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The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey–predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models. [ABSTRACT FROM AUTHOR]

This paper presents the temporal and spatiotemporal dynamics of a delayed prey–predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β , which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Phytoplankton–zooplankton interaction is a topic of high interest among the interrelationships related to marine habitats. In the present manuscript, we attempt to study the dynamics of a three-dimensional system with three types of plankton: non-toxic phytoplankton, toxic producing phytoplankton, and zooplankton. We assume that both non-toxic and toxic phytoplankton are consumed by zooplankton via Beddington–DeAngelis and general Holling type-IV responses, respectively. We also incorporate gestation delay and toxic liberation delay in zooplankton's interactions with non-toxic and toxic phytoplankton correspondingly. First, we have studied the well-posedness of the system. Then, we analyze all the possible equilibrium points and their local and global asymptotic behavior. Furthermore, we assessed the conditions for the occurrence of Hopf-bifurcation and transcritical bifurcation. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also studied. Various time-series, phase portraits, and bifurcation diagrams are plotted to confirm our theoretical findings. From the numerical simulation, we observe that a limited increase in inhibitory effect of toxic phytoplankton against zooplankton can support zooplankton's growth, and rising predator's interference can also boost zooplankton expansion in contrast to the nature of Holling type IV and Beddington–DeAngelis responses. Next, we notice that on variation of toxic liberation delay, the delayed system switches its stability multiple times and becomes chaotic. Furthermore, we draw the Poincaré section and evaluate the maximum Lyapunov exponent in order to verify the delayed system's chaotic nature. Results presented in this article might be helpful to interpret biological insights into phytoplankton–zooplankton interactions. [ABSTRACT FROM AUTHOR]

This study aims to investigate a diffusive predator–prey system incorporating additional food for predators, prey refuge, fear effect, and its carry-over effects. For the temporal model, the well-posedness and persistence of the system have been discussed. We investigated the existence and the stability behavior of the various equilibria. Furthermore, we explored the bifurcations of codimension-1 including transcritical, saddle-node, and Hopf, concerning the crucial parameters. The system also presents codimension-2 bifurcations such as Bogdanov–Takens and cusp bifurcation along with the global homoclinic bifurcation. We observed the bubbling phenomena, which illustrate the fluctuations in the amplitudes of the periodic oscillations. For the spatiotemporal system, we established the non-negativity and boundedness of the solutions. We derived the conditions for the diffusion-driven instabilities in a confined region with Neumann boundary conditions. Extensive numerical simulations have been conducted to depict the various stationary patterns in Turing space. It is observed that incorporating cross-diffusion divides the bi-parametric plane into various sub-regions and dynamic patterns are analyzed in these different regions. The intricate spatiotemporal dynamics exhibited by prey–predator interactions are crucial for unraveling the intricacies within ecological systems. [ABSTRACT FROM AUTHOR]

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter. [ABSTRACT FROM AUTHOR]

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk's work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator's mortality rate to reflect losses incurred during encounters with dangerous prey and prey's trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems. [ABSTRACT FROM AUTHOR]

In this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically stable when reproduction number is less than one. Some conditions on the model parameters are obtained to show the existence as well as nonexistence of limit cycle. Some sufficient conditions for global stability of the endemic equilibrium using Lyapunov function are obtained. The existence of Hopf bifurcation of model is investigated by using Andronov-Hopf bifurcation theorem. Further, numerical simulations are done to exemplify the analytical studies. [ABSTRACT FROM AUTHOR]

In our ecological system, prey species can defend themselves by casting strong and effective defenses against predators, which can slow down the growth rate of prey. Predator has more at stake when pursuing a deadly prey than just the chance of missing the meal. Prey have to "trade off" between reproduction rate and safety and whereas, predator have to "trade off" between food and safety. In this article, we investigate the trade-off dynamics of both predator and prey when the predator attacks a dangerous prey. We propose a two-dimensional prey and predator model considering the logistic growth rate of prey and Holling type-2 functional response to reflect predator's successful attacks. We examine the cost of fear to reflect the trade-off dynamics of prey, and we modify the predator's mortality rate by introducing a new function that reflects the potential loss of predator as a result of an encounter with dangerous prey. We demonstrated that our model displays bi-stability and undergoes transcritical bifurcation, saddle node bifurcation, Hopf bifurcation, and Bogdanov–Taken bifurcations. To explore the intriguing trade-off dynamics of both prey and predator population, we investigate the effects of our critical parameters on both population and observed that either each population vanishes simultaneously or the predator vanishes depending on the value of the handling time of the predator. We determined the handling time threshold upon which dynamics shift, demonstrating the illustration of how predators risk their own health from hazardous prey for food. We have conducted a sensitivity analysis with regard to each parameter. We further enhanced our model by including fear response delay and gestation delay. Our delay differential equation system is chaotic in terms of fear response delay, which is evidenced by the positivity of maximum Lyapunov exponent. We have used numerical analysis to verify our theoretical conclusions, which include the influence of vital parameters on our model through bifurcation analysis. In addition, we used numerical simulations to showcase the bistability between co-existence equilibrium and prey only equilibrium with their basins of attraction. The results that are reported in this article might be useful in interpreting the biological insights gained from studying the interactions between prey and predator. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a system of delay differential equations incorporating prey's refuge, fear, fear-response delay, extra food for predators and their gestation lag. First, we examined the system without delay. The persistence, stability (local and global) and various bifurcations are discussed. We provide detailed analysis for transcritical and Hopf-bifurcation. The existence of positive equilibria and the stability of prey-free equilibrium are interrelated. It is shown that (i) fear can stabilize or destabilize the system, (ii) prey refuge in a specific limit can be advantageous for both species, (iii) at a lower energy level (gained from extra food), the system undergoes a supercritical Hopf-bifurcation and (iv) when the predator gains high energy from extra food, it can survive through a homoclinic bifurcation, and prey may become extinct. The possible occurrence of bi-stability with or without delay is discussed. We observed switching of stability thrice via subcritical Hopf-bifurcation for fear-response delay. On changing some parametric values, the system undergoes a supercritical Hopf-bifurcation for both delay parameters. The delayed system undergoes the Hopf-bifurcation, so we can say that both delay parameters play a vital role in regulating the system's dynamics. The analytical results obtained are verified with the numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of environmental toxicant on the immune response of the body. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the body decreases as the concentration of environmental toxicant increases, and certain criteria are obtained under which it settles down at its equilibrium level. In the absence of toxicant, an oscillatory behavior of immune system and pathogenic growth is observed. However, in the presence of toxicant, oscillatory behavior is not observed. These studies show that the toxicant may have a grave effect on our body's defense mechanism. [ABSTRACT FROM AUTHOR]

A dynamic model for a single-species fishery, which depends partially on a logistically growing resource with functional response, is proposed using taxation as control instrument to protect fish population from overexploitation. The analysis of the model shows that both the equilibrium density of fish population as well as the maximum sustainable yield increase as resource biomass density increases. The optimal harvesting policy is also discussed with the help of Pontryagin's Maximum Principle. It is found that for the optimum equilibrium value of resource biomass density, the total user's cost of harvest per unit effort must be equal to the discounted value of future price at the steady state. [ABSTRACT FROM AUTHOR]

In this investigation, we construct a predator–prey model that distinguishes between immature and mature prey, highlighting group defense strategies within the mature prey. First, we embark on exploring the positivity and boundedness of the solution, unraveling sustainable equilibrium points, and deducing their stability conditions. Upon further investigation, we observe that the system exhibits diverse bifurcations, including Hopf, saddle-node, transcritical, generalized Hopf, cusp, and Bogdanov–Takens bifurcations. The results reveal that heightened fear decreases mature prey density, potentially causing prey extinction beyond a certain threshold. Increased maturation rates lead to the coexistence of immature and mature prey populations and higher predator density. Stronger group defense boosts mature prey density, while weaker defense results in weak persistence. Lower values of the maturation rate of prey and the decline rate of predators sustain only the predator population, reliant on resources other than focal prey. Furthermore, our model demonstrates intriguing and diverse dynamical phenomena, including various forms of bistability across distinct bi-parameter planes. We also explore the dynamics of a related nonautonomous system, where certain parameters are considered to vary with time. In the seasonally forced model, we set out to define criteria regarding the existence and stability of positive periodic solutions. Numerical investigations into the seasonally forced model uncover a spectrum of dynamics, ranging from simple periodic solutions to higher periodicities, bursting patterns, and chaotic behavior. [ABSTRACT FROM AUTHOR]

The interplay of phytoplankton, zooplankton, and fish is one of the most important aspects of the aquatic environment. In this paper, we propose to explore the dynamics of a phytoplankton–zooplankton–fish system, with fear-induced birth rate reduction in the middle predator by the top predator and an additional food source for the top predator fish. Phytoplankton–zooplankton and zooplankton–fish interactions are handled using Holling type IV and II responses, respectively. First, we prove the well-posedness of the system, followed by results related to the existence of possible equilibrium points. Conditions under which a different number of interior equilibria exist are also derived here. We also show this existence numerically by varying the intrinsic growth rate of phytoplankton species, which demonstrates the model's vibrant nature from a mathematical point of view. Furthermore, we performed the local and global stability analysis around the above equilibrium points, and the transversality conditions for the occurrence of Hopf bifurcations and transcritical bifurcations are established. We observe numerically that for low levels of fear, the system behaves chaotically, and as we increase the fear parameter, the solution approaches a stable equilibrium by the route of period-halving. The chaotic behavior of the system at low levels of fear can also be controlled by increasing the quality of additional food. To corroborate our findings, we constructed several phase portraits, time-series graphs, and one- and two-parametric bifurcation diagrams. The computation of the largest Lyapunov exponent and a sketch of Poincaré maps verify the chaotic character of the proposed system. On varying the parametric values, the system exhibits phenomena like multistability and the enrichment paradox, which are the basic qualities of non-linear models. Thus, the current study can also help ecologists to estimate the parameters to study and obtain such important findings related to non-linear PZF systems. Therefore, from a biological and mathematical perspective, the analysis and the corresponding results of this article appear to be rich and interesting. [ABSTRACT FROM AUTHOR]

• A four-dimensional non-autonomous NPZT (nutrient-phytoplankton-zooplankton-environmental toxins) model with group defence is presented. • The change in population densities with respect to some important parameters is explained, and this change also leads to occurrence of Hopf-bifurcation in autonomous model. • We establish the existence and global stability of the periodic solution for the non- autonomous model analytically. • Both autonomous and non-autonomous models show different kinds of multistability. • Non-autonomous model also shows the chaotic nature due to seasonality of parameters. The interaction between phytoplankton and zooplankton has a significant impact on the marine ecology. The interplay between these two species is the building blocks for most of the food webs operating in an aquatic ecosphere. The environmental toxins released by different external sources also affect the phytoplankton-zooplankton dynamics. In the present study, we propose a model to explore the kinetics of a nutrient-phytoplankton-zooplankton-environmental toxins (NPZT) system. The defence mechanism of phytoplankton against zooplankton is reflected through modified Holling type IV response, whereas the consumption of nutrients by phytoplankton is outlined by Holling type II response. The external toxins are assumed to have the capability of reducing the birth rate of phytoplankton species after coming into contact with their cells. To make our model more pragmatic, seasonal variation in the parameters is also taken into account. Firstly, we do the analysis related to the autonomous model (non-seasonal) like; its boundedness, existence of equilibrium points, their stability analysis, and occurrence of Hopf-bifurcation. Further, for the non-autonomous model (seasonal), we analyze the existence of positive periodic solution and its global stability. Through numerical simulations, we observe that for the non-seasonal model, increasing the rate of suppressing phytoplankton's growth by environmental toxin, and rate at which environmental toxin is added to system make it unstable through Hopf-bifurcation. These oscillations can be removed by raising phytoplankton's inhibitory effect against zooplankton, and this increment also leads to the extinction of the zooplankton population, making zooplankton free equilibrium a stable one. Both models, non-seasonal as well as seasonal manifest different types of multistability, and this is an exciting character associated with non-linear models. We also note that the inclusion of seasonality in our system promotes the coexistence of all populations. Further, through numerical simulations, we show that making some of the parameters seasonal can cause the emergence of chaos in the system. To verify chaos, we sketch the Poincaré map and evaluate the maximum Lyapunov exponent. The seasonal model also shows the switching of stability through different periodic and chaotic windows on varying the maximum intrinsic growth rate for phytoplankton, and contact rate between environmental toxin and phytoplankton. To substantiate our results, we picture several time-series graphs, basins of attraction, one and two-parametric bifurcation diagrams. Thus we expect that the present work can assist biologists and mathematicians in studying nutrient-plankton systems in a more detailed and realistic manner. This study can also help researchers in the estimation of non-seasonal as well as seasonal parameters while studying these types of complex non-linear models. Therefore, the present work seems to be enriched from a mathematical and biological point of view. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]