Positive Periodic Solutions Research Articles

Complex Dynamics of Beddington–DeAngelis-Type Predator-Prey Model with Nonlinear Impulsive Control

According to resource limitation, a more realistic pest management is that the impulsive control actions should be adjusted according to the densities of both pest and natural enemy in the field, which result in nonlinear impulsive control. Therefore, we have proposed a Beddington–DeAngelis interference predator-prey model concerning integrated pest management with both density-dependent pest and natural enemy population. We find that the pest-eradication periodic solution is globally stable if the impulsive period is less than the critical value by Floquet theorem. The condition of permanent is established, and a stable positive periodic solution appears via a supercritical bifurcation by bifurcation theorem. Finally, in order to investigate the effects of those nonlinear control strategies on the successful pest control, the bifurcation diagrams showed that the model exists with very complex dynamics. Consequently, the resource limitation may result in pest outbreak in complex ways, which means that the pest control strategies should be carefully designed.

A threshold policy to curb WNV transmission to birds with seasonality

Seasonality is pervasive in nature and WNV is a complex disease which appears to be transmitted periodically. Lacking of vaccine and anti-virus treatment brings culling to be an effective strategy of controlling the spread of WNV. This paper proposes a piecewise smooth model of WNV transmission between mosquitoes and birds with periodic forcing by employing the threshold policy of culling mosquitoes and birds when the number of infected birds exceeds the critical level. We firstly investigate the dynamical behaviours of the periodic subsystems and illustrate the existence and global stability of the disease-free and positive periodic solutions. Then by assuming a periodic variation in the total number of mosquitoes and birds with and without culling strategy respectively in the ecosystem, the subsystems are reduced, respectively. Further, the sliding mode domain and sliding mode dynamics are derived by using the Filippov convex method and the existence of the sliding periodic solution is illustrated. Finally, numerical investigations are carried out to address how the periodic forcing and culling strategy affect the pattern and amplitude of the novel periodic solution. Our results show that a suitable combination of threshold value and culling rates should be properly chosen to ensure that the ideal periodic solution with a low number of infected birds/mosquitoes oscillating in a small range is globally stable and consequently is approximated, which suggests useful strategies of curbing WNV transmission by setting proper threshold level and culling rates.

Dynamics for a tritrophic impulsive periodic plankton–fish system with diffusion in lakes

Impulsive reaction–diffusion system is a novel assemblage of tools for better understanding ecological dynamics. This paper is devoted to mathematical and biological analysis of an impulsive tritrophic periodic plankton–fish system with diffusion and Beddington–DeAngelis functional response in lakes. On the basis of comparison theory and method of upper and lower solution, we have rigorously analyzed ultimate boundedness and permanence of the reaction–diffusion system under impulsive effects. Significantly, the existence and globally asymptotic stability of a unique positive periodic solution are verified by Brouwer's fixed point theorem and auxiliary function method. Additionally, two particular scenarios are chosen to illustrate our results. It is found that impulsive effects can influence lake ecological dynamics and regulate lake biological structure.

Interplay Between Reproduction and Age Selective Harvesting Delays of a Single Population Non-Autonomous System

This paper is concerned with an analysis of the dynamics of a non-autonomous, single population age based growth model with harvesting formulation. First, we derive sufficient conditions for permanence and positive invariance. Then, by constructing a scalar function, namely the Lyapunov function, we arrive at a suitable criterion for global attractivity. With the help of Brouwer fixed point and continuation theorems, we obtain constraints for the existence of a positive periodic solution. Then we prove that there exists only one solution which is almost periodic in nature that is distinct from every other solution. Further, we carry out a numerical simulation to support the analytical findings.

Quantifying the impact of insecticide resistance in the transmission dynamics of malaria

Abstract The fitness cost imposed by insecticide resistance on malaria vectors has appeared as an asset for vector control despite the evolution and spread of insecticide resistance. However, certain levels of insecticide resistance intensity may reduce the efficacy of insecticide-treated nets for certain coverage levels. In this study, we consider a non-autonomous mathematical model for malaria transmission that investigates the impact of the interaction between the intensity of resistance, the coverage of insecticide-treated nets and the fitness cost of resistance on resistant vectors (fitness disadvantage or decreased performance) on the transmission dynamics of malaria. It is shown that the disease-free periodic solution of the model is globally attractive if the time-dependent reproduction number ( R T ( t ) 1 ). The model also admits a unique positive periodic solution when R T ( t ) > 1 . We numerically studied the interaction between the fitness cost imposed by resistance on resistant vectors, the intensity of resistance and bed-net coverage. We observed that the intensity level present in the population may significantly affect the impact of the fitness cost that resistance imposes on resistant mosquitoes as well as reduce the effectiveness of control measures.

Impact of stochastic perturbation on the persistence and extinction risk of a multi-delayed prey–predator system in non-autonomous environment

Dynamical characteristics of a stochastically perturbed system are drastically different from those of a deterministic system. This study deals with the dynamics of a stochastically perturbed time-variant prey–predator model of aquatic animals hilsa (prey) and eel (predator). It is a well-known fact that overfishing and indiscriminate fishing of juvenile hilsa have led to a drastic fall in its population. To save the hilsa population from becoming extinct, we equip our model with age-structured growth and harvest in hilsa and study the effect of environmental noise on both prey and predator populations. We formulate predator’s consumption using Beddington–DeAngelis functional response which also provides access to some alternative food source in the absence of prey. We obtain sufficient conditions for persistence in the mean, extinction, and global attractivity and prove stochastic ultimate boundedness and the existence of a unique positive periodic solution. We have included many numerical examples to validate the analytical results. Through global sensitivity analysis using the partial rank correlation coefficient technique, we observed the environmental noise to be of great influence on the prey–predator population.

Periodicity and positivity in nonlinear neutral integro-dynamic equations with variable delay

Let T be a periodic time scale. We use Krasnoselskii's fixed point theorem for a sum of two operators to show new results on the existence of periodic and positive periodic solutions of a nonlinear neutral integro-dynamic equation with variable delay. We invert this equation to construct a sum of a contraction and a completely continuous map which is suitable for applying Krasnoselskii's theorem. The uniqueness results of this equation are studied by the contraction mapping principle.

Positive periodic solution for inertial neural networks with time-varying delays

Abstract In this paper the problems of the existence and stability of positive periodic solutions of inertial neural networks with time-varying delays are discussed by the use of Mawhin’s continuation theorem and Lyapunov functional method. Some sufficient conditions are obtained for guaranteeing the existence and stability of positive periodic solutions of the considered system. Finally, a numerical example is given to illustrate the effectiveness of the obtained results.

CHAOS IN A NONAUTONOMOUS MODEL FOR THE INTERACTIONS OF PREY AND PREDATOR WITH EFFECT OF WATER LEVEL FLUCTUATION

Water level regulates the dynamics of different populations residing in water bodies. The increase/decrease in the level of water leads to an increase/decrease in the volume of water, which influences the interactions of fishes and catching capability. We examine how seasonal variations in water level and harvesting affect the outcome of prey–predator interactions in an artificial lake. A seasonal variation of the water level is introduced in the predation rate. We derive conditions for the persistence and extinction of the populations. Using the continuation theorem, we determine the conditions for which the system has a positive periodic solution. The existence of a unique globally stable periodic solution is also presented. Moreover, we obtain conditions for the existence, uniqueness and stability of a positive almost periodic solution. We find that if the autonomous system has a stable focus, the corresponding nonautonomous system exhibits a unique stable positive periodic solution. But, whenever the autonomous system shows limit cycle oscillations, the corresponding nonautonomous system exhibits chaotic dynamics. The chaotic behavior of system is confirmed by the positivity of the maximal Lyapunov exponent. For higher values of the assimilation fraction of prey population, the persistent oscillations of the autonomous system are eliminated and this system becomes stable. On the other hand, chaotic nature of the nonautonomous system is converted into periodicity if the assimilation fraction of prey is large. Moreover, populations behave almost periodically if the seasonally varied rate parameters are almost periodic functions of time. Our findings show that water level plays an important role in the persistence of prey–predator system.

Existence of positive periodic solutions for super-linear neutral Li\xe9nard equation with a singularity of attractive type

In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type(x(t)−cx(t−σ))″+f(x(t))x′(t)−φ(t)xμ(t)+α(t)xγ(t)=e(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(x(t)-cx(t-\\sigma)\\bigr)''+f\\bigl(x(t) \\bigr)x'(t)-\\varphi(t)x^{\\mu}(t)+ \\frac{\\alpha(t)}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+infty)rightarrow R, varphi(t)>0 and alpha(t)>0 are continuous functions with T-periodicity in the t variable, c, γ are constants with |c|<1, gammageq1. Many authors obtained the existence of periodic solutions under the condition 0<muleq1, and we extend the result to mu>1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.

Positive periodic solutions for revisited Nicholson's blowflies equation with iterative harvesting term

Nicholson's blowflies differential equation is one of the famous equations in the field of population dynamics. The first steps of the apparition of this equation began in the 1950s with the Nicholson experiments on the sheep blowfly Lucilia cuprina, the hot topic in Australia at that time and since then it has drawn more and more attention of many researchers. The main purpose of the current study is to investigate the existence of periodic positive solutions for Nicholson's blowflies differential equation with an iterative harvesting function by using Schauder's fixed point theorem. Finally, an example is employed to illustrate the effectiveness of our results.

Dynamics of a chemostat with periodic nutrient supply and delay in the growth

This paper introduces a new consideration in the well known chemostat model of a one-species with a periodic input of single nutrient with period ω, which is described by a system of differential delay equations. The delay represents the interval time between the consumption of nutrient and its metabolization by the microbial species. We obtain a necessary and sufficient condition ensuring the existence of a positive periodic solution with period ω. Our proof is based firstly on the construction of a Poincaré type map associated to an ω-periodic integro-differential equation and secondly on the existence of zeroes of an appropriate function involving the fixed points of the above mentioned map, which is proved by using Whyburn’s Lemma combined with the Leray–Schauder degree. In addition, we obtain a uniqueness result for sufficiently small delays.

An exact expression of positive periodic solution for a first-order singular equation

The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

Positive periodic solutions of Friedmann's equation for the acceleration of the cosmological scale factor

Given T>0, Λ∈R, k∈{−1,0,1} and T-periodic function p of bounded variation on [0,T], we obtain conditions that guarantee the existence of a strictly positive T-periodic almost everywhere solution of Friedmann's nonlinear equation a″=(Λ−8πp)a/2−(a′2+k)/2a with singularity. Both a and a′ will be absolutely continuous and a″ Lebesgue integrable on [0,T]. This makes a a scale factor for an FRW cosmology with pressure p, cosmological constant Λ and energy density equal to 3(a′2+k)/8πa2−Λ/8π. By way of Picard iterates, a and a′ can be obtained numerically.

Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations

In this paper, we study the existence of positive periodic solutions for a class of non-autonomous second-order ordinary differential equations $$\begin{aligned} x''+\alpha x' +a(t)x^{n}-b(t)x^{n+1}+c(t)x^{n+2}=0, \end{aligned}$$ where $$\alpha \in {\mathbb {R}} $$ is a constant, n is a finite positive integer, and a(t), b(t), c(t) are continuous periodic functions. By using Mawhin’s continuation theorem, we prove the existence and multiplicity of positive periodic solutions for these equations.

Positive Periodic Solutions of Coupled Singular Rayleigh Systems

This paper mainly aims to investigate the positive periodic solutions for coupled singular Rayleigh systems. In order to establish the coupled structure, the basic framework of graph theory is employed. By means of Lyapunov method, inequality techniques and a classical consequence of Mawhin’s continuation theorem, some sufficient criterion for the positive periodic solutions has been provided. After that, taken the effects of the delays into account and without imposing more conditions, we further study the positive periodic solutions for a kind of coupled singular Rayleigh system with delays. Here not only the structure is more general than the existing works but the conditions imposed are concise. Consequently, compared with the previous results on the singular systems and coupled systems, the results we established are more generalized and some previous ones can been complemented and improved. Finally, the effectiveness of the established results are validated via an numerical example.

Graph-theoretic method on the periodicity of coupled predator–prey systems with infinite delays on a dispersal network

In this paper, infinite delays and prey dispersal on a network are considered in multi-patch periodic predator–prey systems. Novel graph-theoretic method is adopted to estimate the uniform upper bounds of unknown solutions to operator equation Lu=λNu. The existence criterion of positive periodic solutions to coupled predator–prey systems is obtained by using classic coincidence degree theory, which is closely related to prey dispersal network in a patch environment. Finally, two numerical examples are also shown to illustrate the effectiveness of theoretical results.

Modeling seasonal variation for mosquito-borne disease in the tropical monsoon environment

Mosquitoes play an important role in the spread of mosquito-borne diseases. Considering the sensitivity of mosquitoes’ aquatic stage to the seasonal shift, in this paper, we present a seasonally forced mosquito-borne epidemic model by incorporating mosquitoes’ aquatic stage (eggs, larvae, and pupae) and seasonal shift factor, which is a periodic discontinuous differential system. Firstly, some sufficient conditions for the existence and uniqueness of a disease-free solution are obtained. Further, we define the basic reproduction number mathcal{R}_{0}, and obtain the stability of the disease-free solution when mathcal{R}_{0} is less than one. And, if mathcal{R}_{0} is greater than one, the mosquito-borne disease is uniformly persistent and the model admits a positive periodic solution. Finally, some numerical simulations are given to illustrate the main theoretical results. In addition, simulation results also imply that ignoring the effects of seasonal succession can overestimate or underestimate mosquito-borne disease trends.

Positive periodic solution to an indefinite singular equation

In this paper, we prove the existence of a positive periodic solution for nonlinear differential equation with an indefinite singularity by using Green’s function and fixed point theorem in cones. The results are applicable to weak as well as strong singularities. At last, we give one example to show the application of the theorem.

Weak and strong singularities problems to Li\xe9nard equation

This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation: x″+f(x(t))x′(t)+a(t)x=b(t)xα+e(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x''+f\\bigl(x(t)\\bigr)x'(t)+a(t)x= \\frac{b(t)}{x^{\\alpha }}+e(t), $$\\end{document} where the external force e(t) may change sign, α is a constant and alpha >0. The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich–Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.