Interior Equilibrium Point Research Articles

Dynamics of autonomous and nonautonomous Leslie–Gower model for the impacts of fear and its carry‐over effects including predator harvesting

Recently, certain biological field experiments and research findings have revealed that predators not only reduce prey populations through direct predation (lethal effect) but also indirectly affect the growth rate of prey by inducing fear; these nonlethal effects can be carried over seasons or generations. In this paper, we proposed an autonomous and a nonautonomous Leslie–Gower model incorporating some biological factors such as fear and its carry‐over effects, prey refuge, and nonlinear predator harvesting. In the autonomous model, first, we examined the well‐posedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in intrinsic growth rate of predator. Further, we compute and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf bifurcation and direction and stability of limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of fear factor, carry‐over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas for small values of prey birth rate and harvesting effort, the system remains stable. However, when intrinsic growth rate of predator increases from a low value to a higher value, the system shows transition from stability to instability and back to stability. We also explore the effects of seasonal variations in biological parameters in the nonautonomous model. Additionally, we investigate the theoretical existence of positive periodic solutions using the continuation theorem. The influences of seasonal variations in some parameters such as prey's birth rate, fear, and its carry‐over effects; refuge behavior of prey; intrinsic growth rate of predator; and harvesting effort are then numerically investigated. The results reveal the emergence of positive periodic solutions and bursting patterns in the seasonally forced model.

Effect of fear with saturated fear cost and harvesting on aquatic food chain model (plankton–fish model) in the presence of nanoparticles

Studying the interplay of phytoplankton–zooplankton–fish (PP–ZP–F) in an aquatic system is crucial for better understanding of nutrient cycling, assessing ecosystem health, predicting and mitigating harmful algal blooms, and managing fisheries in the water bodies. In order to investigate the effectiveness of nanoparticles (NPs), fear, and harvesting, this paper focuses on exploring the dynamics of a food chain model among PP–ZP–F species. We consider the fear of fish on zooplankton species (which reduces the reproduction rate of ZPs) with saturated fear cost in the presence of nanoparticles (NPs) and harvesting in fish. The system dynamics are studied from the viewpoint of proving positivity, boundedness, and uniqueness, followed by analysing the existence and local stability of biologically feasible equilibria. Conditions for the global stability of the interior equilibrium point are also found. Furthermore, we established the transversality conditions for the occurrence of Hopf, transcritical, and saddle–node bifurcations. To validate our theoretical results, we made numerous phase portraits, time-series graphs, tables showing the extinction of species, and bifurcation diagrams. It is numerically observed that increasing the contact rate of NPs with PPs makes the system stable from chaos, and further increase of contact rate may lead to extinction. Chaos at a low contact rate can also be managed by increasing the fear level, and the chaotic behaviour at a low fear level can again be controlled by enhancing the harvesting of fish species. Over-exploitation may result in the extinction of fish, whereas fear may promote coexistence, stability, and long-term survival of the species. Increased saturated fear cost can make the system chaotic from stable dynamics. Therefore, the theoretical as well as numerical findings of our paper may be of great interest in estimating the behaviour of aquatic systems biologically and practically.

Study on the interior equilibrium point of a special class of 2 × 2 × 2 asymmetric evolutionary games.

Many behavioural interactions in real life involve three individuals. When each individual has two alternative strategies, they can be abstracted into mathematical models by means of asymmetric games. In this paper, we explore a special class of asymmetric games satisfying fixed conditions. Firstly, we analyse two solitary interior equilibrium points and provide the judgement condition for their instability based on the Jacobi matrix local stability analysis method. Secondly, we analyse the interior equilibrium points that are continuously distributed within a line and probe into their stability conditions based on generalized Hamiltonian systems theory. Under the circumstances, the stable interior equilibrium point is surrounded by closed orbits in phase space, which presents an observable stable state where two strategies coexist and fluctuate in each of the three game populations. This work enriches the study of asymmetric games' evolutionary dynamics.

Delay-induced nutrient recycling in plankton system: Application to Sundarban mangrove wetland

The paper discusses the nutrient–plankton system with effect of time delay in nutrient recycling and toxin determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient–plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system.

Dynamic analysis and optimal control strategies of a predator–prey mathematical model for the pest eradication in oil palm plantation

Oil palm cultivation stands as a crucial industry in Indonesia, significantly contributing to the nation’s economy by generating employment opportunities and fostering social welfare for communities residing near plantations. Despite its economic importance, oil palm plantations face various challenges, with one prominent issue being the infestation of nettle caterpillar pests. These pests cause leaf skeletonization, resulting in a staggering 36% reduction in oil palm productivity over a two-year period. This paper explores diverse strategies for pest management in oil palm plantations, encompassing biological control through the stimulation of natural predators, mechanical control involving the collection and incineration of cocoons, and chemical control through pesticide application. The research introduces a predator–prey mathematical model for oil palm plantation pests, where the leaf area serves as the primary food source for caterpillars, acting as prey. Through dynamic model analysis, four equilibrium points are identified, with interconnected conditions dictating their existence and stability. These conditions are visually represented in a bifurcation plane, providing concise information. The study further includes bifurcation diagrams of equilibrium points to elucidate the influence of each parameter on pests, predators, and the leaf area of oil palm plants. Additionally, sensitivity analysis of the stable interior equilibrium point is conducted to understand the impact of individual parameters. The paper extends its investigation to optimal control strategies, evaluating six scenarios categorized into two population conditions: with predators and without predators. Within each population condition, three control strategies are considered—chemical control only, mechanical control only, and a combination of chemical and mechanical control. Simulation results from the optimal control study reveal that the presence of natural predators emerges as a pivotal strategy in effectively managing nettle caterpillars. Notably, the control of resistant pests through pupa incineration has a substantial impact on reducing the caterpillar population in subsequent life cycles.

Spatial dynamics of higher order rock-paper-scissors and generalisations

We introduce and study the spatial replicator equation with higher order interactions and both infinite (spatially homogeneous) populations and finite (spatially inhomogeneous) populations. We show that in the special case of three strategies (rock–paper–scissors) higher order interaction terms allow travelling waves to emerge in non-declining finite populations. We show that these travelling waves arise from diffusion stabilisation of an unstable interior equilibrium point that is present in the aspatial dynamics. Based on these observations and prior results, we offer two conjectures whose proofs would fully generalise our results to all odd cyclic games, both with and without higher order interactions, assuming a spatial replicator dynamic. Intriguingly, these generalisations for N⩾5 strategies seem to require declining populations, as we show in our discussion.

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.

Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.

Eco-epidemiological Model with Infected Prey: Dynamic Behaviour of a Fractional Order Model Including Prey Refuge and Type II Functional Response

In this study, using the Holling type II response function and prey refuge, we bring fractional order into an eco-epidemiological paradigm with diseased prey in which the predator consumes a vast amount of healthy prey in an excessively disproportionately large amount. We show that there are solutions to the fractional order eco-epidemiological paradigm and that these solutions are distinct, non-negativity, and limited. Additionally, we generated a number of equilibrium points and investigated the local and global stability of the interior equilibrium point. We explore the roles that fractional order and the prey shelter play in maintaining the stability of the proposed system’s equilibrium point. Numerous examples are used to illustrate the results, and numerical simulations are used to support our theoretical conclusions.

Co-evolutionary dynamics in optimal multi-agent game with environment feedback

At present, game-based analysis and control problems with environmental feedback significantly contribute to the study of collective cooperation, and pose great challenges for the analysis of multi-agent game systems in complex real world. We consider co-evolutionary dynamics based on an extended multi-agent game with environmental feedback, which is modeled by optional public goods games (OPGGs). Firstly, the co-evolutionary dynamics of OPGGs with environment feedback can be modeled as two parts: the dynamics of OPGGs and synergy coefficient, where the former can represent the group game behavior of three strategies, and the latter can be seen as the evolutionary behavior in changing environments. Subsequently, some possible fixed points will emerge with respect to three types of boundaries for these co-evolutionary dynamics, and the existence and stability for the corresponding boundary fixed points are determined and analyzed. Meanwhile, the results are further extended to interior equilibrium point. It is observed that by combining voluntary participation and environment feedback, cooperation behavior can be still feasible, and the evolving system can oscillatingly converge to the persistent cooperation under some certain threshold conditions. Finally, an example is illustrated to validate the proposed approach. To sum up, environmental feedback mechanism provides a novel perspective to investigate the collective cooperation in a changing world.

Role of Allee and Fear for Controlling Chaos in a Predator–Prey System with Circulation of Disease in Predator

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey’s growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter ([Formula: see text]) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter?

Dynamical study of fractional order Leslie-Gower model of predator-prey with fear, Allee effect, and inter-species rivalry

Employing an improved Leslie-Gower model, the Allee impact on the predator population and the fear influence on the prey population, a mathematical model of the interaction involving two populations, prey and predator, has been explored in the current work. The influence of the memory effect on the evolving behaviour of the model is integrated using a well-known fractional operator known as the Caputo fractional derivative of order (0, 1]. We were inspired to do this study because further research is desired to fully comprehend the dynamics of the Allee impact, fear, and Caputo fractional order. The innovative aspect of the current work is the consideration of the Caputo fractional derivative, which enhances the consistency domain of the system. To support the model's biological resilience and accuracy, the existence, uniqueness, positivity, and boundedness of the solution are provided. On the origin, axial, and interior, four distinct types of equilibrium points are distinguished along with the prerequisites for their existence. We additionally stated about the Hopf bifurcation of our proposed model at the interior equilibrium point in terms of fractional order and the fear influence as the bifurcation factors. To illustrate how various biological characteristics affect the dynamics of the solutions, numerical simulations are offered.

Direction and stability of Hopf bifurcation in an eco-epidemic model with disease in prey and predator gestation delay using Crowley-Martin functional response

<p>In this work, we have studied an eco-epidemic model using the Crowley-Martin functional response that includes disease in prey and gestation delay in the predator population. The model possesses three equilibria, namely the disease-free, Predator-free, and the interior equilibrium point. In addition, we examined the stability of the equilibrium points varying the infection rate and time delay parameter. Detailed analysis of Hopf bifurcation of the interior equilibrium point contains two situations: with delay and without delay. Moreover, we have studied the direction of the Hopf bifurcation and the stability of periodic solutions utilizing normal form theory and the center manifold theorem. It is emphasized that Hopf bifurcation occurs when the time delay exceeds the critical value and that the critical value of the delay is strongly impacted by the infection rate in prey. A detailed numerical simulation is provided to verify the analytical results.</p>

Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

<abstract><p>We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.</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.

Global stability of the interior equilibrium and the stability of Hopf bifurcating limit cycle in a model for crop pest control

<p>Mathematical modeling and analysis of a crop-pest interacting system helps us to understand the dynamical properties of the system such as stability, bifurcations and chaos. In this article, a predator-prey type mathematical model for pest control using bio-pesticides has been analysed to study the global stability property of the interior equilibrium point. Moreover, the occurrence and orbital stability of Hopf bifurcating limit cycle solutions have been studied using ref30's conditions. Analytical and numerical results show that the interior equilibrium of the pest control model is globally asymptotically stable. Also, Hopf bifurcating occurs when the bifurcation parameter crosses the critical value, and the bifurcating periodic solution is found to be stable.</p>

Roles of astrocytes and prions in Alzheimer's disease: insights from mathematical modeling.

We present a mathematical model that explores the progression of Alzheimer's disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer's disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.

Dynamics of an eco-epidemic model with Allee effect in prey and disease in predator

Abstract In this work, the dynamics of a food chain model with disease in the predator and the Allee effect in the prey have been investigated. The model also incorporates a Holling type-III functional response, accounting for both disease transmission and predation. The existence of equilibria and their stability in the model have also been investigated. The primary objective of this research is to examine the effects of the Allee parameter. Hopf bifurcations are explored about the interior and disease-free equilibrium point, where the Allee is taken as a bifurcation point. In numerical simulation, phase portraits have been used to look into the existence of equilibrium points and their stability. The bifurcation diagrams that have been drawn clearly demonstrate the presence of significant local bifurcations, including Hopf, transcritical, and saddle-node bifurcations. Through the phase portrait, limit cycle, and time series, the stability and oscillatory behaviour of the equilibrium point of the model are investigated. The numerical simulation has been done using MATLAB and Matcont.

Dynamical analysis of a chemostat model for 4-chlorophenol and sodium salicylate mixture biodegradation

We consider a mathematical continuous-time model for biodegradation of 4-chlorophenol and sodium salicylate mixture by the microbial strain Pseudomonas putida in a chemostat. The model is described by a system of three nonlinear ordinary differential equations and is proposed for the first time in the paper [Y.-H. Lin, B.-H. Ho, Biodegradation kinetics of phenol and 4-chlorophenol in the presence of sodium salicylate in batch and chemostat systems, Processes, 10:694, 2022], where the model is only quantitatively verified. This paper provides a detailed analysis of the system dynamics. Some important basic properties of the model solutions like existence, uniqueness and uniform boundedness of positive solutions are established. Computation of equilibrium points and study of their local asymptotic stability and bifurcations in dependence of the dilution rate as a key model parameter are also presented. Thereby, particular intervals for the dilution rate are found, where one or three interior (with positive components) equilibrium points do exist and possess different types of local asymptotic stability/instability. Hopf bifurcations are detected leading to the occurrence of stable limit cycles around some interior equilibrium points. A transcritical bifurcation also exists and implies stability exchange between an interior and the boundary (washout) equilibrium. The results are illustrated by lots of numerical examples.

Analyzing Predator-Prey Interaction in Chaotic and Bifurcating Environments

An analysis of discrete-time predator-prey systems is presented in this paper by determining the minimum amount of prey consumed before predators reproduce, as well as by analyzing the system's stability and bifurcation. In order to investigate the local stability of the interior equilibrium point of the proposed model, discrete dynamics system theory is employed first. Moreover, the characteristic equation is analyzed to determine the Neimark-Sacker bifurcation of the system. The normal form and bifurcation theory are used to investigate the NS bifurcation around the interior equilibrium point. Based on its analysis, the system exhibits Neimark-Sacker bifurcation when positive parameters are present and non-negative conditions are met. Develop a feedback control strategy to discover the region of stability of the chaotic behavior. By utilizing the maximum laypanuou exponent, the effect of initial conditions on developed systems is further explored. Finally, a computer simulation illustrates the results of the analysis.