Global Dynamics of a Predator-Prey Model with State-Dependent Maturation-Delay

TL;DR

This paper analyzes a predator-prey model with a state-dependent maturation delay that influences predator dynamics, establishing conditions for predator-prey coexistence, stability of equilibria, and global stability under certain parameters.

Contribution

It introduces a novel predator-prey model incorporating the derivative of the delay, and provides stability conditions for coexistence and equilibrium states.

Findings

Predator coexists with prey if the net reproduction number exceeds one.

Local stability of equilibria depends on functional response types.

Global stability of coexistence occurs under small delay derivatives and high predator interference.

Abstract

In this paper, a stage structured predator-prey model with general nonlinear type of functional response is established and analyzed. The state-dependent time delay (hereafter SDTD) is the time taken from predator's birth to its maturity, formatted as a monotonical (ly) increasing, continuous(ly) differentiable and bounded function on the number of mature predator. The model is quite different from many previous models with SDTD, in the sense that the derivative of delay on the time is involved in the model. First, we have shown that for a large class of commonly used types of functional responses, including Holling types I, II and III, Beddington-DeAngelis-type (hereafter BD-type), etc, the predator coexists with the prey permanently if and only if the predator's net reproduction number is larger than one unit; Secondly, we have discussed the local stability of the equilibria of the model; Finally, for the special case of BD-type functional response, we claim that if the system is permanent, that is, the derivative of SDTD on the state is small enough and the predator interference is large enough, then the coexistence equilibrium is globally asymptotically stable.