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Boyle and Krieger defined sub-matrices for representation matrices of sofic shift. This paper presents some details of relations between integral sub-matrices and representation matrices. Besides, we express a new version of the Decomposition Theorem by sub-matrices. Generally, strong shift equivalence (conjugacy) of sub-matrices does not apply to representation matrices, but we show that this result can be achieved by the fixed diagonal integral sub-matrix.

The HANDY model of Motesharrei, Rivas, and Kalnay examines interactions with the environment by human populations, both between poor and rich people, i.e., "Commoners" and "Elites". The Elites control the society's wealth and consume it at a higher rate than Commoners, whose work produces the wealth. We say a model is "Elite-dominated" when the Elites' per capita population change rate is always at least as large as the Commoners'. We can show the HANDY model always exhibits population crashes for all choices of parameter values for which it is Elite-dominated. But any such model with explicit equations raises questions of how the resulting behaviors depend on the details of the models. How important are the particular design features codified in the differential equations? In this paper, we first replace the explicit equations of HANDY with differential equations that are only described conceptually or qualitatively -using only conditions that can be verified for explicit systems. Next, we discard the equations entirely, replacing them with qualitative conditions, and we prove these conditions imply population collapse must occur. In particular, one condition is that the model is Elite-dominated. We show that the HANDY model with Elite-dominated parameters satisfies our hypotheses and thus must undergo population collapse. Our approach of introducing qualitative mathematical hypotheses can better show the underlying features of the model that lead to collapse. We also ask how societies can avoid collapse.

We investigate the d-dimensional model,where each z j (t) is a time fluctuating resource. This "nonautonomous" model is a generalization of an autonomous Lotka-Volterra d-dimensional model. It is nonautonomous in the sense that it is not specified how the z j (t) are determined. Write S T for the transpose of S = (s ij ). When the rank of S T is equal to d − k for k > 0, we show that at least k species will die out and will do so exponentially fast. For the proof, we invent a family of Lyapunov functions, a "team" of Lyapunov functions that work together to show that k species must die, where k is the dimension of the kernel of S. Each Lyapunov function provides constraints as to which species must die out and together they provide a picture of which are likely to die out. We present a "trophic" condition for that Lotka-Volterra systems that guarantees that there is a trapping region that is globally attracting.

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