Existence Of Positive Periodic Solutions Research Articles

Dynamics of autonomous and nonautonomous Leslie–Gower model for the impacts of fear and its carry‐over effects including predator harvesting

Recently, certain biological field experiments and research findings have revealed that predators not only reduce prey populations through direct predation (lethal effect) but also indirectly affect the growth rate of prey by inducing fear; these nonlethal effects can be carried over seasons or generations. In this paper, we proposed an autonomous and a nonautonomous Leslie–Gower model incorporating some biological factors such as fear and its carry‐over effects, prey refuge, and nonlinear predator harvesting. In the autonomous model, first, we examined the well‐posedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in intrinsic growth rate of predator. Further, we compute and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf bifurcation and direction and stability of limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of fear factor, carry‐over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas for small values of prey birth rate and harvesting effort, the system remains stable. However, when intrinsic growth rate of predator increases from a low value to a higher value, the system shows transition from stability to instability and back to stability. We also explore the effects of seasonal variations in biological parameters in the nonautonomous model. Additionally, we investigate the theoretical existence of positive periodic solutions using the continuation theorem. The influences of seasonal variations in some parameters such as prey's birth rate, fear, and its carry‐over effects; refuge behavior of prey; intrinsic growth rate of predator; and harvesting effort are then numerically investigated. The results reveal the emergence of positive periodic solutions and bursting patterns in the seasonally forced model.

Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, x″(t)+f(x(t))x′(t)+φ(t)xμ(t)−1xγ(t)=e(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x''(t)+f(x(t))x'(t)+\\varphi (t)x^{\\mu}(t)-\\frac{1}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+∞)→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:(0,+\\infty )\\rightarrow R$\\end{document} is continuous, which may have a singularity at the origin, the sign of φ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varphi (t)$\\end{document}, e(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e(t)$\\end{document} is allowed to change, and μ, γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T-periodic solution when μ∈[0,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu \\in [0,+\\infty )$\\end{document}.

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

Existence results for positive periodic solutions to first order neutral differential equations with distributed deviating arguments

We take into account the first order nonlinear neutral differential equation with distributed deviating arguments.
 Using Krasnoselskii's fixed point theorem, we give some new criteria for the existence of positive periodic solutions to this equation. The theorems we have established are illustrated by an example.

Positive periodic solution for enterprise cluster model with feedback controls and time-varying delays on time scales

<abstract><p>This paper aims to study a class of enterprise cluster models with feedback controls and time-varying delays on time scales. Based on periodic time scales theory and the fixed point theorem of strict-set-contraction, some new sufficient conditions for the existence of positive periodic solutions are obtained. Finally, two examples are presented to verify the validity and applicability of the main results in this paper.</p></abstract>

Study of an Epidemiological Model for Plant Virus Diseases with Periodic Coefficients

In the present article, we research the existence of the positive periodic solutions for a mathematical model that describes the propagation dynamics of a pathogen living within a vector population over a plant population. We propose a generalized compartment model of the susceptible–infected–susceptible (SIS) type. This model is derived primarily based on four assumptions: (i) the plant population is subdivided into healthy plants, which are susceptible to virus infection, and infected plants; (ii) the vector population is categorized into non-infectious and infectious vectors; (iii) the dynamics of pathogen propagation follow the standard susceptible–infected–susceptible pattern; and (iv) the rates of pathogen propagation are time-dependent functions. The main contribution of this paper is the introduction of a sufficient condition for the existence of positive periodic solutions in the model. The proof of our main results relies on a priori estimates of system solutions and the application of coincidence degree theory. Additionally, we present some numerical examples that demonstrate the periodic behavior of the system.

Multiplicity results for a class of nonlinear singular differential equation with a parameter

This paper gives sufficient conditions for the existence of positive periodic solutions to general indefinite singular differential equations. Furthermore, under some assumptions we show the existence of two positive periodic solutions. The methods used are Krasnoselski\(\breve{\mbox{i}}\)'s-Guo fixed point theorem and the positivity of the associated Green's function.

Periodic dynamics of predator-prey system with Beddington–DeAngelis functional response and discontinuous harvesting

This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin’s coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified.

Modeling Typhoid Fever Dynamics: Stability Analysis and Periodic Solutions in Epidemic Model with Partial Susceptibility

Mathematical models play a crucial role in predicting disease dynamics and estimating key quantities. Non-autonomous models offer the advantage of capturing temporal variations and changes in the system. In this study, we analyzed the transmission of typhoid fever in a population using a compartmental model that accounted for dynamic changes occurring periodically in the environment. First, we determined the basic reproduction number, R0, for the periodic model and derived the time-average reproduction rate, [R0], for the non-autonomous model as well as the basic reproduction number, R0A, for the autonomous model. We conducted an analysis to examine the global stability of the disease-free solution and endemic periodic solutions. Our findings demonstrated that when R0<1, the disease-free solution was globally asymptotically stable, indicating the extinction of typhoid fever. Conversely, when R0>1, the disease became endemic in the population, confirming the existence of positive periodic solutions. We also presented numerical simulations that supported these theoretical results. Furthermore, we conducted a sensitivity analysis of R0A, [R0] and the infected compartments, aiming to assess the impact of model parameters on these quantities. Our results showed that the human-to-human infection rate has a significant impact on the reproduction number, while the environment-to-human infection rate and the bacteria excretion rate affect long-cycle infections. Moreover, we examined the effects of parameter modifications and how they impact the implementing of efficient control strategies to combat TyF. Although our model is limited by the lack of precise parameter values, the qualitative results remain consistent even with alternative parameter settings.

A non-autonomous approach to study the impact of environmental toxins on nutrient-plankton system

The interaction between phytoplankton and zooplankton has a significant impact on the marine ecology. The interplay between these two species is the building blocks for most of the food webs operating in an aquatic ecosphere. The environmental toxins released by different external sources also affect the phytoplankton-zooplankton dynamics. In the present study, we propose a model to explore the kinetics of a nutrient-phytoplankton-zooplankton-environmental toxins (NPZT) system. The defence mechanism of phytoplankton against zooplankton is reflected through modified Holling type IV response, whereas the consumption of nutrients by phytoplankton is outlined by Holling type II response. The external toxins are assumed to have the capability of reducing the birth rate of phytoplankton species after coming into contact with their cells. To make our model more pragmatic, seasonal variation in the parameters is also taken into account. Firstly, we do the analysis related to the autonomous model (non-seasonal) like; its boundedness, existence of equilibrium points, their stability analysis, and occurrence of Hopf-bifurcation. Further, for the non-autonomous model (seasonal), we analyze the existence of positive periodic solution and its global stability. Through numerical simulations, we observe that for the non-seasonal model, increasing the rate of suppressing phytoplankton’s growth by environmental toxin, and rate at which environmental toxin is added to system make it unstable through Hopf-bifurcation. These oscillations can be removed by raising phytoplankton’s inhibitory effect against zooplankton, and this increment also leads to the extinction of the zooplankton population, making zooplankton free equilibrium a stable one. Both models, non-seasonal as well as seasonal manifest different types of multistability, and this is an exciting character associated with non-linear models. We also note that the inclusion of seasonality in our system promotes the coexistence of all populations. Further, through numerical simulations, we show that making some of the parameters seasonal can cause the emergence of chaos in the system. To verify chaos, we sketch the Poincaré map and evaluate the maximum Lyapunov exponent. The seasonal model also shows the switching of stability through different periodic and chaotic windows on varying the maximum intrinsic growth rate for phytoplankton, and contact rate between environmental toxin and phytoplankton. To substantiate our results, we picture several time-series graphs, basins of attraction, one and two-parametric bifurcation diagrams. Thus we expect that the present work can assist biologists and mathematicians in studying nutrient-plankton systems in a more detailed and realistic manner. This study can also help researchers in the estimation of non-seasonal as well as seasonal parameters while studying these types of complex non-linear models. Therefore, the present work seems to be enriched from a mathematical and biological point of view.

Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat’s lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results.

Positive Periodic Solutions for a First-Order Nonlinear Neutral Differential Equation with Impulses on Time Scales

In this article, we discuss the existence of a positive periodic solution for a first-order nonlinear neutral differential equation with impulses on time scales. Based on the Leggett–Williams fixed-point theorem and Krasnoselskii’s fixed-point theorem, some sufficient conditions are established for the existence of positive periodic solution. An example is given to show the feasibility and application of the obtained results. Since periodic solutions are solutions with symmetry characteristics, the existence conditions for periodic solutions also imply symmetry.

Existence of positive periodic solutions for a periodic predator–prey model with fear effect and general functional responses

This paper investigates the existence of positive periodic solutions for a periodic predator-prey model with fear effect and general functional responses. The general functional responses can cover the Holling types II and III functional response, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, the ratio-dependent type with Michaelis–Menten type functional response, etc. Some new sufficient conditions for the existence of positive periodic solutions of the model are obtained by employing the continuation theorem of coincidence degree theory and some ingenious estimation techniques for the upper and lower bounds of the a priori solutions of the corresponding operator equation. Our results considerably improve and extend some known results.

Existence of positive periodic solutions for a predator-prey model

In this paper, a class of nonlinear predator-prey models with three discrete delays is considered. By linearizing the system at the positive equilibrium point and analyzing the instability of the linearized system, two sufficient conditions to guarantee the existence of positive periodic solutions of the system are obtained. It is found that under suitable conditions on the parameters, time delay affects the stability of the system. The present method does not need to consider a bifurcating equation which is very complex for such a predator-prey model with three discrete delays. Some numerical simulations are provided to illustrate our theoretical prediction.

Global suppression and periodic change of the mosquito population in a sterile release model with delay

In this paper, we study a mosquito population suppression model with a time-delay effect, which includes explicitly the growth stage and sex structure of the mosquito population. According to the release function of sexually active sterile male mosquitoes in the field, we determine a release threshold and obtain sufficient conditions to successfully suppress the wild mosquito population. When these conditions are not satisfied, we use the fixed point theorem to obtain the existence of positive periodic solutions and determine their existence region for the common periodic release case in practice. Finally, we present a numerical example to support our theoretical results.

Positive periodic solutions for a delay model of erythropoiesis with iterative terms

In the present paper, we mainly focus on an iterative erythropoiesis model with a nonlinear harvesting term involving a time-varying delay. Under certain conditions and by the Schauder fixed point theorem, we show that the existence of positive periodic solutions can be guaranteed and based on the Banach fixed point theorem, we establish the existence as well as the continuous dependence on parameters theorems of the unique solution. Moreover, two examples are provided to validate the correctness of our outcomes. Our theoretical findings extend and improve several related ones in the existing literature.

Uniqueness and existence of positive periodic solutions of functional differential equations

<abstract><p>In this paper, some new findings on the uniqueness and existence of positive periodic solutions to first-order functional differential equations are presented. These equations have wide applications in a variety of fields. The most important feature of our argument is that we use the theory of Hilbert's metric to prove the uniqueness of the positive periodic solution when $ q=-1 $ and $ -1 < q < 0 $. In addition, we also investigate the existence results of positive periodic solutions by applying a fixed point theorem for completely continuous maps in a cone. Two examples demonstrate our findings.</p></abstract>

Characterizing the existence of positive periodic solutions in the weighted periodic-parabolic degenerate logistic equation

<p style='text-indent:20px;'>The main goal of this paper is to characterize the existence of positive periodic solutions for a general class of weighted periodic-parabolic logistic problems of degenerate type (see (1)). This result provides us with is an optimal substantial generalization of the previous findings of the authors in [<xref ref-type="bibr" rid="b1">1</xref>] and [<xref ref-type="bibr" rid="b2">2</xref>].</p>

THE DYNAMICAL BEHAVIOR AND PERIODIC SOLUTION IN DELAYED NONAUTONOMOUS CHEMOSTAT MODELS

In this paper, the global dynamics and existence of positive periodic solutions in a general delayed nonautonomous chemostat model are investigated. The positivity and ultimate boundedness of solutions are firstly obtained. The sufficient conditions on the uniform persistence and strong persistence of solutions are established. Furthermore, the criterion on the global attractivity of trivial solution is also established. As the applications of main results, the periodic delayed chemostat model is discussed, and the necessary and sufficient criteria on the existence of positive periodic solutions, and uniform persistence and extinction of microorganism species are obtained. Lastly, the numerical examples are presented to illustrate the main conclusions.

Positive periodic solutions for discrete Nicholson system with multiple time-varying delays

<abstract><p>Fly communities exhibit rich ecological dynamics, and one of the important influencing factors is the interaction between species. A discrete Nicholson-type system with multiple time varying delays which considers the mutualism relationship between two fly species is investigated in this paper. Sufficient conditions for the existence of positive periodic solutions are elucidated. The result is obtained by the well-known continuation theorem of coincidence degree theory. An example is attached to illustrate our result. Moreover, the actual biological descriptions obtained from our main result are explained.</p></abstract>