Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalized‐type top predator

Abstract

In this article, we have considered a continuous tritrophic food chain model subject to the Allee effect on the prey growth function with prey‐dependent Holling type II functional response between the prey and intermediate predator; Crowley‐Martin senses functional response between intermediate predator and top predator, and the top predator is of sexually reproductive type. We have established the positivity and boundedness of the system and the condition of existence of different equilibrium points. The local and global stability of the solutions about the various equilibrium points has been investigated. The center manifold theorem has been used to find the nature of the solution in the neighborhood of the nonhyperbolic equilibrium points and the direction of Hopf bifurcations. Numerical simulation has been carried out to establish the theoretical findings and finally some concluding remarks are given.

Chaotic Dynamics of a Tri-Topic Food Chain Model with Beddington-DeAngelis Functional Response in Presence of Fear Effect

“…In the early nineteenth century Malthus first formulated prey-predator interaction through mathematical models [11,12]. The celebrated Lotka-Volterra model was eventually enhanced by introducing logistic growth function for prey species [13,14], incorporating various functional response and environmental effects and these developments makes prey-predator interplay more and more realistic [15,16,17]. The dynamics of prey-predator interaction can significantly be affected in presence of fear effect in the field of environmental biology and ecology [18].…”

A remark on “ Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalized‐type top predator” [Comp and Math Methods. 2019; E1079, pp. 1–23]

“…[13][14][15][16][17][18][19][20][21][22] Thus, a recent research direction has been to modify the model and variants, via various ecological mechanisms that might prevent the blow-up, such as prey refuge, predator interference, time delay due to gestation, diffusion and mixed boundary conditions to name a few. 20,[23][24][25][26] The most recent in this line of work, 27 considers two new "damping" mechanisms, 1) with initial conditions [7,4,10] at approximately t = 0.8. Here, a 2 = 0.2, 3 = 10, c = 1.6, = 0.8, and all other parameters are the same in (5).…”

“…However, a strong Allee effect in prey, and its subsequent effect on blow-up dynamics on the Upadhyay-Rai class of models, has not been considered to the best of our knowledge, before. 27 The strong Allee effect will clearly cause a decrease in the prey population if one is below the Allee threshold, and this will reduce recruitment into the middle predator class, so the middle predator could perhaps be kept at a lower density, via a strong Allee effect in the prey-which could work favorably in removing a blow-up solution. When the Allee effect is weak, as modeled, the growth rate of prey is lower at lower density-but again the effect on blow-up dynamics is unknown.…”

“…

In Debnath et al (2019), a tritrophic food chain model subject to a Allee effect on the prey growth and Crowley-Martin senses functional response between intermediate predator and top predator, with a top predator of sexually reproductive type is considered. It is claimed that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, and is dissipative.

…”

Chaotic dynamics of a tri-topic food chain model with Beddington–DeAngelis functional response in presence of fear effect

“…In the early nineteenth century Malthus first formulated prey-predator interaction through mathematical models [11,12]. The celebrated Lotka-Volterra model was eventually enhanced by introducing logistic growth function for prey species [13,14], incorporating various functional response and environmental effects and these developments makes prey-predator interplay more and more realistic [15,16,17]. The dynamics of prey-predator interaction can significantly be affected in presence of fear effect in the field of environmental biology and ecology [18].…”