Predator vs prey model4/18/2024 For starters, birds respond with anti-predator defences to the sounds of predators and flee from their nests at the first sign of danger. Compared to direct predation, these forms of behavioral and physiological modifications related to fear of predation are more efficient and longer-lasting evolutionary effects. Predators will actively affect the ecosystem and the prey reproduction process by consuming them and by affecting the behavior and physiology of the survivor indirectly. Since there have been fewer studies related to the double Allee effect in the predator–prey system, in the authors attended the presence of double Allee effect in the ratio-dependent predator–prey system in the prey growth function, where they studied various types of bifurcation behavior and showed bi-stability behavior in the phase plane implying that the considered system dynamics is sensitive in choosing the initial conditions. The authors considered a strong Allee effect in the prey growth function in the Holling type II model with density-dependent death rate for predators and examined various types of bifurcation behavior in the considered model. A Leslie type model with ratio-dependent type interaction term with state impulsive feedback control was considered, and the conditions for existence, uniqueness, and orbital asymptotical stability of periodic solution of order-1 were obtained by applying the geometry theory of differential equations. The same model with discrete time was investigated in, and the authors showed that varying integral step size can arise chaotic orbits via period-doubling and invariant circle. Further, the conditions for occurrence of various bifurcation behaviors were derived and verified numerically with the help of parametric diagrams and phase portraits. In the ratio-dependent predator–prey model, the authors in compared the dynamical properties with and without the Allee effect in logistic prey growth term and showed that the inclusion of Allee effect can remove oscillation behavior of species densities. Recently, there has been greater interest from many researchers to study the effect of incorporating the Allee effect in the predator–prey dynamics. The Allee effect can be caused by various environmental factors including difficulties in finding mating partners at low density, genetic inbreeding, social felicitation of reproduction, low probability of successful mating, depletion in inbreeding rate, etc. The Allee effect is one of the most interesting phenomena in ecology, which is characterized by a positive correlation between the population size and the per capita growth rate for a population. For this reason, a fractional-order predator–prey system has been investigated by several researchers, and numerous outstanding results have been derived, for example, see. In some real situations, fractional-order systems are found to be more accurate than integer-order ones. Despite its rich and varying complexity, there has been greater focus paid by several researchers to modify and build the Leslie type model in several ways. The authors in assumed that the predator growth function is different from the predator’s predation function, and its reduction has a reciprocal relationship with per capita availability of its preferred food, the so-called Leslie type predator–prey model. In real-world situations, clearly confirm that the ratio-dependent interaction has a greater description of predation rates than the prey-dependent interaction. Thus the predation function depends on prey only such as Holling type classification, whereas ratio-dependent, Beddington, and Crowley Martin are the functions of both prey and predator. Many researchers have developed and implemented various forms of interaction terms to understand the more realistic situation. The key factor in a predator–prey relationship is that the growth of predator depends on its functional response to the prey population, which describes the amount of prey each predator eats per unit of time. The typical model of such interactions between two species was first proposed in the form of a simplest mathematical model. One of the prominent topics for ecologists, biologists, and even mathematicians is the study of various mechanisms related to the interaction between prey and predators.
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