Hi group, I'm currently trying to understand a physicists written paper on ecological models. In there, they used the term "quadratic potential" when comparing spatial diffusion with a spring system (see attachment 1). After searching online for this term, I found nothing directly relevant to the material at hand. Can someone tell me what it means intuitively, whether it's represented by a canonical equation, and if so, whether the derivation of that gives the form seen in the paper (attachment 2)? Thanks! For the original equations the variables in attachment 2 refers to, see attachment (3).
hi nigels! "quadratic potential" just means that the potential energy is kx^{2} (or k(x-a)^{2} if the equilibrium position is at x = a) … it results in force being proportional to distance for example, a spring has PE = 1/2 kx^{2} (i doubt there's a rigorous justification for this model … i don't think animals are actually connected to springs! )
Thanks, tiny-tim! That makes much more sense now. However, I noticed that in the attachment, the spring constant is 1/4. Is that because the point is attached by a separate spring on both sides? Somehow that's equivalent to have two parallel springs for some reason? By the way, the paper actually does make the implicit analogy that animals are attached to springs when they move from their den sites. Oh theoreticians...
i don't know what γ θ and φ are i think the hitch-hiker's guide to the galaxy made a similar assumption about humans and their home planets
θ(t) and φ(t) are just functions that modifies the diffusion coefficient K to be time-dependent. γ is defined as the "rate at which territory sizes tend to return to the mean size", which I understand as "the rate at which L_1 returns to its initial state". Yet still, I can't explain the 1/4 spring constant that seems to be used (in attachment #2). I mean, should the constant be 1/2 + 1/2 = 1 since the springs are in a series? It's been a while since I took intro physics.. Here's the open-source paper in case you're curious. http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034033 Ugh..it's always a nightmare when physicists work on biology problems.