Nonmonotonic Functional Response Research Articles

Competitive exclusion and coexistence in a general two-species stochastic chemostat model with Markov switching

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results.

Dynamic analysis of a delayed pest-natural enemy model: Triple effects of non-monotonic functional response, additional food supply and habitat complexity

Considering the food diversity of natural enemy species and the habitat complexity of prey populations, a pest-natural enemy model with non-monotonic functional response is proposed for biological management of Bemisia tabaci. The dynamic characteristics of the proposed model are analyzed. In addition, considering that the conversion from prey to predator has a certain time lag rather than instantaneous, a time delay is introduced into this model, and it is shown that the Hopf bifurcation occurs at the interior equilibrium when the time delay is used as the bifurcation parameter. Furthermore, the values of the parameters that determine the direction of the Hopf bifurcation as well as the stability of the periodic solution are calculated. In order to illustrate the theoretical analysis results, numerical simulations and validation are carried out to demonstrate the effects of non-monotonic functional response, additional food supply and habitat complexity.

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Tumor–immune system interactions are very complicated, being highly nonlinear and not well understood. A large number of tumors can potentially weaken the immune system through various mechanisms such as secreting cytokines that suppress the immune response. In this paper, we propose a tumor–immune system interaction model with a nonmonotonic immune response function and adoptive cellular immunotherapy (ACI). The model has a tumor-free equilibrium and at most three tumor-presence equilibria (low, moderate and high ones). The stability of all equilibria is studied by analyzing their characteristic equations. The consideration of nonmonotonic immune response results in a series of bifurcations such as the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. In addition, numerical simulation results show the coexistence of periodic orbits and homoclinic orbits. Interestingly, along with various bifurcations, we also found two bistable scenarios: the coexistence of a stable tumor-free as well as a high-tumor-presence equilibrium and the coexistence of a stable-low as well as a high-tumor-presence equilibrium, which can show symmetric and antisymmetric properties in a range of model parameters and initial cell concentrations. The new findings indicate that under ACI, patients can possibly reach either a stable tumor-free state or a low-tumor-presence state in the presence of nonmonotonic immune response once the immune system is activated.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

Impulsive Effects and Complexity Dynamics in the Anti-Predator Model with IPM Strategies

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k (k≥2) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations.

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.

Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model

<p>A delayed diffusive predator-prey system with nonmonotonic functional response subject to Neumann boundary conditions is introduced in this paper. First, we analyze the associated characteristic equation to research the conditions for local stability of the positive equilibrium point and the occurrence of Turing instability induced by diffusion in the absence of delay. Second, we provide conditions for the existence of Hopf bifurcation driven by time delay. By utilizing the normal theory and center manifold theorem, we derive explicit formulas for Hopf bifurcation properties such as direction and stability from the positive equilibrium. Third, a hybrid controller is added to the system. By judiciously adjusting the control parameters, we effectively enhance the stability domain of the system, resulting in a modification of the position of the Hopf bifurcation periodic solutions. Numerical simulations demonstrate the presence of rich dynamical phenomena within the system. Moreover, sensitivity analysis was conducted using Latin hypercube sampling (LHS)/partial rank correlation coefficient (PRCC) to explore the impact of parameter variations on the output of prey and predator populations.</p>

Dynamics of a Leslie-Gower type predation model with non-monotonic functional response and weak Allee effect on prey

This research concerns with analysis of a class of modified predator- prey type Leslie-Gower models. The model is described by an autonomous nonlinear ordinary differential equation system. The functional response of predators is Holling IV type or non-monotone, and the growth of prey is affected by the Allee effect. An important aspect is the study of the point (0, 0) since it has a strong influence on the behavior of the system being essential for the existence and extinction of both species, although the proposed system is not define there.

Corona Phase Molecular Recognition of the Interleukin-6 (IL-6) Family of Cytokines Using nIR Fluorescent Single-Walled Carbon Nanotubes

The Interleukin-6 (IL-6) family of cytokines regulates inflammation and plays important roles in numerous biochemical pathways. Typically, cytokine levels are measured using enzyme-linked immunosorbent assay (ELISA) or western blot. However, these techniques usually require substantial processing time, cost, machinery, and specialist training. Understanding the fundamental molecular recognition mechanism of cytokines with synthetic substrates is key to developing new biomedical technologies such as assays, sensors, and therapeutics that overcome the above limitations. Herein, we use the corona phase molecular recognition (CoPhMoRe) approach to engineer new carbon nanotube constructs and study their binding to the inflammatory cytokines: IL-6, interleukin-11 (IL-11), ciliary neurotrophic factor (CNTF), and leukemia inhibitory factor (LIF). Library screening identified two polymer-based CoPhMoRe constructs consisting of single-walled carbon nanotubes complexed with p(AA68-rand-BA16-rand-CD16) polymer (MK2) or p(SS80-rand-BS20) polymer (P14) corona phases. The resulting dissociation constants (KD) were 8.38 ng/mL and 16.7 μg/mL, respectively, compared to that of the natural IL-6 receptor at ∼0.32 ng/mL. In addition, the MK2 constructs showed a nonmonotonic response function upon binding with IL-6. Comparative binding experiments suggest that both constructs appear to recognize the axially aligned α-helical structures present in the Interleukin-6 family. The findings from this study elucidate how nanoparticle interfaces, such as those produced by CoPhMoRe, can be designed to lock onto specific protein features. We find that the α-helical structure of the IL-6 family of cytokines can enable facile molecular recognition, opening the door to new types of label-free, low-cost sensing technologies.

Qualitative structure of a discrete predator-prey model with nonmonotonic functional response

<p style='text-indent:20px;'>In this paper, we study the qualitative structure of a discrete predator-prey model with nonmonotonic functional response near a degenerate fixed point whose eigenvalues are <inline-formula><tex-math id="M2">\begin{document}$ \pm1 $\end{document}</tex-math></inline-formula>. Firstly, the model is transformed into an ordinary differential system by using the normal form theory and the Takens's theorem. Then, the qualitative properties of this ordinary differential system near the degenerate equilibrium are analyzed with the blowing-up method. Finally, according to the conjugacy between the discrete model and the time-one mapping of the vector field, the qualitative structure of this discrete model is obtained. Numerical simulations are also given.</p>

BIFURCATION AND CHAOS ANALYSIS OF A TWO-DIMENSIONAL DISCRETE-TIME PREDATOR–PREY MODEL

The dynamical behavior of a discrete predator-prey system with a nonmonotonic functional response is investigated in this work. We study the local asymptotic stability of the positive equilibrium of the system by examining the characteristic equation of the linearized system corresponding to the model. By choosing the growth rate as a bifurcation parameter, the existence of Neimark-Sacker and period-doubling bifurcations at the positive equilibrium is established. Furthermore, the effects of perturbations on the system dynamics are investigated. Finally, examples are presented to illustrate our main results.

Mathematical model of evasion of immune system by virus

Virus needs to infect cells to spread the disease in the host and to achieve this, they have developed specific tactics to evade the immune system, which is in charge of trying to prevent any infection. In this way, we develop a mathematical model to represent the evasion of immune system by virus using a non-monotonic functional response describing an antipredator behavior, where the virus is the prey and the immune cells are the predator. We found four equilibrium points, the disease free equilibrium, immune evasion equilibrium and two immune activation equilibrium points. The disease free equilibrium is globally asymptotically stable if R0 ≤ 1, the immune evasion equilibrium and two immune activation equilibria are locally asymptotically stable if R0 >1. We conclude that in our model the evasion of immune system is always possible when there is a inoculation of virus in the host, but there is also a chance to control the infection or to activate the immune system when there are a cross-reactive antibodies.

Bifurcation analysis of a two‐dimensional discrete‐time predator–prey model

The dynamical behavior of a discrete predator–prey system with a nonmonotonic functional response is investigated in this work. We study the local asymptotic stability of the positive equilibrium of the system by examining the characteristic equation of the linearized system corresponding to the model. By choosing the growth rate as a bifurcation parameter, the existence of Neimark–Sacker and period‐doubling bifurcations at the positive equilibrium is established. Furthermore, the effects of perturbations on the system dynamics are investigated. Finally, examples are presented to illustrate our main results.

Dynamics analysis of a predator–prey model with nonmonotonic functional response and impulsive control

This paper presents the qualitative analysis of a predator–prey model with nonmonotonic functional response and impulsive effects. Different from previous work, by considering two factors of nonmonotonic functional response and impulsive effects, this paper studies the existence and stability of the periodic semi-trivial solution y=0 first. Then, by constructing an appropriate Poincare map and introducing geometric theory, it is shown that the predator–prey model can exhibit a variety of dynamic phenomena, including orbitally asymptotically stable order-1 periodic solution (O1PS), order-2 periodic solution (O2PS) and globally stable equilibrium point under certain conditions. When there exists an O2PS, its appearance and disappearance as well as the appearance of bifurcation phenomenon are discussed in detail with different selections of the initial value of predator. Finally, numerical simulations illustrate the correctness of the results of the theoretical analysis. The theoretical results presented in this paper can be seen as an advancement to the previous related works.

Mathematical analysis of an ecological system using a non-monotonic functional response: effects of gestation delay and predator harvesting

Mathematical analysis of an ecological system using a non-monotonic functional response: effects of gestation delay and predator harvesting

Dynamics and bifurcations of a Filippov Leslie-Gower predator-prey model with group defense and time delay

In this paper, the Leslie-Gower model with nonmonotonic functional response is extended to a nonsmooth Filippov control system to reflect the integrated pest management. Different from traditional Filippov models, here, we incorporate time delay as to account for predator maturity time. The stability of the equilibria and the existence of Hopf bifurcation of the subsystems are investigated. Moreover, sliding mode dynamics and regular/virtual/pseudoequilibria are analyzed. Numerical simulations indicate that all solutions finally converge to either the regular equilibrium, the pseudoequilibrium or a stable periodic solution according to different values of time delays and threshold levels. A boundary bifurcation that switches a stable regular equilibrium or a stable limit cycle to a stable pseudoequilibrium can occur. Meanwhile, global bifurcations from the standard periodic solution to the sliding switching bifurcation and then to the crossing bifurcation are obtained as time delay is increased. The results show that Filippov control strategies could effectively control the number of pests under the prescribed threshold, however, time delay may challenge pest control by the occurring of the sliding switching and crossing bifurcations.

Dynamics of an eco-epidemiological model with nonmonotonic functional response of susceptible predator on prey species

In this study, a predator–prey population model has been described with disease in the predator species. This is a three-dimensional study comprising of a prey and a predator taken in two different groups such as susceptible and infected predator species. Dynamical behavior of the spread of the disease having the potential to become epidemic has been discussed. Parametric conditions are determined for the control of disease outbreak. Some basic properties like boundedness, persistence of the system have been ensured. Minimal conditions are framed, in such way that the disease can be naturally controlled. Different qualitative behavior like stability, bifurcation and numerical simulations has been performed. Substantial numerical simulations have been carried out in order to validate the obtained theoretical results.

Multiple Periodicity in a Predator–Prey Model with Prey Refuge

We consider a delayed prey–predator model incorporating a refuge with a non-monotone functional response. It is supposed that prey can live in the predatory region and prey refuge, respectively. Based on Mawhin’s coincidence degree and nontrivial estimation techniques for a priori bounds of unknown solutions to the operator equation Lv=λNv, we prove the existence of multiple periodic solutions. Finally, an example demonstrates the feasibility of our main results.

A non-standard finite difference scheme for a class of predator–prey systems with non-monotonic functional response

We design and analyse a non-standard finite difference (NSFD) scheme for a class of predator–prey models with non-monotonic functional response. Such functional response can be used, for instance, to model the prey's group defence ability. The numerical scheme is unconditionally dynamically consistent with respect to the positivity and boundedness of the solutions. We establish analytically the convergence of the scheme and provide numerical simulations for a Holling type-IV functional response that support the theoretical findings. The simulations further show that the NSFD scheme is more efficient than some standard numerical methods. In particular, the standard methods show dynamic inconsistencies when large step-sizes were used. On the other hand, the NSFD scheme replicated different behaviours, including the oscillatory dynamics common to predator–prey systems.

Bifurcation study and pattern formation analysis of a tritrophic food chain model with group defense and Ivlev-like nonmonotonic functional response

In this work, a tritrophic food chain model has been proposed by incorporating Ivlev-like nonmonotonic functional response, where prey is equipped with defense ability. We have performed a detailed dynamical study and pattern formation analysis to obtain complex dynamics of the proposed system. Stability and bifurcation analysis have been performed in the model system. Persistence and permanence are discussed. Bifurcations of codimension-1, in particular, saddle-node, transcritical and Hopf bifurcation are observed. The model system also exhibits bifurcations of codimension-2 such as cusp, Bogdanov-Takens and generalized Hopf bifurcation. Interestingly, it is observed that the middle and top predator population become extinct due to defense ability of prey. Chaotic dynamics is observed via a period-doubling route to chaos with the change in the value of parameter β. The quantification of chaotic dynamics is done, using Lyapunov spectrum and sensitivity analysis. Diffusion induced chaos is studied in the spatiotemporal model system. Hopf bifurcation is seen in the case of a spatially extended system. Further, conditions for Turing instability have been obtained. Pattern formation study is done. In the two-dimensional spatial domain, various non-Turing patterns such as hot-spot, cold-spot, labyrinth patterns are obtained. Ripple and stripe Turing patterns are obtained in case of one-dimensional spatial domain. Also, labyrinth and patchy Turing patterns are obtained in the two-dimensional spatial domain. The spatial distribution of the species shows Turing patterns at the low cost of β, while the increased cost of β changes Turing patterns to non-Turing patterns. Throughout the study, we observe that the parameter β plays an important role in group defense mechanism and is the most sensitive parameter leading to vital change in system dynamics. A wide range of Turing and non-Turing patterns obtained in this work has not been reported so far in literature in any model with group defense.