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Simplified conditions are given for the existence and positivity of wave speed for an integro-difference equation with a strong Allee effect and an unbounded habitat. The results are used to obtain the existence of a critical patch size for an equation with a bounded habitat. It is shown that if the wave speed is positive there exists a critical patch size such that for a habitat size above the critical patch size solutions can persist in space, and if the wave speed is negative solutions always approach zero. An analytical integral formula is developed to determine the critical patch size when the Laplace dispersal kernel is used, and this formula shows existence of multiple equilibrium solutions. Numerical simulations are provided to demonstrate connections among the wave speed, critical patch size, and Allee threshold.

We establish the existence of traveling waves for a Lotka-Volterra competition-diffusion model with a shifting habitat. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and negative near −∞, and shifting rightward at a speed c. We show that under appropriate conditions, for the case that one species is competitively stronger near ∞ and the case that both species coexist near ∞, there exists a critical number c ¯ ( ∞ ) such that for any c > c ¯ ( ∞ ) there exists a forced traveling wave with speed c connecting the origin and a semi-trivial steady state and for c c ¯ ( ∞ ) such a traveling wave does not exist. We also show that when a coexistence steady state exists, for any c > 0 , there is a forced traveling wave with speed c connecting the origin and the coexistence steady state.

We consider a two-species Lotka--Volterra competition-diffusion model with a shifting habitat. The growth rate of each species is nondecreasing along the $x$-axis, and it changes sign and shifts ri...

We study an integro-difference equation that describes the spatial dynamics of a species in a shifting habitat. The growth function is nondecreasing in density and space for a given time, and shifts at a constant speed c. The spreading speeds for the model were previously studied. The contribution of the current paper is to provide sharp conditions for existence of forced traveling waves with speed c. We show the existence of traveling waves with zero value at ∞ or −∞ for c in different value ranges determined by the spreading speeds. We also show the existence of a traveling wave with any speed c for the case that the species can grow everywhere. Our results demonstrate the existence of different types of traveling waves with the same speed.

Previous work involving integro-difference equations of a single species in a homogenous environment has emphasized spreading behaviour in unbounded habitats. We show that under suitable conditions, a simple scalar integro-difference equation incorporating a strong Allee effect and overcompensation can produce solutions where the population persists in an essentially bounded domain without spread despite the homogeneity of the environment. These solutions are robust in that they occupy a region of full measure in the parameter space. We develop bifurcation diagrams showing various patterns of nonspreading solutions from stable equilibria, period two, to chaos. We show that from a relatively uniform initial density with small stochastic perturbations a population consisting of multiple isolated patches can emerge. In ecological terms this work suggests a novel endogenous mechanism for the creation of patch boundaries.AMS subject classification. 92D40, 92D25

We study the spreading speeds of a species described by a reaction–diffusion equation with a shifting habitat on which the species’ growth rate increases in the positive spatial direction. Persistence and the rightward spreading speed have been previously investigated in the case that the species grows near positive infinity and declines near negative infinity. In the present paper we determine the leftward spreading speed for this case and both the rightward and leftward spreading speeds for the case that the species grows everywhere in space. We show that the spreading speeds depend on c, the speed at which the habitat shifts, and numbers $$c^*(\infty )$$ and $$c^*(-\infty )$$ given by the diffusion coefficient and species growth rates at $$\infty $$ and $$-\infty $$ respectively. We demonstrate that if the species declines near negative infinity, the leftward spreading speed is $$\min \{-c, c^*(\infty )\}$$ , and if species grows everywhere, under appropriate conditions, the rightward spreading speed is either $$c^*(\infty )$$ or $$c^*(-\infty )$$ , and the leftward spreading speed is one of $$c^*(\infty )$$ , $$c^*(-\infty )$$ and $$-c$$ . Our results show that it is possible for a solution to form a two-layer wave, with the propagation speeds analytically determined.

In this paper, we develop a phenologically explicit reaction-diffusion model to analyze the spatial spread of a univoltine insect species. Our model assumes four explicit life stages: adult, two larval, and pupa, with a fourth, implicit, egg stage modeled as a time delay between oviposition and emergence as a larva. As such, our model is broadly applicable to holometabolous insects. To account for phenology (seasonal biological timing), we introduce four time-dependent phenological functions describing adult emergence, oviposition and larval conversion, respectively. Emergence is defined as the per-capita probability of an adult emerging from the pupal stage at a particular time. Oviposition is defined as the per-capita rate of adult egg deposition at a particular time. Two functions deal with the larva stage 1 to larva stage 2, and larva stage 2 to pupa conversion as per-capita rate of conversion at a particular time. This very general formulation allows us to accommodate a wide variety of alternative insect phenologies and lifestyles. We provide the moment-generating function for the general linearized system in terms of phenological functions and model parameters. We prove that the spreading speed of the linearized system is the same as that for nonlinear system. We then find explicit solutions for the spreading speed of the insect population for the limiting cases where (1) emergence and oviposition are impulsive (i.e., take place over an extremely narrow time window), larval conversion occurs at a constant rate, and larvae are immobile, (2) emergence and oviposition are impulsive (i.e., take place over an extremely narrow time window), larval conversion occurs at a constant rate starting at a delayed time from egg hatch, and larvae are immobile, and (3) emergence, oviposition, and larval conversion are impulsive. To consider other biological scenarios, including cases with emergence and oviposition windows of finite width as well as mobile larvae, we use numerical simulations. Our results provide a framework for understanding how phenology can interact with spatial spread to facilitate (or hinder) species expansion. This is an important area of research within the context of global change, which brings both new invasive species and range shifts for native species, all the while causing perturbations to species phenology that may impact the abilities of native and invasive populations to spread.

Density dependence plays an important role in population regulation, and has a long history in ecology as a mechanism that can induce local density fluctuations. Yet much less is known about how these endogenous processes affect spatial population dynamics. Biological invasions occur through the combined action of population growth (demography), and movement (dispersal), making them relevant for understanding how density dependence regulates spatial spread. While classical ecological theory suggests that many invasions move at a constant speed, empirical work is illuminating the highly variable nature of biological invasions, which can lead to non-constant spreading speeds. Here, we explore endogenous density dependence as a mechanism for inducing variability in biological invasions. We constructed a set of integrodifference population models that incorporate classic population fluctuation mechanisms to determine how density dependence in demography, including Allee effects, and in dispersal affects the speed of biological invasions. We show that density dependence is a key factor in producing fluctuations in spreading speed when Allee effects are acting on population densities that fluctuate locally. We show that the necessary density fluctuations can arise from either a nonmonotone population growth function where densities fluctuate locally (e.g., overcompensatory population growth), or from density-dependent dispersal when the population growth function results in constant local densities. As density dependence in both demography and dispersal are common, this mechanism of variability may influence many invading organisms.

We consider an integro-difference model to study the effect of a stationary barrier zone on invasion of a population with a strong Allee effect. It is assumed that inside the barrier zone a certain proportion of the population is killed. A Laplace dispersal kernel is used in the model. We provide a formula for the critical width [Formula: see text] of barrier zone. We show that when a barrier zone is set at the front of a population, if the width of barrier zone is bigger than [Formula: see text] then the barrier zone can stop the population invasion, and if the width of barrier zone is less than [Formula: see text] then the population crosses the barrier zone and eventually occupies the entire space. The results are proven by establishing the existence and attractivity of three types of equilibrium solutions. The mathematical proofs involve phase plane analysis and comparison.

We study a lattice differential equation model that describes the growth and spread of a species in a shifting habitat. We show that the long term behavior of solutions depends on the speed of the shifting habitat edge c and a number c ⁎ (∞) c ⁎ ( ∞ ) that is determined by the maximum linearized growth rate and the diffusion coefficient. We demonstrate that if c>c ⁎ (∞) c > c ⁎ ( ∞ ) then the species will become extinct in the habitat, and that if c ⁎ (∞) c c ⁎ ( ∞ ) then the species will persist and spread along the shifting habitat at the asymptotic spreading speed c ⁎ (∞) c ⁎ ( ∞ ) . For our purpose the solutions to the model are formulated in the form of integral equations involving modified Bessel functions, for which new asymptotic estimates are provided. To the best of our knowledge, this is the first time that the classical Bessel functions are used to describe the solutions of lattice differential equations, and this approach possesses its own interest in further studying lattice differential equations.

We study a cooperative system of integro-differential equations. It is shown that the system in general has multiple spreading speeds, and when the linear determinacy conditions are satisfied all the spreading speeds are the same and equal to the spreading speed of the linearized system. The existence of traveling wave solutions is established via integral systems. It is shown that when the linear determinacy conditions are satisfied, if the unique spreading speed is not zero then it may be characterized as the slowest speed of a class of traveling wave solutions. Some examples are presented to illustrate the theoretical results.

We study an integro-difference competition model for the case that two species consecutively invade a habitat. We show that if a species spreads into a traveling wave of its rival, or if two species expand their spatial ranges in both directions, in a direction where open space is available, the species with larger invasion speed can always establish a wave moving into open space with its own speed. We demonstrate that when one species is stronger in competition, under appropriate conditions, the speeds at which the boundaries between two species move can be analytically determined. We find that in general there are multiple invasion speeds in the model. It is possible for a species to develop two separate waves propagating with different invasion speeds. It is also possible for each species to establish a single wave spreading with distinct speeds in both directions. The mathematical analysis relies on linear determinacy and new techniques developed.

We study a model that describes the spatial spread of a species along a habitat gradient on which the species' growth increases. Mathematical analysis is provided to determine the spreading dynamics of the model. We demonstrate that the species may succeed or fail in local invasion depending on the species' growth function and dispersal kernel. We delineate the conditions under which a spreading species may be stopped by poor quality habitat, and demonstrate how a species can escape a region of poor quality habitat by climbing a resource gradient to good quality habitat where it spreads at a constant spreading speed. We show that dispersal may take the species from a good quality region to a poor quality region where the species becomes extinct. We also provide formulas for spreading speeds for the model that are determined by the dispersal kernel and linearized growth rates in both directions.

We study a reaction-diffusion model that describes the growth and spread of a species along a shifting habitat gradient on which the species' growth increases. It is assumed that the linearized species growth rate is positive near positive infinity and is negative near negative infinity. We show that the persistence and spreading dynamics depend on the speed of the shifting habitat edge $c$ and a number $c^*(\infty)$ that is determined by the maximum linearized growth rate and the diffusion coefficient. We demonstrate that if $c>c^*(\infty)$, then the species will become extinct in the habitat, and that if $c

We propose a reaction-advection-diffusion model to study competition between two species in a stream. We divide each species into two compartments, individuals inhabiting the benthos and individuals drifting in the stream. We assume that the growth of and competitive interactions between the populations take place on the benthos and that dispersal occurs in the stream. Our system consists of two linear reaction-advection-diffusion equations and two ordinary differential equations. Here, we provide a thorough study for the corresponding single species model, which has been previously proposed. We next give formulas for the rightward spreading and leftward spreading speed for the model. We show that rightward spreading speed can be characterized as is the slowest speed of a class of traveling wave speeds. We provide sharp conditions for the spreading speeds to be positive. For the two species competition model, we investigate how a species spreads into its competitor's environment. Formulas for the spreading speeds are provided under linear determinacy conditions. We demonstrate that under certain conditions, the invading species can spread upstream. Lastly, we study the existence of traveling wave solutions for the two species competition model.

The spread of an invasive species often results in a decline and subsequent disappearance of native competitors. Several models, primarily based on spatially explicit Lotka–Volterra competition dynamics, have been developed to understand this phenomenon. In general, the goal of these models is to relate fundamental life history traits, for example dispersal ability and competition strength, to the rate of spread of the invasive species, which is also the rate at which the invasive species displaces its native competitor. Stage-structure is often an important determinant of population dynamics, but it has received little attention within the context of Lotka–Volterra invasion models. For many species, behaviors like dispersal and competition depend on life-stage. To describe the processes of invasion in these species, it is important to understand how behaviors that vary as a function of life-stage can impact spread rate. In this paper, we develop a spatially explicit, stage-structured Lotka–Volterra competition model. By comparing spread speed predictions from this model to spread speed predictions from an analogous single-stage model, we are able to determine when stage-structure is important and how stage-dependent behavior can alter the characteristics of an invasion.

The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called ``linear determinacy'' conjecture equates full nonlinear model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading edge of the invasion. A survey that accounts for recent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions covered in this survey are also discussed.

We formulate an integro-difference model to predict the growth and spatial spread of a plant population with an age-structured seed bank and juvenile cohort. We allow the seeds in the bank to be of any age, producing a system of infinitely many equations. We assume that juvenile plants mature into adults at a particular age. The production of new seeds can be density-dependent and so the function describing this growth is allowed to be nonmonotone. The functions describing the seed bank and juvenile plants are linear. We show that when the system has a positive equilibrium, there is a spreading speed that can be computed using model parameters and that this spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. The spreading speed results are obtained through linearization and comparison to an analogous finite system, while the existence of traveling wave solutions is shown by using an asymptotic fixed point theorem. We conduct numerical simulations of a truncate...

A ratio-dependent predator-prey model with a strong Allee effect in prey is studied. We show that the model has a Bogdanov-Takens bifurcation that is associated with a catastrophic crash of the predator population. Our analysis indicates that an unstable limit cycle bifurcates from a Hopf bifurcation, and it disappears due to a homoclinic bifurcation which can lead to different patterns of global population dynamics in the model. We study the heteroclinic orbits and determine all possible phase portraits when the Bogdanov-Takens bifurcation occurs. We also provide the conditions for nonexistence of limit cycle under which the global dynamics of the model can be determined.

The theory of invasions and invasion speeds has traditionally been studied in macroscopic systems. Surprisingly, microbial invasions have received less attention. Although microbes share many of the features associated with competition between larger-bodied organisms, they also exhibit distinctive behaviors that require new mathematical treatments to fully understand invasions in microbial systems. Most notable is the possibility for long-distance interactions, including competition between populations mediated by diffusible toxins and cooperation among individuals of a single population using quorum sensing. In this paper, we model bacterial invasion using a system of coupled partial differential equations based on Fisher's equation. Our model considers a competitive system with diffusible toxins that, in some cases, are expressed in response to quorum sensing. First, we derive analytical approximations for invasion speeds in the limits of fast and slow toxin diffusion. We then test the validity of our analytical approximations and explore intermediate rates of toxin diffusion using numerical simulations. Interestingly, we find that toxins should diffuse quickly when used offensively, but that there are two optimal strategies when toxins are used as a defense mechanism. Specifically, toxins should diffuse quickly when their killing efficacy is high, but should diffuse slowly when their killing efficacy is low. Our approach permits an explicit investigation of the properties and characteristics of diffusible compounds used in non-local competition, and is relevant for microbial systems and select macroscopic taxa, such as plants and corals, that can interact through biochemicals.

We study a time-delayed population system with stage structure for the interaction between two species, the adult members of which are in competition. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. The global stability results are established for each equilibrium. The criteria for global convergence to each equilibrium are sharp and involve these delays. By using lower and upper travelling wave solutions, we show that the model has travelling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.

We study the existence of traveling wave solutions for partially degenerate cooperative reaction–diffusion systems that can have three or more equilibria. We show via integral systems that there exist traveling wave solutions in a partially degenerate reaction–diffusion system with speeds above two well-defined extended real numbers. We prove that the two numbers are the same and may be characterized as the spreading speed as well as the slowest speed of a class of traveling wave solutions provided that the linear determinacy conditions are satisfied. We demonstrate our theoretical results by examining a partially degenerate Lotka–Volterra competition model with advection terms.

We study the existence of traveling wave solutions for the two-species Lotka-Volterra competition model in the form of integro-differential equations. The model incorporates asymmetric dispersal kernels that describe long distance dispersal processes of competing species in space. Using lower and upper traveling wave solutions, we show that the model has traveling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.

Habitat loss and climate change jointly threaten a large fraction of earth's biodiversity. A key goal is to understand how these threats play out differentially across species. Focusing on insects that undergo an ontogenetic shift in habitat requirements, we use critical patch size models to examine how breeding strategy influences the abilities of different kinds of species to persist in small habitat patches. In general, we find that income breeders require larger habitat patches for population persistence than do capital breeders. However, increases in patch size requirements as a result of factors that limit oviposition (e.g., resource availability, weather conditions) are more severe for capital breeders than for income breeders. From a conservation perspective, our work suggests that a species' sensitivity to habitat loss, both today and in the future, can depend critically on evolved behavioral strategies. Explicit consideration of such behavioral strategies, including a careful accounting of their relationship with dispersal and survival, provides a map linking life-history spectra, spatial requirements, and management.

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.

We establish the existence of travelling wave solutions for delayed cooperative recursions that are allowed to have more than two equilibria. We define an important extended real number that is used to determine the speeds of travelling wave solutions. The results can be applied to a large class of delayed cooperative reaction?diffusion models. We show that for a delayed Lotka?Volterra reaction?diffusion competition model, there exists a finite positive number that can be characterized as the slowest speed of travelling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium.

We study the existence of traveling wave solutions for competition models in the form of integro-difference equations. We show that for a two-species competition model it is possible that two species spread at different speeds, and there exists a traveling wave solution. For an $m$-species competition model, under the assumption that species have the same dispersal and growth properties but have different competition abilities, we establish the existence of traveling wave solutions.

We study a chemostat model that describes competition between two species for two essential resources based on storage. The model incorporates internal resource storage variables that serve the direct connection between species growth and external resource availability. Mathematical analysis for the global dynamics of the model is carried out by using the monotone dynamical system theory. It is shown that the limiting system of the model basically exhibits the familiar Lotka-Volterra alternatives: competitive exclusion, coexistence, and bi-stability, and most of these results can be carried over to the original model.

The existence of a heteroclinic bifurcation for the Michaelis–Menten-type ratio-dependent predator-prey system is rigorously established. Limit cycles related to the heteroclinic bifurcation are also discussed. It is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin, and the bifurcation triggers a catastrophic shift from the state of large oscillations of predator and prey populations to the state of extinction of both populations. It is also shown that the limit cycles related to the heteroclinic bifurcation originally bifurcate from the Hopf bifurcation.

We study an integro-difference equation model that describes the spatial dynamics of a species in an expanding or contracting habitat. We give conditions under which the species disperses to a region of poor quality where the species eventually becomes extinct. We show that when the species persists in the habitat, the rightward and leftward spreading speeds are determined by c, the speed at which the habitat quality increases or decreases in time, as well as [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], which are formulated in terms of the dispersal kernel and species growth rates in both directions. We demonstrate that in the case that the species grows everywhere in space, the rightward spreading speed is [Formula: see text] if c is relatively small and is [Formula: see text] if c is large, and the leftward spreading speed is one of [Formula: see text], [Formula: see text], or [Formula: see text]. We also show that it is possible for a solution to form a two-layer wave, with the propagation speeds of the two layers analytically determined.

We provide rigorous analysis for a discrete-time model composed of the Ricker function and Beverton–Holt function. This model was proposed by Lewis and Li [Bull. Math. Biol. 74 (2012) 2383–2402] in the study of a population in which reproduction occurs at a discrete instant of time whereas death and competition take place continuously during the season. We show analytically that there exists a period-doubling bifurcation curve in the model. The bifurcation curve divides the parameter space into the region of stability and the region of instability. We demonstrate through numerical bifurcation diagrams that the regions of periodic cycles are intermixed with the regions of chaos. We also study the global stability of the model.

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.

One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on 'linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper [11].

In the last three decades, several models on the interaction of glucose and insulin have appeared in the literature, the mostly used one is generally known as the "minimal model" which was first published in 1979 and modified in 1986. Recently, this minimal model has been questioned by De Gaetano and Arino [4] from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called "dynamic model". Their model makes use of certain simple and specific functions and introduces time delay in a particular way. The outcome is that the model always admits a globally asymptotically stable steady state. The objective of this paper is to find out if and how this outcome depends on the specific choice of functions and the way delay is incorporated. To this end, we generalize the dynamical model to allow more general functions and an alternative way of incorporating time delay. Our findings show that in theory, such models can possess unstable positive steady states. However, for all conceivable realistic data, such unstable steady states do not exist. Hence, our work indicates that the dynamic model does provide qualitatively robust dynamics for the purpose of clinic application. We also perform simulations based on data from a clinic study and point out some plausible but important implications.

A chemostat model of n species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered. Sufficient conditions are given for there to be a single winning species and for two species to coexist, driving the others to extinction. In the case when n=3, it is shown that every solution converges to one of the single-species or two-species steady states, and hence the dynamics of the model is completely determined. The results generalize those of Hsu, Cheng, and Hubbell [SIAM J. Appl. Math., 41 (1981), pp. 422--444] as well as Butler and Wolkowicz [ Math. Biosci., 83 (1987), pp. 1--48] who considered two species.

A simple model for an autonomous pulsing drug delivery system was previously introduced. This model involves negative feedback action, with hysteresis, of an enzyme on the permeability of a membrane through which substrate, at constant external concentration, must diffuse to reach the enzyme. The qualitative dynamics of this model permit, depending on system parameters and external driving substrate concentration, two separate single steady state, double steady state, and permanently alternating (oscillatory) behaviors. The present contribution is concerned with rigorous proofs regarding the global stability of steady states when permanent alternation is precluded, and the existence and globally asymptotic stability of a limit cycle in the permanently alternating case. Also, we prove that more restrictive but often realistic conditions on the system parameters imply limitations on the number of alternations the system can undergo before reaching steady state. (c) 2000 American Institute of Physics.

A model of the chemostat involving two species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered. The model incorporates distributed time delay in the form of integral differential equations in order to describe the time involved in converting nutrient to biomass. The delays are included in the nutrient and species concentrations simultaneously. A general class of monotone increasing functions is used to describe nutrient uptake. Sufficient conditions based on biologically meaningful parameters in the model are given that predict competitive exclusion for certain parameter ranges and coexistence for others. We prove that the global asymptotic attractivity of steady states of the model is similar to that of the corresponding model without time delays. However, our results indicate that when the inherent delays are in fact large, ignoring them may result in incorrect predictions.

This is a correction to the article [B. Li, S. Bewick, J. Shang, and W. F. Fagan, SIAM J. Appl. Math., 74 (2014), pp. 1397--1417].

We obtain some multiple integral average conditions for oscillation and nonoscillation of the first order nonautonomous delay equations. Our theorems improve several previous well-known results.

In this paper, we consider a competition model between n species in a chemostat that incorporates both monotone and nonmonotone general response functions and distinct removal rates. We show that only the species with the lowest break-even concentration survives, provided that the variation of distinct removal rates relative to the flow rate of the chemostat can be controlled by either the difference between the two lowest break-even concentrations or by a parameter based on the structure of response functions. LaSalle's extension theorem of the Lyapunov stability theory and fluctuation lemma are the main tools.

We introduce a new technique to analyze the generalized characteristic equations to obtain some infinite integral conditions for oscillation of the nonautonomous delay differential equations.

Consider the delay differential equation ẋ(t) + p(t)x(t − τ) = 0, where p(t) ∈ C[[t0, ∞), R+] and τ is a positive constant. We show that every solution of this equation oscillates if ∫tt−τp(t) dt ≥ 1/e for sufficiently large t and ∫∞t0+τp(t)[exp(∫tt−τp(s) ds − 1/e) − 1] dt = ∞.

A model of competition in a turbidostat between two species for an inhibitory growth-limiting nutrient is considered. It is shown that the model has rich dynamics. A coexistence equilibrium and the washout equilibrium can be asymptotically stable simultaneously so that coexistence may depend on initial conditions. Under certain conditions, periodic coexistence of the two species occurs. There is a possibility that two species coexist, whereas one species dies out in the absence of its rival.

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.

In this study, we expand on the susceptible-infected-susceptible (SIS) heterosexual mixing setting by including the movement of individuals of both genders in a spatial domain in order to more comprehensively address the transmission dynamics of competing strains of sexually-transmitted pathogens. In prior models, these transmission dynamics have only been studied in the context of nonexplicitly mobile heterosexually active populations at the demographic steady state, or, explicitly in the simplest context of SIS frameworks whose limiting systems are order preserving. We introduce reaction-diffusion equations to study the dynamics of sexually-transmitted diseases (STDs) in spatially mobile heterosexually active populations. To accomplish this, we study a single-strain STD model, and discuss in what forms and at what speed the disease spreads to noninfected regions as it expands its spatial range. The dynamics of two competing distinct strains of the same pathogen on this population are then considered. The focus is on the investigation of the spatial transition dynamics between the two endemic equilibria supported by the nonspatial corresponding model. We establish conditions for the successful invasion of a population living in endemic conditions by introducing a strain with higher fitness. It is shown that there exists a unique spreading speed (where the spreading speed is characterized as the slowest speed of a class of traveling waves connecting two endemic equilibria) at which the infectious population carrying the invading stronger strain spreads into the space where an equilibrium distribution has been established by the population with the weaker strain. Finally, we give sufficient conditions under which an explicit formula for the spreading speed can be found.

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94-121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers.