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The complex nonlinear Fokas–Lenells (FL) equation to obtain the explicit travelling wave solutions under different values parameters such as solitary wave solutions, kink solitary wave solutions and periodic wave solutions and others solutions are studied using the dynamical system approach. Also, the optical solitons for the FL equation, as well as the plane-wave, complex dark-singular, and complex periodic-singular solutions are obtained via the e x p (- Φ (ζ)) expansion method. In conclusion, graphical representations of these solutions are provided so that the dynamics of these waves can be viewed. [ABSTRACT FROM AUTHOR]

The investigation revolved around the study of the time fractional Gardner equation, which was examined in terms of the conformable derivative. The reduction of the Gardner equation to an integer order nonlinear ordinary differential equation was carried out, and subsequently, the resulting equations were solved using the Jacobi elliptic function method. The construction of exact solutions, including solitary wave, periodic, and shock wave solutions, for the fractional order of the Gardner equation was performed. A comparison between the exact solutions and the fractional solutions was presented. This work is important because the suggested technique offers a simple and efficient way to examine a wide range of nonlinear fractional differential equations. By employing this approach, it becomes possible to solve several nonlinear time-fractional differential equations that involve conformable derivatives. The graphical representation of the resulting data simplifies the process of determining the physical significance of the equation. [ABSTRACT FROM AUTHOR]

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and 1 : 2 , 1 : 3 , 1 : 4 strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM. [ABSTRACT FROM AUTHOR]

In this paper, we use new methods to investigate different bifurcations of fixed points in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect. The nonstandard discretization scheme produces a discrete analog of the continuous-time toxic-phytoplankton–zooplankton model with Allee effect. The local stability for proposed system around all of its fixed points is derived. We obtain the codimension-1 conditions of various bifurcations such as period doubling and Neimark–Sacker. Moreover, the system produces codimension-2 bifurcations such as resonance 1:1, 1:2, 1:3, and 1:4. Furthermore, the system can produce very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexisting periodic orbits, and chaotic behavior. Theoretical analysis is validated by numerical methods. [ABSTRACT FROM AUTHOR]

The development of reliable and efficient techniques for protecting modern communications systems becomes a very active research point which stimulates major challenges in the twenty-first century. Several encryption schemes, including chaos-based encryption systems, have been proposed for this purpose. In this paper, we design a new hybrid technique based on asymmetrically clipped optical orthogonal frequency-division multiplexing (ACO-OFDM) and discrete memristive chaos for physical layer encryption and watermarking (PL-EW). The proposed technique is secured in the underlying physical layer of visible light communication (VLC) systems. The hybrid PL-EW ACO-OFDM technique uses two-stage encryption process to modulate both order and values of pixels in the plain image. Moreover, the introduced hybrid technique achieves the authentication property via employing two-dimensional discrete wavelet transform for the digital image watermarking process in encrypted images. A new suggested two-dimensional hyperchaotic modular cascaded memristive (2D HC-MCM) map is proposed and employed as an enhanced chaos generator capable of overcoming the observed deficiencies in similar discrete chaos systems. The proposed 2D HC-MCM map extends the secret space of original memristive map by a factor of 100. The positive Lyapunov exponents (LEs) of 2D HC-MCM map are 10 times greater than the corresponding LEs in conventional memristive map, which indicates that the complexity of chaotic attractors is boosted. The 15 statistical tests of NIST SP 800-22 are applied to examine the randomness quality of the generated chaotic sequences. The bit error rate (BER) of the proposed secure system is studied over the dispersive VLC channel. The results show that all statistical tests are passed for wide range of secret parameters, and a 10−4 BER is achievable at SNR = 18 dB. The best extracted digital watermark image with an ideal NCC can be achieved at SNR = 24 dB. Finally, a plethora of security attacks is applied to verify the immunity of hybrid PL-CEW ACO-OFDM technique against possible types of attacks. It is shown that the proposed scheme can resist brute force, differential, statistical, and different signal processing attacks. [ABSTRACT FROM AUTHOR]

Forecasting the dynamical behaviors of nonlinear systems over long time intervals represents a great challenge for scientists and has become a very active area of research. The employment of the well-known artificial recurrent neural networks (RNNs)-based models requires a high computational cost, and they usually maintain adequate accuracy for complicated dynamics over short intervals only. In this work, an efficient reservoir-computing (RC) approach is presented to predict the time evolution of the complicated dynamics of a fractional order hyperchaotic finance model. Compared with the well-known deep learning techniques, the suggested RC-based forecasting model is faster, more accurate for long-time prediction, and has a smaller execution time. Numerical schemes for fractional order systems are generally time-consuming. The second goal of the present study is to introduce a faster, more efficient, and simpler simulator to the fractional order chaotic/hyperchaotic systems. The RC model is utilized in a proposed RC-based digital image encryption scheme. Security analysis is carried out to verify the performance of the proposed encryption scheme against different types of statistical, KPA, brute-force, CCA, and differential attacks. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map's fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map's parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work. [ABSTRACT FROM AUTHOR]

This work is devoted to present a generalized complex discrete fractional Gaussian map. Analytical and numerical analyses of the proposed map are conducted. The dynamical behaviors and stability of fixed points of the map are explored. The existence of fractal Mandelbrot and Julia sets is examined along with the corresponding fractal characteristics. The influences of the key parameters of the map and fractional order are examined. Moreover, nonlinear controllers are designed in the complex domain to control Julia sets generated by the map or to achieve synchronization between two Julia sets in master/slave configuration. Numerical simulations are provided to attain a deep understanding of nonlinear behaviors of the proposed map. Then, a suggested efficient chaos-based encryption technique is introduced by integrating the complicated dynamical behavior and fractal sets of the proposed map with the pseudo-chaos generated from the modified lemniscate hyperchaotic map. [ABSTRACT FROM AUTHOR]

A discreet fractional-order Cournot–Bertrand competition duopoly game is introduced based on the fractional-order difference calculus of the Caputo operator. The model is designed when players can make long memory decisions. The local stability of equilibrium points is discussed for the proposed model. Some numerical simulations explore the model's bifurcation and chaos by employing bifurcation diagrams, phase portraits, maximal Lyapunov exponents, and time series. According to our findings, the fractional-order parameter has an effect on the game's stability and dynamics. [ABSTRACT FROM AUTHOR]

This work is devoted to explore the dynamics of the proposed discrete fractional-order prey–predator model. The model is the generalization of the conventional discrete prey–predator model to its corresponding fractional-order counterpart. The fixed points of the proposed model are first found and their stability analyses are carried out. Then, the nonlinear dynamical behaviors of the model, including quasi-periodicity and chaotic behaviors, are investigated. The influences of fractional order and different parameters in the model are examined using several techniques such as Lyapunov exponents, bifurcation diagrams, phase portraits and C 0 complexity. The feedback control method is suggested to suppress the chaotic dynamics of the model and stabilize any selected unstable fixed point of the system. [ABSTRACT FROM AUTHOR]

A field experiment was carried out during the two successive summer seasons of 2017and 2018 in private sector farm at Kom Hamada city, Buhera Governorate, Egypt to investigate the effect of organic fertilizers tea (Chicken manure tea, Rabbit manure tea, Compost tea, Biochar tea and without organic fertilizers tea) as soil additions and foliar spray with boron at 0, 50 and 100 mg L-1 as well as their interaction on growth, yield and quality of potato tubers (Solanum tuberosum L.) cv. Beleny. Obtained results showed that, soil addition with chicken manure tea and foliar spray with high concentration of boron (100 mg L-1) gave the highest values in vegetative growth parameters. Moreover, application of either chicken manure or rabbit manure tea with the recommended rate of NPK and foliar spray with boron at 100 mg L-1 recorded the highest increases in the tuber yield of potato. But the best chemical components of potato tubers (N, P, K, carbohydrates and starch) verified when using chicken manure tea and foliar spray with boron at 100 mg L-1 compared with all other treatments and in both seasons. [ABSTRACT FROM AUTHOR]

In this paper, we study the competition between two firms whose outputs are quantities. The first firm considers maximization of its profit while the second firm considers maximization of its social welfare. Adopting a gradient-based mechanism, we introduce a nonlinear discrete dynamic map which is used to describe the dynamics of this game. For this map, the fixed points are calculated and their stability conditions are analyzed. This includes investigating some attracting set and chaotic behaviors for the complex dynamics of the map. We have also investigated the types of the preimages that characterize the phase plane of the map and conclude that the game's map is noninvertible of type Z 4 − Z 2 . [ABSTRACT FROM AUTHOR]

The study focuses on the impact of foliar spraying cyanobacterium Spirulina platensis extract and the inoculation with the endophyte N2-fixing Pseudomonas stutzeri, and their mixture in the presence of different nitrogen doses on growth and yield of onion under field conditions. Bioactive compounds of Spirulina and Pseudomonas were analyzed by GC–MC and amino acid production of Spirulina by the amino acid analyzer. Hydrogen cyanide (HCN), indole acetic acid (IAA), ammonia (NH3), pectinase activity, and N2-fixation of Pseudomonas were measured. Plant height (cm), leaf length (cm), number of green leaves, bulb diameter (cm), fresh and dry weight of plant (g), chlorophyll a, b of leaves, bulb weight (g), marketable bulb yield (t. ha−1), cull bulb weight (t. ha−1), total bulb yield (t. ha−1), bulb diameter (cm), total soluble solids (TSS%), dry matter content (DM%), evaluation of storage behavior, and economic feasibility were estimated. Spirulina extract has several bioactive compounds. Pseudomonas can produce HCN, NH3, IAA, pectinase, and nitrogen fixation. The application of mixture with recommended dose of nitrogen increases the onion plant parameters, marketable yield, total bulb yield, bulb weight, bulb diameter, TSS%, DM%, net return, benefit–cost ratio (B:C), lowest cumulative weight loss% of bulbs during storage, and reduce culls weight compared with other treatments in two seasons. Application of S. platensis extract and inoculation with endophyte nitrogen-fixing P. stutzeri enhance the growth and productivity of the onion under different doses of nitrogen fertilizer. [ABSTRACT FROM AUTHOR]

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter. [ABSTRACT FROM AUTHOR]

This paper investigates the difference between price and quantity competition in a mixed duopoly game. We describe the behavior of a duopolistic Bertrand competition market with environmental taxes. There are two cases. In the first, the public firm is privatized and in the second, it is not privatized. In case I, private duopoly (postprivatization) where players use different production methods and choose their prices with (bounded rationality and naive). In case II, mixed duopoly (preprivatization) in this case there are two levels for the market including standard objective of the private firm is to maximize profits and including another objective function of the public firm namely "private welfare maximization". We study numerically the dynamical behaviors of the models. The Nash equilibrium loses stability through a period-doubling bifurcation and the market in the end gets to be disordered. The disordered behavior of the market has been controlled by using feedback control method. [ABSTRACT FROM AUTHOR]

In this work, we extend further the analytical study of complex dynamics exist in two coupled logistic maps. New results about the occurrence of various types of bifurcation in the system, including flip bifurcation, pitchfork bifurcation and Neimark–Sacker bifurcation are presented. To the best of authors’ knowledge, the presence of chaotic dynamics in system’s behavior has been investigated and proved analytically via Marotto’s approach for first time. Numerical simulations are carried out in order to verify theoretical results. Furthermore, chaos based encryption algorithm for images is presented as an application for the coupled logistic maps. Different scenarios of attacks are considered to demonstrate its immunity and effectiveness against the possible attacks. Finally, a circuit realization for the coupled logistic maps is proposed and utilized in a suggested real time text encryption system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a host–parasitoid model with a Holling type-II functional response and incorporate harvest effort. The Holling type-II response leads to saturation in parasitized hosts, creating a potential economic harvesting opportunity. To address overexploitation risks, we integrate a harvest effort function, determining an optimal threshold to prevent depletion. We explore model dynamics and bifurcations, including co-dimension one behaviors such as flip and Neimark–Sacker bifurcations, we provide numerical examples for validation. Our suggested difference-algebraic model, compared to continuous-time models, exhibits rich dynamics within the Nicholson–Bailey host–parasitoid framework. [ABSTRACT FROM AUTHOR]

Faba bean (Vicia faba L.) is the main source of plant proteins in Egypt. This crop is attacked by numerous of pests as mites, Aphids, white fly and leaf hoppers. Unfortunately these pests contribute to transmit viral diseases . The experiment was carried out Sakha Agricultural Research Station, Kafr El-Sheikh Governorate and conducted in 2013/2014 season to determine the age distribution patterns of pests and predators as a method for predicting the reproductive capabilities and probability of the continuation of arthropods species in the future .In addition to the relationships between phenols leaf content and occurrence of arthropods was studied. Populations of sucking pests and their natural enemies on leaflets of different cultivars of faba bean; Giza 716, Sakha 3 and Giza 40 were recorded. The age distribution patterns of the above mentioned arthropods were represented by three patterns are expanding population, decline population and stationary population. Decline age distribution was found for phytophagous mite; Tetranychus urticae Koch on Giza 716 cultivar which means that the mite population is decreasing . The population of age distribution of Aphis Giza 716 spp. stages appeared as expanding population on the three faba bean cultivars. The population of age distribution of Empoasca sp. was expanding on Giza 716 and Giza 40 cultivars of faba bean, but it declining on Sakha 3 cultivar. However the three age patterns were found for Chrysoperla carnea Steph., expanding for Aphidoletes aphidimyza (Rond.). Data revealed that the correlation between total phenols in faba bean leaflets were significantly positive only in Sakha 3 cultivar. [ABSTRACT FROM AUTHOR]

This work is related to dynamics of a second-order rational difference equation. We investigate the parametric conditions for local asymptotic stability of equilibria. Center manifold theorem and bifurcation theory are implemented to discuss the parametric conditions for existence and direction of period-doubling bifurcation and pitchfork bifurcation at trivial equilibrium point. Moreover, the parametric conditions for existence and direction of Neimark–Sacker bifurcation at positive steady state are investigated with the help of bifurcation theory. The chaos control in the system is discussed through implementation of OGY feedback control method. In particular, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control. Finally, numerical simulations are provided to illustrate theoretical results. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to address the codimension-one bifurcation of Marotto’s map and its utility in image encryption. First of all, local stability analysis and local bifurcation analysis of fixed points of the considered map are investigated in details. According to the classical bifurcation theory and the center manifold theorem, the map exhibits various bifurcation types such as transcritical, flip and Neimark–Sacker bifurcations. Second of all, the map is proven to be chaotic in the sense of Marotto. Since image encryption based on chaotic maps is very promising for cryptography, Marotto’s map, compound chaos, and spatiotemporal chaos are combined to encrypt and decrypt images. Numerical simulations agree with the analytical framework for the complex dynamics of the map. Furthermore, different test images are used to demonstrate the effectiveness of the method implemented for encryption. [ABSTRACT FROM AUTHOR]

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model. [ABSTRACT FROM AUTHOR]

The dynamics of a Cournot duopoly with relative profits maximizations and costs function with externalities is considered. Results concerning the equilibria of the economic model and their stability are presented and the occurrence of bifurcations is stated. A double route to chaotic dynamics, via flip bifurcations and via Neimark–Sacker bifurcations for game is studied. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

We are delighted to announce the publication of this Special Issue devoted to fresh problems in discrete fractional calculus (DFC) and its applications in engineering and natural sciences. Conflicts of Interest The Guest Editors declare that they have no conflicts of interest regarding the publication of this Special Issue. Discrete fractional calculus (DFC) research is gaining a lot of attention, both theoretically and practically. [Extracted from the article]

In this paper we try to solve a paradox related to the results of Theocharis (1960). When the number of competitors increases the Cournot–Nash equilibrium loses stability. We relax the assumption about homogeneity in the decision mechanism and show that if we admit heterogeneity than by increasing the number of competitors the stability region on the parameters’ space may enlarge instead of shrinking. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we discuss the stability conditions for a nonlinear fractional-order hyperchaotic system. The fractional-order hyperchaotic Novel and Chen systems are introduced. The existence and uniqueness of solutions for two classes of fractional-order hyperchaotic Novel and Chen systems are investigated. On the basis of the stability conditions for nonlinear fractional-order hyperchaotic systems, we study synchronization between the proposed systems by using a new nonlinear control technique. The states of the fractional-order hyperchaotic Novel system are used to control the states of the fractional-order hyperchaotic Chen system. Numerical simulations are used to show the effectiveness of the proposed synchronization scheme. [Copyright &y& Elsevier]

Highlights: [•] The dynamical behaviors of a discrete Lorenz system are investigated. [•] The stability conditions of the fixed points are analyzed. [•] The normal form theorem is applied to investigate dynamics of the system. [•] Numerical simulations are presented to verify the theoretical results. [Copyright &y& Elsevier]

Abstract: The dynamical system of four heterogeneous firms is derived. Existence and stability conditions of the fixed points are investigated and also complex dynamics is studied. Numerical simulations are used to illustrate the complex behaviors of the proposed dynamic game. The chaotic behavior of the game has been controlled by using feedback control method. [Copyright &y& Elsevier]

This work presents two forms of coupled fractional-order memristor chaotic systems. The existence and uniqueness of solutions are studied. Moreover, the range of parameters and time span at which the proposed two models exhibit continuous dependence on initial conditions are examined. The unique equilibrium point for each system is found, and the corresponding stability analysis is carried out. The regions of stability in the space of parameters are obtained, whereas numerical simulations are employed to confirm theoretical results. The bifurcation diagrams, in addition to Lyapunov exponents, are utilized to examine the effects of key parameters in two models. A chaos-based encryption scheme is presented as an application to utilize complicated chaotic behaviors in coupled circuits. [ABSTRACT FROM AUTHOR]

Abstract: A bounded rationality duopoly game with delay is formulated. Its dynamical evolution is analyzed. The existence of an economic equilibrium of the game is derived. The local stability analysis has been carried out. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. Numerical simulations were used to show bifurcation diagrams and phase portraits. [Copyright &y& Elsevier]

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have heterogeneous expectations. Two players with different expectations are considered; one is boundedly rational and the other thinks with adaptive expectations. The stability conditions of the equilibria are discussed. We show how the dynamics of the game depend on the model parameters. We demonstrate that as some parameters of the game are varied, the stability of Nash equilibrium is lost through period doubling bifurcation. The chaotic features are justified numerically via computing Lyapunov exponents, sensitive dependence on initial conditions and the fractal dimension. [Copyright &y& Elsevier]

We analyze a nonlinear discrete-time Cournot duopoly game, where players have heterogeneous expectations. Two types of players are considered: boundedly rational and naive expectations. In this study we show that the dynamics of the duopoly game with players whose beliefs are heterogeneous, may become complicated. The model gives more complex chaotic and unpredictable trajectories as a consequence of increasing the speed of adjustment of boundedly rational player. The equilibrium points and local stability of the duopoly game are investigated. As some parameters of the model are varied, the stability of the Nash equilibrium point is lost and the complex (periodic or chaotic) behavior occurs. Numerical simulations are presented to show that players with heterogeneous beliefs make the duopoly game behave chaotically. Also, we get the fractal dimension of the chaotic attractor of our map which is equivalent to the dimension of Henon map. [Copyright &y& Elsevier]

This work introduces a discrete modified Leslie–Gower prey–predator system with Holling type-II functional response. The persistence of the discrete model under certain conditions is discussed. The conditions assuring the existence of fixed points are derived and nonlinear dynamics of system are explored at these fixed points. It has been shown that the system exhibits transcritical bifurcation and flip bifurcation at semi-trivial fixed point under certain bifurcation values. In addition, the center manifold and bifurcation theories are employed to attain the conditions for existence of flip and Neimark–Sacker bifurcations at coexistence fixed point. The system is found to exhibit periodic solutions along with bifurcations leading to wide range of chaotic dynamics. The numerical simulations are performed to confirm the analytical analysis. [ABSTRACT FROM AUTHOR]

Background: Tomato mosaic virus (ToMV) is a dangerous disease of tomato (Lycopersicon esculentum) that reduces dramatically the yield. To reduce ToMV infection, microalgal isolates were utilized. Microalgal species (Chlorella vulgaris, Anabaena oryzae, Spirulina platensis, Nostoc linckia and Dunaliella salina) were shown to be responsible for the stimulation of tomato resistance against ToMV. Results: Initial signs of discoloration and mosaic in ToMV-inoculated plants were detected and identified on inoculated leaves at 6 and 12 dpi in control and treated plants, respectively, suggesting that microalgae may inhibit ToMV growth. Treatment with microalgae resulted in a significant decrease in symptoms (up to 63% reduction in disease severity) and negative ELISA readings, indicating that the microalgae induced resistance in tomato against ToMV infection. The isolates also enhanced the activity of pathogenesis-related enzymes (PPO and POX reaching to 0.033 and 0.054 in D. salina, respectively), as well as tomato growth characters in comparison with the control. Microalgal treatments demonstrated that the salicylic acid (SA) and jasmonic acid (JA) pathways were involved in tomato plant defense responses. The relative gene expressions of PR1 and phenylalanine ammonia lyase (PAL), which are involved in the SA and JA pathways, respectively, were improved in treated plants compared to the control. Conclusion: The findings indicated that algal-induced ToMV resistance was mediated via several defense pathways in tomato. The antiviral mechanism was described, which provides a light on the potential of algae in plant viral disease management. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to discuss stability and bifurcation analysis for a class of discrete predator-prey interaction with Holling type II functional response and harvesting effort. Firstly, we establish a discrete singular bioeconomic system, which is based on the discretization of a system of differential algebraic equations. It is shown that the discretized system exhibits much richer dynamical behaviors than its corresponding continuous counterpart. Our investigation reveals that, in the discretized system, two types of bifurcations (i.e., period-doubling and Neimark–Sacker bifurcations) can be studied; however, the dynamics of the continuous model includes only Hopf bifurcation. Moreover, the state delayed feedback control method is implemented for controlling the chaotic behavior of the bioeconomic model. Numerical simulations are presented to illustrate the theoretical analysis. The maximal Lyapunov exponents (MLE) are computed numerically to ensure further dynamical behaviors and complexity of the model. [ABSTRACT FROM AUTHOR]

This paper focusses on developing two species, where only prey species suffers by a contagious disease. We consider the logistic growth rate of the prey population. The interaction between susceptible prey and infected prey with predator is presumed to be ruled by Holling type II and I functional response, respectively. A healthy prey is infected when it comes in direct contact with infected prey, and we also assume that predator-dependent disease spreads within the system. This research reveals that the transmission of this predator-dependent disease can have critical repercussions for the shaping of prey–predator interactions. The solution of the model is examined in relation to survival, uniqueness and boundedness. The positivity, feasibility and the stability conditions of the fixed points of the system are analysed by applying the linearization method and the Jacobian matrix method. [ABSTRACT FROM AUTHOR]

This study introduces a newly modified Lorenz model capable of demonstrating bifurcation within a specified range of parameters. The model demonstrates various bifurcation behaviors, which are depicted as distinct structures in the diagram. The study aims to discover and analyze the existence and stability of fixed points in the model. To achieve this, the center manifold theorem and bifurcation theory are employed to identify the requirements for pitchfork bifurcation, period-doubling bifurcation, and Neimark–Sacker bifurcation. In addition to theoretical findings, numerical simulations, including bifurcation diagrams, phase pictures, and maximum Lyapunov exponents, showcase the nuanced, complex, and diverse dynamics. Finally, the study applies the Ott–Grebogi–Yorke (OGY) method to control the chaos observed in the reduced modified Lorenz model. [ABSTRACT FROM AUTHOR]

In this study, a mathematical Omicron model has been studied in view of the densities of a susceptible population without a vaccine, with the first dose of vaccine, with the second dose of vaccine and with a booster dose of vaccine respectively, at the time t. The impact of the Omicron variant virus reservoir has been analysed. The basic reproduction number ( R 0 ) of the system at the disease-free equilibrium for all positive parameters has been discussed. An analysis of local and global stability under various conditions at disease-free and endemic equilibrium points has been conducted. The sensitivity analysis is performed and the most sensitive parameter of the system is identified. On the basis of the effect of various parameters on the system, we perform numerical simulation of the Omicron variant system in Indian population. [ABSTRACT FROM AUTHOR]

This paper presents a discrete predator–prey interaction model involving hibernating vertebrates, with detailed analysis and simulation. Hibernation contributes to the survival and reproduction of organisms and species in the ecosystem as a whole. In addition, it also constitutes a wise sharing of time, space, and resources with others. We have created a new predator–prey model by integrating the two species, Holling-III and Holling-I, which have a bifurcation within a specified parameter range. We discovered that this system possesses the stability of fixed points as well as several bifurcation behaviors. To accomplish this, the center manifold theorem and bifurcation theory are applied to create existence conditions for period-doubling bifurcations and Neimark–Sacker bifurcations, which are depicted in the graph as distinct structures. Examples of numerical simulations include bifurcation diagrams, maximum Lyapunov exponents, and phase portraits, which demonstrate not just the validity of theoretical analysis but also complex dynamical behaviors and biological processes. Finally, the Ott–Grebogi–Yorke (OGY) method and phases of chaos control bifurcation were used to control the chaos of predator–prey model in hibernating vertebrates. [ABSTRACT FROM AUTHOR]

We consider the model of interaction of predator and prey with omnivore using three different waiting time distributions. The first waiting time is induced by the power law distribution which is the generator of Pareto statistics. The second waiting time is induced by exponential decay law with a particular property of Delta Dirac distribution when the fractional order tends to 1, this distribution is link to the Poison distribution. While the last waiting distribution, induced by the Mittag-Leffler distribution, presents a crossover from exponential to power law. For each model, we presented the conditions under which the existence of unique set of exact solutions is reached using the fixed-point Picard's method. Making use of a recent suggested numerical scheme, we solved each model numerically and some numerical simulations were generated for different values of fractional orders. We noticed a new attractor which can be considered as a combination of the Brownian motion and power law distribution in the model with the Atangana-Baleanu fractional derivative. With the aim to capture more attractors, we modified the model and presented also some numerical simulations. Our new model provides more attractors than the existing one even for fractional differential cases. We presented finally the Maximal Lyapunov exponent and the bifurcation diagrams. The comparative study shows that modeling with non-local and non-singular kernel fractional derivative leads to more attractors as this kernel is able to capture more physical problems. [ABSTRACT FROM AUTHOR]