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Bm . Thus we permit a graded response, supposing that if h(x) ∈ B0 (7) the pattern is not recognized, and if h(x) ∈ B1 (8) the pattern is recognized and some action bj , k + 1 ≤ j ≤ m takes place. The principal focus of interest is those composite paths x which trigger pattern recognition-and-response. That is, given a fixed initial state a0 , such that h(a0 ) ∈ B0 , we examine all possible subsequent paths x beginning with a0 and leading to the event h(x) ∈ B1 . , am ) ∈ B1 . For each positive integer n let N (n) be the number of grammatical and syntactic high probability paths of length n which begin with some particular a0 having h(a0 ) ∈ B0 and lead to the condition h(x) ∈ B1 .

Let X represent the appropriately scaled number of predators, Y the scaled number of prey, t the time, and ω a parameter defining their interaction. The model assumes that the ecologically dominant relation is an interaction between predator and prey, so that 26 R. Wallace and D. Wallace dX/dt = ωY (1) dY /dt = −ωX Thus the predator populations grows proportionately to the prey population, and the prey declines proportionately to the predator population. After differentiating the first and using the second equation, we obtain the differential equation (2) d2 X/dt2 + ω 2 X = 0 having the solution X(t) = sin(ωt); Y (t) = cos(ωt).

We suppose it possible to define a retranslation from the B-language into ˆ the translated the Y-language through a deterministic code book, and call Y information source, as mirrored by B. Define some distortion measure comparing paths y to paths yˆ, d(y, yˆ) [21]. We invoke the Rate Distortion Theorem’s mutual information I(Y, Yˆ ), which is the splitting criterion between high and low probability pairs of paths. Impose, now, a parametization by an inverse coupling strength K, and a renormalization symmetry representing the global structure of the system coupling.