Stage-structured population systems with temporally periodic delay (original) (raw)
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Bulletin of Mathematical Biology, 2018
Mosquito-borne diseases remain a significant threat to public health and economics. Since mosquitoes are quite sensitive to temperature, global warming may not only worsen the disease transmission case in current endemic areas but also facilitate mosquito population together with pathogens to establish in new regions. Therefore, understanding mosquito population dynamics under the impact of temperature is considerably important for making disease control policies. In this paper, we develop a stage-structured mosquito population model in the environment of a temperature-controlled experiment. The model turns out to be a system of periodic delay differential equations with periodic delays. We show that the basic reproduction number is a threshold parameter which determines whether the mosquito population goes to extinction or remains persistent. We then estimate the parameter values for Aedes aegypti, the mosquito that transmits dengue virus. We verify the analytic result by numerical simulations with the temperature data of Colombo, Sri Lanka where a dengue outbreak occurred in 2017. Keywords Mosquito • Climate change • Periodic delay • Dengue • Basic reproduction ratio • Population dynamics This work is supported in part by the NSERC of Canada.
Journal of Differential Equations, 2009
We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.
A Theoretical Approach to Understanding Population Dynamics with Seasonal Developmental Durations
Journal of Nonlinear Science, 2016
There is a growing body of biological investigations to understand impacts of seasonally changing environmental conditions on population dynamics in various research fields such as single population growth and disease transmission. On the other side, understanding the population dynamics subject to seasonally changing weather conditions plays a fundamental role in predicting the trends of population patterns and disease transmission risks under the scenarios of climate change. With the host-macroparasite interaction as a motivating example, we propose a synthesised approach for investigating the population dynamics subject to seasonal environmental variations from theoretical point of view, where the model development, basic reproduction ratio formulation and computation, and rigorous mathematical analysis are involved. The resultant model with periodic delay presents a novel term related to the rate of change of the developmental duration, bringing new challenges to dynamics analysis. By investigating a periodic semiflow on a suitably chosen phase space, the global dynamics of a threshold type is established: all solutions either go to zero when basic reproduction ratio is less than one, or stabilise at a positive periodic state when the reproduction ratio is greater than one. The synthesised approach developed here is applicable to broader contexts of investigating biological systems with seasonal developmental durations.
International Journal of Differential Equations, 2011
A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings. Results indicate that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable for the considered parameter values. It is also observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value.
Ecological Informatics, 2020
Mathematical models are a powerful tool when used to describe ectotherms' life cycles, above all for their suitability in being utilised for decision support systems. In particular, two models continue to arouse the interest of the scientific community and inspire new developments: the Manetsch-VanSickle Distributed Delay Model and the Von Foerster equation. Even though these models have been widely studied, discussed and applied, some aspects relating to their different points of view in representations of the same life cycle are yet to be explored. One of the main issues open for ongoing investigation is the different modes of division in preimaginal stages, which leads to different interpretations of the concept of age between the two models. The Distributed Delay Model considers a subdivision in h chained preimaginal stages with the same size, based on the concept of physiological time, in which the development of the species is related to the daily average temperature. On the other hand, the Von Foerster equation considers chronological age, defined commonly as a time with a different scale. This work highlights the analogies between the two models and shows, using the case study of L. botrana, how to obtain the number of the h stages considered by the Distributed Delay Model, from the number of observed preimaginal stages of the Von Foerster equation. To make the models comparable, the upwind scheme has been applied to the Von Foerster equation, leading to a system of ordinary differential equations that is similar to the Distributed Delay Model.
Impacts of Incubation Delay on the Dynamics of an Eco-Epidemiological System—A Theoretical Study
Bulletin of Mathematical Biology, 2008
Parasite and predator play significant role in trophic interaction, productivity and stability of an ecosystem. In this paper, we have studied a host-parasite-predator interaction that incorporates incubation delay. How the qualitative and quantitative behaviors of the system alter with the incubation delay have been discussed both from mathematical and biological point of views. It is observed that for a lower infection rate, the system is stable for all delays; but for a higher infection rate, there exists a threshold value of the delay above which the system is unstable and below which the system is stable leading to the persistence of all the species. Also, the instability arising from the incubation delay may be controlled if somehow the growth rate of predator population is increased. Numerical studies have also been performed to illustrate different analytical findings.
Applied Mathematics and Computation, 2010
A delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function: the Verhulst-Pearl logistic growth function and the Maynard-Smith-Slatkin function. It is shown that, for any birth rate function satisfying some assumptions, the trivial equilibrium of the model is globally-asymptotically stable if the associated vectorial reproduction number is less than unity. Further, the model has a non-trivial equilibrium which is locally-asymptotically stable under a certain condition. The non-trivial equilibrium bifurcates into a limit cycle via a Hopf bifurcation. It is shown, by numerical simulations, that the amplitude of oscillating solutions increases with increasing maturation delay. Numerical simulations suggest that the Maynard-Smith-Slatkin function is more suitable for modelling the vector population dynamics than the Verhulst-Pearl logistic growth model, since the former is associated with increased sustained oscillations, which (in our view) is a desirable ecological feature, since it guarantees the persistence of the vector in the ecosystem.
Dynamics of a stage-structured population model incorporating a state-dependent maturation delay
Nonlinear Analysis: Real World Applications, 2005
In this paper we derive a stage-structured model for a single species on a finite onedimensional lattice. There is no migration into or from the lattice. The resulting system of equations, to be solved for the total adult population on each patch, is a system of delay equations involving the maturation delay for the species, and the delay term is nonlocal involving the population on all patches. We prove that the model has a positivity preserving property. The main theorems of the paper are comparison principles for the cases when the birth function is increasing and when the birth function is a nonmonotone function. Using these theorems we prove results on the global stability of a positive equilibrium.
A mathematical study on the dynamics of an eco-epidemiological model in the presence of delay
In the present work a mathematical model of the prey-predator system with disease in the prey is proposed. The basic model is then modified by the introduction of time delay. The stability of the boundary and endemic equilibria are discussed. The stability and bifurcation analysis of the resulting delay differential equation model is studied and ranges of the delay inducing stability as well as the instability for the system are found. Using the normal form theory and center manifold argument, we derive the methodical formulae for determining the bifurcation direction and the stability of the bifurcating periodic solution. Some numerical simulations are carried out to explain our theoretical analysis.
Dynamics of a Time-Delayed Lyme Disease Model with Seasonality
SIAM Journal on Applied Dynamical Systems, 2017
In this paper, we propose a time-delayed Lyme disease model incorporating the climate factors. We obtain the existence of a disease-free periodic solution under some additional conditions. Then we introduce the basic reproduction ratio R0 and show that under the same set of conditions, R0 serves as a threshold parameter in determining the global dynamics of the model; that is, the disease-free periodic solution is globally attractive if R0 < 1; the system is uniformly persistent and admits a positive periodic solution if R0 > 1. Numerically, we study the Lyme disease transmission in Long Point, Ontario, Canada. Our simulation results indicate that Lyme disease is endemic in this region if no further intervention is taken. We find that Lyme disease will die out in this area if we decrease the recruitment rate of larvae, which implies that we can control the disease by preventing tick eggs from hatching into larvae.