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We offer two improvements to prior results concerning global stability of the 2D Ricker Equation. We also offer some new methods of approach for the more pathological cases and prove some miscellaneous results including a special nontrivial case in which the mapping is conjugate to the 1D Ricker map along an invariant line and a proof of the non-existence of period-2 points.

We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to 1 2 √ T L T R where T L and T R are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier's Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.

The dynamics of the 2D Ricker equation depend on the positioning of its isoclines. We give a complete description for the parameter values where the two isoclines overlap. In this case, we show the plane is covered by a family of invariant curves and that the restriction to each is conjugate to a family of one‐dimensional population maps. Properties of these 1D maps, including global stability, are established, and we discuss the implications for the 2D Ricker equation.