Dynamic analysis and control of a rice-pest system under transcritical bifurcations

World-wide rice consummation constantly grows constantly with non-proportional yield as large amounts of rice are lost due to pest infestations. Cultural methods were widely applied at an early stage of agricultural pest management but then replaced over time through insecticides. To describe a rice-pest system and to control the corresponding pests applying cultural methods and/or insecticides, statistical analyses have been used, and also other mathematical models using an Integrated Pest Management (IPM) strategy considering long time period and more parametric values. Considering the limitations of IPM, we have developed a mathematical model for a rice-pest system found in agricultural management. The mathematical model consists of two non-linear differential equations (NDEs) to illustrate the interrelation of rice and its corresponding agricultural pests. This model is extended to become an optimal control problem (minimization problem), considering both, cultural methods and pesticides, to minimize the density of agricultural pests and to increase the production of rice, reducing gross annual losses. Pesticides have been applied only in emergencies to reduce environmental pollution and damage to nearby ecosystems such as aquatic ecosystems, and a decision model has also been developed to mitigate potential risks. To compare the effectiveness of the considered controls, the ratio of annual production of rice is studied for both controls and without control. This study contributes to building a relationship between NDEs and agricultural management as well as connecting mathematically rice-pest relationships to global food security.

Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families Information

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis, moreover Algebraic aspects are also included such that hamiltonian cases and Galois differential groupes. It should be noted that these families have associated oscillating type problems given their similarity to the Liénard equations.

Exotic bifurcations in three connected populations with Allee effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) increase by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and find the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis

<p style='text-indent:20px;'>In the present study, a nonlinear model is formulated to demonstrate crop - weed interactions, when they both grow together on agricultural land and compete with each other for the same resources like sunlight, water, nutrients etc., under the aegis of herbicides. The developed model is mathematically analyzed through qualitative theory of differential equations to demonstrate rich dynamical characteristics of the system, which are important to be known for maximizing crop yield. The qualitative results reveal that the system not only exhibits stability of more than one equilibrium states, but also undergoes saddle - node, transcritical and Hopf bifurcations, however, depending on parametric combinations. The results of saddle - node and transcritical bifurcations help to plan strategies for maximum crop yield by putting check over the parameters responsible for the depletion of crops due to their interaction with weeds and herbicides. Hopf - bifurcation shows bifurcation of limit cycle through Hopf - bifurcation threshold, which supports that crop - weed interactions are not always of regular type, but they can also be periodic.</p>

Interspecific density-dependent model of predator–prey relationship in the chemostat

The objective of this study is to analyze a model of competition for one resource in the chemostat with general interspecific density-dependent growth rates, taking into account the predator–prey relationship. This relationship is characterized by the fact that the prey species promotes the growth of the predator species which in turn inhibits the growth of the first species. The model is a three-dimensional system of ordinary differential equations. With the same dilution rates, the model can be reduced to a planar system where the two models have the same local and even global behavior. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. Using the nullcline method, we present a geometric characterization of the existence and stability of all equilibria showing the multiplicity of coexistence steady states. The bifurcation diagrams illustrate that the steady states can appear or disappear only through saddle-node or transcritical bifurcations. Moreover, the operating diagrams describe the asymptotic behavior of this system by varying the control parameters and show the effect of the inhibition of predation on the emergence of the bistability region and the reduction until the disappearance of the coexistence region by increasing this inhibition parameter.

Time-delayed predator–prey interaction with the benefit of antipredation response in presence of refuge

A field experiment on terrestrial vertebrates observes that direct predation on predator–prey interaction can not only affect the population dynamics but the indirect effect of predator’s fear (felt by prey) through chemical and/or vocal cues may also reduce the reproduction of prey and change their life history. In this work, we have described a predator–prey model with Holling type II functional response incorporating prey refuge. Irrespective of being considering either a constant number of prey being refuged or a proportion of the prey population being refuged, a different growth rate and different carrying capacity for the prey population in the refuge area are considered. The total prey population is divided into two subclasses: (i) prey x in the refuge area and (ii) prey y in the predatory area. We have taken the migration of the prey population from refuge area to predatory area. Also, we have considered a benefit from the antipredation response of the prey population y in presence of cost of fear. Feasible equilibrium points of the proposed system are derived, and the dynamical behavior of the system around equilibria is investigated. Birth rate of prey in predatory region has been regarded as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the interior equilibrium point. Moreover, the conditions for occurrence of transcritical bifurcations have been determined. Further, we have incorporated discrete-type gestational delay on the system to make it more realistic. The dynamical behavior of the delayed system is analyzed. Finally, some numerical simulations are given to verify the analytical results.

Complexities and Bifurcations Induced by Drug Responses in a Pulsed Tumour-Immune Model

Responses to drugs play key roles in exploring how drug toxicity affects the evolution of tumour cells. We model pulsed comprehensive therapies using an impulsive tumour-immune model, in which the application of comprehensive therapies is dependent on a threshold tumour size. By employing the definitions and properties of the Poincaré map, we show that the effector cell eradication periodic solution is globally stable under threshold conditions. In the light of bifurcation theorems, it is shown that transcritical bifurcations can occur with respect to many treatment parameters including depletion rate, chemotherapeutic drug concentration, a medicine toxicity coefficient and the accumulation rate of effector cells. Then we provide conditions for the existence of order-[Formula: see text] [Formula: see text] periodic solutions. The results indicate that the threshold [Formula: see text] is sensitive to treatment parameters and the proposed system exists with very complex dynamics when treatment parameters are chosen as bifurcation parameters. Moreover, a therapeutic protocol with a smaller chemotherapy drug dosage and more frequent applications is more effective for maintaining a high tumour cell depletion rate.

Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map

Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.