Non-delayed System Research Articles

Study of Bifurcation and Delay-Driven Chaos in a Prey–Predator Model with Fear in Prey Reproduction and Two Forms of Harvesting

In this paper, we delve into a predator–prey model incorporating a fear effect in prey reproduction, influenced by both delay and harvesting. The model accounts for delayed fear dynamics to capture more realistic dynamics. Initially, our focus lays on the nondelayed model, examining each biologically plausible equilibrium points and assessing their stability concerning the parameters of the model. Next, detailed mathematical results are provided, encompassing the asymptotic stability of all equilibria, Hopf bifurcation, and the direction and stability of bifurcated periodic solutions. Also, the stability analysis of the Hopf-bifurcating periodic solution is confirmed through the computation of first Lyapunov coefficient. Furthermore, we observed that the nondelayed system experiences Bogdanov–Takens bifurcation in a two-parameter space. Subsequently, we analyzed the corresponding delayed system, establishing the existence of a stable limit cycle through Hopf bifurcation concerning the delay parameter. Additionally, the inclusion of delay can prompt critical dynamics within the system, resulting in period-doubling routes toward chaotic oscillations. To validate our analytical findings, we conducted comprehensive and meticulous numerical simulations. The findings of the numerical simulations suggest that the impact of fear can be used as a measure of chaos control.

Deciphering two delay dynamics of ecological system with generalist predator incorporating competitive interference

In this manuscript, an attempt has been made to understand the delay induced (gestation and carry-over effect delay) dynamics of an ecological system with generalist predator exerted fear and its carry-over effect with competitive interference. The designed model exhibits finite time blow up depending on large initial data. The stability of both the delayed and non-delayed systems have been analyzed along with Hopf-bifurcation analysis. It has been observed that carry-over and fear effects act in opposite way in context of stability control for non-delayed system. The two delay (carry-over effect and gestation delay) have significant impact on the dynamics. The former exhibits both stabilizing and destabilizing role while the latter has a destabilizing tendency on the system dynamics. The blow up phenomena for predator species have been shown numerically by verifying the analytical conditions. Our study incorporates a diverse array of figures and diagrams to illustrate and support our findings. Through the exploration of non-linear models, our research unveils several intriguing characteristics. These insights can prove invaluable for biologists seeking a more detailed and pragmatic understanding of generalist predator–prey systems. The visual representations provided in our study contribute to a comprehensive analysis, enhancing the accessibility and applicability of the findings for researchers and practitioners in the field.

Forward completeness does not imply bounded reachability sets and global asymptotic stability is not necessarily uniform for time-delay systems

An example of a time-invariant time-delay system that is uniformly globally attractive and exponentially stable, hence forward complete, but whose reachability sets from bounded initial conditions are not bounded over compact time intervals is provided. This gives a negative answer to two current conjectures by showing that (i) forward completeness is not equivalent to robust forward completeness (i.e. boundedness of reachability sets) and (ii) global asymptotic stability is not equivalent to uniform global asymptotic stability. In addition, a novel characterization of robust forward completeness for systems having a finite number of discrete delays is provided. This characterization relates robust forward completeness of the time-delay system with the forward completeness of an associated nondelayed finite-dimensional system.

DYNAMICS OF DELAYED AUTONOMOUS MODEL WITH CARRY-OVER EFFECTS, INCLUDING PREY HARVESTING

This paper proposes a two-dimensional modified Leslie–Gower predator–prey model with carry-over effects, fear, prey protection and additional food, including prey harvesting. For this, it assumes the carry-over effect with delay incorporated in the birth rate of the prey population and gestation delay in Leslie–Gower term of a predator, and supposes that the predator consumes the prey according to Holling type II functional response. This dynamical system is split into two groups, namely non-delayed and delayed systems, respectively. In the non-delayed system, first, we discuss positivity, boundedness solutions, and the existence of equilibria, especially finding the coexistence of interior equilibrium points under certain conditions. After that, local and global stabilities are systematically analyzed. In addition, we obtain the conditions of Hopf-bifurcation in terms of the growth rate of prey species, prey protection and preference rate of predator for the non-delayed system. Whereas in the delayed system, both the delays have an essential role in governing the dynamical system discussed in three cases, and each case found the Hopf-bifurcation parameters for which the system changes stability behavior into instability. At last, the numerical simulations have been carried out to verify the theoretical findings.

Spatiotemporal patterns and bifurcations of a delayed diffusive predator-prey system with fear effects

In this paper, we consider a diffusive predator-prey system with fear response delay, where a benefit from the ani-predation response in addition to the cost is taken into account. As a step toward understanding the underlying mechanism of the spatiotemporal pattern formations by rigorous mathematical analysis, we are interested in how the joint effects of time delay, the fear effect strength, the carrying capacity, as well as the diffusion rates could affect the complex spatiotemporal pattern formations of the system. In particular, Turing instability of the Hopf bifurcating periodic solutions is investigated in details. To that end, a general formula is derived to determine Turing instability of the Hopf bifurcating periodic solutions for general 2-component delayed-diffusive system. This extends our earlier general results for non-delayed diffusive system to its delayed-diffusive counterparts. Our general formula tends to be new and can be applied to a variety of 2-component delayed-diffusive systems.

A mathematical model for tumor-immune competitive system with multiple time delays

A mathematical model is proposed to study the complicated dynamics of tumor-immune interactions with three discrete time delays. The proposed model consists of nine coupled ordinary differential equations (ODEs) whose components are tumor cells, tumor-specific CD8+T cells, dendritic cells, macrophages, regulatory T-cells (Tregs), interleukin-10 (IL-10), IL-12, transforming growth factor-β (TGF-β) and interferon-γ (IFN-γ). By utilizing the quasi-steady-state approximation for cytokine concentrations, we obtain a system of four coupled ODEs. We introduce three discrete time delays into our deterministic system to better understand the tumor-immune dynamics. Basic properties of the system, including existence, uniqueness, positivity, boundedness, and uniform persistence are discussed. Stability analysis of both the delayed and non-delayed system has been performed and the conditions for stability and direction of Hopf bifurcation have been determined. Furthermore, we assess the length of time delay required to maintain the stability of period-1 limit cycle. Parameter estimation techniques are discussed and numerical simulations are provided to support our theoretical analysis. Notably, we observe that time delays do not significantly influence the system behavior for the existing set of parameters value.

Time delays in skill development and vacancy creation: Effects on unemployment through mathematical modelling

One of the major macroeconomic issues that plagues any economy and impedes its evolution is unemployment. To address this major macroeconomic concern, we have developed and investigated a delayed mathematical model of unemployment with four dynamic variables. The three dynamic variables represent the segregation of the population into three classes: the unemployed class, the employed class, and the retired class. Also, an additional variable for new vacancies available for recruitment is considered in the model as the fourth dynamic variable. The development of skills and the creation of jobs for the unemployed segment of society are two key factors that greatly aid in the treatment of this important macroeconomic issue. Since these procedures require time, we have included time delays in skill development and vacancy creation. The existence of a unique employment equilibrium point has been established, and the stability of the non-delayed system around the equilibrium point has been analysed. Also, the system dynamics for various delay combination scenarios have been examined. It has been proven in each scenario that when the respective time delay exceeds a certain threshold value, the considered model experiences a Hopf bifurcation around the employment equilibrium point. Additionally, an examination of the Lyapunov stability and Hopf bifurcation properties has been carried out. Lastly, numerical simulations are performed to confirm the analytical results. Our analysis shows that it is not possible to take large values for both the time delays without causing system instability. Our work emphasizes the necessity of including time delays in the unemployment model to better understand the system dynamics, which will help devise and adopt the necessary strategies to reduce the unemployment rate. It is deduced from the quantitative observations that one of the key strategies for reducing unemployment can be to encourage and raise public awareness about skill development. Additionally, it is advised that policymakers ensure that policies for vacancy creation are designed by considering that the rate of vacancy creation is commensurate with the number of unemployed individuals.

Multistability and bifurcation analysis for a three-strategy game system with public goods feedback and discrete delays

The interaction between evolution and public goods exists in social and biological systems. Exploring mathematical models that combine evolution and public goods facilitates insights and discoveries about the evolution of cooperation. We propose a third strategy called the encouragement strategy, which can reward and punish, and bear the related costs, in addition to the classical cooperation and defection strategies. We incorporate strategy and public goods dynamics to formulate a high-dimensional and high-order three-strategy evolutionary game system with public goods feedback. For this non-delayed system, we evaluate the existence of equilibria and the conditions for the occurrence of transcritical and Hopf bifurcations. Interestingly, the non-delayed system exhibits multiple multistability due to different payoffs and initial conditions. The results suggest that the environment with rich public goods is suitable for cooperators to survive. Surprisingly, raising reward does not always promote the survival of cooperators. Cooperators prevail in environments with moderate reward and severe punishment. Further, we develop a delayed system by considering information delay and production delay. For the delayed system, we investigate the properties of Hopf bifurcation, including existence, direction and stability. We find that small delays do not affect the evolutionary outcome of the strategies. However, delays larger than the critical value lead to periodic oscillations of the strategies, in which case the cooperators still have a dominant advantage. Unfortunately, larger delays can cause the disappearance of cooperators. Theoretical and numerical results show that public goods feedback, reward, punishment, payoffs, initial conditions, and delays are all significant factors in determining the outcome of strategic evolution.

Maturation delay induced stability enhancement and shift of bifurcation thresholds in a predator–prey model with generalist predator

Discrete time delay has been widely used in models of interacting populations to capture various biological processes such as gestation, maturation, food digestion, disease transmission and so on. In most of the works, it has been demonstrated that a longer maturation delay is responsible for the destabilization of the coexistence steady-state. The constructed model system, however, may produce different phenomena if the maturation delay is included in an ecologically justified way, preventing the model from producing inconsistent results. Previous studies focus on the influence of delayed maturation in predator–prey dynamics with specialist predators mostly. The primary goal of this article is to explore how delayed maturity in generalist predators qualitatively affects the dynamics of a predator–prey system. To illustrate, we consider a predator–prey model with generalist predators and Holling type II functional response to characterize the generalist predator’s grazing behavior toward their primary prey. We find various parameter regimes for the non-delayed system where we see either no coexistence, single coexistence, or multiple coexistence of prey and their generalist predators. Rich dynamics, induced by various local and global bifurcations, are displayed for the non-delayed system. The inclusion of a maturation delay in the model system causes the system’s qualitative properties to alter, such as shifting bifurcation thresholds and a reduction in the region of multi-coexistence in a parametric domain. Our analysis demonstrates that the delayed maturation of generalist predators does not act as a destabilizing factor; instead, it promotes stable coexistence. Visual depictions of all the analytical findings have been presented through extensive numerical simulation and the ecological implications of the obtained results are summarized in the concluding section.

CHAOTIC DYNAMICS OF A STAGE-STRUCTURED PREY–PREDATOR SYSTEM WITH HUNTING COOPERATION AND FEAR IN PRESENCE OF TWO DISCRETE DELAYS

Depending on behavioral differences, reproductive capability and dependency, the life span of a species is divided mainly into two classes, namely immature and mature. In this paper, we have studied the dynamics of a predator–prey system considering stage structure in prey and the effect of predator-induced fear with two discrete time delays: maturation delay and fear response delay. We consider that predators cooperate during hunting of mature prey and also include its impact in fear term. The conditions for existence of different equilibria, their stability analysis are carried out for non-delayed system and bifurcation results are presented extensively. It is observed that the fear parameter has stabilizing effect whereas the cooperative hunting factor having destabilizing effect on the system via occurrence of supercritical Hopf-bifurcation. Further, we observe that the system exhibits backward bifurcation between interior equilibrium and predator free equilibrium and hence the situation of bi-stability occurs in the system. Thereafter, we differentiate the region of stability and instability in bi-parametric space. We have also studied the system’s dynamics with respect to maturation and fear response delay and observed that they also play a vital role in the system stability and occurrence of Hopf-bifurcation is shown with respect to both time delays. The system shows stability switching phenomenon and even higher values of fear response delay leads the system to enter in chaotic regime. The role of fear factor in switching phenomenon is discussed. Comprehensive numerical simulation and graphical presentation are carried out using MATLAB and MATCONT.

Dynamics of a Coccinellids-Aphids Model with Stage Structure in Predator Including Maturation and Gestation Delays

This work studies a three-dimensional predator–prey model with gestation delay and stage structure between aphidophagous coccinellids and aphid pests, where the interaction between mature coccinellids and aphids is inscribed by Crowley–Martin functional response function, and immature coccinellids and aphids act in the form of Holling-I type. We prove the positivity and boundedness of the solution of the nondelayed system and analyze its equilibrium point, local asymptotic stability, and global stability. In addition to the delays, the critical values of Hopf bifurcation occurring for different parameters are also found from the numerical simulation. The stability of the delayed system and Hopf bifurcation with different delays as parameters are also discussed. Our model analysis shows that the time delay essentially governs the system’s dynamics, and the stability of the system switches as delays increase. We also investigate the direction and stability of the Hopf bifurcation using the normal form theory and center manifold theorem. Finally, we perform computer simulations and depict diagrams to support our theoretical results.

Strong stability with impact of maturation delay and diffusion on a toxin producing phytoplankton–zooplankton model

Plankton species is backbone for any healthy aquatic system, but a global increment in toxin phytoplankton blooms is a major threat to aquatic systems. As toxic phytoplankton Microcystis aeruginosa becomes too high, Artemia salina a zooplankton species avoids to consume it and more inclines towards a non-toxic phytoplankton Chaetoceros gracilis for food. Considering this fact, additional food is provided to zooplankton species to survive when non-toxic species is insufficient for the zooplankton and toxin phytoplanktons are harvested linearly to control toxicity level in the system. Also, increasing of phytoplankton may increase competitive behaviour between toxic and non-toxic phytoplanktons due to limited resources. Keeping these in mind, we propose a three populations reaction–diffusion delay model. In this study, existence of feasible equilibria has been studied. Conditions for Turing instability of delayed and non-delayed system have been discussed analytically. Strong stability analysis with respect to both delay and diffusion is performed and excitable nature through Hopf bifurcation is also examined. It has been investigated that if the delay spatial system shows diffusion driven instability, then it is also shown by the delay free system for the same parameters. The study explores that the model depicts strong stability with respect to both delay and diffusion. Movements of the species are responsible for the occurrence of planktonic bloom via Hopf bifurcation due to diffusion and affect the critical value of the time delay. This study suggests that a specific amount of harvesting ensures the long-term sustainability. It is also inspected that considerable amount of alternative food supply supports the zooplankton for its growth. Turing patterns like spots are observed. The behaviour of the system depends on the both variation of delay and choice of initial population. This study says that maturation delay stabilizes the system more quickly in the presence of diffusion than in the absence of diffusion.

The impact of fear effect on the dynamics of a delayed predator–prey model with stage structure

In this paper, a stage structure predator–prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed. The prey populations are divided into two parts: juvenile prey and adult prey. From extensive experimental data, it has been found that prey fear of predators can alter the physiological behavior of individual prey, and the fear effect reduces their reproductive rate and increases their mortality. In addition, we also consider the presence of constant ratio refuge in adult prey populations. Moreover, we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors. We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria. Next by choosing the intra-specific competition coefficient of adult prey as bifurcation parameter, we demonstrate that Hopf bifurcation may occur near the positive equilibrium point. Then by taking the gestation delay as bifurcation parameter, the sufficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given. And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory. What’s more, numerical experiments are performed to test the theoretical results obtained in this paper.

Consequences of fear effect and prey refuge on the Turing patterns in a delayed predator-prey system.

This study presents a qualitative analysis of a modified Leslie-Gower prey-predator model with fear effect and prey refuge in the presence of diffusion and time delay. For the non-delayed temporal system, we examined the dissipativeness and persistence of the solutions. The existence of equilibria and stability analysis is performed to comprehend the complex behavior of the proposed model. Bifurcation of codimension-1, such as Hopf-bifurcation and saddle-node, is investigated. In addition, it is observed that increasing the strength of fear may induce periodic oscillations, and a higher value of fear may lead to the extinction of prey species. The system shows a bistability attribute involving two stable equilibria. The impact of providing spatial refuge to the prey population is also examined. We noticed that prey refuge benefits both species up to a specific threshold value beyond which it turns detrimental to predator species. For the non-spatial delayed system, the direction and stability of Hopf-bifurcation are investigated with the help of the center manifold theorem and normal form theory. We noticed that increasing the delay parameter may destabilize the system by producing periodic oscillations. For the spatiotemporal system, we derived the analytical conditions for Turing instability. We investigated the pattern dynamics driven by self-diffusion. The biological significance of various Turing patterns, such as cold spots, stripes, hot spots, and organic labyrinth, is examined. We analyzed the criterion for Hopf-bifurcation for the delayed spatiotemporal system. The impact of fear response delay on spatial patterns is investigated. Numerical simulations are illustrated to corroborate the analytical findings.

Composite fuzzy voltage-based command-filtered learning control of electrically-driven robots with input delay using disturbance observer

This paper presents an improved composite fuzzy learning control for uncertain electrically-driven robot manipulators with input delay and the external disturbances. In the framework of the backstepping algorithm, fuzzy systems are employed to approximate the unknown terms where the accuracy of fuzzy learning is also considered by defining prediction errors. With the aid of integral technique and the dynamic surface control, a variable is engendered for the system in such a way that the input-delayed robotic system is converted to the non-delayed robotic system. Besides, the command-filtered control is used to cope with the complexity explosion of the backstepping-based design. In order to improve the robust behavior of the control system, the proposed control scheme is equipped with disturbance observers (DOBs). Different from the previous works, the information of the input-delayed, the compensated error surfaces (obtained from the command-filtered approach), the prediction errors and the disturbance estimations (derived from DOBs) are unified to construct the proposed control framework. The stability of the overall system is verified by the Lyapunov theorem. The efficiency of the proposed concept is illustrated using various simulations for an electrically-driven robot manipulator in the presence of uncertainties.

Hopf bifurcation of a financial dynamical system with delay

The aim of this work is to investigate the qualitative behaviour of a financial dynamical system which contains a time delay. We investigate the dynamic response of this system of which variables are interest rate, investment demand, price index and average profit margin. As a plus to the available literature, the model investigated takes into account a timed delayed feedback in the investment demand. We perform a stability analysis at the fixed points and show that the system undergoes a Hopf bifurcation using well-known methods of stability analyses for delayed systems. The bifurcation analyses are supported by numerical simulations. The analysis reveals that for a set of parameters for which the non-delayed system is stable, a delay in the investment demand may drive the system to instability.

A model for pest control using integrated approach: impact of latent and gestation delays

In this article, we have established a mathematical model using impulsive differential equations for the dynamics of crop pest management in the presence of a pest with its predator and bio-pesticides. The pest population is divided into two subpopulations, namely, the susceptible pests and the infected pests. In this control process, bio-pesticides (generally virus) infect the susceptible pest through viral infection within the pest and make it infected so that predators can consume it quickly. We assume that pest controlling, using this integrated approach, is a delayed process and thus incorporated latent time of susceptible pest and gestation delay of predator in the model as time delay parameters. The system dynamics have been analyzed using qualitative theory: the existence of the equilibrium points and their stability properties has been derived. Hopf bifurcation of the coexisting equilibrium point is presented for both the delayed and non-delayed system. Detail numerical simulations are performed in support of analytical results and illustrate the different dynamical regimes observed in the system. We have observed that the system becomes free of infection when the latent time of the pest is large. Coexisting equilibrium exists for the lower value of latent delay, and it can change the stability properties from stable to unstable when it crosses its critical value. In contrast, gestation delay affects the stability switches of coexisting equilibrium only. The combined effect of the two delays on the system is shown numerically. Also, viral replication rate, infection rate (from virus to pest) is also significant from the pest management perspective. In summary, both the delay are essential for crop pest management, and pest control will be successful with tolerable delays.

Robust adaptive backstepping control of uncertain fractional-order nonlinear systems with input time delay

In the present work, a fractional adaptive backstepping control approach is proposed for controlling a class of uncertain fractional-order nonlinear systems subjected to unknown external disturbance and input delay. In control design, by introducing fractional command-filter, the key issue of the common backstepping method called the “explosion of complexity” is addressed. Furthermore, the Szász–Mirakyan operator is used as a simple but efficient universal approximator to approximate uncertain or unknown terms and reduce the design complexity. In addition, to enhance the capability of the proposed controller in dealing with the approximation error and the unknown disturbance, an efficient robust control term is also incorporated into it. One advantage of the suggested control approach is that the input-delay fractional-order system is effectively treated by introducing a new variable as a non-delayed fractional-order system. The other advantage is that the number of tuning variables is considerably low; specifically, for an n-order fractional nonlinear system, only n adaptive parameters are adjusted online. This makes the implementation of the proposed approach simpler. The Lyapunov stability theorem is used to prove that all of the closed-loop system’s signals are semi-globally uniformly ultimately bounded. The feasibility of the suggested control approach is shown by comparing the simulation results obtained with the fuzzy system.

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

Impact of fear on delayed three species food-web model with Holling type-II functional response

This paper deals with the investigation of the three species food-web model. This model includes two logistically growing interaction species, namely [Formula: see text] and [Formula: see text], and the third species [Formula: see text] behaves as the predator and also host for [Formula: see text]. The species [Formula: see text] predating on the species [Formula: see text] with the Holling type-II functional response, while the first species [Formula: see text] is benefited from the third species [Formula: see text]. Further, the effect of fear is incorporated in the growth rate of species [Formula: see text] due to the predator [Formula: see text] and time lag in [Formula: see text] due to the gestation process. We explore all the biologically possible equilibrium points, and their local stability is analyzed based on the sample parameters. Next, we investigate the occurrence of Hopf-bifurcation around the interior equilibrium point by taking the value of the fear parameter as a bifurcation parameter for the non-delayed system. Moreover, we verify the local stability and existence of Hopf-bifurcation for the corresponding delayed system. Also, the direction and stability of the bifurcating periodic solutions are determined using the normal form theory and the center manifold theorem. Finally, we perform extensive numerical simulations to support the evidence of our analytical findings.