It depends on what you mean by “escape”, and what you view as the alternative.
I suspect that the pursuer could never converge on the same instantaneous point, given sufficient initial distance (and orientation). At a certain distance, the prey could enter a stable orbit around the pursuer. I don’t have a mathematical proof but I strongly suspect this to be the case,and I can envision the structure of a proof.
Could the prey infinitely extend the gap between themselves and the pursuer? No. I don’t have the tooling to actually present such a proof, but of that one I am confident.
I think if you introduced concepts of obstacles and a “radius of escape” (where if the gap meets a threshold the predator is permanently foiled), then there are almost certainly scenarios where the prey could escape.
We actually see this scenario play out in nature all the time
Although, I’m realizing that for completeness, there probably are mathematical constraints around the relationship between the required absolute values of turning speeds and movement speeds. They’re kinda egde-casey for any practically imagined scenarios, but would come into play for a rigorous proof.
It depends on what you mean by “escape”, and what you view as the alternative.
I suspect that the pursuer could never converge on the same instantaneous point, given sufficient initial distance (and orientation). At a certain distance, the prey could enter a stable orbit around the pursuer. I don’t have a mathematical proof but I strongly suspect this to be the case,and I can envision the structure of a proof.
Could the prey infinitely extend the gap between themselves and the pursuer? No. I don’t have the tooling to actually present such a proof, but of that one I am confident.
I think if you introduced concepts of obstacles and a “radius of escape” (where if the gap meets a threshold the predator is permanently foiled), then there are almost certainly scenarios where the prey could escape.
We actually see this scenario play out in nature all the time
WOW, really nice explanation.
Thank you!
Although, I’m realizing that for completeness, there probably are mathematical constraints around the relationship between the required absolute values of turning speeds and movement speeds. They’re kinda egde-casey for any practically imagined scenarios, but would come into play for a rigorous proof.