Interior Equilibrium Research Articles

Optimal control of susceptible mature pest concerning disease-induced pest-natural enemy system with cost-effectiveness

Abstract This article addresses the pressing issue of pest outbreaks in India, which poses significant challenges for farmers and ecologists. A novel system is proposed for effective control that leverages natural enemies. Here, the pests are classified into juveniles and mature individuals, further categorized as susceptible or infected. The study introduces harvesting, incorporating external efforts and natural phenomena, in a pest-epidemic prey–predator system featuring a prey-stage structure. The model reveals three equilibria: trivial, boundary (indicating the absence of natural enemies), and interior equilibria. Notably, the trivial equilibrium is consistently unstable. As demonstrated by stability analysis, the survival or extinction of natural enemies hinges on control variables, including the harvesting rate, disease transmission rate, and natural death rate. Local stability is assessed using the Routh–Hurwitz criterion, while global stability is explored through the Lyapunov method. Furthermore, optimal control theory and Pontryagin’s maximum principle are applied for model optimization, unveiling crucial optimality conditions and determining the optimal harvesting rate for susceptible mature prey. Numerical computations validate theoretical insights, offering valuable guidance for formulating policies that optimize the control of susceptible adult pests within a disease-induced pest-natural enemy system, ensuring sustained cost-effectiveness.

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.

Optimal water tariffs for domestic, agricultural and industrial use

Consider a water supplier who determines sales rates with the goals of maximizing profits, protecting consumer welfare, and ensuring adequate future water supplies. Buyers are differentiated and can use the water for domestic, agricultural, and industrial purposes. We propose a leader-follower finite-horizon differential game. The leader (the water supplier) determines the selling price and the followers (consumers) react by requesting their optimal amount of water. We calculate a feedback Stackelberg equilibrium assuming that all user demand is satisfied (interior equilibrium). We compare two different tariff schemes: linear tariffs (the price paid is a multiple of the volume of water purchased), and increasing block tariffs (the unit price is lower for quantities of water that do not exceed a fixed threshold). We show that block pricing is never optimal and linear pricing is always preferred.

Impact of Fear and Group Defense on the Dynamics of a Predator–Prey System

To reduce the chance of predation, many prey species adopt group defense mechanisms. While it is commonly believed that such defense mechanisms lead to positive feedback on prey density, a closer observation reveals that it may impact the growth rate of species. This is because individuals invest more time and effort in defense rather than reproductive activities. In this study, we delve into a predator–prey system where predator-induced fear influences the birth rate of prey, and the prey species exhibit group defense mechanism. We adopt a nonmonotonic functional response to govern the predator–prey interaction, which effectively captures the group defense mechanism. We present a detailed mathematical analysis, encompassing the determination of feasible equilibria and their stability conditions. Through the analytical approach, we demonstrate the occurrence of Hopf and Bogdanov–Takens (BT) bifurcations. We observe two distinct types of bistabilities in the system: one between interior and predator-free equilibria, and another between limit cycle and predator-free equilibrium. Our findings reveal that the parameters associated with group defense and predator-induced fear play significant roles in the survival and extinction of populations.

A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

The human inclination to value rarity contributes to the exploitation of many wildlife species by driving up their market value when their numbers are scarce. This research investigates a model to explore how the value attached to rarity impacts the harvesting of predator species and subsequently influences the dynamics of predator–prey interactions in an ecological system. To broaden the model's scope, it's extended to its nonautonomous form by allowing the harvesting rate of predator species and the degree of neutrality in consumers to respond for the rarity of predator species to change over time. Both the autonomous and nonautonomous systems are explored through analytical and numerical investigation. Numerical findings reveal that as the prey population's growth rate gradually increases, the unforced system shifts from stable boundary equilibrium (harvesting absent) to stable interior equilibrium, then to oscillatory interior equilibrium, followed by period doubling oscillations leading to chaotic behavior. The autonomous system exhibits limit cycle oscillations around the boundary equilibrium when predator species are minimally harvested. Moderate harvesting stabilizes the system, while excessive harvesting induces instability. However, the chaotic oscillations can be eliminated through period‐halving route upon decreasing the consumers' response to predator rarity. Furthermore, the seasonally forced model demonstrates periodic solutions, higher order periodic solutions, and chaotic dynamics contingent upon different settings of prey growth rate, the strength of seasonality in predator harvesting, and the consumers' response to predator rarity.

FILIPPOV TYPE PREY–PREDATOR SYSTEM FOR SELECTIVE HARVESTING OF PREY

In this paper, the dynamics of a Filippov system with logistically growing prey and Holling Type II predation is explored. The selective nonlinear harvesting of prey is carried out when the ratio of prey to predator is above a specified threshold value. In order to prevent the extinction of both the species and sustainable economic gains, no harvesting is allowed when the ratio is below a threshold value. The equilibrium states and their stability for the individual smooth are analyzed. Filippov convex method is applied to determine the dynamics on the boundary. The sliding mode dynamics is scrutinized, and it is observed that there is no escaping region. The existence of boundary points, tangent points and pseudo-equilibrium points are established. The conditions are obtained for the visibility of the tangent points. The complex behavior within the sliding region is studied by experimenting with different initial conditions. The increase in harvesting effort may lead to extinction of both the species through the sliding region of the boundary. The change in the stability behavior of both the interior equilibrium states is investigated with respect to model parameters. Border cross bifurcates with threshold value as the bifurcation parameter is shown for both the interior equilibrium states.

Dynamics of a plant-herbivore system with Ricker plant growth and the strong Allee effects on plant population

We examine a discrete-time model of plant-herbivore interaction, in which the population of plants follows the Ricker law and is affected by strong Allee effects. We discuss the equilibrium points, including their characteristics and number. Additionally, we analyze in detail the local behavior of the solutions around the equilibrium points. Under certain initial conditions, both populations may go extinct. We conduct research demonstrating the occurrence of transcritical and period-doubling bifurcations, resulting in a stable two-cycle at the boundary equilibrium. However, the interior equilibrium becomes unstable due to the emergence of the Neimark-Sacker bifurcation and period-doubling. We confirm the latter's existence through analytical methods and illustrate the period-doubling bifurcation through numerical simulations.

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>

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>

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 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.

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.

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.

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.

Exploring dynamics of plant-herbivore interactions: bifurcation analysis and chaos control with Holling type-II functional response.

In this study, we examine the plant-herbivore discrete model of apple twig borer and grape vine interaction, with a particular emphasis on the extended weak-predator response to Holling type-II response. We explore the dynamical and qualitative analysis of this model and investigate the conditions for stability and bifurcation. Our study demonstrates the presence of the Neimark-Sacker bifurcation at the interior equilibrium and the transcritical bifurcation at the trivial equilibrium, both of which have biological feasibility. To avoid unpredictable outcomes due to bifurcation, we employ chaos control methods. Furthermore, to support our theoretical and mathematical findings, we develop numerical simulation techniques with examples. In summary, our research enhances the comprehension of the dynamics pertaining to interactions between plants and herbivores in the context of discrete-time population models.

Modeling the effects of baculovirus to control insect population in agricultural fields

Baculoviruses are special kind of pathogens that attack insects and thus have a potential to reduce the density of insect population in agricultural fields and may be an alternate to chemical insecticides. In this work, we present a model consisting of non-linear differential equations to assess the effects of baculovirus as a bio-insecticide on insect population and thus crop production. In the modeling phenomenon, we consider one-time spray per season. It is shown that under certain conditions, one-time spray is sufficient to form self-sustained cycle of baculovirus replication in the agriculture fields. For the formulated model, the equilibrium solutions are obtained and their stability behavior is discussed. It is observed that parameters related to virus infection rate and virus replication rate are important to reduce the density of insects and increasing crop production. This study reveals that in the case of one-time spray, to achieve the stability of interior equilibrium solution, the infection rate must lie within its two threshold values. The effect of some other parameters on the dynamics of insect population and crop production is also shown by means of numerical simulation.

Modeling the role of renewable energy to mitigate the atmospheric level of carbon dioxide along with sustainable development

In the past few decades, the use of fossil fuels has increased dramatically due to industrialization in developing nations. The elevation of carbon dioxide (CO2) has become a serious concern for the entire world. Therefore, most countries want to reduce the use of fossil fuels by transitioning to renewable energy sources. In this research work, we formulate a nonlinear mathematical model to study the interplay between atmospheric CO2, human population, and energy production through traditional energy sources (coal, oil, and gas) and renewable energy sources (solar, wind, and hydro). For the model formulation, we consider that the atmospheric level of CO2 increases due to human activities and energy production through traditional energy sources. Additionally, we consider that the dependency of the human population shifts from traditional to renewable sources of energy as the atmospheric level of CO2 increases. Through the model analysis, we have obtained a condition that implies a change in the equilibrium level of CO2 by increasing the deployment rate of renewable energy sources. This condition demonstrates that the atmospheric level of CO2 can be reversed from its current level through sufficient deployment of renewable energy sources. Moreover, for a certain critical value of the growth rate parameter of renewable energy, three interior equilibria may exist. This situation indicates the presence of hysteresis, which implies that the system may undergo discontinuous jumps between different stable steady states. Furthermore, we discuss an optimal control problem to minimize atmospheric CO2, simultaneously minimizing the cost of the considered strategy.

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.