On the explosive instability in a three‐species food chain model with modified Holling type IV functional response

Abstract

In earlier literature, a version of a classical three‐species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd.

Patterns formation in hyperbolic reaction-diffusion models with cross-diffusion

“…Summarizing, from (38) and (39 ) it is easy to see that the hyperbolicity of the system has no effect on the unstable modes having k 2 1 , k 2 2 , d c and k c exactly the same expressions as for the standard reaction-diffusion model. Neverthless, owing to (40) 6 , the hyperbolic character of the governing model modifies the Turing regions through the constitutive parameters µ * u , ν * v .…”

“…Our goal is to elucidate how hyperbolicity affects the pattern formation as well as the transient dynamics from an homogeneous steady state to a patterned one. In particular, we point out the occurrence of wave instability in our hyperbolic model for two interacting species whereas, as well known, in the parabolic case one needs at least three reaction-diffusion equations [37]- [39].…”

Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

“…More details as regards the functional responses can be found in [23,[39][40][41]. However, the generalized Holling type IV functional response or the Monod-Haldane function [5,[42][43][44][45]…”

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]

“…12,13 This property has been proven for many variants of the model as well. [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.…”