Dedication

It is with great pleasure and respect that we dedicate this special issue of the Journal of Difference Equations and Applications (JDEA) to our esteemed colleague and friend, Jim M. Cushing, in celebration of his 80th birthday. Jim grew up in the small town of Cheyenne, Wyoming, which he left in 1960 for university studies. He graduated from the University of Colorado, Boulder, in 1964 with a Bachelor of Arts degree in mathematics. Four years later he received a Ph.D. in applied mathematics at the University of Maryland, College Park. From 1968 to 2021 he held a professorship at the University of Arizona, Tucson, in the Department of Mathematics, where he was also an affiliate member of the Interdisciplinary Programme in Applied Mathematics since its creation in 1976. This long stint was interrupted by post-doctoral fellowships at the T. J. Watson IBM Research Center in Yorktown Heights, New York, and an Alexander von Humboldt Fellowship at the University of Tübingen, Germany. Jim retired in May 2021 and now holds an emeritus professorship at the University of Arizona.

Professor Cushing has made foundational contributions to difference equations and their applications, particularly in the field of mathematical biology with a focus on structured population dynamics. His prolific career spans over 47 years, with significant contributions to difference equations for more than 27 years. He has authored six books and more than 180 papers, laying the groundwork for many recent advances in the theory of difference equations applied to structured population dynamics. In recognition of his substantial impact, Jim was honoured as a Fellow of the American Mathematical Society in 2013 and received the Bernd Aulbach Prize from the ISDE in 2021. Below, we highlight some of Jim's major research contributions that have profoundly influenced the theory and application of difference equations.

Professor Cushing's work in difference equations includes significant advancements in asymptotic theory and bifurcation dynamics. His early papers on nonlinear matrix population models in the late 1980s and early 1990s, such as ‘Nonlinear matrix models and population dynamics’ (1988), ‘A strong ergodic theorem for nonlinear matrix models for the dynamics of structured populations’ (1989), and ‘The net reproductive number and stability in matrix population models’ (1994), set the stage for subsequent research. In these works, Professor Cushing generalized the strong ergodic theorem of demography in linear systems $X(t+1)=\mathrm{AX}(t)$ to nonlinear systems of the form $X(t+1)=h(X(t))\mathrm{AX}(t),$ where A is nonnegative, irreducible and primitive and $h:{\mathbb{R}}^n\to (0,1]$, $h(0)=1$, to show: $\underset{t\to \mathrm{\infty }}{lim}X(t)/||X(t)\Vert =\eta ,$ where η is the positive unit eigenvector associated with the dominant eigenvalue r of A. It was also shown that these conditions do not imply convergence of the total population size to $\Vert \eta \Vert$ (periodic behaviour is possible). In the linear case, for the general age-structured population models, $X(t+1)=\mathrm{AX}(t)$, Professor Cushing established an important relation between the population growth rate r and the basic reproduction number $R_0$ (defined in terms of the matrix of fertilities F and the matrix of transition probabilities, T, A = F + T): $r<(=,>)1\mathrm{iff}{R}_0<(=,>)1.$In addition, an ordered relation between $R_0$ and r was established: $R_0<1\mathrm{implies}{R}_0\le r<1\mathrm{and}{R}_0>1\mathrm{implies}1<r\le R_0.$The value $R_0$ in demography has an important biological interpretation as the expected number of offspring per individual over its lifetime. The value of $R_0$ can be directly calculated from T and F. The importance of r and $R_0$ as bifurcation values was also established in the 1988 and 1989 papers, where the bifurcation behaviour could be subcritical or supercritical. This early work with extensions and numerous applications to more general nonlinear matrix equations, uniform persistence and periodic habitats are summarized in his 1998 book published by SIAM, An Introduction to Structured Population Dynamics. This book serves as a classic reference in discrete-time structured population dynamics.

Professor Cushing applied his theoretical insights to various biologically significant settings, including cannibalism, competition, Allee effects, semelparous populations, and evolution. A notable application is his collaborative work on flour beetle (Tribolium castaneum) populations, known as the LPA model (Larvae, Pupae, Adults), a system of three difference equations. His book Chaos in Ecology: Experimental Nonlinear Dynamics (2003) and related publications, such as “Chaotic dynamics in an insect population” (1997), have significantly impacted the scientific community. A significant contribution of this work was the rigorous experimental confirmation of the existence of numerous nonlinear phenomena in populations, including cycles, phase switching, and chaos. Another notable application is Jim's interdisciplinary collaboration which links field studies with mathematical models to describe and predict animal behaviour, such as egg cannibalism and reproductive synchrony in gulls. This work is summarized in his recent book published by Springer Nature, Modeling Behavior and Population Dynamics: Seabirds, Seals, and Marine Iguanas (2023).

More recently, Professor Cushing's research has focussed on evolutionary dynamics. He extended the classical dynamic dichotomy for discrete population models to evolutionary settings. Utilizing game theory and a juvenile-adult population model, he demonstrated that cannibalism can be an evolutionarily stable strategy (ESS) in his 2015 paper ‘An evolutionary game theoretic model of cannibalism.’ Additionally, he explored the phenomenon of partial migration, attributing it to negative density dependence alone, in his 2017 paper ‘The puzzle of partial migration: adaptive dynamics and evolutionary game theory perspectives.’ He also established significant findings on Allee effects in evolutionary settings in ‘The evolutionary dynamics of a population model with a strong Allee effect’ (2015).

Professor Cushing has also made significant contributions to the field through his mentorship. He is well know as an ideal mentor who, through his inclusiveness, compassion, and pursuit of mathematical rigor, has helped his students to become strong, independent researchers. In total, he has trained 10 postdoctoral scholars and 16 graduate students, in addition to involving numerous undergraduate students in his research. Many of his former students are women, minorities or international students.

In summary, Professor Cushing's significant research contributions have advanced the theory of difference equations and their applications to population dynamics. His work has laid the foundation for much of the recent research in structured population dynamics and inspired many researchers.