommatidia are activated. To maintain this angle as constant the insect must ascend or descend as it spirals toward the light and form the helical shape characteristic of real insect paths. In later cases, insects will begin their approach at a point in the negative z-direction and at an initial radius from the z-axis, evident in Figure 8.

The Equations of the Trajectory were used in conjunction with the differential equation to obtain three-dimensional plots. These equations constitute Equation 7.

The constants r₀ and m are defined in Equation 8.

Like Figure 5, Figure 9 has few revolutions and approaches the light quickly because the angle a is small. Later figures show increased revolutions and more evident conic shape with larger a angles.

With more revolutions, Figure 10 shows a clearer depiction of the cone and the fixed angle b that is held throughout the flight path.

Figure 11 has even more revolutions around the light source because the insect closes more slowly with such a large angle. The outline of a cone formed by the constant angle, b, can be seen more clearly.

Conclusion

    One may be curious of the behavior of the model in cases that have been neglected. The case at a=90 degrees is interesting to explore. At this angle the insect will neither converge nor diverge from the source because tangent vectors to the path are perpendicular to the position vectors resulting in a circular trajectory. In the case 90<a<180 degrees, the insect will spiral outward as time passes. In approaching a light source insects use angles 0<a<90 degrees . A downfall of our model is that it predicts the insect to keep approaching the light source indefinitely, when in reality the insect can only approach until physical contact is made with the light source.

Bibliography

Boyadzhiev, Khristo N.  "Spirals and Conchospirals in the Flight of Insects."  The College Mathematics Journal 30:1 (1999):  23-31.