@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_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{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} }