The Dynamics of Modified Leslie-Gower Predator-Prey Model Under the Influence of Nonlinear Harvesting and Fear Effect

Abstract

A modified Leslie-Gower predator-prey model with fear effect and nonlinear harvesting is developed and investigated in this study. The predator is supposed to feed on the prey using Holling type-II functional response. The goal is to see how fear of predation and presence of harvesting affect the model's dynamics. The system's positivity and boundlessness are demonstrated. All conceivable equilibria's existence and stability requirements are established. All sorts of local bifurcation occurrence conditions are presented. Extensive numerical simulations of the proposed model are shown in form of Phase portraits and direction fields. That is to guarantee the correctness of the theoretical results of the dynamic behavior of the system and to confirm the existence of various forms of bifurcations. The fear rate is observed to have a stabilizing effect up to a threshold value, after which it leads to prey extinction. The harvesting coefficients, on the other hand, serve as control parameters that, when exceeded, trigger the system to extinction.

Fear and Hunting Cooperation's Impact on the Eco-Epidemiological Model's Dynamics

“…The study of the bifurcation of the system (2) is vital because the parameters are not constant in actual reality and are constantly changing according to the conditions of the environment containing the organisms of the system. Now, rewrite the system (2) in the vector form as: 𝑑𝐗 𝑑𝑡 = 𝐅(𝐗, 𝜇), 𝐗 = (𝑥 1 , 𝑥 2 , 𝑥 3 ) T , 𝐅 = (𝑥 1 𝑓 1 (𝐗, 𝜇), 𝑥 2 𝑓 2 (𝐗, 𝜇), 𝑥 3 𝑓 3 (𝐗, 𝜇)) T , (39) where 𝜇 ∈ ℝ represents a bifurcation parameter. Hence the second and third directional derivatives for ( 39) can be written respectively as:…”

“…However, the presence of a predator may drastically alter the prey's physiological and behavioral characteristics to the point where it may have a greater impact on the prey population than direct predation [17] and the mentioned references therein. There are many research studies using mathematical modeling considered the impact of hunting cooperation [1,[17][18][19][20][21][22][23][24][25] and the impact of fear [16,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] in the prey-predator models separately. Numerous prey species alter their behavior when predators are around because of the possibility of being eaten.…”

Mathematical Modeling of Tumors Growth: Competition based on Gompertz model in Two Dimensions

“…As a result, the model can be predicted as to how it will behave given a variety of initial conditions but for a predetermined set of parameter values. Following the curve (a trajectory) from the initial values in the direction of the arrows is shown in Fig 1 . The direction field generated by the differential equation for each value of X and Y is where the arrows originate by using MATHEMATICA 25,26 . The green points represent the stable points and the red points refer to unstable points 27 .…”

Dynamics of a Fractional-Order Prey-Predator Model with Fear Effect and Harvesting

“…The concept of the optimal harvesting has an important significant effect in treating the renewable stocks due the economic feature and to maintain and prevent the species away from the extinction, so that many authors and researchers are widely considered and discussed this subject in their papers [9][10][11][12][13][14].…”