Positive Equilibrium Research Articles

Turing–Hopf bifurcation in a general Selkov–Schnakenberg reaction–diffusion system

The dynamics of a general Selkov–Schnakenberg reaction–diffusion model is investigated. We first study the globally stability of the positive equilibrium and the existence of Hopf bifurcation for the corresponding ordinary differential equation (ODE). Based on that, Turing and Turing–Hopf bifurcations due to diffusion of reaction–diffusion model are demonstrated. Using the center manifold theory and the normal form method, the bifurcation diagram of Turing–Hopf bifurcation is obtained and the spatial–temporal patterns in the different regions are considered. Numerical simulations show that the spatially homogeneous periodic, the spatially inhomogeneous and the spatially inhomogeneous periodic solutions appear.

DYNAMICAL ANALYSIS OF TB MODEL WITH NONLINEAR INCIDENCE RATE AND OPTIMAL CONTROL

We propose and analyze a TB transmission model with nonlinear incidence rate, immunization and medical treatment. First, the existences and stabilities of the equilibrium are studied. The results indicate the basic reproduction number [Formula: see text] is the threshold of disease extinction and persistence. The disease-free equilibrium is globally asymptotically stable when [Formula: see text], and the disease will gradually disappear. The unique positive equilibrium is local stability and the disease is uniformly persistent when [Formula: see text]. Second, optimal control is added to the original model because of limited resources. Finally, the stability of the equilibrium and the theoretical results of optimal control are verified by numerical simulations, and the sensitivity of the parameters is analyzed by the PRCC method.

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains.

Mathematical modeling of the unemployment problem in a context of financial crisis

In this paper, we formulate a new system of nonlinear ordinary differential equations to study the unemployment problem in a context of financial crisis. We first prove the existence of a unique positive equilibrium. Then, using an appropriate Lyapunov function under some specified conditions, we prove the global stability of the unique positive equilibrium. We also propose and compare two control strategies with the objective to improve, at the lowest cost, the employment rate. Our results suggest that, in order to reduce the unemployment rate, it is better for a government to assist unemployed people in building their own business which will allow them to further create new vacancies than to assist self-employed individuals to create new vacancies. Numerical simulations are presented to substantiate the theoretical results.

Diffusion-Induced Instability of the Periodic Solutions in a Reaction-Diffusion Predator-Prey Model with Dormancy of Predators

A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and stability of the equilibrium solutions and derive precise conditions on system parameters so that the system can undergo Hopf bifurcations around the positive equilibrium solution. Moreover, the properties of Hopf bifurcation are studied in detail. For the reaction-diffusion system, we are able to derive conditions on the diffusion coefficients so that the spatially homogeneous Hopf bifurcating periodic solutions can undergo diffusion-triggered instability. To support our theoretical analysis, we also include several numerical results.

Global stability of a networked predator–prey model

A network is introduced to Lotka predator–prey model, where the network structure describes the movement directions between every two nodes. By using monotone iterative approach, we study the role of hunting strength on persistence and extinction. In the case of strong hunting, solutions converge uniformly to the semipositive equilibrium such that the predator survives while the prey becomes extinct. In the case of weak hunting, solutions converge uniformly to the positive equilibrium such that both predator and prey coexist.

The Influence of Nonlinear Cannibalism to Logistic Equation

A single species model with Holling II type cannibalism term is proposed and studied in this paper. Local and global stability property of the system are investigated. By applying the iterative method, we show that the system always admits the unique globally asymptotically stable positive equilibrium. A threshold value R0, which depends on the cannibalism rate and the transform rate, is obtained. Depending on R0 > 1, R0 = 1 or R0 < 1, the final density of the species will smaller or equal to or bigger than the system without cannibalism. Our study shows that if the cannibalism rate is too large, and transform rate is too small, then R0 > 1 and cannibalism has negative effect on the final density of the species, which increase the extinction property of the species.

Fear Effect on a Predator–Prey Model with Non-Differential Fractional Functional Response

In this paper, we study the factor of the fear effect in a predator–prey model with prey refuge and a non-differentiable fractional functional response due to the group defense. Since the functional response is non-differentiable, the dynamics of this system are considerably different from the dynamics of a classical predator–prey system. The persistence, the stability and the existence of the steady states are investigated. We examine the Hopf bifurcation at the unique positive equilibrium. Direct Hopf bifurcation is studied via the central manifold theorem. When the value of the fear factor decreases and is less than a threshold κH, the limit cycle appears, and it disappears through a loop of heteroclinic orbits when the value of the fear factor is equal to a value κhet.

A Reaction-Diffusion Model with Spatially Inhomogeneous Delays.

Motivated by population growth in a heterogeneous environment, this manuscript builds a reaction-diffusion model with spatially dependent parameters. In particular, a term for spatially uneven maturation durations is included in the model, which puts the current investigation among the very few studies on reaction-diffusion systems with spatially dependent delays. Rigorous analysis is performed, including the well-posedness of the model, the basic reproduction ratio formulation and long-term behavior of solutions. Under mild assumptions on model parameters, extinction of the species is predicted when the basic reproduction ratio is less than one. When the birth rate is an increasing function and the basic reproduction ratio is greater than one, uniqueness and global attractivity of a positive equilibrium can be established with the help of a novel functional phase space. Permanence of the species is shown when the birth function is in a unimodal form and the basic reproduction ratio is greater than one. The synthesized approach proposed here is applicable to broader contexts of studies on the impact of spatial heterogeneity on population dynamics, in particular, when the delayed feedbacks are involved and the response time is spatially varying.

Effects of Groundwater Level Control on Soil Salinity Change in Farmland around Wetlands in Arid Areas: A Case Study of the Lower Reaches of the Shiyang River Basin, China

The farmland around wetlands in the lower reaches of an arid area is susceptible to salinization. To explore the effects of the groundwater level control at an irrigation cycle scale on the salt concentration of the topsoil solution, this study carried out groundwater level control and irrigation experiments using the intelligent groundwater control and in-situ field monitoring system (also referred to as the groundwater control system) in the experimental base for groundwater control of the Shiyang River basin. On this basis, this study compared and analyzed the changes in groundwater depth, soil salinity, soil moisture content, and total water potential in zones with and without groundwater control (also referred to as the control and non-control zones, respectively). Results show: (1) When the groundwater depth increased by about 50 cm under the influence of the groundwater control system, the salt accumulation layer of the soil bulk shifted downward by about 20 cm, and the topsoil bulk salt (at a depth of less than 40 cm) decreased to below 5.0 g/kg; (2) In summer, the pore water electrical conductivity (ECp) of the topsoil in the control and non-control zones exhibited alternating rapid decreases and slow increases. In the concentration stage of the soil solution, the ECp of the topsoil in the non-control zone had significantly higher increased amplitude than that in the control zone, especially 3–8 days after irrigation. At this stage, the ECp of the topsoil in the control and non-control zones increased in two (slow and rapid increase) and three (slow, rapid, and fairly rapid increase) periods, respectively; (3) At the concentration stage of the topsoil solution, both the moisture content and solution salt content of the topsoil in the control zone were in a negative equilibrium state, with the absolute values of the equilibrium values gradually increasing. In contrast, the moisture content and solution salt content of the topsoil in the non-control zone were in negative and positive equilibrium, respectively, with the absolute values of their equilibrium values gradually increasing. The groundwater control system can mitigate the concentration rate of the topsoil solution by increasing the groundwater depth and influencing the water and salt equilibrium of the topsoil solution, which can create a suitable topsoil salt environment for crop growth. This study is of great significance for determining an appropriate ecological water level interval and optimizing groundwater control strategies for farmland around wetlands.

Existence of positive periodic solutions for a predator-prey model

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

Dynamic analysis of a stage-structured forest population model with non-smooth continuous threshold harvesting

In this article, we develop a delayed tree population model with non-smooth continuous threshold harvesting. By carefully addressing the main difficulty caused by the nonlinear term and the time delay together, the existence and number of the positive equilibrium points are classified. Taking full use of the domain-decomposition method and Hopf bifurcation theory, the stability of the model, including local and global asymptotic stability of the equilibria, bistability and stability switches occurring at the positive equilibrium, is analyzed relatively completely. This is followed by an investigation of the influence of the maturation delay of trees on the optimal harvesting. Numerical analysis illustrates that the model gives rise to chaotic-like oscillations through period doubling with the increase of the maturation delay or the harvesting rate and undergoes long transients and regime shifts. Even, the maturation delay can induce multi-types of tristability.

Dynamics of a delayed population patch model with the dispersion matrix incorporating population loss

AbstractIn this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.

Economic incentives and political inequality in the management of environmental public goods

We study how the allocation of power in a voting procedure affects the regulation and extraction of environmental public goods. In an appropriation game experiment, different endowments induce heterogeneous preferences among the three group members regarding their preferred quota, aimed at increasing social efficiency by restricting aggregate extraction. The players vote by submitting a proposal; one among the submitted proposals is implemented, selected at random, but across treatments, we vary the odds that a type sets the regulation.Participants vote for their selfishly preferred quota between half and two-thirds of the time. Although setting quotas decreased aggregate extraction by roughly 10%, we do not find differences between treatments. Once quotas are removed, extraction rises back to inefficient levels. Our results suggest that, although participants are reciprocal, they neglect the positive equilibrium effects of a stricter regulation.

Dynamics of a Delayed Predator–Prey Model with Prey Refuge, Allee Effect and Fear Effect

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results.

Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect

In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help to understand the changing patterns of biological populations from a completely new perspective. The existence and stability of the three equilibria are analyzed, and the formation conditions of Neimark–Sacker bifurcation around the unique positive equilibrium point are established using the center manifold theorem and bifurcation theory. An attracting closed invariant curve appears, which corresponds to the periodic oscillations between predators and prey over a long period of time. Finally, some numerical simulations and their biological meanings are given to reveal the complex dynamical behavior.

The role of tumor activation and inhibition with saturation effects in a mathematical model of tumor and immune system interactions undergoing oncolytic viral therapy

We propose and study a mathematical model governing interactions between cancer and immune system with an oncolytic viral therapy (OVT), wherein cancer cells can activate and inhibit immune cells simultaneously with saturations. When the therapy is not applied, it is shown that the interaction can support at most three hyperbolic positive equilibria where two of them are always asymptotically stable and the other is a saddle point. The reachable stable tumor burden can be either small or large depending on initial tumor size. We analyze the full model by proving global asymptotic stability of the virus‐free equilibrium that corresponds to OVT failure. Sufficient conditions based on model parameters are derived under which the model is uniformly persistent. The proposed system is validated using a mouse model of human pancreatic cancer carried out by Koujima et al. Global sensitivity analysis indicates that the rates of tumor‐mediated killing and immune cell exhaustion are critical for tumor progression and therapy success. Numerical bifurcation analysis reveals that the saddle point can be utilized to estimate the maximum tumor load for eradication by OVT. Moreover, an immunosuppressive microenvironment may enhance viral therapy efficacy.

Complex dynamical study of a delayed prey–predator model with fear in prey and square root harvesting of both species

In the current study, the dynamics of predator-prey systems under the influence of fear effect on the reproduction of prey population and harvesting on both species has been proposed. Assessing the dynamics of the system with the combined influence of fear and harvesting for various values of n is our central objective. We present comprehensive mathematical findings that cover fundamental dynamical features, the presence of positive equilibria, and the stability of all equilibria. Hopf-bifurcating periodic solutions have been demonstrated to emerge around the positive equilibrium point, and the direction of the Hopf-bifurcating limit cycle is determined using the first Lyapunov coefficient. Furthermore, in two-parameter space, we have seen that the system experiences the Bogdanov-Takens bifurcation. Moreover, we have included predator gestation delay and noticed some chaotic dynamics in the system. In addition, we run through numerical simulations to numerically validate our mathematical findings. The article is concluded with a conclusion at the end.

Stability Analysis and Control Strategy of a Generalized Food Chain Model

The stability analysis and control strategy for a generalized food chain model is considered in this paper. Firstly, we investigate the existence of the positive equilibrium and give the conditions of local asymptotic stability. Meanwhile, we design a feedback controller and prove the stability of the proposed model with the controller by using the Lyapunov function. Due to the addition of control, the unstable food chain system can become stable. At last, the analysis results are verified by numerical simulations. Our results may help to maintain the stability of the biological systems and avoid ecosystem imbalance.

Global stability of Leslie-Gower Predator-prey Model with Density Dependent Birth Rate on Prey Species and Prey Refuge

A Leslie-Gower predator prey model with density dependent birth rate on prey species and prey refuge is proposed and studied in this paper. Sufficient condition which ensure the global stable of the positive equilibrium is obtained. Our study indicates density dependent birth rate of prey species has negative effect on the final density of both prey and predator species. Density dependent birth rate may lead to the Allee effect of prey species and enhance the extinction chance of the species. Numeric simulations are carried out to show the feasibility of the main results.