Interaction of maturation delay and nonlinear birth in population and epidemic models

Stability switches, periodic oscillations and global stability in an infectious disease model with multiple time delays

A delay differential equation model of an infectious disease is considered and analyzed. In this model, the impact of information due to the presence of infection is considered explicitly. As information propagation is dependent on the prevalence of the disease, the delay in reporting the prevalence is an important factor. Further, the time lag in waning immunity related to protective measures (such as vaccination, self-protection, responsive behaviour etc.) is also accounted. Qualitative analysis of the equilibrium points of the model is executed and it is observed that when the basic reproduction number is less unity, the local stability of the disease free equilibrium (DFE) depends on the rate of immunity loss as well as on the time delay for the waning of immunity. If the delay in immunity loss is less than a threshold quantity, the DFE is stable, whereas, it loses its stability when the delay parameter crosses the threshold value. When, the basic reproduction number is greater than unity, the unique endemic equilibrium point is found locally stable irrespective of the delay effect under certain parametric conditions. Further, we have analyzed the model system for different scenarios of both delays (i.e., no delay, only one delay, and both delay present). Due to these delays, oscillatory nature of the population is obtained with the help of Hopf bifurcation analysis in each scenario. Moreover, at two different time delays (delay in information's propagation), the emergence of multiple stability switches is investigated for the model system which is termed as Hopf-Hopf (double) bifurcation. Also, the global stability of the endemic equilibrium point is established under some parametric conditions by constructing a suitable Lyapunov function irrespective of time lags. In order to support and explore qualitative results, exhaustive numerical experimentations are carried out which lead to important biological insights and also, these results are compared with existing results.

The Existence of Periodic Solutions for Second-Order Delay Differential Systems

In this paper, we consider a kind of second-order delay differential system. By taking some transforms, the property of delay is reflected in the boundary condition. The wonder is that the corrseponding first-order system is exactly the so-called P-boundary value problem of Hamiltonian system which has been studied deeply by many mathematicians, including the authors of this paper. Firstly, we define the relative Morse indexμ Q ( A , B ) for the delay system and give the relationship with the P-indexi P ( γ R ) of Hamiltonian system. Secondly, by this index, topology degree and saddle point reduction, the existence of periodic solutions is established for this kind of delay differential system.

An Epidemic Model with Time Delay Determined by the Disease Duration

Immuno-epidemiological models with distributed recovery and death rates can describe the epidemic progression more precisely than conventional compartmental models. However, the required immunological data to estimate the distributed recovery and death rates are not easily available. An epidemic model with time delay is derived from the previously developed model with distributed recovery and death rates, which does not require precise immunological data. The resulting generic model describes epidemic progression using two parameters, disease transmission rate and disease duration. The disease duration is incorporated as a delay parameter. Various epidemic characteristics of the delay model, namely the basic reproduction number, the maximal number of infected, and the final size of the epidemic are derived. The estimation of disease duration is studied with the help of real data for COVID-19. The delay model gives a good approximation of the COVID-19 data and of the more detailed model with distributed parameters.

Population growth and competition models with decay and competition consistent delay

We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of Arino et al. (J Theor Biol 241(1):109-119, 2006) by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition between two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, [Formula: see text], such that the evolutionary trend is for the delay to approach [Formula: see text].

Effects of Vector Maturation Time on the Dynamics of Cassava Mosaic Disease

Many plant diseases are caused by plant viruses that are often transmitted to plants by vectors. For instance, the cassava mosaic disease, which is spread by whiteflies, has a significant negative effect on plant growth and development. Since only mature whiteflies can contribute to the spread of the cassava mosaic virus, and the maturation time is non-negligible compared to whitefly lifetime, it is important to consider the effects this maturation time can have on the dynamics. In this paper, we propose a mathematical model for dynamics of cassava mosaic disease that includes immature and mature vectors and explicitly includes a time delay representing vector maturation time. A special feature of our plant epidemic model is that vector recruitment is negatively related to the delayed ratio between vector density and plant density. We identify conditions of biological feasibility and stability of different steady states in terms of system parameters and the time delay. Numerical stability analyses and simulations are performed to explore the role of various parameters, and to illustrate the behaviour of the model in different dynamical regimes. We show that the maturation delay may stabilise epidemiological dynamics that would otherwise be cyclic.

Kinetic models for epidemic dynamics with social heterogeneity

We introduce a mathematical description of the impact of the number of daily contacts in the spread of infectious diseases by integrating an epidemiological dynamics with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann-type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical SIR-type compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to moments of the contact distribution. Furthermore, the kinetic model allows to clarify how a selective control can be assumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very large number of daily contacts. We conduct numerical simulations which confirm the ability of the model to describe different phenomena characteristic of the rapid spread of an epidemic. Motivated by the COVID-19 pandemic, a last part is dedicated to fit numerical solutions of the proposed model with infection data coming from different European countries.

Hopf bifurcation and chaos in a Leslie–Gower prey–predator model with discrete delays

In this work, a Leslie–Gower prey-predator model with two discrete delays has been investigated. The positivity, boundedness and persistence of the delayed system have been discussed. The system exhibits the phenomenon of Hopf bifurcation with respect to both delays. The conditions for occurrence of Hopf bifurcation are obtained for different combinations of delays. It is shown that delay induces the complexity in the system and brings the periodic oscillations, quasi-periodic oscillations and chaos. The properties of periodic solution have been determined using central manifold and normal form theory. Further, the global stability of the system has been established for different cases of discrete delays. The numerical computation has also been performed to verify analytical results.

Modeling the effect of temperature on dengue virus transmission with periodic delay differential equations

Dengue fever is a re-emergent mosquito-borne disease, which prevails in tropical and subtropical regions, mainly in urban and peri-urban areas. Its incidence has increased fourfold since 1970, and dengue fever has become the most prevalent mosquito-borne disease in humans now. In order to study the effect of temperature on the dengue virus transmission, we formulate a dengue virus transmission model with maturation delay for mosquito production and seasonality. The basic reproduction number $\mathbb{R}_0$ of the model is computed, and results suggest that the dengue fever will die out if $\mathbb{R}_0$ < 1, and there exists at least one positive periodic solution and the disease will persist if $\mathbb{R}_0$ > 1. Theoretical results are applied to the outbreak of dengue fever in Guangdong province, China. Simulations reveal that the temperature change causes the periodic oscillations of dengue fever cases, which is good accordance with the reported cases of dengue fever in Guangdong province. Our study contributes to a better understanding of dengue virus transmission dynamics and proves beneficial in preventing and controlling of dengue fever.

An almost periodic Ross-Macdonald model with structured vector population in a patchy environment

An almost periodic Ross-Macdonald model with age structure for the vector population in a patchy environment is considered. The basic reproduction ratio [Formula: see text] for this model is derived and a threshold-type result on its global dynamics in terms of [Formula: see text] is established. It is shown that the disease is uniformly persistent if [Formula: see text], while the disease will die out if [Formula: see text]. Numerical simulations show that the biting rate greatly affects the disease transmission, and human migration sometimes could reduce the transmission risk. We further obtain a condition numerically to determine whether a control strategy on migration is necessary. Moreover, numerical results indicate that prolonging the length of maturation period of vector is beneficial to the disease control, and the threshold length of the maturation period for disease outbreak can be computed. Finally, the comparison between the almost periodic and periodic models shows that the periodic model may overestimate or underestimate the disease transmission risk.

Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models

In this paper, we perform qualitative analysis to two SIS epidemic models in a patchy environment, without and with linear recruitment. The model without linear recruitment was proposed and studied by Allen et al. (SIAM J Appl Math 67(5):1283-1309, 2007). This model possesses a conserved total population number, whereas the model with linear recruitment has a varying total population. However, both models have the same basic reproduction number. For both models, we establish the global stability of endemic equilibrium in a special case, which partially solves an open problem. Then we investigate the asymptotic behavior of endemic equilibrium as the mobility of infected and/or susceptible population tends to zero. Though the basic reproduction number is a well-known critical index, our theoretical results strongly suggest that other factors such as the variation of total population number and individual movement may also play vital roles in disease prediction and control. In particular, our results imply that the variation of total population number can cause infectious disease to become more threatening and difficult to control.

Implementation of control strategies for sterile insect techniques

In this paper, we propose a sex-structured entomological model that serves as a basis for design of control strategies relying on releases of sterile male mosquitoes (Aedes spp) and aiming at elimination of the wild vector population in some target locality. We consider different types of releases (constant and periodic impulsive), providing sufficient conditions to reach elimination. However, the main part of the paper is focused on the study of the periodic impulsive control in different situations. When the size of wild mosquito population cannot be assessed in real time, we propose the so-called open-loop control strategy that relies on periodic impulsive releases of sterile males with constant release size. Under this control mode, global convergence towards the mosquito-free equilibrium is proved on the grounds of sufficient condition that relates the size and frequency of releases. If periodic assessments (either synchronized with the releases or more sparse) of the wild population size are available in real time, we propose the so-called closed-loop control strategy, under which the release size is adjusted in accordance with the wild population size estimate. Finally, we propose a mixed control strategy that combines open-loop and closed-loop strategies. This control mode renders the best result, in terms of overall time needed to reach elimination and the number of releases to be effectively carried out during the whole release campaign, while requiring for a reasonable amount of released sterile insects.

Adaptive dispersal effect on the spread of a disease in a patchy environment

During outbreaks of a communicable disease, people intensely follow the media coverage of the epidemic. Most people attempt to minimize contact with others, and move themselves to avoid crowds. This dispersal may be adaptive regarding the intensity of media coverage and the population numbers in different patches. We propose an epidemic model with such adaptive dispersal rates to examine how appropriate adaption can facilitate disease control in connected groups or patches. Assuming dependence of the adaptive dispersal on the total population in the relevant patches, we derived an expression for the basic reproduction number R 0 to be related to the intensity of media coverage, and we show that the disease-free equilibrium is globally asymptotically stable if R 0 < 1 , and it becomes unstable if R 0 > 1 . In the unstable case, we showed a uniform persistence of disease by using a perturbation theory and the monotone dynamics theory. Specifically, when the disease mildly affects the dispersal of infectious individuals and rarely induces death, a unique endemic equilibrium exists in the model, which is globally asymptotically stable in positive states. Moreover, we performed numerical calculations to explain how the intensity of media coverage causes competition among patches, and influences the final distribution of the population.

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics

A new stage-structured model for the population dynamics of the mosquito (a major vector for numerous vector-borne diseases), which takes the form of a deterministic system of non-autonomous nonlinear differential equations, is designed and used to study the effect of variability in temperature and rainfall on mosquito abundance in a community. Two functional forms of eggs oviposition rate, namely the Verhulst-Pearl logistic and Maynard-Smith-Slatkin functions, are used. Rigorous analysis of the autonomous version of the model shows that, for any of the oviposition functions considered, the trivial equilibrium of the model is locally- and globally-asymptotically stable if a certain vectorial threshold quantity is less than unity. Conditions for the existence and global asymptotic stability of the non-trivial equilibrium solutions of the model are also derived. The model is shown to undergo a Hopf bifurcation under certain conditions (and that increased density-dependent competition in larval mortality reduces the likelihood of such bifurcation). The analyses reveal that the Maynard-Smith-Slatkin oviposition function sustains more oscillations than the Verhulst-Pearl logistic function (hence, it is more suited, from ecological viewpoint, for modeling the egg oviposition process). The non-autonomous model is shown to have a globally-asymptotically stable trivial periodic solution, for each of the oviposition functions, when the associated reproduction threshold is less than unity. Furthermore, this model, in the absence of density-dependent mortality rate for larvae, has a unique and globally-asymptotically stable periodic solution under certain conditions. Numerical simulations of the non-autonomous model, using mosquito surveillance and weather data from the Peel region of Ontario, Canada, show a peak mosquito abundance for temperature and rainfall values in the range [Formula: see text]C and [15-35] mm, respectively. These ranges are recorded in the Peel region between July and August (hence, this study suggests that anti-mosquito control effects should be intensified during this period).

Discrete or distributed delay? Effects on stability of population growth

The growth of a population subject to maturation delay is modeled by using either a discrete delay or a delay continuously distributed over the population. The occurrence of stability switches (stable-unstable-stable) of the positive equilibrium as the delay increases is investigated in both cases. Necessary and sufficient conditions are provided by analyzing the relevant characteristic equations. It is shown that for any choice of parameter values for which the discrete delay model presents stability switches there exists a maximum delay variance beyond which no switch occurs for the continuous delay model: the delay variance has a stabilizing effect. Moreover, it is illustrated how, in the presence of switches, the unstable delay domain is as larger as lower is the ratio between the juveniles and the adults mortality rates.

The effect of delayed death in HIV/AIDS models

HIV-infected patients who receive treatment survive for some years after they have acquired the disease. The received treatment causes sustained reduction of viral reproduction by improving the immune function, leading to prolonged progression period to AIDS development. This prolonged progression period has created variability in survival times that affects estimates produced using mathematical models that do not include delay in disease related mortality. This paper investigates the effect of including delay in AIDS death occurrence in HIV/AIDS transmission models. A simple mathematical model with two stages of HIV progression is developed and extended to include time delay in the occurrence of AIDS deaths. Numerical simulations indicate that time delay changes the mortality curves considerably but has less effect on the proportion of infectives. The study highlights the importance of incorporating delay in models of HIV/AIDS for the production of accurate HIV/AIDS estimates.

A new model with delay for mosquito population dynamics

In this paper, we formulate a new model with maturation delay for mosquito population incorporating the impact of blood meal resource for mosquito reproduction. Our results suggest that except for the usual crowded effect for adult mosquitoes, the impact of blood meal resource in a given region determines the mosquito abundance, it is also important for the population dynamics of mosquito which may induce Hopf bifurcation. The existence of a stable periodic solution is proved both analytically and numerically. The new model for mosquito also suggests that the resources for mosquito reproduction should not be ignored or mixed with the impact of blood meal resources for mosquito survival and both impacts should be considered in the model of mosquito population. The impact of maturation delay is also analyzed.

Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.

TWO TYPES OF CONDITION FOR THE GLOBAL STABILITY OF DELAYED SIS EPIDEMIC MODELS WITH NONLINEAR BIRTH RATE AND DISEASE INDUCED DEATH RATE

We study global asymptotic stability for an SIS epidemic model with maturation delay proposed by K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol.39(4) (1999) 332–352. It is assumed that the population has a nonlinear birth term and disease causes death of infective individuals. By using a monotone iterative method, we establish sufficient conditions for the global stability of an endemic equilibrium when it exists dependently on the monotone property of the birth rate function. Based on the analysis, we further study the model with two specific birth rate functions B1(N) = b e -aN and B3(N) = A/N + c, where N denotes the total population. For each model, we obtain the disease induced death rate which guarantees the global stability of the endemic equilibrium and this gives a positive answer for an open problem by X. Q. Zhao and X. Zou, Threshold dynamics in a delayed SIS epidemic model, J. Math. Anal. Appl.257(2) (2001) 282–291.

Global analysis on delay epidemiological dynamic models with nonlinear incidence

In this paper, we derive and study the classical SIR, SIS, SEIR and SEI models of epidemiological dynamics with time delays and a general incidence rate. By constructing Lyapunov functionals, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is shown. This analysis extends and develops further our previous results and can be applied to the other biological dynamics, including such as single species population delay models and chemostat models with delay response.

The impact of maturation delay of mosquitoes on the transmission of West Nile virus

We formulate and analyze a delay differential equation model for the transmission of West Nile virus between vector mosquitoes and avian hosts that incorporates maturation delay for mosquitoes. The maturation time from eggs to adult mosquitoes is sensitive to weather conditions, in particular the temperature, and the model allows us to investigate the impact of this maturation time on transmission dynamics of the virus among mosquitoes and birds. Numerical results of the model show that a combination of the maturation time and the vertical transmission of the virus in mosquitoes has substantial influence on the abundance and number of infection peaks of the infectious mosquitoes.

Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition

In this paper, we consider a class of delay reaction–diffusion equations (DRDEs) with a parameter ε >0. A homogeneous Neumann boundary condition and non-negative initial functions are posed to the equation. By letting , such an equation is formally reduced to a scalar difference equation (or map dynamical system). The main concern is the relation of the absolute (or delay-independent) global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation. By employing the idea of attracting intervals for solution semiflows of the DRDEs, we prove that the globally stable dynamics of the map indeed ensures the delay-independent global stability of a constant steady state of the DRDEs. We also give a counterexample to show that the delay-independent global stability of DRDEs cannot guarantee the globally stable dynamics of the map. Finally, we apply the abstract results to the diffusive delay Nicholson blowfly equation and the diffusive Mackey–Glass haematopoiesis equation. The resulting criteria for both model equations are amazingly simple and are optimal in some sense (although there is no existing result to compare with for the latter).

Perverse consequences of infrequently culling a pest

There are potentially many situations in which creatures will be subject to infrequent but regular culling. In terms of controlling crop pests, some farmers may only be able to afford to apply pesticides occasionally. Alternatively, pesticides may be applied only occasionally to limit their unwelcome side effects, which include pesticide resistance, chemical poisoning of agricultural workers, and environmental degradation. In terms of conservation, some species (such as the red deer in the UK) may be culled occasionally to maintain balances within their ecosystem. However, in this paper we discover, as the culmination of an exploration of adult-stage culling of a creature with juvenile and adult life stages, that, in certain circumstances, regular but infrequent culling will, perversely, increase the average population of the creature.

Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate

In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.

Threshold dynamics in a time-delayed epidemic model with dispersal

The global dynamics of a time-delayed model with population dispersal between two patches is investigated. For a general class of birth functions, persistence theory is applied to prove that a disease is persistent when the basic reproduction number is greater than one. It is also shown that the disease will die out if the basic reproduction number is less than one, provided that the initial size of the infected population is relatively small. Numerical simulations are presented using some typical birth functions from biological literature to illustrate the main ideas and the relevance of dispersal.

An SIR epidemic model with partial temporary immunity modeled with delay

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.

Modeling spruce budworm population revisited: impact of physiological structure on outbreak control

Understanding the dynamics of spruce budworm population is very important for the protection of spruce and balsam fir trees of North American forests, and a full understanding of the dynamics requires careful consideration of the individual physiological structures that is essential for outbreak control. A model as a delay differential equation is derived from structured population system, and is validated by comparing simulation results with real data from the Green River area of New Brunswick (Canada) and with the periodic outbreaks widely observed. Analysis of the equilibrium stability and examination of the amplitudes and frequencies of periodic oscillations are conducted, and the effect of budworm control strategies such as mature population control, immature population control and predation by birds are assessed. Analysis and simulation results suggest that killing only budworm larvae might not be enough for the long-term control of the budworm population. Since the time required for development during the inactive stage (from egg to second instar caterpillar) causes periodic outbreak, a strategy of reducing budworms in the inactive stage, such as removing egg biomass, should also be implemented for successful control.

Modeling relapse in infectious diseases

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.

Basic Knowledge and Developing Tendencies in Epidemic Dynamics

Infectious diseases have been a ferocious enemy since time immemorial. To prevent and control the spread of infectious diseases, epidemic dynamics has played an important role on investigating the transmission of infectious diseases, predicting the developing tendencies, estimating the key parameters from data published by health departments, understanding the transmission characteristics, and implementing the measures for prevention and control. In this chapter, some basic ideas of modelling the spread of infectious diseases, the main concepts of epidemic dynamics, and some developing tendencies in the study of epidemic dynamics are introduced, and some results with respect to the spread of SARS in China are given.

On the population dynamics of the malaria vector

A deterministic differential equation model for the population dynamics of the human malaria vector is derived and studied. Conditions for the existence and stability of a non-zero steady state vector population density are derived. These reveal that a threshold parameter, the vectorial basic reproduction number, exist and the vector can established itself in the community if and only if this parameter exceeds unity. When a non-zero steady state population density exists, it can be stable but it can also be driven to instability via a Hopf Bifurcation to periodic solutions, as a parameter is varied in parameter space. By considering a special case, an asymptotic perturbation analysis is used to derive the amplitude of the oscillating solutions for the full non-linear system. The present modelling exercise and results show that it is possible to study the population dynamics of disease vectors, and hence oscillatory behaviour as it is often observed in most indirectly transmitted infectious diseases of humans, without recourse to external seasonal forcing.

An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.

A stage structured predator-prey model and its dependence on maturation delay and death rate

Many of the existing models on stage structured populations are single species models or models which assume a constant resource supply. In reality, growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computational study. Our work indicates that if the juvenile death rate (through-stage death rate) is nonzero, then for small and large values of maturation time delays, the population dynamics takes the simple form of a globally attractive steady state. Our linear stability work shows that if the resource is dynamic, as in nature, there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics.

Bifurcation diagram of a complex delay-differential equation with cubic nonlinearity

We reduce the Lang-Kobayashi equations for a semiconductor laser with external optical feedback to a single complex delay-differential equation in the long delay-time limit. The reduced equation has a time-delayed linear term and a cubic instantaneous nonlinearity. There are only two parameters, the real linewidth enhancement factor and the complex feedback strength. The equation displays a very rich dynamics and can sustain steady, periodic, quasiperiodic, and chaotic regimes. We study the steady solutions analytically and analyze the periodic solutions by using a numerical continuation method. This leads to a bifurcation diagram of the steady and periodic solutions, stable and unstable. We illustrate the chaotic regimes by a direct numerical integration and show that low frequency fluctuations still occur.