[MS10] Model Bifurcations and Ecological Relevance

[MS10-10] Multiple Stability Transitions, Basin Evolution, and Stock Benefits in a Delayed Predator-prey Model

*Bapan Ghosh¹, Yovan Singh¹ (1. Indian Institute of Technology Indore, India (India))

Keywords:

Delay Differential Equations,Population dynamics,Quasipolynomial function,Hopf bifurcation,Stability switching

Time delays naturally occur in most of the processes in physical, biological, chemical, economic systems due to non-instantaneous feedback . In population dynamics, gestation delay, maturation delay, dispersal delay, etc. are incorporated in modeling to capture more realistic features. In this presentation we present a predator-prey system with Cosnor functional response and discrete time delay in numerical response. The delay induced stability analysis and basin structure will be analyzed. We first define several delay-induced stability classifications such as (i) saddle invariance, (ii) stability invariance, (iii) stability change, (iv) stability switching, (v) instability invariance, (vi) instability switching. Stability of equilibrium is determined by the movement of eigenvalues from the left half complex plane to the right half complex plane and vice-versa. Extensive numerical demonstrations have been provided to draw the variation of eigenvalues and delay threshold curves. We have shown that our minimal model (two-specie with a single time delay parameter) exhibits all six stability classifications. Further we have shown a degenerate situation where equilibrium remains stable for all delays (possibly except at the critical delay). Finally we explore the structure of the basin of stable fixed point and limit cycles when delay is increased. It is found that delay often could be beneficial in stock conservation of predator populations. We expect that this study helps understanding delay-induced stability theory and population dynamics in general.