The discrete-time recursion system u_[n+1]=Q[u_n] with u_n(x) a vector of population distributions of species and Q an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on Q are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

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One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on 'linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper [11].

In the last three decades, several models on the interaction of glucose and insulin following the intra venous glucose tolerance test (IVGTT) have appeared in the literature. One of the mostly used one is generally known as the "minimal model" which was first published in 1979 and modified in 1986. Recently, this minimal model has been challenged by De Gaetano and Arino [4] from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called "dynamic model". Their model makes use of certain simple and specific functions and introduces time delay in a particular way. The outcome is that the model always admits a globally asymptotically stable steady state. The objective of this paper is to find out if and how this outcome depends on the specific choice of functions and the way delay is incorporated. To this end, we generalize the dynamical model to allow more general functions and an alternative way of incorporating time delay. Our findings show that in theory, such models can possess unstable positive steady states. However, for all conceivable realistic data, such unstable steady states do not exist. Hence, our work indicates that the dynamic model does provide qualitatively robust dynamics for the purpose of clinic application. We also perform simulations based on data from a clinic study and point out some plausible but important implications.

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c * , and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme [10] for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers.