Playing with models: quantitative exploration of life.

Speed bested by agility

Posted in Dynamics, Game theory by Alexander Lobkovsky Meitiv on November 14, 2010

The rabbit relies on superior maneuverability to escape predators.

Next time you see a squirrel fleeing from a cat, notice it’s escape strategy. Instead of bolting straight for the safety of a tree, the squirrel follows a zig-zag trajectory with sharp, frequent turns punctuated by its thrashing tail. The pursuing cat, unable to execute the sharp turns, overshoots and looses ground as it adjusts its trajectory.

The flight behavior of a prey is instinctual and highly sophisticated–after all, predation is a major selective pressure in a Darwinian universe. The variation of escape strategies reflects the intrinsic abilities of the prey. Some prey, like the antelope, rely on superior speed and safety in numbers. Others, like the rabbit, or the squirrel, rely on superior maneuverability. In this post I will illustrate how superior maneuverability can be used to escape from a faster predator.

What is maneuverability? We will define it in a narrow sense as the ability to quickly change the course or direction of motion. The rate of change of the direction of motion is also known as acceleration. The word acceleration is used colloquially when the rate of change of velocity is in the same direction as velocity. More generally, the vector of acceleration could point in any direction. If the acceleration vector points in the direction opposite to velocity, we would say that the moving object decelerates. Circular motion is sustained by a centripetal acceleration which points toward the center of the circle, perpendicular to the velocity vector.

To illustrate how a slower animal can successfully evade a faster one let us consider a simple model of the predator-prey pursuit. Suppose, the only constraint on the motion is the velocity and acceleration caps. The predator’s maximum velocity is greater than that of the prey. Vice versa, the prey can accelerate faster (in any direction) which means, among other things, that it can make sharper turns.

Here are the model pursuit strategies:

Predator:

  • If traveling slower than the maximum speed, accelerate in the instantaneous direction of the prey at maximum acceleration.
  • When traveling at maximum velocity, project the acceleration vector on the direction perpendicular to the instantaneous velocity. This will insure that speed does not exceed the maximum.

Prey:

  • If traveling slower than the maximum speed, accelerate away from the predator at maximum acceleration.
  • If traveling at maximum velocity and the predator is a certain distance D away, stay the course.
  • If traveling at maximum speed and the predator is within striking distance, execute a turn away from the predator at the tightest turning radius possible.

Even without doing the simulations of this model we can foresee the qualitative features of the trajectories it yields. When the prey is further than D away from the predator, it will run along a straight trajectory which means that the speedier predator will eventually catch up with it and draw within the distance of caution D. At that point the prey will commence a sharp turn away from the predator. The predator, being less agile, will not be able to turn as sharply and will overshoot the prey and the distance between them will grow and might exceed D. When that happens, the prey will stop turning and run along a straight line again. The cycle will repeat ad infinitum the predator not being able to get closer to the prey than some finite fraction of D.

The movie below shows the trajectories of the prey (green) and predator (red) produced by the simple model when the predator is 50% faster, but the prey is able to achieve twice the acceleration.

Finally, let me point out that the strategies in the simple model are far from optimal. For example, one can imagine that if the predator could anticipate the direction of the prey’s turn (which is possible in the above scenario in which the prey always turns away from the predator), it could potentially intercept the prey. The optimality of a particular escape — pursuit strategies is usually hard to prove and the methods of such proofs are still subject of current research.

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