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.

Mathematical Modeling of Impact of Pollutants on Industries Based on Resource Biomass

Depletion of resources such as forestry, minerals etc. and resource-based industries such as wood and paper etc., due to rising pollution, is one of the biggest challenges which the humankind is facing today. In this paper, a mathematical model has been designed to give an insight into the effect of pollutants on natural resources which in turn affects the growth and stability of industries dependent on such biomass. The model is analyzed using stability theory of differential equations. Five dependent variables are considered in the model and some important assumptions are made. Two equilibria are found in the equilibrium analysis and conditions of local and global stability of interior equilibrium are obtained. Numerical simulation is also done to demonstrate the analytical findings. It is found in the study that as we impose an environmental tax on the polluters, the concentration of pollutants in the environment is controlled and the stable equilibrium shifts in such a way that the densities of resource biomass and dependent industries are close to the densities which correspond to the pollution free ecosystem.

An Eco-Epidemiological Model Incorporating Harvesting Factors

The biological system relies heavily on the interaction between prey and predator. Infections may spread from prey to predators or vice versa. This study proposes a virus-controlled prey-predator system with a Crowley–Martin functional response in the prey and an SI-type in the prey. A prey-predator model in which the predator uses both susceptible and sick prey is used to investigate the influence of harvesting parameters on the formation of dynamical fluctuations and stability at the interior equilibrium point. In the analytical section, we outlined the current circumstances for all possible equilibria. The stability of the system has also been explored, and the required conditions for the model’s stability at the equilibrium point have been found. In addition, we give numerical verification for our analytical findings with the help of graphical illustrations.

Stability and Hopf Bifurcation Analysis of a Delayed Innovation Diffusion Model with Intra-Specific Competition

This article is concerned with the diffusion of a sport in a region, and the innovation diffusion model comprising of population classes, viz. nonadopters class, information class and adopters class. A qualitative analysis is carried out to assess the global asymptotic stability of the interior equilibrium for null delay. It has also been proved that the parameter [Formula: see text] (age gaps among sportspersons) in the intra-specific competition between the new players and the senior players can even destabilize the otherwise globally stable interior equilibrium state and the coexistence of all the populations is possible through periodic solutions due to Hopf bifurcation. With the help of normal form theory and center manifold arguments, the stability of bifurcating periodic orbits is determined. Numerical simulations have been executed in support of the analytical findings.

Modelling the Impacts of Early Intervention on Desert Locust Outbreak Prevention

To preserve crop production losses, monitoring of desert locust attacks is a significant feature of agriculture. In this paper, a mathematical model was formulated and analyzed to protect crops against desert locust attack via early intervention tactics. We consider a triple intervention approach, namely, proaction, reaction, and outbreak prevention. The model integrates a stage-structured locust population, logistics-based crop biomass, and blended early intervention via pesticide spray. We assume that the amount of pesticide spray is proportional to the density of the locust population in the infested area. Conventional short residual pesticides within ultralow volume formulation and equipment control operations are considered. The trivial and locust-free equilibrium of the model is unstable, whereas the interior equilibrium is asymptotically stable. Numerical simulations validate the theoretical results of the model. In the absence of intervention measures, desert locust losses are approximately 71% of expected crop production. The model projection shows that effective proactive early intervention on hopper stage locust contained locust infestation and subdued public health and environmental hazards. Relevant and up-to-date combined early interventions control desert locust aggression and crop production losses.

Consistent Conjectural Variations Equilibrium in the Semi-Mixed Oligopoly

We study a variant of the mixed oligopoly model with conjectural variations equilibrium, in which one of the producers maximizes not his net profit but the convex combination of the latter with the domestic social surplus. The coefficient of this convex combination is named socialization level. The producers’ conjectures concern the price variations depending upon their production output variations. In this work, we extend the models studied before, considering the case of the producers’ cost functions being convex but not necessarily quadratic. The notion of exterior and interior equilibrium is introduced (similarly to previous works), developing a consistency criterion for the conjectures. Existence and uniqueness theorems are formulated and proven. Results concerning the comparison between conjectural variations, perfect competition, and Cournot equilibriums are provided. Based on these results, we formulate an optimality criterion for the election of the socialization level. The existence of the optimal socialization level is proven under the condition that the public company cannot be too weak as compared to the private firms.

Dynamics of a stage–structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator

A kind of stage-structure Rosenzweig–MacArthur model with linear harvesting in prey and cannibalism in predator is investigated in this paper. By analyzing the model, local stability of all possible equilibrium points is discussed. Moreover, the model undergoes a Hopf–bifurcation around the interior equilibrium point. Numerical simulations are carried out to illustrate our main results.

Dynamical of Ratio-Dependent Eco-epidemical Model with Prey Refuge

This paper discusses the dynamic analysis of three species in the eco-epidemiology model by considering the ratio-dependent function and prey refuge. The prey refuge is applied under the fact that infected prey has protection instincts that allow it to reduce predation risk. Here, we get the boundedness and three equilibrium points where are existence under certain conditions. In the model, three equilibrium points are locally asymptotically stable and one of the equilibrium points is globally asymptotically stable. We find that the system undergoes Hopf bifurcation around the interior equilibrium point by choosing as a bifurcation parameter. We also find a condition for uniform persistence. Finally, several simulations of numerical are performed not only to illustrate the analytical results but also to illustrate the effect of the prey refuge.

Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.

Local Stability Analysis On Lotka-Volterra Predator-Prey Models With Prey Refuge And Harvesting

We propose a predator-prey model by incorporating a constant harvesting rate into a Lotka-Volterra predator-prey model with prey refuge which was studied recently. All the positive equilibria and the local stability of the proposed model are studied and analyzed by sorting out the intervals of the parameters involved in the model. These intervals of the parameters exhibit the effects on the dynamical behaviors of prey and predators. The emphasis is put on the ranges of the prey refuge constant and harvesting rate. We show that the model has two positive boundary equilibria and one equilibrium. By using the qualitative theory for planar systems, we show that the two positive boundary equilibria can be saddles, saddle-nodes, topological saddles or stable or unstable nodes, and the interior positive equilibrium is locally asymptotically stable. Under suitable restrictions on the parameters, we prove that the positive interior equilibrium is a stable node.