Two competing models currently offer to explain empirical regularities observed in food webs. The Lotka-Volterra model describes population dynamics; the cascade model describes trophic structure. In a real ecological community, both population dynamics and trophic structure are important. This paper proposes and analyses a new hybrid model that combines population dynamics and trophic structure: the Lotka-Volterra cascade model (LVCM). The LVCM assumes the population dynamics of the Lotka-Volterra model when the interactions between species are shaped by a refinement of the cascade model. A critical surface divides the three-dimensional parameter space of the LVCM into two regions. In one region, as the number of species becomes large, the limiting probability that the LVCM is qualitatively globally asymptotically stable is positive. In the region on the other side of the critical surface, and on the critical surface itself, this limiting probability is zero. Thus the LVCM displays an ecological phase transition: gradual changes in the probabilities of various kinds of population dynamical interactions related to feeding can have sharp effects on a community’s qualitative stability. The LVCM shows that an inverse proportionality between connectance and the number of species, and a direct proportionality between the number of links and the number of species, as observed in data on food webs, need not be directly connected with the qualitative global asymptotic stability or instability of population dynamics. Empirical testing of the LVCM will require field data on the population dynamical effects of feeding relations.

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Abstract. In this paper, the problem on periodic solutions of the bidirectional associative memory neural networks with both periodic coefficients and periodic time-varying delays is discussed. By using degree theory, inequality technique and Lyapunov functional, we establish the existence, uniqueness, and global asymptotic stability of a periodic solution. The obtained results of stability are less restrictive than previously known criteria, and the hypotheses for the boundedness and monotonicity on the activation functions are removed.

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This paper investigates the problem of robustness analysis for descriptor systems with parameter uncertainties in both the derivative and state matrices. Using a parameter dependent Lyapunov function, we derive a linear matrix inequality (LMI) based sufficient condition for the admissibility of the system. Unlike the existing results, our criterion has no restriction on the rank of the derivative matrix. Further, we use the obtained method to study interval descriptor systems and multi-parameter singular perturbed systems. The proposed approaches overcome some drawbacks of the existing results. Finally, we present two numerical examples to show the effectiveness of the main results.