Feasible Equilibrium Points Research Articles

Effective Pest Control Through Natural Predators: A Dynamical Plant–Pest–Predator Model

The impact of pests on crop yield is a significant worry for farmers, and finding an effective strategy to control insect population growth has become a pressing matter. This study explores the dynamic analysis of a model that incorporates natural predators to control crop pests. In the research, an analysis is conducted on the model’s positivity and boundedness, followed by an examination of the existence and stability of the equilibrium points. The study focused on analyzing bifurcations at a biologically feasible equilibrium point. The parameters considered for bifurcation are the predation rate by pests and the consumption rate of natural predators. In addition, an optimal control technique has been employed to enhance the growth of the plant population through the utilization of Pontryagin’s maximum principle. Moreover, the impact of the nine parameters on the model is examined through sensitivity analysis using the partial rank correlation coefficient method. Numerical simulations are conducted to validate the analytical findings, revealing the occurrence of bifurcation and the positive impact of control on plant population growth.

Mathematical modeling and bifurcation analysis of a delayed eco-epidemiological model with disease in predator and linear harvesting

ABSTRACT This study proposes a three-dimensional prey-predator model with disease in predators, including the time delay due to the gestation of the predator population. The positive invariance of the solutions and the existence of equilibria in the proposed system have been explained. We also analyse the local stability of all feasible equilibrium points for the delayed and non-delayed systems. Additionally, the system’s Hopf bifurcation with regard to several parameters has been investigated. Further, the existence of periodic solutions through Hopf bifurcation is shown with respect to time delay, and it is also shown that time delay is crucial in order to control the dynamics of the system. Explicit formulas are presented to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution using the normal form theory and the centre manifold theorem. The outcome of the theoretical investigation is ultimately validated through numerical simulation.

A Stage-structure Leslie-Gower Model with Linear Harvesting and Disease in Predator

The growth dynamics of various species are affected by various aspects. Harvesting interventions and the spread of disease in species are two important aspects that affect population dynamics and it can be studied. In this work, we consider a stage-structure Leslie–Gower model with linear harvesting on the both prey and predator. Additionally, we also consider the infection aspect in the predator population. The population is divided into four subpopulations: immature prey, mature prey, susceptible predator, and infected predator. We analyze the existences and stabilities of feasible equilibrium points. Our results shown that the harvesting in prey and the disease in predator impacts the behavioral of system. The situation in the system is more complex due to disease in the predator population. Some numerical simulations are given to confirm our results.

An intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model

Intuitionistic fuzzy sets cannot consider the degree of indeterminacy (i.e., the degree of hesitation). This study presents an intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model. We examine the proposed model by assuming generalized trapezoidal intuitionistic fuzzy numbers for the initial condition. Feasible equilibrium points, along with their stability criteria, are evaluated. We describe the characteristics of intuitionistic fuzzy solutions and clarify the difference between strong and weak intuitionistic fuzzy solutions. Numerical simulations are performed in MATLAB to validate the model results.

Simulation and forecasting CTLs response effectualness against global dynamics of SARS-CoV-2/HIV coinfection

Quantifying the dynamical traits and harshness of SARS-CoV-2/HIV coinfection, a modified deterministic seven compartmental mathematical model is framed at cellular level encapsulating the interrelationships between epithelial cells (predominant target cells of COVID-19 infection), CD4+ T cells (target cells of HIV/AIDS infection), SARS-CoV-2 virions, HIV virions and HIV-specific CTL response. As a way to comprehend the kinetics of SARS-CoV-2/HIV coinfection and to provide prediction of model outcomes, the positivity, boundedness and existence conditions of the biologically feasible equilibrium points have been studied. The criteria of local stability and global stability of the coinfection system around the non-negative equilibrium points have been investigated and the analytical results are enhanced numerically. To measure the influence of the model parameters in SARS-CoV-2/HIV transmission dynamics, sensitivity analysis has been performed. Semi-relative sensitivity analysis demonstrates the maximum deviation of the concentration of infected epithelial cells due to changes in the model parameters and states that six parameters are most sensitive. The numerical simulation stipulates that the COVID-19 infection is likely to be died out from a HIV patient body by increasing the apoptosis of infected epithelial cells and that is possible through the proliferation of the CD4+ T cells count of an individual living with HIV via antiretroviral therapy (ART). Additionally, elevated concentration of CD4+ T cells in people living with HIV assists in heightening the CTLs response to combat COVID-19 and other opportunistic infections.

A nonlinear mathematical model on the dynamical study of a fractional-order delayed predator–prey scheme that incorporates harvesting together and Holling type-II functional response

In the present investigation, the functional response of Holling type II and the impact of harvesting together are used to analyse the dynamical behaviour of a fractional-order delayed predator–prey scheme. In addition to the existence and uniqueness of the solution, the suggested design’s boundedness has also been proven. Additionally, both locally and globally applicable stability criteria for each feasible equilibrium point as well as their requirements for existence were also looked into. The Magtinon’s statement is used to obtain the local stability conditions, and the creation of an adequate Lyapunov function is used to demonstrate the global stability. Furthermore, we discussed about the potential for a Hopf bifurcation close to a given interior point while considering the delay to be a bifurcation factor. The Predictor–Corrector system is then put to use to carry out certain numerical simulations. Fractional order enhances the stability of the solutions and enriches the dynamics of the delayed differential system with harvesting. Our study reveals that delayed fractional order includes harvesting with type-II functional control can be used to adjust the biomass of prey species and predator species such that prey species and predator species finally each a better state level.

Analysis of interval-valued model for interaction between plankton-fish population in marine ecosystem

The crux of present investigation is to develop an interval-valued population model for the interaction between phytoplankton, zooplankton and fish population under cyrtoid type functional response. The present study is also concerned to analyze the mathematical model under impreciseness and the parametric functional form for an interval valued model parameters. The boundedness, existence of the solution, stability analysis and all possible feasible equilibrium points have been examined. The optimal harvesting strategy has been implemented and obtained the optimal solution with the help of the Pontryagin maximum principle. The significant impact of interval valued biological parameters has been analyzed and portrayed by means of graph for given different fit values, approximate to the model system in real scenario.

A study on imprecise mathematical model for optimal management and utilization of renewable resource by population

The development and survival of the biological population depend heavily on the availability of food and other materials that may be produced from renewable resources. In real life situation the information reported about the parameters of ecological model system may not be precise due to various issues like insufficient data, error in data collection, wrongly measurement etc. To address all of these challenges, an ecological model system must be developed in an ambiguous setting where the ecological parameters are handled as different kinds of fuzzy numbers. In this article, we present an imprecise mathematical model for optimal management and utilization of renewable resources by population. We introduce the model system in interval environment and apply a parametric form of interval number which makes different form of crisp ecological model. The positivity and boundedness of the proposed model system are checked. The stability criterion of feasible equilibrium points and bionomical equilibrium have been discussed. Moreover maximum sustainable strategy and optimal harvesting strategy are analyzed using interval number. Numerical discussion are provided to verify the result of the proposed system.

Stability and bifurcation analysis of delayed tri‐trophic food chain model in poisoned environment with fear effect and additional food

One of the most important goals of theoretical ecologists is to find a strategy for controlling the chaos in ecological models to maintain healthy ecosystems. We investigate the influence of fear and the supply of additional food to predators in a delayed tri‐trophic food chain model, where interference among the species is determined by Holling type‐II functional response. The proposed model incorporates the fact that the growth rates of prey and middle predators are decreased due to the fear of middle and special predators, respectively. The additional food is supplied to both predators by either nature or external agencies. Furthermore, the poisoned prey in the predator's environment is taken into account. In the proposed model, we use two time delays: one for the growth term of the middle predator and another for the gestation delay of the special predator. We determine the conditions for the existence of ecologically feasible equilibrium points and their local and global stability. In addition, we establish the conditions for the existence of Hopf bifurcation around interior equilibrium to seek periodic behavior in nondelayed and delayed models. Numerical investigations are performed to justify the proposed theoretical findings through phase portraits, time series of solutions, and bifurcation diagrams. We observe that the chaotic dynamics of a tri‐trophic food chain model can be controlled by the proper choice of the fear effect and by supplying additional food parameters. As a result, our findings provide more ecological insights into the dynamics of the delayed tri‐trophic food chain model with the fear effect and additional food supply.

Chaos Control of a Delayed Tri-Trophic Food Chain Model with Fear and Its Carry Over Effects

One of the main objectives of theoretical ecologists involves finding mechanisms to control the chaos in ecological models to maintain positive densities of the species. Numerous researchers have suggested that, apart from the direct killing in the prey–predator relationship, there are some indirect effects, such as fear of predation. Induced fear can lead to slowing down the growth rate of the prey species, and this non-chemical strategy can be carried over to successive seasons or upcoming generations. In this work, we explore the impact of fear due to predation and its carry-over effect (COE) in a delayed tri-trophic food chain model, whereas the Holling type-II functional response is used to determine the interference among the species. The proposed model is an asymmetric interaction food chain model since the species in this model only kills other species. The growth rate of prey and middle predators is affected due to the respective fear of predation by middle and special predators. The non-delayed model considered in this paper generalizes the models developed by Hastings–Powell and Panday et al. The gestation delay in the special predator’s growth term is incorporated into the proposed model. We determined the essential conditions for the existence of ecologically feasible equilibrium points and their local and global stability. Furthermore, we developed the conditions for the occurrence of the Hopf bifurcation around an interior equilibrium to seek periodic behaviors of delayed and non-delayed models. Numerical examples were performed to justify the proposed theoretical findings and to show the impacts of fear and its COE parameters on the system dynamics through phase portraits, the time series of solutions, and bifurcation diagrams. We discovered that the chaotic behavior of the food chain model can be controlled by using the fear effect and its COE parameters. The dynamics of the delayed food chain model with the fear effect and its COEs are further explored in our findings. Our theoretical findings clearly provide a mechanism to protect and control species populations in ecological systems. It is also essential for developing optimized harvesting strategies in fisheries and pest management in agriculture.

ANALYSIS OF TUMOR-IMMUNE RESPONSE MODEL BY USING CONFORMABLE FRACTIONAL ORDER DERIVATIVE

In this research paper, the authors propose a generalized three-dimensional fractional order tumor-immune response model. The generalization of the model is made by introducing interleukin-2 (IL2) cell population as the third variable in the proposed system. The study of the proposed model is performed by using a new concept of fractional-order derivatives called as conformable fractional-order derivative. The authors aim to study, analyze, and compare the dynamical be- havior of both the three-dimensional fractional order model and the conformable fractional order version of the proposed model. The stability analysis is done for both versions of the model at the biologically feasible equilibrium points. To vali- date the theoretical results numerically, numerical simulation is performed by using a piecewise constant approximation process.

Modeling and Analysis of Predator-Prey Model with Fear Effect in Prey and Hunting Cooperation among Predators and Harvesting

In this paper, we present and analyze predator-prey system where prey population is linearly harvested and affected by fear and the prey population has grown logistically in the absence of predators. The predator population follows hunting cooperation, and it predates the prey population in the Holling type II functional responses. Based on those assumptions, a two-dimensional mathematical model is derived. The positivity, boundedness, and extinction of both prey and predator populations of the solution of the system are discussed. The existence, stability (local and global), and the Hopf bifurcation analysis of the biologically feasible equilibrium points are investigated. The aim of this research is to explore the effect of fear on the prey population and hunting cooperation on the predator population, and both prey and predator populations are harvested linearly and taken as control parameters of the model. If the values of c 1 > 1 , then both prey and predator populations are extinct and also fear parameter has a stabilizing effect on system 4. From the numerical simulation, it was found that the fear effect, hunting cooperation, prey harvesting, and predator harvesting change the dynamics of system 4.

Fractional order system dynamical behaviors with Beddington-DeAngelis functional response

The present study proposes a fractional order prey-predator model with Beddington-DeAngelis functional response, that the Caputo fractional derivative is applied. There is exploration of the solutions' existence, uniqueness, non-negativity, and boundedness. Stability of all feasible equilibrium points is determined locally by the use of Matignon's condition. Moreover, we also provide sufficient conditions to assure global asymptotic stability for both the predator-extinction equilibrium point and the positive equilibrium point, with selecting a relevant Lyapunov function and the incidence of Hopf-bifurcation is also displayed. Finally, the fractional order effect on the stability behavior of systems is investigated theoretically and also illustrated numerically to support theoretical results.

Qualitative Analysis of Food-Web Model through Diffusion-Driven Instability

Many food webs exist in the ecosystem, and their survival is directly dependent on the growth rate of primary prey; it balances the entire ecosystem. The spatiotemporal dynamics of three species' food webs was proposed and analyzed in this paper, where the intermediate predator's predation term follows Holling Type IV and the top predator's predation term follows Holling Type II. To begin, we examine the system's stability using linear stability analysis. We first obtained an equilibrium solution set and then used a Jacobian method to investigate the system's stability at a biologically feasible equilibrium point. We investigate random movement in species in the presence of diffusion, establish conditions for system stability, and derive the Turing instability condition. Following that, the Turing instability condition for a spatial food web system is calculated. Finally, numerical simulations are used to validate the findings. We discovered several intriguing spatial patterns (spots, strip, and mixed patterns) that help us understand the dynamics of the real-world food web. As a result, the Turing instability analysis used in the complex food web system is especially relevant experimentally because the associated consequences can be researched and applied to a wide range of mathematical, ecological, and biological models.

Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect

In an ecological system, omnivores are often significantly important as they feed upon more than one trophic level in a food chain model. Intraguild predation is a special kind of omnivory that is ubiquitous in many ecological communities. A lot of field experiments suggest that the experience over a time duration in the recent past affects the growth rate of all species at the present time. Here, we have considered the Caputo type fractional ordered three-species food chain model. The memory effect is explained in terms of the order of the fractional derivative. Intermediate, top predators feed upon prey by Holling type I functional response and the top predator consumes intermediate predator by Holling type II functional response. Here, we established the existence and uniqueness of the solution of the considered system. Also, we prove the positivity and boundedness of the system solution. All possible feasible equilibrium points with required parametric conditions are discussed in the text. The local stability of all feasible equilibrium points and the parametric conditions of global stability of all non-trivial equilibrium points are discussed. The memory effect can stabilize the system from the periodic oscillatory situation which has been proved through the Hopf bifurcation analysis. Finally, some numerical simulation has been carried out to make significant comments on the dynamics of the considered system about the memory effect and other system parameters. It is clear from the numerical simulations that the order of fractional derivative can be taken as the indicator of chaos control. The biological interpretations make the article most significant from the ecological point of view.

An eco-epidemiological model with the impact of fear.

In this study, we propose and analyze an eco-epidemiological model with disease in prey and incorporated the effect of fear on prey species due to predator population. We assume that the prey population grows logistically in the absence of predator species, and the disease is limited to the prey population only. We divide the total prey population into two distinct classes: susceptible prey and infected prey. Predator populations are not infected by the diseases, though feed both the susceptible and infected prey. Due to the fear of predators, the prey population becomes more vigilant and moves away from suspected predators. Such a foraging activity of prey reduces the chance of infection among susceptible prey by lowering the contact with infected prey. We assume that the fear of predators has no effect on infected prey as they are more vigilant. Positivity, boundedness, and uniform persistence of the proposed model are investigated. The biologically feasible equilibrium points and their stability are analyzed. We establish the conditions for the Hopf bifurcation of the proposed model around the endemic steady state. As the level of fear increases, the system moves toward the steady state from a limit cycle oscillation. The increasing level of fear cannot wipe out the diseases from the system, but the amplitude of the infected prey decreases as the level of fear is increased. The system changes its stability as the rate of infection increases, and the predator becomes extinct when the rate of infection in prey is high enough though predators are not infected by the disease.

Untangling role of cooperative hunting among predators and herd behavior in prey with a dynamical systems approach

In the real world, group formation is ubiquitous. This group behavior is according to the species' needs; some species use it for hunting and some for protection from predators. In this paper, we proposed a prey-predator model with cooperative hunting among predators and herd behavior in prey, and we studied the temporal and spatiotemporal analysis. We first examine the system's positivity, boundedness, and then analyze the local stability criterion of equilibrium points. We perform Hopf–bifurcation analysis at feasible equilibrium points by considering s and α as the bifurcation parameters. With the incorporation of diffusion into the system, we derive the diffusion-driven instability of the prey-predator system. Then we performed a weakly nonlinear analysis and derived amplitude equations by considering self-diffusion as Turing bifurcation parameters. The stability analysis of these amplitude equations leads to the identification of various Turing patterns, such as spots, stripes, and mixed. Employing numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. Our findings reveal that cooperative hunting, interspecific competition, and self-diffusion have important implications in the prey-predator system.

A Delayed Eco-Epidemiological Model with Weak Allee Effect and Disease in Prey

In this paper, we have proposed and analyzed a mathematical model for the eco-epidemiological system with disease in prey population, placing particular emphasis on the effects of an incubation time delay and the influence of weak Allee effect on the growth of the predator population. We investigate basic properties of the proposed model including positivity, boundedness, uniform persistence and the stability of the biologically feasible equilibrium points. Using the time delay as bifurcation parameter, we explore the stability of the interior equilibrium point and observe that Hopf bifurcation can occur when the incubation time delay crosses some threshold value. We have identified the stability regions associated with the extinction of populations, stability of the steady states, and high-periodic oscillations in disease transmission. Analytical findings are supported by numerical illustrations that demonstrate the behavior of the proposed model in different dynamical regimes. Stability criteria for the biologically feasible equilibrium points were obtained and validated with numerical simulations. We found most sensitive system parameters using the PRCC sensitivity analysis. We observe that our model simulation exhibits high-periodic oscillations due to the increasing value of time delay parameter [Formula: see text] and the disease transmission rate [Formula: see text].

Global dynamics of a fractional-order HFMD model incorporating optimal treatment and stochastic stability

Hand, foot and mouth disease (HFMD) is highly contagious and occurs primarily among children under the age of five. Analysis of transmission dynamics of infectious diseases is essential to prevent the adverse effects caused by the diseases. The current study presents a fractional-order SEIR-type epidemic model to investigate the dynamics of HFMD transmission. The biological feasibility of the proposed model system is demonstrated from an epidemiological perspective. The basic reproduction number, R0, is obtained through the next-generation matrix approach. Around the feasible equilibrium points, the asymptotic dynamics of the proposed model system are examined, both at local and global levels. It is found that, the model undergoes transcritical bifurcation at R0 = 1. As a result, R0 plays the role of threshold in determining the future course of the disease. The optimal treatment control of fractional epidemic models are less explored. Here, an optimal control problem is formulated considering a time-dependent treatment measure u(t) both in exposed and infected classes. The findings are also visualized and verified by simulating the model using some feasible parameter values, from which it can be concluded that fractional-order gives better results. A gradual decrease in the total cases as well as in the peak is observed in the presence of treatment control. Further, we extend the study in a stochastic environment with the help of white noise and investigate the stochastic stability of the endemic equilibrium point. Finally, parameters defining the threshold quantity R0 are scaled with the normalized forward sensitivity index.

Chaotic dynamics of a plankton-fish system with fear and its carry over effects in the presence of a discrete delay

Plankton-fish interactions are the central topic of interest related to marine ecology. Apart from the direct predation (lethal effect) of zooplankton by fish, there are some non-lethal implications in the zooplankton-fish relationship. Due to induced fear of predation, there can be a reduced reproduction rate of zooplankton species, and the effect of this non-lethal interconnection can be carried over to subsequent seasons or generations. In the current study, we tend to analyse the role of fish-induced fear in zooplankton with its carry-over effects (COEs) and a corresponding discrete delay (COE delay) in a phytoplankton-zooplankton-fish population model. We use Holling type IV and II functional responses to model the phytoplankton-zooplankton and zooplankton-fish interplay, respectively. In the well-posedness of the present biological system, firstly, we evaluate an invariant set in which the solutions of the model remain bounded. Then we prove its persistence under some ecologically well-behaved conditions. Next, we establish the conditions under which the different feasible equilibrium points exist; the existence of various interior equilibria is also set up here. To study the system's dynamical behavior, local and global stability analyses for the equilibria mentioned above are also discussed. Further, the theoretical conditions for Hopf and transcritical bifurcations in non-delayed and delayed models are determined. Impacts of non-lethal parameters, fear, and its carry-over effects, on the population densities are studied analytically and supported numerically. For intermediate values of COEs parameter, we notice that the system behaves chaotically, and decreasing (or increasing) it to low (or high) values, solution converges to interior equilibrium point through period-halving. Calculation of the largest Lyapunov exponent and drawing of Poincaré map validate the chaotic nature of the system. The chaos for medium values of COEs parameter can also be controlled by decreasing the fear parameter. Next, we numerically validate the theoretical result for transcritical bifurcation. We also note that our system shows the phenomenon of enrichment of paradox, and the attribute of multistability. In the delayed model, we observe that increasing delay can eliminate chaotic oscillations through amplitude death phenomenon. We draw various types of graphs and diagrams to assist our results. Thus we can say that the present study has various interesting characters related to non-linear models and can help biologists to study the plankton-fish models in a more detailed and pragmatic manner.