Finite Time Blow-up in a Delayed Diffusive Population Model with Competitive Interference

Abstract

In the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In [1], we have seen that the model does possess globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition, even under the parametric restrictions derived in [1]. We prove blow-up in the delayed model, as well as the non delayed model, providing sufficient conditions on the largeness of data, required for finite time blow-up. Numerical simulations show, that actually the initial data does not have to be very large, to induce blow-up. The spatially explicit system is seen to possess Turing instability. We have also studied Hopfbifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory.1991 Mathematics Subject Classification. Primary: 35B36, 37C75, 60H35; Secondary: 92D25, 92D40. , and I is a 2 × 2 identity matrix. For the non-trivial solution of (20), we require thatThis gives us a polynomial in k, and via standard theory [16], we derive the following necessary and sufficient conditions for Turing Instability;

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

“…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.…”