Feasible Equilibrium Research Articles

From lignocellulosic biomass to chemical precursors: Simultaneous valorization of furfural and levulinic acid over mesoporous acid catalysts

This paper analyses the furfural and levulinic acid aldol condensation over mesoporous acid materials. This process is a promising route to obtain chemical feedstock and biofuel precursors from two of the most available bio-platform molecules produced from nonedible lignocellulosic biomass. Different acid solids were tested (ion-exchange resins, MCM, γ-Al2O3), obtaining the best results with alumina (γ-Al2O3). This material reaches a feasible equilibrium between sustainability (heterogeneous, available, and cheap catalysis, soft temperatures), activity (45 % of levulinic acid conversion, C10s yield of 44 %) and stability (no deactivation observed after four cycles). These results significantly improve those reported in the literature using microporous materials since the adsorption is significantly limited, as demonstrated by the good carbon balance (>95 %), the kinetic model and spent catalyst characterization.

An integrative framework of cooperative advertising with reference price effects

The importance of reference price effects in consumer behavior and marketing decisions is now well established in the literature. However, research on the impact of these effects on cooperative advertising decisions in marketing channels remains very limited. A two-period model is developed to analyze how members of a bilateral monopoly channel should set pricing and advertising decisions in a context where first-period price serves as the reference price of second period. By solving a Stackelberg game where the manufacturer is the leader, nine feasible equilibria are endogenously obtained. These equilibria correspond to different combinations (scenarios) of the respective decisions of the retailer and manufacturer to undertake or not and to support or not local advertising in each period. The profitability of each of these scenarios for the players and their pricing and advertising strategies over time depend, among others, on how sensitive consumers are to price changes over time.

Dynamical response of a reaction–diffusion predator–prey system with cooperative hunting and prey refuge

The present research is concerned with the combined outcome of the cooperative hunting and prey refuge in a spatiotemporal predator–prey model. Firstly, the problem is confirmed to be well-posed and some basic preliminaries are provided within the context of the temporal environment. Subsequently, both the local and the global stability of the temporal system including permanence are thoroughly investigated so as to emerge the fact that the competition between the hunting cooperation factor a and the refuge coefficient r can resolve the dynamics of the system. More precisely, global stability for all of the feasible non-negative equilibria corresponding to the temporal environment and the coexistence equilibrium in the spatiotemporal domain are explored in the event of the hunting cooperation factor a not exceeding the prey refuge coefficient r. However, the moment a exceeds r, where both the Hopf bifurcation and the Turing bifurcation are induced by hunting cooperation. Nevertheless, a distinct Turing instability mechanism is emerged when the prey diffusivity exceeds that of predator but interestingly, the opposite is customarily a reasonable constraint in many predator–prey models. Later on, the diffusion coefficient is chosen as a bifurcation parameter interpreting pattern transition and the amplitude equations close to the onset are thereby derived. The stability analysis is made use of to explain the selection of patterns among hot spot patterns, the mixture of hot spots and stripes patterns and the stripe patterns themselves. Finally, numerical simulations are performed to explore pattern selection influenced by the hunting cooperation factor, the prey refuge coefficient and the diffusivity as well. Some interesting dynamical complexities including the variation of the number of equilibria, the bifurcation scenario, etc, also emerge out from such quantitative simulations.

Stability criteria for the consumption and exchange of essential resources.

Models of consumer effects on a shared resource environment have helped clarify how the interplay of consumer traits and resource supply impact stable coexistence. Recent models generalize this picture to include the exchange of resources alongside resource competition. These models exemplify the fact that although consumers shape the resource environment, the outcome of consumer interactions is context-dependent: such models can have either stable or unstable equilibria, depending on the resource supply. However, these recent models focus on a simplified version of microbial metabolism where the depletion of resources always leads to consumer growth. Here, we model an arbitrarily large system of consumers governed by Liebig's law, where species require and deplete multiple resources, but each consumer's growth rate is only limited by a single one of these resources. Resources that are taken up but not incorporated into new biomass are leaked back into the environment, possibly transformed by intracellular reactions, thereby tying the mismatch between depletion and growth to cross-feeding. For this set of dynamics, we show that feasible equilibria can be either stable or unstable, again depending on the resource environment. We identify special consumption and production networks which protect the community from instability when resources are scarce. Using simulations, we demonstrate that the qualitative stability patterns derived analytically apply to a broader class of network structures and resource inflow profiles, including cases where multiple species coexist on only one externally supplied resource. Our stability criteria bear some resemblance to classic stability results for pairwise interactions, but also demonstrate how environmental context can shape coexistence patterns when resource limitation and exchange are modeled directly.

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.

Equilibrium in a large Lotka–Volterra system with pairwise correlated interactions

Consider a Lotka–Volterra (LV) system of coupled differential equations: ẋk=xk(rk−xk+(Bx)k),x=(xk),1≤k≤n,where r=(rk) is a n×1 vector and B a n×n matrix. Assume that the interaction matrix B is random and follows the elliptic model: B=1αnA+μn1n1nT,where A=(Aij) is a n×n matrix with N(0,1) entries satisfying the following dependence structure (i) the entries Aij on and above the diagonal are i.i.d., (ii) for i<j each vector (Aij,Aji) is standard Gaussian with covariance ρ, and independent from the other entries; vector 1n stands for the n×1 vector of ones. Parameters α,μ are deterministic and may depend on n.Leveraging on Random Matrix Theory, we analyze this LV system as n→∞ and study the existence of a positive equilibrium. This question boils down to study the existence of a (componentwise) positive solution to the linear equation: x=r+Bx,depending on B’s parameters (α,μ,ρ), a problem of independent interest in linear algebra.In the case where no positive equilibrium exists, we provide sufficient conditions for the existence of a unique stable equilibrium (with vanishing components), and following Bunin (2017), present a heuristics estimating the number of positive components of the equilibrium and their distribution.The existence of positive equilibria for large Lotka–Volterra systems has been raised in Dougoud et al. (2018), and addressed in various contexts by Bizeul and Najim (2021) and Akjouj and Najim (2022).Such LV systems are widely used in mathematical biology to model populations with interactions, and the existence of a positive equilibrium known as a feasible equilibrium corresponds to the survival of all the species xk within the system.

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].

Asymptotic Properties and Stability Switch of a Delayed-Within-Host-Dengue Infection Model with Mitosis and Immune Response

In this paper, a delayed-within-host-dengue infection model with mitosis and immune response is analyzed. The basic reproduction number is calculated and a detailed discussion on the local and global dynamics of the model is conducted. By using comparison arguments, it is shown that when the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, the existence of Hopf bifurcation and stability switch at the immunity-activated infection equilibrium of the model with or without delay is established. Furthermore, by means of Lyapunov functional and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the immunity-activated infection equilibrium. Numerical simulations are given to illustrate the main theoretical results. The normal form is calculated to analyze some properties of the bifurcation periodic solution when the time delay is absent. Moreover, we carry out sensitivity analysis on basic reproduction number to determine crucial parameters that affect the stability of each of feasible equilibrium.

Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior

This article considers the impact of competitive response to interfering time and anti-predator behavior of a three species system in which one predator consumes both the competing prey species. Here one of the competing species shows anti-predator behavior. We have shown that its solutions are non-negative and bounded. Further, we analyze the existence and stability of all the feasible equilibria. Conditions for uniform persistence of the system are derived. Applying Bendixson’s criterion for high-dimensional ordinary differential equations, we prove that the coexistence equilibrium point is globally stable under specific conditions. The system admits Hopf bifurcation when anti-predator behavior rate crosses a critical value. Analytical results are verified numerically.

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.

Bifurcation induced by delay parameter in plant growth dynamics

In this paper, a mathematical model is proposed for plant growth dynamics using delay differential equations. The state variables considered are: plant biomass and nutrient concentration. It is assumed that the delayed nutrient concentration adversely affects the plant growth. The positivity of solutions is established and the feasible interior equilibrium point is calculated. The stability of the system around the interior equilibrium is checked. Hopf-Bifurcation occurred around the critical value of the delay parameter. Analytical findings are supported using MATLAB simulation.

Modelling the Effect of Toxin Producing Prey on Predator Population using Delay Differential Equations

A mathematical model is proposed to study the effect of toxin producing prey on predator population using delay differential equations. The associated state variables are Prey populations and predator populations. The assumption is that the toxicity released by prey population adversely affects the predator population. The feasible interior equilibrium is calculated. Hopf bifurcation is observed about the critical value of delay parameter. Analytical findings are supported using MATLAB simulation.

Complex Dynamics of a Three-Species Food Chain Model with Fear and Allee Effect

In ecology, predator–prey interactions are very complex in nature. Apart from direct killing, predator induces fear among their prey which affects the life-history, behavioral changes, and reproduction potential of the prey population. On the other hand, the Allee effect has a great impact on regulating the population size, community structure and population dynamics. In the present investigation, we modify the Hastings–Powell (HP) [1991] model by considering the cost of fear in middle predator and the Allee effect in top predator. The stability conditions for the biologically feasible equilibria are derived using linear stability analysis. Considering the cost of fear and the Allee effect as key parameters, the Hopf bifurcation analysis is carried out around the interior equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Our numerical results suggest that the fear effect can stabilize the system. It is observed that high levels of fear among middle predator decrease the population density of top predator. We also observe that if the Allee parameter is increased, then the system becomes stable from chaotic oscillations. However, further increase in the Allee parameter leads to population extinction. We have also drawn several one- and two-parameter bifurcation diagrams which explore rich dynamical behaviors.

Hopf-Bifurcation and Pattern Selections in a Three Trophic Level Food Web System

The ecosystem comprises many food webs, and their existence is directly dependent upon the growth rate of primary prey; it balances the whole ecosystem. This paper studies the temporal and spatiotemporal dynamics of three trophic levels of food web system, consisting of two preys and two predators. We first obtained an equilibrium solution set and studied the system’s stability at a biological feasible equilibrium point using a Jacobian method. We show the occurrence of Hopf-bifurcation by considering the growth rate of prey as the bifurcation parameter for the temporal model. In the presence of diffusion, we study random movement in species, establish conditions for the system’s stability, and derive the Turing instability condition. A multiple-scale analysis is used to determine the amplitude equations in the neighborhood of the Turing bifurcation point. After applying amplitude equations, the system has a rich dynamical behavior. The stability analysis of these amplitude equations leads to the development of various Turing patterns. Finally, with numerical simulations, the analytical results are verified. Within this framework, our study through the dynamical behavior of the complex system and bifurcation point based on the prey growth rate can serve as a baseline for numerous researchers working on ecological models from diverse perspectives. As a result, the Hopf-bifurcation and multiple-scale analysis used in the complex food web system is particularly relevant experimentally because the linked consequences may be researched and applied to many mathematical, ecological, and biological models.

Discretization, Bifurcation, and Control for a Class of Predator-Prey Interactions

The present study focuses on the dynamical aspects of a discrete-time Leslie-Gower predator-prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark-Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings.

Dynamics of SIS-epidemic model with competition involving fractional-order derivative with power-law kernel

Infectious disease and competition play important roles in the dynamics of a population due to their capability to increase the mortality rate for each organism. In this paper, the dynamical behaviors of a single species population are studied by considering the existence of the infectious disease, intraspecific competition, and interspecific competition. The fractional-order derivative with a power-law kernel is utilized to involve the impact of the memory effect. The population is divided into two compartments namely the susceptible class and the infected class. The existence, uniqueness, non-negativity, and boundedness of the solution are investigated to confirm the biological validity. Three types of feasible equilibrium points are identified namely the origin, the disease-free, and the endemic points. All biological conditions which present the local and global stability are investigated. The global sensitivity analysis is given to investigate the most influential parameter to the basic reproduction number and the density of each class. Some numerical simulations including bifurcation diagrams and time series are also portrayed to explore more the dynamical behaviors.

Dynamical behavior of a coupling SEIR epidemic model with transmission in body and vitro, incubation and environmental effects.

In this paper, a coupling SEIR epidemic model is proposed to characterize the interaction of virus spread in the body of hosts and between hosts with environmentally-driven infection, humoral immunity and incubation of disease. The threshold criteria on the local (or global) stability of feasible equilibria with or without antibody response are established. The basic reproduction number $ R_{b0} $ is obtained for the SEIR model without an antibody response, by which we find that the disease-free equilibrium is locally asymptotically stable if $ R_{b0} < 1 $. Two endemic equilibria exist if $ R_{b0} < 1 $, in which one is locally asymptotically stable under some additional conditions but the other is unstable, which means there is backward bifurcation. In addition, the uniform persistence of this model is discussed. For the SEIR model with an antibody response, the basic reproduction number $ R_{0} $ is calculated, from which the disease-free equilibrium is globally asymptotically stable if $ R_0\leq1 $, and the unique endemic equilibrium is globally asymptotically stable if $ R_0 > 1 $. Antibody immunity in the host plays a great role in the control of disease transmission, especially when the diseases between the hosts are entirely extinct once antibody cells in the host reach a proper level. Finally, the main conclusions are illustrated by some special examples and numerical simulations.