
Jim Michael CushingUniversity of Arizona | UA · Department of Mathematics
Jim Michael Cushing
Ph.D.
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250
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Introduction
Jim M Cushing is professor emeritus in the Department of Mathematics at the University of Arizona, Tucson, Arizona. He does research in Applied Mathematics with applications to population, evolutionary, and ecological dynamics.
Additional affiliations
September 1968 - present
Education
September 1964 - June 1968
September 1961 - August 1964
Publications
Publications (250)
In Chapter 1, we studied both linear and nonlinear single-species population models represented by one-dimensional difference equations. In Chapter 2, we studied linear age- or size-structured population models represented by systems of linear difference equations.
The main focus of this chapter is on the introduction of models that exhibit multiple attractors. Most single-species density-dependent population models without structure that are widely studied, such as the discrete logistic (Beverton-Holt) equation (1.25), the Leslie-Gower competition model (3.34), and the Ricker (exponential) competition model...
The models studied in the previous chapters, the \(k\times k\) matrix model
for structured populations in Chapter 6 and its special \(k=1\) case
In Chapter 1, we considered single (scalar) difference equations as models for the discrete-time dynamics of a biological population.
In Chapter 2 we studied systems of linear difference equations the form
The main focus of this chapter is on the introduction of single-species population models and their analysis. Of fundamental interest in population dynamics is how population size changes in time. Is the population increasing, decreasing, fluctuating, or even chaotic? Is it in danger of extinction or will it persist? What are the characteristics of...
Infectious diseases are among the leading causes of mortality worldwide, presenting complex challenges in public health, economic stability, and social structures. The continuous emergence and re-emergence of pathogens, driven by factors such as global mobility, environmental changes, and microbial adaptation, necessitate dynamic and robust respons...
In the previous chapters, we used autonomous population models with constant model parameters (coefficients) to study fundamental principles in theoretical population biology. These parameters account for a variety of vital rates, mechanisms, and processes that affect a population’s growth or decay, including fertility rates, survival probabilities...
The theory of the dynamics of infectious diseases is one of the oldest branches of mathematical biology. In 1927, Kermack and McKendrick raised mathematical epidemiology to a new level when they introduced a continuous-time mathematical model of infectious diseases [187]. The seminal deterministic continuous-time infectious disease model framework...
This book presents contributions related to new research results presented at the 27th International Conference on Difference Equations and Applications, ICDEA 2022, that was held at CentraleSupélec, Université Paris-Saclay, France, under the auspices of the International Society of Difference Equations (ISDE), July 18–22, 2022. The book aims not o...
This article is devoted to the first steps of nine mathematicians
from five countries on their path to mathematics, chaos and discrete
dynamical systems, some from early childhood. In these life
stories, the names of outstanding mathematicians arise, crisscrossing
the nine stories in unexpected ways. These mathematicians also
interacted with each o...
We consider a Darwinian (evolutionary game theoretic) version of a standard susceptible-infectious SI model in which the resistance of the disease causing pathogen to a treatment that prevents death to infected individuals is subject to evolutionary adaptation. We determine the existence and stability of all equilibria, both disease-free and endemi...
Our purpose in this chapter is to address a significant time-scale discrepancy between the models studied in Chaps. 11 and 12 and the gull populations that motivated them. This discrepancy arises because, unlike in the models, reproductive synchrony and juvenile maturation in gulls occur on different time scales.
In this chapter we apply the differential equation-based modeling techniques developed in Chap. 1 to a marine mammal behavioral system. This research demonstrated that the same modeling techniques developed for bird behavior can be applied more broadly to other organisms. James Hayward, Shandelle Henson, and Brian Dennis, with undergraduate student...
Research reported in this chapter was the initial project carried out by the Seabird Ecology Team. It provided one of the first rigorous demonstrations that a differential equation model using environmental variables can be used to describe, explain, and accurately predict the aggregate behavior of marine vertebrates. It also served as the impetus...
Motivated by the empirical findings in Chaps. 7–9 that egg-laying synchrony is adaptive in the presence of egg cannibalism in gulls, we now extend the proof-of-concept cannibalism model in Chap. 11 to include the possibility of reproductive synchrony.
The study reported in this chapter was carried out in the Galá pagos Islands in parallel with the work on flightless cormorants reported in Chap. 5. Here the application of differential equation modeling to a marine reptilian system further demonstrates the broad applicability of this technique. This work formed the basis of a Master of Science the...
Chapters 1–3 report studies that focus on single state variables within behavioral systems. Behavioral systems, however, involve multiple state variables. In this chapter we demonstrate the efficacy of differential equation-based modeling which focuses on two state variables, behavior and habitat occupancy. This research was initially published in...
Climate change results in a broad range of changes in animals including physiological responses, predator-prey relations, and feeding tactics. In this chapter we explore the impact of changes in sea surface temperature on egg cannibalism, a feeding tactic used by gulls. We use logistic regression and information-theoretic methods of model selection...
In this chapter we use a modification of the classic Lotka-Volterra predator-prey ordinary differential equation model to test whether rebounding bald eagle numbers were primarily responsible for declining numbers of gulls nesting on Protection Island, Washington.
In this chapter we employ logistic regression and Darwinian dynamics to explore how a behavior with a crucial physiological function might be co-opted to function as a coping behavior in response to stress generated by the presence of predators. This study was initially reported in the Journal of Biological Dynamics in 2012 by senior researchers Sh...
A wide variety of natural systems exhibit spontaneous oscillator synchrony. In this chapter we use discrete-time mathematical modeling and Monte Carlo methods to provide the first demonstration of synchronous, every-other-day egg laying in seabirds. Along with Chap. 7, this chapter provides background for empirical and theoretical analyses that app...
This chapter is based on work carried out in the Galápagos Islands, Ecuador in 2011. James Hayward and Susana Velastegui Chávez, with two graduate students, Libby Megna and Brianna Payne, collected the data. Shandelle Henson carried out the data analysis, which involved Poisson regression, logistic regression, and multi-model inference based on the...
In Chap. 7 we described the occurrence of egg cannibalism by gulls, and in Chap. 8 we reported that under certain conditions female gulls lay their eggs on a synchronous, every-other-day schedule. Here we use logistic regression, chi-square analysis, and Monte Carlo simulations to show that every-other-day egg-laying synchrony functions as an adapt...
In this chapter we set up the modeling framework for a study of cannibalism by means of low-dimensional structured population models that distinguish only between cannibals and victims. Our specific interest is on adult cannibalism of juveniles . In Sect. 11.2 we use matrix modeling methodology [5, 7, 8] to develop and analyze a general discrete-ti...
We derive and study a Darwinian dynamic model based on a low-dimensional discrete- time population model focused on two features: density-dependent fertility and a trade-off between inherent (density free) fertility and post-reproduction survival. Both features are assumed to be dependent on a phenotypic trait subject to natural selection. The mode...
We prove bifurcation theorems for evolutionary game theoretic (Darwinian dynamic) versions of nonlinear matrix equations for structured population dynamics. These theorems generalize existing theorems concerning the bifurcation and stability of equilibrium solutions when an extinction equilibrium destabilizes by allowing for the general appearance...
For nonlinear scalar difference equations that arise in population dynamics the geometry of the graph obtained by plotting the population growth rate as a function of inherent fertility leads to information about the number of positive equilibria and about the local stability of positive equilibria. Specifically, equilibria on decreasing segments o...
The classic Ricker equation \(x_{t+1}=bx_{t}\exp \left( -cx_{t}\right) \) has positive equilibria for \(b>1\) that destabilize when \(b>e^{2}\) after which its asymptotic dynamics are oscillatory and complex. We study an evolutionary version of the Ricker equation in which coefficients depend on a phenotypic trait subject to Darwinian evolution. We...
We prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the...
We derive and analyze a Darwinian dynamic model based on a general difference equation population model under the assumption of a trade-off between fertility and survival. Both inherent and density dependent terms are functions of a phenotypic trait (subject to Darwinian evolution) and its population mean. We prove general theorems about the existe...
We describe the evolutionary game theoretic methodology for extending a difference equation population dynamic model in a way so as to account for the Darwinian evolution of model coefficients. We give a general theorem that describes the familiar transcritical bifurcation that occurs in non-evolutionary models when theextinction equilibrium destab...
For structured populations with an annual breeding season, life-stage interactions and behavioral tactics may occur on a faster time scale than that of population dynamics. Motivated by recent field studies of the effect of rising sea surface temperature (SST) on within-breeding-season behaviors in colonial seabirds, we formulate and analyze a gene...
1 Abstract
Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class‐dependent. The classic derivation of such models from partial differential equations produces time delays in...
One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projection matrix is primitive by showing the existence of a...
I discuss one dimensional maps as discrete time models of population dynamics from an extinction-versus-survival point of view by means of bifurcation theory. I extend this approach to a version of these population models that incorporates the dynamics of a single phenotypic trait subject to Darwinian evolution. This is done by proving a fundamenta...
We consider the phenomenon of partial migration which is exhibited by populations in which some individuals migrate between habitats during their lifetime, but others do not. First, using an adaptive dynamics approach, we show that partial migration can be explained on the basis of negative density dependence in the per capita fertilities alone, pr...
The basic reproduction number _R_0 is, by definition, the expected life time number of offspring of a newborn individual. An operationalization entails a specification of what events are considered as “reproduction” and what events are considered as “transitions from one individual-state to another”. Thus, an element of choice can creep into the co...
Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extincti...
We study a discrete time, structured population dynamic model that is motivated by recent field observations concerning certain life history strategies of colonial-nesting gulls, specifically the glaucous-winged gull (Larus glaucescens). The model focuses on mechanisms hypothesized to play key roles in a population's response to degraded environmen...
The following article describes the joint research the three authors have conducted for many years at Protection Island, a federally protected National Wildlife Refuge in the Strait of Juan de Fuca in Washington State, and at Galapagos National Park, Ecuador. Their research, which is sponsored by the National Science Foundation, includes a great nu...
The book contains recent developments and contemporary research in mathematical analysis and in its application to problems arising from the biological and physical sciences. The book is of interest to readers who wish to learn of new research in such topics as linear and nonlinear analysis, mathematical biology and ecology, dynamical systems, grap...
These proceedings of the 18th International Conference on Difference Equations and Applications cover a number of different aspects of difference equations and discrete dynamical systems, as well as the interplay between difference equations and dynamical systems. The conference was organized by the Department of Mathematics at the Universitat Autò...
As exemplified by classic Lotka–Volterra theory, there are several canonical outcomes possible to a two species (interference) competitive interaction: coexistence, initial condition-dependent competitive exclusion of one species, or the global exclusion of one species. Evolutionary versions of Lotka–Volterra dynamics have been investigated in orde...
Cannibalism, which functions as a life history trait in at least 1300 species of both invertebrates and vertebrates, plays important ecological and evolutionary roles in populations. During times of low resource availability, cannibalism of juveniles by adults can redirect reproductive energy to times of higher resource availability. For example, p...
We consider a multi-trait evolutionary (game theoretic) version of a two class (juvenile-adult) semelparous Leslie model. We prove the existence of both a continuum of positive equilibria and a continuum of synchronous 2-cycles as the result of a bifurcation that occurs from the extinction equilibrium when the net reproductive number R-0 increases...
No abstract is available for this article.
An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evol...
Professors Jim Hayward, Shandelle Henson and Jim Cushing form part of an interdisciplinary group of biologists and mathematicians who focus their studies on the behavioural ecology of marine organisms. Here they discuss several aspects of their research and how they enable underrepresented groups to become part of it.
This brief survey of nonlinear Leslie models focuses on the fundamental bifurcation that occurs when the extinction equilibrium destabilizes as R
0 increases through 1. Of particular interest is the bifurcation that occurs when only the oldest age class is reproductive, in which case the Leslie projection matrix is not primitive. This case is disti...
The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple....
In nonlinear matrix models, strong Allee effects typically arise when the fundamental bifurcation of positive equilibria from the extinction equilibrium at r=1 (or R0=1) is backward. This occurs when positive feedback (component Allee) effects are dominant at low densities and negative feedback effects are dominant at high densities. This scenario...
Papers by Invited Speakers: Competitive exclusion through discrete time models: Azmy S. Ackleh and Paul L. Salceanu.- Benford solutions of linear difference equations: Arno Berger and Gideon Eshun.- Harvesting and dynamics in some one-dimensional population models: Eduardo Liz and Frank M. Hilker.- Chaos and wild chaos in Lorenz-type systems: Hinke...
The bifurcation that occurs from the extinction equilibrium in a basic discrete time, nonlinear juvenile-adult model for semelparous populations, as the inherent net reproductive number R0 increases through 1, exhibits a dynamic dichotomy with two alternatives: an equilibrium with overlapping generations and a synchronous 2-cycle with non-overlappi...
In this paper, we consider nonlinear Leslie models for the dynamics of semelparous age-structured populations. We establish stability and instability criteria for positive equilibria that bifurcate from the extinction equilibrium at R0=1. When the bifurcation is to the right (forward or super-critical), the criteria consist of inequalities involvin...
We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and...
There is evidence for density dependent dispersal in many stage-structured species, including flour beetles of the genus Tribolium. We develop a bifurcation theory approach to the existence and stability of (non-extinction) equilibria for a general class of structured integrodifference equation models on finite spatial domains with density dependen...
A system of difference equations that arises in population dynamics is studied. Criteria are given for the existence of equilibria lying in the positive cone and for the existence of periodic cycles lying on the boundary of the cone. These equilibria and cycles arise from a bifurcation that occurs as a fundamental parameter R 0 increases through th...
In many stage-structured species, different life stages often occupy separate spatial niches in a heterogeneous environment. Life stages of the giant flour beetle Tribolium brevicornis (Leconte), in particular adults and pupae, occupy different locations in a homogeneous habitat. This unique spatial pattern does not occur in the well-studied stored...
Spontaneous oscillator synchrony is a form of self-organization in which populations of interacting oscillators ultimately cycle together. This phenomenon occurs in a wide range of physical and biological systems. In rats and humans, oestrous/menstrual cycles synchronize through social stimulation with pheromones acting as synchronizing signals. In...
Spatial segregation among life-cycle stages has been observed in many stage-structured species, including species of the flour beetle Tribolium. We investigate density-dependent dispersal of life-cycle stages as a possible mechanism responsible for this separation. We explore this hypothesis using stage-structured, integrodifference equation (IDE)...
When evolution plays a role, population dynamic models alone are not sufficient for determining the outcome of multi-species interactions. As an expansion of Maynard Smith's concept of an evolutionarily stable strategy, evolutionary game theory combines population and evolutionary dynamics so that natural selection becomes a dynamic game with pheno...
If the demographic parameters in a matrix model for the dynamics of a structured population are dependent on a parameter u, then the population growth rate r=r(u) and the net reproductive number R 0=R 0(u) are functions of u. For a general matrix model, we show that r and R 0 share critical values and extrema at values u=u* for which r(u*)=R 0(u*)=...
We give a definition of a net reproductive number R0 for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits...
Integrodifferential equations appear quite early in the mathematical development of theoretical population dynamics in the
pioneering work of such mathematicians as V. Volterra and V. A. Kostitzin. In their attempts to model the growth of populations
by means of differential equations these early investigators were quick to point out that the curre...
Spontaneous oscillator synchrony has been documented in a wide variety of electrical, mechanical, chemical, and biological systems, including the menstrual cycles of women and estrous cycles of Norway Rats (Rattus norvegicus). In temperate regions, many colonial birds breed seasonally in a time window set by photoperiod; some studies have suggested...
Matrix models are widely used to describe the discrete time dynamics of structured populations (i.e., biological populations in which individuals are classified into discrete categories such as age, size, etc.). A fundamental biological question concerns population extinction and persistence, i.e., the stability or instability of the extinction sta...
Using the methodology of evolutionary game theory (EGT), I study a class of Darwinian matrix models which are derived from a class of nonlinear matrix models for structured populations that are known to possess stable (normalized) distributions. Utilizing the limiting equations that result from this ergodic property, I prove extinction and stabilit...
Aim: Our aim is to show the utility of evolutionary game theory (EGT) methods in describing and predicting the outcome of experiments for which genetic data are available in the absence of phenotypic data. As an example we use experimental data from genetically perturbed cultures of the flour beetle Tribolium castaneum. Using natural selection, the...
Negative frequency-dependent selection is a well known microevolutionary process that has been documented in a population of Perissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika (Africa). Adult P. microlepis are lepidophages, feeding on the scales of other living fish. As an adaptation for this feeding behavior P. microlepis...
Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R
0 = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall...
We perform sensitivity analyses on a mathematical model of malaria transmission to determine the relative importance of model parameters to disease transmission and prevalence. We compile two sets of baseline parameter values: one for areas of high transmission and one for low transmission. We compute sensitivity indices of the reproductive number...
We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in which both species are present) and exclusion attractors (in which one species is absent). Recen...
Many species show considerable variation in behaviour among individuals. We show that some behaviours are largely deterministic and predictable with mathematical models. We propose a general differential equation model of behaviour in field populations and use the methodology to explain and predict the dynamics of sleep and colony attendance in sea...
The Leslie-Gower model is a discrete time analog of the competition Lotka-Volterra model and is known to possess the same dynamic scenarios of that famous model. The Leslie-Gower model played a historically significant role in the history of competition theory in its application to classic laboratory experiments of two competing species of flour be...
In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevent information to G.Ladas.
Theoretical studies of population dynamics and ecological interactions tend to focus on asymptotic attractors of mathematical
models. Modeling and experimental studies show, however, that even in controlled laboratory conditions the attractors of mathematical
models are likely to be insufficient to explain observed temporal patterns in data. Instea...
We explore the persistence of corn oil sensitivity in a population of the flour beetle Tribolium castaneum using evolutionary game methods that model population dynamics and changes in the mean strategy of a population over time.
The strategy in an evolutionary game is a trait that affects the fitness of the organisms. Corn oil sensitivity represen...
In previous studies we developed a general compartmental methodology for modeling animal behavior and applied the methodology to marine birds and mammals. In this study we used the methodology to construct a system of two differential equations to model the dynamics of territory attendance and preening in a gull colony on Protection Island, Strait...
A global branch of positive cycles is shown to exist for a general discrete time, juvenile-adult model with periodically varying coefficients. The branch bifurcates from the extinction state at a critical value of the mean, inherent fertility rate. In comparison to the autonomous system with the same mean fertility rate, the critical bifurcation va...
A scaling rule of ecological theory, accepted but lacking experimental confirmation, is that the magnitude of fluctuations in population densities due to demographic stochasticity scales inversely with the square root of population numbers. This supposition is based on analyses of models exhibiting exponential growth or stable equilibria. Using two...
We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, and recovered classes, before reentering the susceptible class. Susceptible mosquitoes can...
In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates...
This chapter discusses the message that in order to understand population fluctuations, deterministic and stochastic forces must be viewed as an integral part of the ecological system. The chapter explains the models, both animal and mathematical, and discusses how model parameters are estimated and the model validated. With the parameterized model...
This chapter highlights one of the chaotic treatments in the route-to-chaos experiment. A study of complex dynamics—in particular, chaotic dynamics—requires a sufficiently long time series of data. In this experiment, chaotic treatment is considered in more detail by examining some temporal patterns predicted by the chaotic attractor. Sensitivity t...
This chapter discusses a discrete time LPA model for the dynamics of a population with three life-cycle stages (larval, pupal, and adult). The chapter provides statistical methods for parameterizing and evaluating the model using data and applies these methods to data obtained from a controlled, replicated laboratory experiment involving flour beet...
The success of the LPA model in describing and predicting flour beetle dynamics is not restricted to the bifurcation experiment. Several other studies of an entirely different kind also support the accuracy of the LPA model and its ability to predict and explain dynamic patterns observed in flour beetle populations. This chapter focuses on chaos an...
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