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This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient σ(r,t) through blood vessels, and the tumor attracts blood vessels at a rate proportional to α(t). So, the boundary value condition ∂σ ∂r + α(t)(σ − ψ(t)) = 0,r = R(t),t > 0, holds on the boundary, where the function ψ(t) is the concentration of nutrient externally supplied to the tumor. Considering that the nutrients provided by the outside world are often periodic, the research in this paper assumes that ψ(t) is a periodic function. Sufficient conditions for the global stability of zero steady state are presented. Under certain conditions, we prove that there exists at least one periodic solution to the model. The results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

Zika is a mosquito-transmitted viral disease which may spread directly by the vector or sexual transmission. Zika virus may persist in semen and urine for a long time after disappearance from the blood, those persons are known as the convalescent humans. It can also be transmitted vertically among mosquitoes. In this paper, we have considered an eight-compartment Zika model to study the effect of all the said aspects on the virus dynamics in deterministic as well as stochastic environment. In analytic part, we have computed basic reproduction number and discussed stability of different equilibria. We have shown the proposed model undergoes through transcritical bifurcation when the reproduction number is unity and validate the model with real infection data of Dominican Republic in 2016. To study the model in stochastic environments, the additive noise is taken into consideration which is formulated considering the standard Brownian white noises proportional to each class. We have obtained the condition for disease extinction and persistence in mean. All theoretical findings are justified by numerical simulations. Lastly, the paper is ended with some conclusions. [ABSTRACT FROM AUTHOR]

Pine wilt disease is a destructive forest disease with strong infectivity, a wide spread range and high difficulty in prevention and control. Since controlling Monochamus alternatus, the vector of pine wood nematode (Bursaphelenchus xylophilus) can reduce the occurrence of pine wilt disease efficiently, the parasitic natural enemy of M. alternatus, Dastarcus helophoroides, is introduced in this paper. Considering the influence of parasitic time of D. helophoroides on the control effect, based on the mutualistic symbiosis and parasitic relationship among pine wood nematode, M. alternatus and D. heloporoides, this paper establishes a pine wood nematode prevention and control model with delay. Then, the stability of positive equilibrium and the existence of Hopf bifurcation are discussed. Besides, we obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Finally, numerical simulations with two sets of meaningful parameters selected by means of statistical analysis are carried out to support the theoretical findings. Through the comparative analysis of numerical simulations, the factors affecting the control effect of pine wilt disease are obtained, and some suggestions are put forward for practical control in the forest. [ABSTRACT FROM AUTHOR]

In this paper, a new delayed fractional-order model including susceptible migratory birds, infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds. Fear of predators is considered in the model, as fear can reduce the reproduction rate and disease transmission rate among prey. First, some basic mathematical results of the proposed model are discussed. Then, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such an uncontrolled system are established. A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon. It is found that the control scheme can successfully suppress the bifurcation behavior of the system, and the pulse width can be arbitrarily selected on the premise of ensuring the control effect. Compared with the traditional time-delay feedback controller, the control scheme proposed in this paper has more advantages in practical application, which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment. The proposed control scheme, in particular, remains effective even after the system has been disrupted by a constant. Numerical simulation verifies the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper handles with a time-space SIR model in two regions which have a common interface. The model is defined as coupled reaction–diffusion system with a robin boundary in interface to predict the immigration between the two regions. As known, the immigration has an influence on the spread of disease, this is caused by immigrants who could carry the disease from their areas to other places. For this reason, this paper deals with the existence of solutions to SIR model with disease spread between different regions by the Fadeo–Galerkin method and validate this result by the numerical simulations. [ABSTRACT FROM AUTHOR]

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ. 2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

For a prime p and a positive integer m, let 픽pm be the finite field of characteristic p, and ℜl := 픽pm[v]/〈vl − v〉 be a non-chain ring for a positive integer l > 1. In this paper, we study the (σ,δ)-cyclic codes over ℜl. Further, we study the application of these codes in finding DNA codes. Toward this, we first define a Gray map to find classical codes over 픽pm using codes over the ring ℜl. Later, we find the conditions for a (σ,δ)-cyclic code to be reversible. Finally, we provide DNA codes using this algebraic method with some classical codes of better parameters. [ABSTRACT FROM AUTHOR]

In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model's local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with the sustained periodic oscillation phenomenon in a three-species delayed predation system with Holling type II functional response and age structure in top predator. The top predator fertility function is regarded as a piecewise continuous smooth function with regard to their maturation period τ 2 . The complicated dynamic behavior is proved by integrated semigroup theory and Hopf bifurcation theorem for semilinear equations with non-dense domain. Through qualitative analysis and bifurcation study of the system, we yield that this system has a nontrivial periodic solution that bifurcates from the positive equilibrium age distribution when bifurcation parameter τ passes through some critical values. Numerical simulations are also provided to illustrate theoretically analytical results. [ABSTRACT FROM AUTHOR]

This paper deals with the host–parasitoid model, where the logistic equation governs the host population growth, and a proportion of the host population can find refuge. The equilibrium points' existence, number, and local character are discussed. Taking the parameter regulating the parasitoid's growth as a bifurcation parameter, we prove that Neimark–Sacker and period-doubling bifurcations occur. Despite the complex behavior, it can be proved that the system is permanent, ensuring the long-term survival of both populations. Furthermore, it was observed that the presence of the proportional refuge does not significantly influence the system's behavior compared to the system without a proportional refuge. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical ordinary differential tumor-immune model is proposed based on an immune checkpoint inhibitor, which is an innovative method for tumor immunotherapies. Two important factors in tumor-immune response are the programmed cell death protein 1 (PD-1) and its ligand PD-L1. The model consists of three populations: tumor cells, activated T cells and anti-PD-1. By analyzing the dynamics of the model, it is found that there is always a unique tumor-free equilibrium and at most two tumor interior equilibria. The nonexistence of nontrivial positive periodic orbits is established by using the new Dulac function, and then a global dynamics of the model is obtained. The conclusions of our analysis show that increasing the possibility of T cells killing tumor cells (p), early detection of tumor cells, or the use of PD-1 inhibitors to activate T cells are effective in eliminating tumor cells. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem of tumor growth with necrotic core. The model is a parabolic-hyperbolic partial differential equations, which is composed of three first-order nonlinear hyperbolic equations, a parabolic equation and an ordinary differential equation. First, we obtained the approximation model by polishing the Heaviside function, and then proved the existence and uniqueness of the solution of the approximation model. In addition, we improved the regularity of solution of the approximate problem by using the characteristic curves method, and finally proved the global existence of the weak solution of the original problem by the convergence. [ABSTRACT FROM AUTHOR]

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed c ∗ and the basic reproduction number R 0 are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder's fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when ℛ 0 > 1 , for every speed c > c ∗ there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any 0 < c < c ∗ when ℛ 0 > 1 or c > 0 when ℛ 0 < 1. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions and analyze the large-time behavior for Gurtin–Maccamy population model involving conformable fractional derivatives. As a preliminary step, we construct a generic structure of the solution associated with our proposed model by utilizing some basic properties and tools of conformable fractional calculus. We establish the existence of a unique solution of the given model with the given initial conditions. At last, by using the upper and lower solutions for the characteristic equation, we define the upper and lower boundaries for the obtained solution and describe the large-time behavior of the total population. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]

AIDS is a chronic infectious disease that has been having a major impact on human health and social stability. In this paper, based on the deterministic SIATR model, a stochastic SIATR model is developed to account for the spread of AIDS by introducing the Ornstein–Uhlenbeck process. First, the existence and uniqueness of the global positive solution of the model are investigated. Second, a threshold R0E is set: when R0E < 1, the disease will become extinct; when R0E > 1, the disease will persist. Then, it is shown that there exists a stationary distribution for the model when R0E > 1. On this basis, we derive the exact expression for the probability density function of the model in the neighborhood of the quasi-equilibrium state. Finally, the results of the previous proof are verified by several numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic cholera model that incorporates media coverage and two time delays driven by Lévy noise in order to deeply understand the propagation process of cholera in the real world, generalized nonlinear incidence rates β1(M) and β2(M) are also introduced. First, we discuss the existence and uniqueness of the global positive solution of the stochastic model by using the Lyapunov method. Moreover, the dynamic properties of stochastic solution around the disease-free and endemic equilibria are demonstrated. At last, we present numerical simulation results to reveal how Lévy jumps, time delays and media coverage affect the asymptotic properties of the stochastic model. [ABSTRACT FROM AUTHOR]

This paper investigates dynamical behaviors of a stochastic plant–pollinator model with degenerate diffusion. We derive a sufficient and almost necessary condition for the permanence and extinction for plant–pollinator model, which characterizes the long-time behavior of the dynamic systems. Examples and numerical results are provided to illustrate our findings. [ABSTRACT FROM AUTHOR]

An eco-epidemiological predator–prey system containing two separate prey patches has been assessed in this paper with disease in prey population involving ratio-dependent functional responses and incidence rates. The model dynamics is studied by employing environmental perturbations to develop more genuine as well as realistic dynamics. This comprehensive study offers analytical insights into the positivity and uniform boundedness of the model system. Through numerical investigations, the study reveals the occurrence of local bifurcations, bistable regions, hysteresis phenomena, and parametric regions where saddle-node and transcritical bifurcation curves exist. Utilizing appropriate Lyapunov functions, it is demonstrated that a unique globally positive solution emerges from positive initial values. It has also been proved that the suggested model is stochastically ultimate bounded. Afterward, certain sufficient criteria illustrate the extermination of disease and persistence in mean. Notably, stochastic perturbations are shown to potentially impede disease propagation, suggesting strategies for dynamically controlling the spread of the illness. Additionally, the study derives a set of requirements for the existence of an ergodic stationary distribution. Numerous numerical simulations, conducted using MATLAB and MATCONT, effectively illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose an Echinococcosis model with logistic growth. After giving the basic reproductive number R0, we prove that R0 can be used to govern the threshold dynamics of the model: if R0 < 1, the disease will go to extinction, while if R0 > 1, the disease will persist. Based on the data of Echinococcosis in Ürümqi, Xinjiang, China during 2006–2016, we estimate the parameters in the model and calculate that R0 = 1.42 (95% CI [0.767, 4.327]). The results show that Echinococcosis is endemic in Ürümqi, China. In addition, we obtain that MAPE = 3.54% and RMSPE = 3.87%, which indicates that the model has certain reliability and rationality. Furthermore, we carry out the sensitivity analysis of the model parameters to identify the key factors affecting the prevalence of Echinococcosis, and the effective control efforts are suggested focusing on reducing the proportion rate from sheep to dogs and increasing the recovery rate of dogs to curb the prevalence of Echinococcosis in Ürümqi, China. [ABSTRACT FROM AUTHOR]

In this paper, we present and assess a predator–prey Leslie–Gower model including disease, refuge and treatment in prey population. There are two groups of prey: those who are susceptible and infected. It is hypothesized that prey population is affected by diseases and refuge, and grows logistically in the absence of predators. Infected prey population receives treatment. The predators’ growth rate is governed by the modified Leslie–Gower dynamics. The dynamical attributes of the resulting system are boundedness, positivity of solutions, extinction criteria, existence and (local and global) stability. Biology uses mathematical analysis to identify the possible attributes of equilibrium points. The focus of this study is to assess how treatment and refuge affect the populations of ill prey, susceptible prey, predators and treated prey. The numerical simulation indicates that the influence of treatment, and refuge change the dynamics of the system (2.1). Extensive numerical simulations were performed to validate our analytical findings by using the Mathematica and MATLAB software. [ABSTRACT FROM AUTHOR]

The SOS/ERK cascades are key signaling pathways that regulate cellular processes ranging from cellular proliferation, differentiation and apoptosis to tumor formation. However, the properties of these signaling pathways are not well understood. More importantly, how stochastic perturbations of internal and external cellular environment affect these pathways remains unanswered. To answer these questions, we, in this paper, propose a stochastic model according to the biochemical reaction processes of the SOS/ERK pathways, and, respectively, research the dynamical behaviors of this model under the four kinds of noises: Gaussian noise, colored noise, Lévy noise and fraction Brown noise. Some important results are found that Gaussian and colored noises have less effect on the stability of the system when the strength of the noise is small; Lévy and fractional Brownian noises significantly change the trajectories of the system. Power spectrum analysis shows that Lévy noise induces a system with quasi-periodic trajectories. Our results not only provide an understanding of the SOS/ERK pathway, but also show generalized rules for stochastic dynamical systems. [ABSTRACT FROM AUTHOR]

Therapeutic agents acting on RNA, including RNA modification, RNA editing and RNA interference (RNAi), play a vital role in gene function study, signaling pathway, drug discovery, disease treatment, vaccine development and so on. Therein, RNAi as an emerging gene therapy has been widely applied in many cancer studies by silencing oncogenes or specific mRNA in malignant tumor cells. Mechanism and efficiency of RNAi are the key issues of RNAi technology. RNA silencing involves dynamic modeling, analyses and optimal control of RNAi gene system. Physiological delay and Hill response describing off-target are considerable elements involving RNAi efficiency. In this paper, we first formulate a four-dimensional RNAi model with time delays and Hill functions, and then investigate the complex dynamic behaviors including the number, existence and stability of internal equilibria and Hopf bifurcations of single delay and two delays. Furthermore, based on the specific mRNA degradation adopted in impulsive patterns, we build an optimal problem by adding exogenous dsRNA at alterable time points in variable dosages in a treatment session. By the method of gradient formula, we can find the optimal impulsive time and proportion of dsRNA. Finally, simulation indicates that (1) physiological lags not only raise the oscillations of mRNA but also cut down the levels of cost; (2) smaller delays and larger rates of siRNA–mRNA complex formation and dsRNA synthesis imply the rapid composition of RISC and fast synthesis of dsRNA leading to more desirable therapeutic schedule, which affords evidence for gene regulation and RNAi; (3) a larger half-saturation coefficient characterizes a unique and stable higher targeted mRNA, whereas a smaller half-saturation coefficient generates bistability in which the higher and lower targeted mRNAs simultaneously emerge; and (4) the bistability will provide a good guidance to control, suppress and degrade targeted mRNA. [ABSTRACT FROM AUTHOR]

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed. [ABSTRACT FROM AUTHOR]

This paper is concerned with a classical two-species prey–predator reaction–diffusion system with ratio-dependent functional response and subject to homogeneous Neumann boundary condition in a two-dimensional rectangle domain. By analyzing the associated eigenvalue problem, the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the constant coexistence equilibrium are established. Then when the bifurcation parameter is in the interior of range for Turing instability and near Turing bifurcation curve, the amplitude equations of the original system near the constant coexistence equilibrium are obtained by multiple-scale time perturbation analysis. On the basis of the obtained amplitude equations, the stability and classifications of spatiotemporal patterns of the original system at the constant coexistence equilibrium are discussed. Finally, to verify the validity of the obtained theoretical results, numerical simulations are also carried out. [ABSTRACT FROM AUTHOR]

In this paper, a population system with cross-diffusion and habitat complexity is selected as study object. We investigate that how cross-diffusion and habitat complexity destabilize the otherwise stable periodic solutions of the ODEs to generate the new abundant spatial Turing patterns. By utilizing the local Hopf bifurcation theorem and perturbation theory, we establish a formula to determine the Turing instability of periodic solutions of the population system with cross-diffusion and habitat complexity. Finally, numerical simulations are performed to verify theoretical analysis, simultaneously, we verify the formation process of spatial Turing patterns when the cross-diffusion coefficients and habitat complexity change. [ABSTRACT FROM AUTHOR]

The menstrual cycle of fertile female is regulated by multiple organs and hormones, controlling the reproductive health and fertility of females. This paper is dedicated to exploring how two negative feedback loops regulate and influence the dynamic behavior of the menstrual cycle. We simplified the regulation circuit of the menstrual cycle into a nonlinear differential equation with two time delays. This simplified mathematical model integrates the hypothalamus–pituitary–ovary primary negative feedback loop and the negative feedback auxiliary loop between the ovary and the pituitary. We investigated the delay-dependent conditions for the stability of the model’s equilibrium points and also demonstrated the presence of Hopf bifurcations. Theoretical analysis suggests that the primary feedback loop is the major factor driving oscillation in the system. Further numerical simulations indicate that the negative feedback auxiliary loop allows the system to adjust its amplitude while maintaining the robustness of the cycle length. Consequently, the results obtained from this model provide new insights into the regulation of oscillations in menstrual cycle irregularities. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dynamics of the modified Leslie–Gower model with a weak Allee effect and fear effect on predators. Assuming that the inherent growth rate of prey is much faster than that of predators, this becomes a singular perturbation problem. Based on the geometric singular perturbation theory, we prove the existence of canard cycles and relaxation oscillation and provide an asymptotic expression of the relaxation oscillation period. Furthermore, it demonstrates that, under certain conditions, the degenerate transcritical bifurcation point is a global attractor using geometric singular perturbation theory, the center manifold theorem and the blow-up method. Numerical simulations verify the theoretical results. [ABSTRACT FROM AUTHOR]

Due to the random search of species and from the economic point of view, combined harvesting is more suitable than selective harvesting. Thus, we have developed and analyzed a prey–predator model with the combined effect of nonlinear harvesting in this research paper. Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system. We have examined the local stability analysis of all the equilibrium points. Besides these various types, rich and complex dynamical behaviors such as backward, saddle-node, Hopf and Bogdanov–Takens (BT) bifurcations, homo-clinic loop and limit cycles appear in this model. Furthermore, interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points. Also, we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species. We have observed that the extinction of predator species may not happen due to backward bifurcation, although a stable predator-free equilibrium (PFE) exists. Finally, numerical simulations are discussed using MATLAB to verify all the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a stochastic host-generalist parasitoid model with Holling II functional response. First, by constructing a suitable Lyapunov function, we prove that the stochastic system has a unique ergodic stationary distribution under certain condition. Second, we analyze the stationary probability density when the corresponding deterministic system has a unique stable positive equilibrium. The original stochastic system is transformed to an Itô average diffusion system by using Taylor expansions at the positive equilibrium point, polar coordinate transformation and stochastic average method. Then the stationary probability density is obtained by solving the Fokker–Planck equation of the average diffusion system. Finally, we explore noise-induced state transitions between two random attractors in the bistable regions: (i) between two internal stable equilibria; (ii) between an internal equilibrium and a limit cycle. The critical noise intensity for which the state switching occurs is estimated by calculating confidence ellipses. The results show that large environmental fluctuations can obviously affect the stability of ecosystems, and cause transitions between high and low biomass of the hosts and the generalist parasitoids. [ABSTRACT FROM AUTHOR]

This paper proposes a fishery model with price fluctuations and predators. Under the assumption that the price changes much faster than other variables in the system, the fishery model can be considered as a fast–slow system. Using the approximate aggregation method, a simplified three-dimensional model is used for further research. The results indicate that due to overfishing by fishermen and the presence of predators, fish populations may become extinct, which is also known as catastrophic equilibrium. To avoid this situation, we propose two solutions. The first is to establish marine protected areas (MPAs) that prohibit fishing. We propose a fish population dynamics model with MPA, assuming that fish and predators move freely in protected areas and fishing areas. The results show that it is necessary to establish MPA. The second type is that regulatory agencies can control overfishing by fishermen by increasing taxes, thereby avoiding a catastrophic equilibrium. The ultimate goal is to achieve maximum economic benefits while maintaining balanced development of the ecosystem. [ABSTRACT FROM AUTHOR]

In this paper, we proposed and analyzed a five-dimensional system of ordinary differential equations modeling the competition of two competing bacteria in a chemostat under the influence of the leachate recirculation and in the presence of a pathogen associated only with the bacteria 1. We suppose that the nutriment is present into two forms, soluble and insoluble nutriment, and both two forms are continuously added to the chemostat. The proposed model takes the form of an “SI” epidemic model and uses general increasing growth functions and general increasing incidence rate. It admits multiple equilibria that we give the conditions under which we assure both the existence and the local stability of each equilibrium point. The possibility of periodic trajectory was excluded, and the uniform persistence of both types of bacteria was proved. Finally, several numerical examples confirming the theoretical findings are given. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a new cytokine-enhanced HIV infection model with intracellular delay τ1, virus replication delay τ2 and immune response delay τ3 are investigated. The positivity and boundedness of all solutions for the model with non-negative initial values have been proved. Moreover, two important biological parameters, called the virus reproductive number R0 and the antibody immune reproductive number R1 are established. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, the global dynamics of the equilibria is completely determined by R0 and R1. On the one hand, the results show that intracellular delay τ1, viral replication delay τ2 and immune response delay τ3 have no effect on the stability of E0, E1, and if τ1 ≥ 0, τ2 ≥ 0, τ3 = 0, the endemic equilibrium with the presence of antibody response E2 is globally asymptotically stable. On the other hand, when τ1 ≥ 0, τ2 ≥ 0, τ3 > 0, numerical analysis confirms the theorems and suggests that time delay play a positive role in virus infection, with the increase of τ3, the dynamic behavior of the equilibrium E2 will change as follows: locally asymptotically stable → unstable; Hopf bifurcation appears. [ABSTRACT FROM AUTHOR]

During the outbreak of emerging infectious diseases, a complex interplay of random events, media messages and human emotions inevitably affects disease transmission. In this paper, we propose a new stochastic epidemic model incorporating emotional influence under three-tier media alerts. Itô’s formula, Lyapunov function method and ergodicity of Markov chain are employed to research the extinction–persistence behavior of diseases. Our findings show that early detection and reporting contribute to effective prevention. However, the intensity of reporting rates and warning thresholds will need to be chosen according to the characteristic of each epidemic. Otherwise, media coverage may have a negative impact on the human psychology of disease prevention and control. A biological paradox emerges: high reporting rate and premature activation of the third-tier media warning may increase the number of infected individuals. [ABSTRACT FROM AUTHOR]

This paper centers on the analysis of the dynamics of a modified Leslie–Gower predator–prey model employing Holling-type II schemes, with the prey exhibiting pure random diffusion and the predator undergoing a mixed form of movement. The extinction of species and uniform persistence of this system are explored, and several conditions for the stability, uniqueness and multiplicity of positive steady-state solutions are derived. In contrast to the specialist and generalist predator–prey systems in open advective environments, the dynamics of this system are more intricate. It emerges that multiple positive steady-state solutions and the bistable phenomenon exist for this system when a small advection rate and a moderate predation rate are imposed. Numerical simulations reveal that the increase of diffusion rate for prey disadvantages the survival of itself and has no impact on predator invasion, while the increase of diffusion rate for predators favors the invasion of itself. [ABSTRACT FROM AUTHOR]

The sign of the speed of bistable traveling wave solution is an important focus in the field of population dynamics, since it can be used to predict which species will win the competition and ultimately survive. In this paper, the sign of bistable wave speed of a plankton model with cubic nonlinearity is studied. By a delicate analysis, a sufficient condition for bistable wave to be stationary is first obtained. Symmetric and multiplication properties for stationary bistable wave are established. By upper-lower solution method and comparison principle, two sufficient conditions are found so that the bistable wave speed is negative or positive. Moreover, numerical simulations are performed to illustrate our theoretic conclusions. [ABSTRACT FROM AUTHOR]

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIS epidemic infectious diseases model with double stochastic perturbations is proposed. First, the existence and uniqueness of the positive global solution of the model are proved. Second, the controlling conditions for the extinction and persistence of the disease are obtained. Besides, the effects of the intensity of volatility ξ 1 and the speed of reversion 1 on the dynamical behaviors of the model are discussed. Finally, some numerical examples are given to support the theoretical results. The results show that if the basic reproduction number ℛ 0 s < 1 , the disease will be extinct, that is to say that we can control the threshold ℛ 0 s to suppress the disease outbreak. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the dynamic behaviors of fractional- order logistic model with Allee effects in Caputo–Fabrizio sense. First of all, we apply the two-step Adams–Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system. The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur–Chon criterion. Moreover, we discuss the existence and direction for Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. Numerical simulations are provided to illustrate theoretical discussion. It is observed that Allee effect plays an important role in stability analysis. Strong Allee effect in population enhances the stability of the coexisting steady state. In additional, the effect of fractional-order derivative on dynamic behavior of the system is also investigated. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a discrete-time fractional-order population model are considered. The stability analysis and the topological classification of the model at the fixed point have been investigated. It is shown that the model undergoes flip and Neimark–Sacker bifurcations around the co-existence fixed point by using the bifurcation and the normal form theory. These bifurcations lead to chaos when the parameter changes at critical point. In order to control chaotic behavior in the model result from Neimark–Sacker bifurcation, the OGY feedback method has been used. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the presented model are plotted to support the correctness of the analytical results. The positive Lyapunov exponents demonstrate that chaotic behavior exists in the considered model. [ABSTRACT FROM AUTHOR]

Szemerédi’s Regularity Lemma (SRL) is a crucial tool in the analysis of large graphs, having made significant contributions in the proof of some sensational results in mathematics. Traditional methods for studying proteins in Protein–Protein Interaction (PPI) networks typically only extract the first-order or second-order neighbor information of proteins, ignoring the potential third-order or higher-order neighbor information between proteins, which may hide certain relationships between proteins. To explore more in-depth insights for PPI networks, we take into account the fourth-order neighbor information of proteins and reconstruct the network in this paper, naming it the weighted dense PPI network. We then partition it using SRL, which primarily utilizes the structural information and corresponds to a unique partition of the original network. Bioinformatics analyses such as those for pathway enrichment analysis and multiple sequence alignment show that our method can classify interacting protein pairs, grouping proteins with functional association, disease association, and sequence similarity together. Overall, this paper has three essential contributions: (1) we present a new model to overcome the astronomically large demand of vertices in applying SRL, and achieve protein classification; (2) we reconstruct a weighted dense PPI network which can make SRL work and mine potential interactions more efficiently; and (3) proteins in the same class partitioned by our method not only have sequence similarity, but also have functional associations. [ABSTRACT FROM AUTHOR]

In this paper, we study the pulsating traveling fronts for a spatially discrete periodic reaction–diffusion system with a quiescent stage. It is known that there exists a critical number c∗ > 0 (called minimal wave speed) such that a pulsating traveling front exists if and only if its speed is above c∗. In this paper, we derive a uniqueness theorem for supercritical pulsating traveling fronts. Further, we show that all supercritical pulsating traveling fronts are exponentially stable. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle. [ABSTRACT FROM AUTHOR]

In this paper, SIR model with feedback control has been considered. By constructing the Lyapunov function, using the Routh–Hurwitz criterion and the positive definite quadratic form theory, some sufficient conditions have been obtained, and some well-known results have been generalized. It is worth mentioning that if the feedback control variables have an inhibitory effect on the susceptible population and the infected population, the number of basic regeneration of the disease will be reduced. The reduction depends only on the control of the susceptible population, but not on the control of the infected population. This is a very interesting conclusion. [ABSTRACT FROM AUTHOR]

Whenever a disease spreads in the population, people have a tendency to alter their behavior due to the availability of knowledge concerning disease prevalence. Therefore, the incidence term of the model must be suitably changed to reflect the impact of information. Furthermore, a lack of medical resources affects the dynamics of disease. In this paper, a mathematical model of malaria of type ShIhRh − SvIv with media information and saturated treatment is considered. The analysis of the model is performed and it is established that when the basic reproduction number, ℛ0, is less than unity, the disease may or may not die out due to saturated treatment. Furthermore, it is pointed out that if medical resources are accessible to everyone, disease elimination in this situation is achievable. The global asymptotic stability of the unique endemic equilibrium point (EEP) is established using the geometric approach under parametric restriction. The sensitivity analysis is also carried out using the normalized forward sensitivity index (NFSI). It is difficult to derive the analytical solution for the governing model due to it being a system of highly nonlinear ordinary differential equations. To overcome this challenge, a specialized numerical scheme known as the non-standard finite difference (NSFD) approach has been applied. The suggested numerical method is subjected to an elaborate theoretical analysis and it is determined that the NSFD scheme maintains the positivity and conservation principles of the solutions. It is also established that the disease-free equilibrium (DFE) point has the same local stability criteria as that of continuous model. Our proposed NSFD scheme also captures the backward bifurcation phenomena. The outcomes of the NSFD scheme are compared to two well-known standard numerical techniques, namely the fourth-order Runge–Kutta (RK4) method and the forward Euler method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a partial differential equation (PDE) model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans. The model features the variable infectiousness during the infectious period, which we call the infection age of the infectious host. We define two thresholds ℛ1j and ℛ2j,j = 1, 2, and show that the strain j can not invade the system if ℛ1j +ℛ 2j < 1. Further, the disease-free equilibrium of the system is globally asymptotically stable if maxj{ℛ1j +ℛ 2j} < 1. When ℛ1j > 1, strain j dominance equilibrium ℰj exists, and is locally asymptotically stable when ℛ1j > 1, ℛ1i < ςℛ 1j,i,j = 1, 2,i≠j, ς ∈ (0, 1). Then, by applying Lyapunov–LaSalle techniques, we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain j. This implies strain j eliminates the other strain as long as ℛ1i/ℛ 1j < b i/bj < 1,i≠j, where bj denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain j. Finally, we study the existence of the coexistence equilibria under some conditions. [ABSTRACT FROM AUTHOR]

This paper studies the spatial propagation dynamics in diffusive disease transmission models with infectious stages, which may be nonmonotonic and partially degenerate. Using the solutions of linear heat equations, we transform the system into a new system that contains nonlocal delays and two unknown functions. From this new system, it is easy to obtain an important threshold and the numerical estimation for this threshold is easy. When the initial infection satisfies proper decaying behavior, we find that this threshold is the spreading speed for all infectious subgroups. Then the minimal wave speed of traveling wave solutions is proved to be the threshold, which connects the disease-free state to the endemic case. Moreover, some numerical examples are presented to explore the possible extension. [ABSTRACT FROM AUTHOR]

Insect populations, which are diverse and widespread, provide a principal area of utilization of the stage-structured modeling approach. In this paper, housefly populations incorporating a stage-structured model are investigated theoretically and graphically. First, stability charts and rightmost characteristic roots of the positive equilibrium are elucidated analytically and numerically. Furthermore, the Hopf bifurcation at the positive equilibrium is derived employing geometric stability switch criterion. Second, the properties of Hopf bifurcation are determined using the center manifold theorem and by reducing the equation to the Poincaré normal form. Finally, the correctness of the theoretical derivation is confirmed using a numerical simulation based on specific parameter values. Our results show that with an increase in delay τ, the unique positive equilibrium may undergo two stability switches: from stable to unstable, and from unstable to stable. Interestingly, the characteristic equation has pure imaginary roots at the second pair and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also simulate the unstable periodic solutions at the second pair of critical values through a bifurcation diagram. Therefore, a pair of supercritical Hopf bifurcations appear around the positive equilibrium of the housefly population stage-structured model. [ABSTRACT FROM AUTHOR]