Finite time Blow up in a population model with competitive interference and time delay

TL;DR

This paper investigates finite time blow-up in a delayed predator-prey model with modified functional responses, showing that large initial data can cause solutions to become unbounded, contrasting previous results for small initial conditions.

Contribution

It demonstrates that solutions to the predator-prey system can blow up in finite time for large initial data, even under conditions previously thought to ensure boundedness, and analyzes stability and bifurcations.

Findings

Finite time blow-up occurs for large initial data.

Numerical simulations indicate blow-up can happen with relatively small initial data.

The system exhibits Turing instability and Hopf bifurcation phenomena.

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 \cite{RK15}, 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, \emph{even} under the parametric restrictions derived in \cite{RK15}. 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 Hopf-bifurcation direction in the spatial system, as well as stability of the spatial Hopf-bifurcation using the central manifold theorem and normal form theory.