Hopf Bifurcation and Control for the Bioeconomic Predator–Prey Model with Square Root Functional Response and Nonlinear Prey Harvesting

Abstract

In this essay, we introduce a bioeconomic predator–prey model which incorporates the square root functional response and nonlinear prey harvesting. Due to the introduction of nonlinear prey harvesting, the model demonstrates intricate dynamic behaviors in the predator–prey plane. Economic profit serves as a bifurcation parameter for the system. The stability and Hopf bifurcation of the model are discussed through normal forms and bifurcation theory. These results reveal richer dynamic features of the bioeconomic predator–prey model which incorporates the square root functional response and nonlinear prey harvesting, and provides guidance for realistic harvesting. A feedback controller is introduced in this paper to move the system from instability to stability. Moreover, we discuss the biological implications and interpretations of the findings. Finally, the results are validated by numerical simulations.

Trends and Paradoxes of Competitive Evolution in the Predation Mechanism

“…-ii-Of the significant parameters of this predation model containing e, b, τ, c, a, and P, only e, b, τ, and c, are involved in (17). The other two, a and P, are not involved, as they relate to prey behavior.…”

“…Let us consider the case where only one of the parameters e, τ, b and c is different for the two attractors. Expression (17) has constant sign everywhere, so the attractor (whatever its nature, point-like or not) is necessarily on one of the coordinate planes (x,y 1 ) or (x,y 2 ), the one corresponding to the larger e, b, τ or the smaller c. This is elementary Darwinism, as long as we interpret "advantage" and "disadvantage" in the sense of larger or smaller q and not the predator population or anything else large or small. The same applies when there are several parameters with different values, provided that the differences in the values of these parameters are all in the same direction of increasing or decreasing q (examples: increasing τ and b or decreasing e and increasing c).…”

“…In this context, one can refer to [17], which describes a bioeconomic predator-prey model derived from the model proposed in [18], which incorporates the square root functional response and nonlinear prey harvesting. Due to the last condition, the model demonstrates intricate dynamic behaviors in the predator-prey plane, mainly via the occurrence of Hopf bifurcation utilizing economic profit as a bifurcation parameter.…”

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

“…From a biological perspective, we know that both species u and v coexist and probably enjoy permanence when the catching efforts can be found in either the first or second quadrants; the predator population could be on the verge of extinction (that is, washout stage) when the harvesting efforts are found in the third and fourth quadrants. We see that both species would become extinct if the prey were over-harvested (Barman and Upadhyay [15], Guo et al [16], Hasibuan et al [17]). One can secure the survival of the prey by managing the amount of harvesting, but when the degree of harvesting is in the third quadrant, one cannot guarantee the life of the predator.…”

Research on Group Behavior Modeling and Individual Interaction Modes with Informed Leaders

“…This led to the development of the influential Cucker-Smale (CS) population model. Recent studies have delved into more intricate and realistic dynamics, such as molecular physics [25], local interactions [26], predator-prey [27], and metric distance models [28,29]. Despite discussions of hierarchical relationships in the literature, most simulations assume an equal status among individuals.…”