John E. Franke

Mutual exclusion versus coexistence for discrete competitive systems

Using discrete competition models where the density dependent growth functions are either all exponential or all rational, notwithstanding the complex interactions of the species, we establish an exclusion principle. Moreover, in a 2-species discrete competition model where the growth functions are exponential and rational, an example is given illustrating coexistence when our conditions are satisfied. We obtain an exclusion principle for this 2-species model for some choice of parameters.

Geometry of exclusion principles in discrete systems

Abstract Using an n-species system of difference equations with very general density dependent growth functions, we prove that weak dominance and invariance gives the exclusion of all the (n − 1)-dominated species in the system. This result is applied to very specific competition models. An example of coexistence in a competition model with no invariance is studied. A notion of strong dominance is developed in a general setting and shown to imply exclusion of the (n − 1)-dominated species. An example of a competition model illustrating that weak dominance plus invariance does not imply strong dominance is given.

Discrete-Time SIS EpidemicModel in a Seasonal Environment

We study the combined effects of seasonal trends and diseases on the extinction and persistence of discretely reproducing populations. We introduce the epidemic threshold parameter, R0 , for predicting disease dynamics in periodic environments. Typically, in periodic environments, R0 > 1 implies disease persistence on a cyclic attractor, while R0 < 1 implies disease extinction. We also explore the relationship between the demographic equation and the epidemic process. In particular, we show that in periodic environments, it is possible for the infective population to be on a chaotic attractor while the demographic dynamics is nonchaotic.

Attractors for discrete periodic dynamical systems

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.

Global attractions in competitive systems

We establish the ecological principle of mutual exclusion as a mathematical theorem in a discrete system modeling the interactions of pioneer species

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark–Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.

Global attractivity and convergence to a two-cycle in a difference equation

We obtain conditions under which every positive solution of a difference equation of the form y n+1=y n-1 f(y n-1, y n ), n=0, 1, 2, … is attracted to its positive equilibrium. We also obtain conditions under which every positive solution approaches a two-cycle, which may be an equilibrium. The results apply to a population model with two age classes

The chain recurrent set, attractors, and explosions

Charles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x . In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Population models with periodic recruitment functions and survival rates

We study the combined effects of periodically varying carrying capacity and survival rate on populations. We show that our populations with constant recruitment functions do not experience either resonance or attenuance when either only the carrying capacity or the survival rate is fluctuating. However, when both carrying capacity and survival rate are fluctuating the populations experience either attenuance or resonance, depending on parameter regimes. In addition, we show that our populations with Beverton–Holt recruitment functions experience attenuance when only the carrying capacity is fluctuating.

Multiple attractors via CUSP bifurcation in periodically varying environments

Periodically forced (non-autonomous) single species population models support multiple attractors via tangent bifurcations, where the corresponding autonomous models support single attractors. Elaydi and Sacker obtained conditions for the existence of single attractors in periodically forced discrete-time models. In this paper, the Cusp Bifurcation Theorem is used to provide a general framework for the occurrence of multiple attractors in such periodic dynamical systems.