Investigation of an explosive food chain model with interference and inhibitory effects

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).…”

Some mathematical problems in population ecology

“…The feeding behavior of the organisms in this sequence determines their trophic levels. Tritrophic (three level) food chain models have been extensively studied for their complex dynamics in ecology and applied mathematics [126,120,169,4,3]. Some examples of a tritrophic food chain are vegetation-antelope-tiger, crayfish-catfish-humans and mouse-snake-owl.…”

“…Later, Upadyhay and Rai developed a three species food chain model different from Hastings and Powell where there is a Holling type II functional response and a generalist top predator which can switch its food source when its favorite prey is nonexistent [170]. These models have been well investigated as they exhibit a variety of rich dynamics ranging from extinction to periodic dynamics to chaos and most recently finite time blow up [4,81,77,129,114,124,40,130,169].…”

“…This model is based on a modified Leslie-Gower formulation [80], and considers the interactions between a generalist top predator p, depredating on a specialist middle predator v, that in turn is depredating on a prey u, where (u, v, p) are solutions to the above system (5.1)-(5.3). Similar models have generated a plethora of past and current research interest [4,81,77,129,114,168,123,165,166,40,116,122,43,119,117,130,169,132,120]. The novelties in the current model are that the prey u is equipped with defense ability, via a Monod-Haldane functional response for the interaction between prey u and middle predator v.…”

“…These include temporal chaos, spatio-temporal chaos, Turing patterns, rich bifurcation structure and finite time blow-up of solutions for large and small initial conditions [124,125]. 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 [119,126,169,132,120]. The most recent in this line of work [28], considers two new "damping" mechanisms, (1) a Allee effect (both weak and strong) is considered in the prey and (2) the top predator's functional response is considered to be of Crowley-Martin type, although its explicit dynamics are modeled via a modified Leslie-Gower scheme.…”