Consider the differential equation y′=Ay where A has many positive eigenvalues. This is the simplest case of
a dynamical system containing multiple instabilities (i.e. positive feedback loops),
Where is the selection? It isn’t there. You have multiple independent exponential growth rates.
Consider y′=f(y) a chaotic system like a double pendulum. Fix y to a particular typical solution.
consider y′+z′=f(y+z) as a differential equation in z. Here z represents the difference between y and some other solution to x′=f(x). If you start at z=0 then z stays at 0. However, small variations will grow exponentially. After a while, you just get a difference between 2 arbitrary chaotic paths.
I can’t see a way of meaningfully describing these as optimizing processes with competing subagents. Arguably y′=Ay could be optimising |y|. However, this doesn’t seem canonical, as for any invertable B. z(0)=By(0) and z′=BAB−1z describes an exactly isomorphic system, but dosen’t preserve modulus. This isomorphism does preserve yTAy. That could be the thing being optimised.
+1. The multiple feedback loops have to be competing in some important sense; it’s just not true that “whenever there’s a dynamical system containing multiple instabilities (i.e. positive feedback loops) … there should be a canonical way to interpret that system as multiple competing subsystems...”
In the OP’s case study, the molecules are competing for scarce resources. More abstractly, perhaps we can say that there are multiple feedback loops such that when the system has travelled far enough in the direction pushed by one feedback loop, it destroys or otherwise seriously inhibits movement in the directions pushed by the other feedback loops.
Consider a pencil balanced on its point. It has multiple positive feedback loops, (different directions to fall in) and falling far in one direction prevents falling in others. But once it has fallen, it just sits there. That said, evolution can settle into a strong local minimum, and just sit there.
Consider the differential equation y′=Ay where A has many positive eigenvalues. This is the simplest case of
Where is the selection? It isn’t there. You have multiple independent exponential growth rates.
Consider y′=f(y) a chaotic system like a double pendulum. Fix y to a particular typical solution.
consider y′+z′=f(y+z) as a differential equation in z. Here z represents the difference between y and some other solution to x′=f(x). If you start at z=0 then z stays at 0. However, small variations will grow exponentially. After a while, you just get a difference between 2 arbitrary chaotic paths.
I can’t see a way of meaningfully describing these as optimizing processes with competing subagents. Arguably y′=Ay could be optimising |y|. However, this doesn’t seem canonical, as for any invertable B. z(0)=By(0) and z′=BAB−1z describes an exactly isomorphic system, but dosen’t preserve modulus. This isomorphism does preserve yTAy. That could be the thing being optimised.
+1. The multiple feedback loops have to be competing in some important sense; it’s just not true that “whenever there’s a dynamical system containing multiple instabilities (i.e. positive feedback loops) … there should be a canonical way to interpret that system as multiple competing subsystems...”
In the OP’s case study, the molecules are competing for scarce resources. More abstractly, perhaps we can say that there are multiple feedback loops such that when the system has travelled far enough in the direction pushed by one feedback loop, it destroys or otherwise seriously inhibits movement in the directions pushed by the other feedback loops.
Consider a pencil balanced on its point. It has multiple positive feedback loops, (different directions to fall in) and falling far in one direction prevents falling in others. But once it has fallen, it just sits there. That said, evolution can settle into a strong local minimum, and just sit there.
Mmm, good point. My hasty generalization was perhaps too hasty. Perhaps we need some sort of robust-to-different-initial-conditions sort of criterion.