The population dynamics of a three-species ecological system with impulsive effect are investigated. Using the theories of impulsive equations and small-amplitude perturbation scales, the conditions for the system to be permanent when the number of predators released is less than some critical value can be obtained. Furthermore, because the predator in the system follows the predictions of optimal foraging theory, it follows that optimal foraging promotes species coexistence. In particular, the less beneficial prey can support the predator alone when the more beneficial prey goes extinct. Moreover, the influences of the impulsive effect and optimal foraging on inherent oscillations are studied using simulation, which reveals rich dynamic behaviors such as period-halving bifurcations, a chaotic band, a periodic window, and chaotic crises. In addition, the largest Lyapunov exponent and the power spectra of the strange attractor, which can help analyze the chaotic dynamic behavior of the model, are investigated. This information will be useful for studying the dynamic complexity of ecosystems.

A delay SIR epidemic model with difference in immunity and successive vaccination is proposed to understand their effects on the disease spread. From theorems, it is obtained that the basic reproduction number governs the dynamic behavior of the system. The existence and stability of the possible equilibria are examined in terms of a certain threshold condition about the basic reproduction number. By use of new computational techniques for delay differential equations, we prove that the system is permanent. Our results indicate that the recovery rate and the vaccination rate are two factors for the dynamic behavior of the system. Numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease, to support the analytical conclusion, and to illustrate possible behavioral scenarios of the model.

We focus on a spatially extended Holling-type IV predator-prey model that contains some important factors, such as noise (random fluctuations), external periodic forcing, and diffusion processes. By a brief stability and bifurcation analysis, we arrive at the Hopf and Turing bifurcation surface and derive the symbolic conditions for Hopf and Turing bifurcation on the spatial domain. Based on the stability and bifurcation analysis, we obtain spiral pattern formation via numerical simulation. Additionally, we study the model with a color noise and external periodic forcing. From the numerical results, we know that noise or external periodic forcing can induce instability and enhance the oscillation of the species density, and the cooperation between noise and external periodic forces inherent to the deterministic dynamics of periodically driven models gives rise to the appearance of a rich transport phenomenology. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.

A hybrid SIR vector disease model with incubation is established, where susceptible host population satisfies the logistic equation and the recovered host individuals are commercially harvested. It is utilized to discuss the transmission mechanism of infectious disease and dynamical effect of commercial harvest on population dynamics. Positivity and permanence of solutions are analytically investigated. By choosing economic interest of commercial harvesting as a parameter, dynamical behavior and local stability of model system without time delay are studied. It reveals that there is a phenomenon of singularity induced bifurcation as well as local stability switch around interior equilibrium when economic interest increases through zero. State feedback controllers are designed to stabilize model system around the desired interior equilibria in the case of zero economic interest and positive economic interest, respectively. By analyzing corresponding characteristic equation of model system with time delay, local stability analysis around interior equilibrium is discussed due to variation of time delay. Hopf bifurcation occurs at the critical value of time delay and corresponding limit cycle is also observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied. Numerical simulations are carried out to show consistency with theoretical analysis.

The authors introduce stochasticity into a predator-prey system with Beddington-DeAngelis functional response and stage structure for predator. We present the global existence and positivity of the solution and give sufficient conditions for the global stability in probability of the system. Numerical simulations are introduced to support the main theoretical results.

We establish the existence of traveling wave solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model incorporating a prey refuge. By using the shooting argument, invariant manifold theory, and the Hopf bifurcation theorem, we analyze the dynamic behavior of this model in the three-dimensional phase space. Numerical results are also presented to illustrate the theoretical results.

Peer-to-peer (P2P) botnets have emerged as one of the serious threats to Internet security. To prevent effectively P2P botnet, in this paper, a mathematical model which combines the scale-free trait of Internet with the formation of P2P botnet is presented. Explicit mathematical analysis demonstrates that the model has a globally stable endemic equilibrium when infection rate is greater than a critical value. Meanwhile, we find that, in scale-free network, the critical value is very little. Hence, it is unrealistic to completely dispel the P2P botnet. Numerical simulations show that one can take effective countermeasures to reduce the scale of P2P botnet or delay its outbreak. Our findings can provide meaningful instruction to network security management.

The global stability of a multigroup SEIR epidemic model with general latency distribution and general incidence rate is investigated. Under the given assumptions, the basic reproduction number ${\Re }_0$ is defined and proved as the role of a threshold; that is, the disease-free equilibrium $P_0$ is globally asymptotically stable if ${\Re }_0\le 1$, while an endemic equilibrium $P^{*}$ exists uniquely and is globally asymptotically stable if ${\Re }_0>1$. For the proofs, we apply the classical method of Lyapunov functionals and a recently developed graph-theoretic approach.

Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point $(u_2^{\mathrm{*}},v_2^{\mathrm{*}},w_2^{\mathrm{*}})$, and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameter $c_1$, the sequence “holes $\to$ holes-stripe mixtures $\to$ stripes $\to$ spots-stripe mixtures $\to$ spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point $(u_1^{\mathrm{*}},v_1^{\mathrm{*}},0)$, and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.

A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.

We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.

A delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence is investigated. By constructing a pair of upper-lower solutions and using Schauder's fixed point theorem, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.

We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratio , we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. If ${ℛ}_0>1$, we obtain sufficient conditions under which the endemic equilibrium $E^{*}$ of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.

We establish the interaction model of two cell populations following the concept of the random-walk, and assume the cell movement is constrained by space limitation primarily. Furthermore, we analyze the model to obtain the behavior of two cell populations as time is closed to initial state and far into the future.

Two virus infection models with antibody immune response and chronically infected cells are proposed and analyzed. Bilinear incidence rate is considered in the first model, while the incidence rate is given by a saturated functional response in the second one. One main feature of these models is that it includes both short-lived infected cells and chronically infected cells. The chronically infected cells produce much smaller amounts of virus than the short-lived infected cells and die at a much slower rate. Our mathematical analysis establishes that the global dynamics of the two models are determined by two threshold parameters $R_0$ and $R_1$. By constructing Lyapunov functions and using LaSalle's invariance principle, we have established the global asymptotic stability of all steady states of the models. We have proven that, the uninfected steady state is globally asymptotically stable (GAS) if , the infected steady state without antibody immune response exists and it is GAS if , and the infected steady state with antibody immune response exists and it is GAS if $R_1>1$. We check our theorems with numerical simulation in the end.

A class of neural networks described by nonlinear impulsive neutral nonautonomous differential equations with delays is considered. By means of Lyapunov functionals and differential inequality technique, criteria on global exponential stability of this model are derived. Many adjustable parameters are introduced in criteria to provide flexibility for the design and analysis of the system. The results of this paper are new and they supplement previously known results. An example is given to illustrate the results.

By considering bacterium death and general functional response we develop previous model of bacterial colony which focused on the traveling speed of bacteria. The minimal wave speed for our model is expressed by parameters and the necessary and sufficient conditions for traveling wave solutions (TWSs) are given. To prove the existence of TWSs, an auxiliary system is introduced and the existence of TWSs for this auxiliary system is proved by Schauder’s fixed point theorem. The limit arguments show the existence of TWSs for original system. By introducing negative one-sided Laplace transform, we prove the nonexistence of TWSs.

A mathematical model is proposed to consider the effects of saturated diagnosis and vaccination on HIV/AIDS infection. By employing center manifold theory, we prove that there exists a backward bifurcation which suggests that the disease cannot be eradicated even if the basic reproduction number is less than unity. Global stability of the disease-free equilibrium is investigated for appropriate conditions. When the basic reproduction number is greater than unity, the system is uniformly persistent. The proposed model is applied to describe HIV infection among injecting drug users (IDUs) in Yunnan province, China. Numerical studies indicate that new cases and prevalence are sensitive to transmission rate, vaccination rate, and vaccine efficacy. The findings suggest that increasing vaccination rate and vaccine efficacy and enhancing interventions like reducing share injectors can greatly reduce the transmission of HIV among IDUs in Yunnan province, China.

In this paper we established a mathematical model for national fitness in China. Based on a questionnaire and data of the General Administration of Sport of China and the National Bureau of Statistics of China, the dynamics for three classes of people are expressed by a system of three-dimensional ordinary equations. Model parameters are estimated from the data. This study indicated that national fitness put out by the Chinese government is reasonable. By finding the key parameter, the best measure to promote national fitness is put forward. In order to increase the number of people who frequently participate in sport exercise in a short period of time, if only one measure can be chosen, guiding people who never take part in physical exercise will be the best measure.

We apply the Krasnoselskii fixed-point theorem to investigate the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with a parameter; some verifiable sufficient results are established easily. In particular, our results extend and improve some previous results.

To investigate the spreading speed of cholera, Codeço’s cholera model (2001) is developed by a reaction-diffusion model that incorporates both indirect environment-to-human and direct human-to-human transmissions and the pathogen diffusion. The two transmission incidences are supposed to be saturated with infective density and pathogen density. The basic reproduction number $R_0$ is defined and the formula for minimal wave speed $c^{*}$ is given. It is proved by shooting method that there exists a traveling wave solution with speed $c$ for cholera model if and only if $c\ge c^{*}$.

The dynamics of a delayed stochastic model simulating wastewater treatment process are studied. We assume that there are stochastic fluctuations in the concentrations of the nutrient and microbes around a steady state, and introduce two distributed delays to the model describing, respectively, the times involved in nutrient recycling and the bacterial reproduction response to nutrient uptake. By constructing Lyapunov functionals, sufficient conditions for the stochastic stability of its positive equilibrium are obtained. The combined effects of the stochastic fluctuations and delays are displayed.

We investigate the spatiotemporal dynamics of a bacterial colony model. Based on the stability analysis, we derive the conditions for Hopf and Turing bifurcations. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by parameters in the model and find that the model dynamics exhibit a diffusion controlled formation growth to spots, holes and stripes pattern replication, which show that the bacterial colony model is useful in revealing the spatial predation dynamics in the real world.

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction number $R_0$ which is defined through the spectral radius of a linear integral operator. If $R_0$ < 1, then the disease-free periodic solution is globally asymptotically stable and if $R_0$ > 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.

According to integrated pest management strategies, we construct and investigate the dynamics of a Holling-Tanner predator-prey system with state dependent impulsive effects by releasing natural enemies and spraying pesticide at different thresholds. Applying the Dulacs criterion, the global stability of the positive equilibrium in the system without impulsive effect is discussed. By using impulsive differential equation geometry theory and the method of successor functions, we prove the existence of periodic solution of the system with state dependent impulsive effects. Furthermore, the stability conditions of periodic solutions are obtained. Some simulations are exerted to illustrate the feasibility of our main results.

A patch model for echinococcosis due to dogs migration is proposed to explore the effect of dogs migration among patches on the spread of echinococcosis. We firstly define the basic reproduction number $R_0$. The mathematical results show that the dynamics of the model can be completely determined by $R_0$. If , the disease-free equilibrium is globally asymptotically stable. When $R_0>1$, the model is permanence and endemic equilibrium is globally asymptotically stable. According to the simulations, it is shown that the larger diffusion of dogs from the lower epidemic areas to the higher prevalence areas can intensify the spread of echinococcosis. However, the larger diffusion of dogs from the higher prevalence areas to the lower epidemic areas can reduce the spread and is beneficial for disease control.

We study a class of discrete SIRS epidemic models with nonlinear incidence rate $F(S)G(I)$ and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction number , then the disease-free equilibrium is globally asymptotically stable, and if ${ℛ}_0>1$, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only when ${ℛ}_0>1$, the disease in the model is permanent. Some special cases of $F(S)G(I)$ are discussed. Particularly, when $F(S)G(I)=\beta SI/(1+\lambda I)$, it is obtained that the endemic equilibrium is globally asymptotically stable if and only if ${ℛ}_0>1$. Furthermore, the numerical simulations show that for general incidence rate $F(S)G(I)$ the endemic equilibrium may be globally asymptotically stable only as ${ℛ}_0>1$.

We study the global property in a delayed periodic predator-prey model with stage-structure in prey and density-independence in predator. The sufficient conditions on the ultimate boundedness of all positive solutions are obtained, and the sufficient conditions of the integrable form for the permanence and extinction are further established, respectively. Some well-known results on the predator density-dependency are improved and extended to the predator density-independent cases. The theoretical results are confirmed by the special examples and the numerical simulations.

We investigate a stochastic SI epidemic model in the complex networks. We show that this model has a unique global positive solution. Then we consider the asymptotic behavior of the model around the disease-free equilibrium and show that the solution will oscillate around the disease-free equilibrium of deterministic system when $R_0\le 1$. Furthermore, we derive that the disease will be persistent when $R_0>1$. Finally, a series of numerical simulations are presented to illustrate our mathematical findings. A new result is given such that, when $R_0\le 1$, with the increase of noise intensity the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system.

We investigate a predator-prey model with dispersal for both predator and prey among n patches; our main purpose is to extend the global stability criteria by Li and Shuai (2010) on a predator-prey model with dispersal for prey among n patches. By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive coexistence equilibrium of this model is unique and globally asymptotically stable if it exists.

This paper is concerned with the shunting inhibitory cellular neural networks (SICNNs) with time-varying delays in the leakage (or forgetting) terms. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of pseudo almost periodic solutions by using Lyapunov functional method and differential inequality techniques. We also provide numerical simulations to support the theoretical result.

A nonautonomous predator-prey model with infertility control in the prey is formulated and investigated. Threshold conditions for the permanence and extinction of fertility prey and infertility prey are established. Some new threshold values of integral form are obtained. For the periodic cases, these threshold conditions act as sharp threshold values for the permanence and extinction of fertility prey and infertility prey. There are also mounting concerns that the quantity of biological sterile drug is obtained in the process of the prevention and control of pest in the grasslands and farmland. Finally, two examples are given to illustrate the main results of this paper. The numerical simulations shown that, when the pest population is permanet, different dynamic behaviors may be found in this model, such as the global attractivity and the chaotic attractor.

Based on the real observation data in Tongcheng city, this paper established a mathematical model of schistosomiasis transmission under flood in Anhui province. The delay of schistosomiasis outbreak under flood was considered. Analysis of this model shows that the disease free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. The stability of the unique endemic equilibrium may be changed under some conditions even if the basic reproduction number is larger than one. The impact of flood on the stability of the endemic equilibrium is studied and the results imply that flood can destabilize the system and periodic solutions can arise by Hopf bifurcation. Finally, numerical simulations are performed to support these mathematical results and the results are in accord with the observation data from Tongcheng Schistosomiasis Control Station.

In order to reflect the population covered by institutional medical services, the concept of hospital potential capacity is proposed and a formula for its estimation is developed based on a population dynamic model. Using the collected data on hospital outpatient and inpatient services and the demographical information on Chongqing as an example, the demand for medical resource allocation in Chongqing is dynamically estimated. Moreover, the proposed formula is also useful in the estimation of the basic reproduction number in epidemiology. The results can be contributed to the improvement of decision-making in the allocation of medical resources and the evaluation of the interventions and control efforts of the infectious disease.

The paper investigates the behavior of price differential equation model based on economic theory with two delays. The primary aim of this thesis is to provide a research method to explore the undeveloped areas of the price model with two delays. Firstly, we modify the traditional price model by considering demand function as a downward opening quadratic function, and supply and demand functions both depending on the price of the past and the present. Then the price model with two delays is established. Secondly, by considering the price model with one delay, we get the stable interval. Regarding another delay as a parameter, we studied the linear stability and local Hopf bifurcation. In addition, we pay attention to the direction and stability of the bifurcating periodic solutions which are derived by using the normal form theory and center manifold method. Afterwards, the study turns to simulate the results through numerical analysis, which shows that the provided method is valid.

This paper presents a deterministic SATQ-type mathematical model (including susceptible, alcoholism, treating, and quitting compartments) for the spread of alcoholism with two control strategies to gain insights into this increasingly concerned about health and social phenomenon. Some properties of the solutions to the model including positivity, existence and stability are analyzed. The optimal control strategies are derived by proposing an objective functional and using Pontryagin’s Maximum Principle. Numerical simulations are also conducted in the analytic results.

We investigate an SIR epidemic model with stochastic perturbations. We assume that stochastic perturbations are of a white noise type which is directly proportional to the distances of three variables from the steady-state values, respectively. By constructing suitable Lyapunov functions and applying Itô’s formula, some qualitative properties are obtained, such as the existence of global positive solutions, stochastic boundedness, and permanence. A series of numerical simulations to illustrate these mathematical findings are presented.

A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction number $R_0$ is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable if $R_0\le 1$, and the system is uniformly persistence if $R_0>1$. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.

An SIRS model incorporating a general nonlinear contact function is formulated and analyzed. When the basic reproduction number , the disease-free equilibrium is locally asymptotically stable. There is a unique endemic equilibrium that is locally asymptotically stable if ${ℛ}_0>1$. Under some conditions, the endemic equilibrium is globally asymptotically stable. At last, we conduct numerical simulations to illustrate some results which shed light on the media report that may be the very effective method for infectious disease control.

To study the relationship between antimicrobial resistance and the concentration of antibiotics, a competitive population dynamical model is proposed between the susceptible strain and the resistant strain with antibiotic exposure. The strict mathematical analysis is performed, and the results indicate that long-term high strength antibiotic treatment and prevention can induce the extinction of susceptible strain. Thus, the prescribed dose of antibiotics must be strictly controlled during the treatment and prevention of the infections in clinics.

An SIR epidemic model with nonlinear incidence rate and time delay is investigated. The disease transmission function and the rate that infected individuals recovered from the infected compartment are assumed to be governed by general functions $F(S,I)$ and $G(I)$, respectively. By constructing Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is obtained. It is shown that the global properties of the system depend on both the properties of these general functions and the basic reproductive number $R_0$.

This paper investigates the dynamic behavior of a viral infection model with general contact rate between susceptible host cells and free virus particles. If the basic reproduction number of the virus is less than unity, by LaSalle’s invariance principle, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number of the virus is greater than unity, then the virus persists in the host and the endemic equilibrium is locally asymptotically stable.

We analyze the influence of stochastic perturbations on a single-species logistic model with the population’s nonlinear diffusion among $n$ patches. First, we show that this system has a unique positive solution. Then we obtain sufficient conditions for stochastic permanence and persistence in mean, stationary distribution, and extinction. Finally, we illustrate our conclusions through numerical simulation.

We consider an SIR endemic model in which the contact transmission function is related to the number of infected population. By theoretical analysis, it is shown that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. Furthermore, we find that the threshold value of disease spreading will be increased, when the half-saturation coefficient is more than zero, which means that it is an effective intervention policy adopted for disease spreading. However, when the endemic equilibria exist, we find that the disease can be controlled as long as we let the initial values lie in the certain range by intervention policy. This will provide a theoretical basis for the prevention and control of disease.

A discrete-time-delay differential mathematical model that described HIV infection of CD4⁺ T cells with drugs therapy is analyzed. The stability of the two equilibria and the existence of Hopf bifurcation at the positive equilibrium are investigated. Using the normal form theory and center manifold argument, the explicit formulas which determine the stability, the direction, and the period of bifurcating periodic solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.

We investigated a nonlinear model of the interaction between nutrients and plankton, which was addressed using a pair of reaction-advection-diffusion equations. Based on numerical analysis, we studied a model without diffusion and sinking terms, and we found that the phytoplankton density (a stable state) increased with the increase of nutrient density. We analyzed the model using a linear analysis technique and found that the sinking of phytoplankton could affect the system. If the sinking velocity exceeded a certain critical value, the stable state became unstable and the wavelength of phytoplankton increased with the increase of sinking velocity. Furthermore, band patterns were also produced by our model, which was affected by the diffusion and sinking of phytoplankton. Thus, the change in the diffusion and sinking of phytoplankton led to different spatial distributions of phytoplankton. All of these results are expected to be useful in the study of plankton dynamics in aquatic ecosystems.

Based on Mawhin's coincidence degree theory, sufficient conditions are obtained for the existence of at least two positive periodic solutions for a plant-hare model with toxin-determined functional response (nonmonotone). Some new technique is used in this paper, because standard arguments in the literature are not applicable.