Positive Equilibrium Point Research Articles

The dynamics of a delayed predator-prey model with square root functional response and stage structure

<abstract> <p>In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results.</p> </abstract>

Bifurcation analysis and chaos control of a discrete fractional-order Leslie-Gower model with fear factor

<p>This study focused on the dynamical behavior analysis of a discrete fractional Leslie-Gower model incorporating antipredator behavior and a Holling type Ⅱ functional response. Initially, we analyzed the existence and stability of the model's positive equilibrium points. For the interior positive equilibrium points, we investigated the parameter conditions leading to period-doubling bifurcation and Neimark-Sacker bifurcation using the center manifold theorem and bifurcation theory. To effectively control the chaos resulting from these bifurcations, we proposed two chaos control strategies. Numerical simulations were conducted to validate the theoretical results. These findings may contribute to the improved management and preservation of ecological systems.</p>

Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model

<p>A delayed diffusive predator-prey system with nonmonotonic functional response subject to Neumann boundary conditions is introduced in this paper. First, we analyze the associated characteristic equation to research the conditions for local stability of the positive equilibrium point and the occurrence of Turing instability induced by diffusion in the absence of delay. Second, we provide conditions for the existence of Hopf bifurcation driven by time delay. By utilizing the normal theory and center manifold theorem, we derive explicit formulas for Hopf bifurcation properties such as direction and stability from the positive equilibrium. Third, a hybrid controller is added to the system. By judiciously adjusting the control parameters, we effectively enhance the stability domain of the system, resulting in a modification of the position of the Hopf bifurcation periodic solutions. Numerical simulations demonstrate the presence of rich dynamical phenomena within the system. Moreover, sensitivity analysis was conducted using Latin hypercube sampling (LHS)/partial rank correlation coefficient (PRCC) to explore the impact of parameter variations on the output of prey and predator populations.</p>

A double time-delay Holling Ⅱ predation model with weak Allee effect and age-structure

<abstract><p>A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation.</p></abstract>

Stability Analysis of Prey-Predator Model Migration with Holling Type-III Response Function in The Presence of Competition and Toxicity

In this paper, we propose and investigate a prey-predator model with two zones contaminated with heavy metal toxicity, especially copper (Cu), which enter the reservation zone and the unreserved zone in the aquatic environment. The dynamics of the prey population in the ecosystem can migrate from the reservation zone to the unreserved zone or vice versa, while predators are assumed to look for prey in the unreserved zone. The dynamic behavior of the population is expressed as a system of differential equations based on food intake capacity and other factors. We introduce a predator population with a Holling type III predation response function, coupled with inter-specific competition among prey due to overlapping diets and assuming the prey is contaminated with copper metal toxicity. The presence of a positive equilibrium point, namely the interior equilibrium point, is analyzed and investigated for its stability using the Routh-Hurwitz stability test. Numerical simulations are carried out to verify the results of the analysis and dynamics of the system solution. The results of the analysis of the interior equilibrium point T3 in each case is a stable point. This indicates a change in the balance of prey populations and predator populations.

Dynamical properties of a new SIR epidemic model

Taking into account the depletion of food supply by all individuals, and the fact that chronic infectious diseases will not cause the infected individuals to lose their fertility completely, we first propose a new SIR epidemic model of ODE. For this model, we derive its basic reproduction number $ \mathcal{R}_0 $, and show that the disease-free equilibrium point is globally asymptotically stable when $ \mathcal{R}_0\le1 $, while the unique positive equilibrium point is globally asymptotically stable when $ \mathcal{R}_0>1 $. Then we incorporate the spatial dispersion and free boundary condition into this ODE model. The well-posedness and longtime behaviors are obtained. Particularly, we find a spreading-vanishing dichotomy in which the basic reproduction number $ \mathcal{R}_0 $ plays a crucial role.

Qualitative analysis and traveling wave solutions of a predator-prey model with time delay and stage structure

<abstract><p>In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model.</p></abstract>

Transient Dynamics of a Food Chain Model with Vigilance in the Biparameter Spaces

This article focuses on the asymptotic and transient behavior of a food chain model with vigilance in the biparameter spaces. Firstly, the local asymptotic stability conditions of the positive equilibrium point are obtained. Then, the transient dynamical behavior of the positive equilibrium point is analyzed based on eight parameters. Using numerical simulation, the stable regions of the model with single and double parameters are plotted, as well as the value changes of resilience, reactivity, and amplification envelope within the stable regions. The results show that the coexistence equilibrium point is always reactive, and the corresponding parameter ranges for system stability are large, but the resilience values are small. The vigilance parameter of the middle- and lower-level organisms has the greatest impact on the system’s resilience and reactivity, and the lowest vigilance corresponds to the maximum resilience and reactivity. Our results provide important references for effective management and protection of wild animal populations by the potential for transient amplification, disturbance growth time, and amplitude.

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

In this paper, we propose a new Leslie-Gower predator-prey model with predator-dependent prey refuge. Firstly, we obtain the positivity and boundedness of the system solution. Secondly, we prove that the origin is unstable using blow-up method, analyze the existence and local stability of the boundary equilibrium point and positive equilibrium point, and prove that the unique positive equilibrium point of the system is globally asymptotically stable by constructing a suitable Dulac function. Finally, mathematic analysis and numerical simulation show that: (1) when the strength of the predator-dependent prey refuge k = 0 , the dynamics of the predator-prey system without predator-dependent prey refuge are consistent with the results obtained from the traditional Leslie-Gower predator-prey system; (2) when k tends to positive infinity, the predator-dependent refuge lead to prey population densities fall somewhere between without prey refuge and with proportional refuge. However, the predator densities within this new form of the predator-dependent prey refuge is greater than the densities of predators without prey refuge and with proportional refuge; (3) increasing the strength k of the predator-dependent prey refuge can increase the densities of predator and prey populations respectively.

Bifurcation Analysis and Chaos Control for Prey-Predator Model With Allee Effect

The main purpose of this work is to discuss the dynamics of a predator-prey dynamical system with Allee effect. The conformable fractional derivative is applied to convert the fractional derivatives which appear in the governing model into ordinary derivatives. We use the piecewise-constant approximation method to discritize the considered model. We also investigate the occurrence of positive equilibrium points. Moreover, we analyse the stability of the equilibrium point using some stability theorems. This work also explores a Neimark-Sacker bifurcation and a period-doubling bifurcation using the theory of bifurcations. The distance between the obtained equilibrium point and some closed curves is examined for various values for the considered bifurcation parameter. The chaos control is nicely analysed using the hybrid control approach. Furthermore, we present the maximum Lyapunov exponents for different values for the bifurcation parameter. Numerical simulations are utilized to ensure that the obtained theoretical results are correct. The used techniques can be applied to deal with predator-prey models in various versions.

BIFURCATION ANALYSIS AND CHAOS CONTROL OF DISCRETE PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT

This paper presents a study of a discrete prey–predator model with additive Allee effect. The model is discretized using the forward Euler method, and the system is analyzed using bifurcation theory and chaos control. The equilibrium point of the discrete system is obtained through equilibrium analysis, and the stability of the equilibrium point is determined by analyzing the parameter conditions. The study also establishes the existence and direction of Neimark–Sacker bifurcation at the positive equilibrium point. The paper proposes two control strategies to manage chaotic behavior and Neimark–Sacker bifurcation: exponential control and hybrid feedback control. These control methods are demonstrated to be effective in controlling the chaotic behavior and bifurcation of the system through numerical simulation. Overall, the results of the study provide important insights into the dynamics of the discrete prey–predator model with additive Allee effect, as well as effective methods for controlling chaos and bifurcation in the system.

Pattern dynamics in a reaction–diffusion predator–prey model with Allee effect based on network and non-network environments

In this paper, we establish a predator–prey model with Beddington–Deangelis functional response. For this model, firstly, the existence of the positive equilibrium point and the conditions for the Turing instability are studied. Then the amplitude equation is derived through weakly nonlinear analysis, and the relationship between the selection of pattern and the coefficients of the amplitude equation is obtained. At the same time, through a large number of numerical simulations, we verify the accuracy of the theoretical analysis. Actually, we mainly choose to change the values of parameters r and d2 to study the sensitivity of the pattern to them. When the pattern tends to be stable, there could be the pattern of spots, coexistence of spots and stripes or stripes. Even if they are spot pattern, the spot density will also be different due to the selection of parameters. Finally, we simulate and compare that the network structure(mainly BA and WS) has a certain influence on the time required for pattern stabilization and the distribution of node density. The final results show that the growth rate of prey, diffusion coefficients and network structure all play an important role in the formation of Turing pattern.

Bifurcation analysis of impulsive fractional-order Beddington–DeAngelis prey–predator model

In this paper, a fractional density-dependent prey–predator model has been considered. Certain reading of local and global stabilities of an equilibrium point of a system was extracted and conducted by applying fractional systems’ stability theorems along with Lyapunov functions. Meanwhile, the persistence of the aforementioned system has been discussed and claimed to imply a local asymptotic stability for the given positive equilibrium point. Moreover, the presented model was extended to a periodic impulsive model for the prey population. Such an expansion was implemented through the periodic catching of the prey species and the periodic releasing of the predator population. By studying the effect of changing some of the system’s parameters and drawing their bifurcation diagram, it was observed that different periodic solutions appear in the system. However, the effect of an impulse on the system subjects the system to various dynamic changes and makes it experience behaviors including cycles, period-doubling bifurcation, chaos and coexistence as well. Finally, by comparing the fractional system with the classic one, it has been concluded that the fractional system is more stable than its classical one.

Dynamic analysis of a fractional-order delayed SIQR epidemic model for COVID-19 pandemic

In this paper, we build a fractional-order delayed SIQR epidemic model for COVID-19 pandemic. By leveraging the linearization method and the Laplace transform, the characteristic equation of the linearized system corresponding to the model is derived. The relationship between time delay and the stability of the positive equilibrium point is then explored through the examination of the roots of the characteristic equation and the cross-sectional conditions. Furthermore, a formula for calculating the critical value of time delay for the Hopf bifurcation is derived. Finally, suitable system parameters are chosen for numerical simulation to validate the soundness of the theoretical analysis.

Dynamical analysis of a Lotka–Volterra competition model with both Allee and fear effects

Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.

Lyapunov functions for studying global asymptotic stability of two rumor spreading models

In a previous work (2018, Commun. Theor. Phys. 70, 795–802), a new compartment model for the spreading of rumors was introduced and analyzed. However, only the local asymptotic stability of this model was discussed. In the present work, we first provide a rigorous mathematical analysis for the global asymptotic stability (GAS) of the above-mentioned rumor spreading model. By constructing suitable Lyapunov candidate functions, we obtain the GAS of a rumor-free (boundary) equilibrium point and a unique rumor-spreading (positive) equilibrium point. After that, we utilize the approach based on the Lyapunov candidate functions to study the GAS of another rumor spreading model with control strategies, which was proposed in (2022, Physica A 606, 128157). As an important consequence, the GAS of the rumor spreading model with control strategies is determined fully without resorting to technical hypotheses used in the benchmark work. Lastly, the theoretical findings are supported by a set of illustrative numerical examples. The obtained results not only improve the ones constructed in the two above-mentioned benchmark papers but also can be extended to study the global dynamics of other rumor propagation models in the context of both integer-order and fractional-order derivatives.

Role reversal in a stage-structured prey-predator model with fear, delay, and carry-over effects.

The present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov-Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmony.

Hopf Bifurcation Analysis of a Continuous Investment Update Project Model

Some investment projects aim not only to produce goods but more importantly, to update and efficiently supply products in accordance with market demand. A double time delay differential dynamics model is formulated for continuous renewal investment projects based on the flowchart of the capital appreciation process and the assumed transfer functions. By analyzing the mathematical model, it can be determined that a unique local asymptotically stable positive equilibrium point exists for the continuous investment project. In accordance with the Hopf branching theorem, the model displays periodic behavior in proximity to its positive equilibrium point under certain conditions. The simulation results are compared under various conditions, and the validity of the relevant conclusions is confirmed.

Research on the Dynamic Mechanism of Technological Innovation Diffusion in Enterprise Communities Based on a Predation Diffusion Model

In order to study the dynamic mechanism of the impact of technological innovation diffusion on enterprise population networks, a corresponding relationship between enterprise population networks and predatory models was established based on a predatory model. Without considering the impact of technological innovation diffusion, the stability of the enterprise population network was analyzed, and the results showed that it has the characteristic of local asymptotic stability at a positive equilibrium point. Considering the influence of technological innovation diffusion, the stability of the enterprise population network becomes complex, and its stability at the positive equilibrium point is also affected by the eigenvalue of the Laplacian matrix and technological innovation diffusion coefficient. The simulation experimental results indicate that in addition to the influence of technological innovation diffusion coefficient, the connection probability density of enterprise population networks has an important impact on stability. Only when the connection probability density is very small can the enterprise population network resist the impact of technological innovation diffusion and maintain stability.

Time delays and economic profit in fractional differential‐algebraic predator–prey model

In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored.