Spatial Pattern of Ratio-Dependent Predator–Prey Model with Prey Harvesting and Cross-Diffusion

Abstract

This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.

Effect of multiple gestation delays in a prey–predator food chain system with infected class of prey

“…The remaining roots of (15) given by equation ( 16) are negative from the Case I. Hence, predatorfree equilibrium E 2 ( Ŝ, X, 0, 0) is locally asymptotically stable for all τ 1 ≥ 0 under the condition (14).…”

“…However, a more reliable mathematical model needs to incorporate functional responses that measure the prey encountered per predator in unit time. A number of analysts incorporated Holling type I to III, ratio-dependent functional responses, etc [9][10][11][12][13][14]. The researcher examined the tri-trophic food chain dynamics with gestation delay performing positivity, boundedness, stability, and occurrence of Hopf bifurcation [7].…”

Modeling escaping behavior from the herd in different ecological interactions

“…The eigenvalues of the Jacobian matrix (7) are the solution of the following characteristic equation…”

“…The Holling II interaction functional [6] represents the saturation of the consumption of the prey by a predator. The ratio-dependent interaction functional [7] represents the interaction that depends on the two populations. More precisely, the hunted prey depends on the ratio of the prey density to the predator density.…”

Dynamical analysis of a delayed food chain model with additive Allee effect

“…To derive a reliable mathematical model, the functional response term plays a vital role, which usually measures the quantity of prey intake by predators per unit time. Various functional responses have been derived and utilized, such as Holling type I to III [3][4][5], ratiodependent [6][7][8], Beddington-DeAngelis [9][10][11], and Crowley-Martin function response [12,13]. In modeling of population dynamics, two types of models are popular, namely discrete [14,15] and continuous-time models [11,13].…”