@article{morena_franke_2012, title={Predicting attenuant and resonant 2-cycles in periodically forced discrete-time two-species population models}, volume={6}, ISSN={["1751-3766"]}, DOI={10.1080/17513758.2012.710338}, abstractNote={Periodic environments may either enhance or suppress a population via resonant or attenuant cycles. We derive signature functions for predicting the responses of two competing populations to 2-periodic oscillations in six model parameters. Two of these parameters provide a non-trivial equilibrium and two provide the carrying capacities of each species in the absence of the other, but the remaining two are arbitrary and could be intrinsic growth rates. Each signature function is the sign of a weighted sum of the relative strengths of the oscillations of the perturbed parameters. Periodic environments are favourable for populations when the signature function is positive and are deleterious if the signature function is negative. We compute the signature functions of four classical, discrete-time two-species populations and determine regions in parameter space which are either favourable or detrimental to the populations. The six-parameter models include the Logistic, Ricker, Beverton–Holt, and Hassell models.}, number={2}, journal={JOURNAL OF BIOLOGICAL DYNAMICS}, author={Morena, Matthew A. and Franke, John E.}, year={2012}, pages={782–812} }
 @article{franke_yakubu_2011, title={PERIODICALLY FORCED DISCRETE-TIME SIS EPIDEMIC MODEL WITH DISEASE INDUCED MORTALITY}, volume={8}, ISSN={["1551-0018"]}, DOI={10.3934/mbe.2011.8.385}, abstractNote={We use a periodically forced SIS epidemic model with disease induced mortality to study the combined effects of seasonal trends and death on the extinction and persistence of discretely reproducing populations. We introduce the epidemic threshold parameter, R0 , for predicting disease dynamics in periodic environments. Typically, R0 <1 implies disease extinction. However, in the presence of disease induced mortality, we extend the results of Franke and Yakubu to periodic environments and show that a small number of infectives can drive an otherwise persistent population with R0 >1 to extinction. Furthermore, we obtain conditions for the persistence of the total population. In addition, we use the Beverton-Holt recruitment function to show that the infective population exhibits period-doubling bifurcations route to chaos where the disease-free susceptible population lives on a 2-cycle (non-chaotic) attractor.}, number={2}, journal={MATHEMATICAL BIOSCIENCES AND ENGINEERING}, author={Franke, John E. and Yakubu, Abdul-Aziz}, year={2011}, month={Apr}, pages={385–408} }
 @article{franke_yakubu_2008, title={Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models}, volume={57}, ISSN={["1432-1416"]}, DOI={10.1007/s00285-008-0188-9}, abstractNote={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.}, number={6}, journal={JOURNAL OF MATHEMATICAL BIOLOGY}, author={Franke, John E. and Yakubu, Abdul-Aziz}, year={2008}, month={Dec}, pages={755–790} }
 @article{franke_yakubu_2007, title={Using a signature function to determine resonant and attenuant 2-cycles in the Smith-Slatkin population model}, volume={13}, ISSN={["1563-5120"]}, DOI={10.1080/10236190601078987}, abstractNote={We study the responses of discretely reproducing populations to periodic fluctuations in three parameters: the carrying capacity and two demographic characteristics of the species. We prove that small 2-periodic fluctuations of the three parameters generate 2-cyclic oscillations of the population. We develop a signature function for predicting the responses of populations to 2-periodic fluctuations. Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the three parameters. Periodic environments are deleterious for populations when the signature function is negative, while positive signature functions signal favorable environments. We compute the signature function for the Smith–Slatkin model, and use it to determine regions in parameter space that are either favorable or detrimental to the species.}, number={4}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Franke, John E. and Yakubu, Abdul-Aziz}, year={2007}, month={Apr}, pages={289–308} }
 @article{ernel'yanov_wolff_2006, title={Asymptotic behavior of Markov semigroups on preduals of von Neumann algebras}, volume={314}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2005.04.016}, abstractNote={We develop a new approach for investigation of asymptotic behavior of Markov semigroup on preduals of von Neumann algebras. With using of our technique we establish several results about mean ergodicity, statistical stability, and constrictiviness of Markov semigroups.}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Ernel'yanov, EY and Wolff, MPH}, year={2006}, month={Feb}, pages={749–763} }
 @article{franke_yakubu_2006, title={Discrete-time sis epidemic model in a seasonal environment}, volume={66}, ISSN={["1095-712X"]}, DOI={10.1137/050638345}, abstractNote={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.}, number={5}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Franke, John E. and Yakubu, Abdul-Aziz}, year={2006}, pages={1563–1587} }
 @article{franke_yakubu_2006, title={Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models}, volume={204}, ISSN={["0025-5564"]}, DOI={10.1016/j.mbs.2006.08.016}, abstractNote={We use a periodically forced density-dependent compensatory Leslie model to study the combined effects of environmental fluctuations and age-structure on pioneer populations. In constant environments, the models have globally attracting positive fixed points. However, with the advent of periodic forcing, the models have globally attracting cycles. We derive conditions under which the cycle is attenuant, resonant, and neither attenuant nor resonant. These results show that the response of age-structured populations to environmental fluctuations is a complex function of the compensatory mechanisms at different life-history stages, the fertile age classes and the period of the environment.}, number={1}, journal={MATHEMATICAL BIOSCIENCES}, author={Franke, John E. and Yakubu, Abdul-Aziz}, year={2006}, month={Nov}, pages={1–20} }
 @article{franke_yakubu_2006, title={Signature function for predicting resonant and attenuant population 2-cycles}, volume={68}, ISSN={["1522-9602"]}, DOI={10.1007/s11538-006-9086-8}, abstractNote={Populations are either enhanced via resonant cycles or suppressed via attenuant cycles by periodic environments. We develop a signature function for predicting the response of discretely reproducing populations to 2-periodic fluctuations of both a characteristic of the environment (carrying capacity), and a characteristic of the population (inherent growth rate). Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the carrying capacity and the demographic characteristic. Periodic environments are deleterious for populations when the signature function is negative. However, positive signature functions signal favorable environments. We compute the signature functions of six classical discrete-time single species population models, and use the functions to determine regions in parameter space that are either favorable or detrimental to the populations. The two-parameter classical models include the Ricker, Beverton-Holt, Logistic, and Maynard Smith models.}, number={8}, journal={BULLETIN OF MATHEMATICAL BIOLOGY}, author={Franke, John E. and Yakubu, Abdul-Aziz}, year={2006}, month={Nov}, pages={2069–2104} }
 @article{franke_yakubu_2005, title={Multiple attractors via CUSP bifurcation in periodically varying environments}, volume={11}, ISSN={["1023-6198"]}, DOI={10.1080/10236190412331335436}, abstractNote={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.}, number={4-5}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Franke, JE and Yakubu, AZ}, year={2005}, month={Apr}, pages={365–377} }
 @article{franke_yakubu_2005, title={Periodic dynamical systems in unidirectional metapopulation models}, volume={11}, ISSN={["1563-5120"]}, DOI={10.1080/10236190412331334563}, abstractNote={In periodically varying environments, population models generate periodic dynamical systems. To understand the effects of unidirectional dispersal on local patch dynamics in fluctuating environments, dynamical systems theory is used to study the resulting periodic dynamical systems. In particular, a unidirectional dispersal linked two patch nonautonomous metapopulation model is constructed and used to explain the qualitative dynamics of linked versus unlinked independent patches. As in single-patch, single-species population models, unidirectional nonautonomous models support multiple attractors where local population models support single attractors.}, number={7}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Franke, JE and Yakubu, AA}, year={2005}, month={Jun}, pages={687–700} }
 @article{franke_yakubu_2005, title={Population models with periodic recruitment functions and survival rates}, volume={11}, ISSN={["1563-5120"]}, DOI={10.1080/10236190500386275}, abstractNote={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.}, number={14}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Franke, JE and Yakubu, AA}, year={2005}, month={Dec}, pages={1169–1184} }
 @article{chan_franke_2004, title={Probabilities of extinction, weak extinction permanence, and mutual exclusion in discrete, competitive, Lotka-Volterra systems}, volume={47}, DOI={10.1016/S0898-1221(04)00018-5}, number={03-Feb}, journal={Computers & Mathematics With Applications}, author={Chan, D. M. and Franke, J. E.}, year={2004}, pages={365–379} }
 @article{chan_franke_2004, title={Probabilities of extinction, weak extinction, permanence, and mutual exclusion in discrete, competitive, Lotka-Volterra systems that involve invading species}, volume={40}, ISSN={["1872-9479"]}, DOI={10.1016/j.mcm.2004.10.013}, abstractNote={The probabilities of various biological asymptotic dynamics are computed for a stable system that is invaded by another competing species. The asymptotic behaviors studied include extinction, weak extinction, permanence, and mutual exclusion. The model used is a discrete Lotka-Volterra system that models species that compete for the same resources. Among the results found are that the chance of permanence occurring in the invaded system is significantly higher than the probability of permanence in a purely random system, and that multiple extinctions that include the invading species and one of the original species are impossible.}, number={7-8}, journal={MATHEMATICAL AND COMPUTER MODELLING}, author={Chan, DM and Franke, JE}, year={2004}, month={Oct}, pages={809–821} }
 @article{franke_selgrade_2003, title={Attractors for discrete periodic dynamical systems}, volume={286}, ISSN={["0022-247X"]}, DOI={10.1016/S0022-247X(03)00417-7}, abstractNote={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.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Franke, JE and Selgrade, JF}, year={2003}, month={Oct}, pages={64–79} }
 @article{chan_franke_2001, title={Multiple extinctions in a discrete competitive system}, volume={2}, ISSN={["1468-1218"]}, DOI={10.1016/S0362-546X(99)00282-5}, number={1}, journal={NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS}, author={Chan, DM and Franke, JE}, year={2001}, month={Mar}, pages={75–91} }
 @article{francke_yakubu_1999, title={Exclusionary population dynamics in size-structured, discrete competitive systems}, volume={5}, ISSN={["1023-6198"]}, DOI={10.1080/10236199908808185}, abstractNote={A discrete multi-species size-structured competition model is considered. By using decreasing growth functions, we achieve the self-regulation of species. We develop various biologically significant conditions for global convergence to the extinction state of the dominated species in the competitive system. With an example we illustrate coexistence in a chaotic supr transient. The chaotic attractor has an unusual pulsating nature.}, number={3}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Francke, JE and Yakubu, AA}, year={1999}, pages={235–249} }
 @article{franke_hoag_ladas_1999, title={Global attractivity and convergence to a two-cycle in a difference equation}, volume={5}, ISSN={["1023-6198"]}, DOI={10.1080/10236199908808180}, abstractNote={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}, number={2}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Franke, JE and Hoag, JT and Ladas, G}, year={1999}, pages={203–209} }
 @article{franke_yakubu_1997, title={Principles of competitive exclusion for discrete populations with reproducing juveniles and adults}, volume={30}, ISSN={["0362-546X"]}, DOI={10.1016/S0362-546X(97)00299-X}, abstractNote={We invent notions of dominance and weak dominance for discrete multi-species systems with competing juveniles and adults. In this model, both the juveniles and adults are allowed to reproduce. We prove that, a dominant species drives the dominated species to extinction. In discrete juvenile-adult systems that do not allow juveniles to reproduce, it is known that weak dominance is equivalent to dominance, provided all the growth functions are exponentials. We show that if juveniles are allowed to reproduce, then a weakly dominant species could be driven to extinction. It all the growth functions are exponential functions, we obtain sufficient conditions that guarantee the equivalence of weak dominance and dominance.}, number={2}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Franke, JE and Yakubu, AA}, year={1997}, month={Dec}, pages={1197–1205} }