The stability and Hopf bifurcation of the diffusive Nicholson's blowflies model in spatially heterogeneous environment

TL;DR

This paper analyzes the stability and Hopf bifurcation phenomena in a spatially heterogeneous diffusive Nicholson's blowflies model, revealing how large diffusion rates and delays influence population dynamics and stability.

Contribution

It provides new insights into the stability criteria and bifurcation behavior of the model, especially regarding the effects of large diffusion and time delays.

Findings

Unique positive steady state stability depends on the egg production to death rate ratio.

Large delays can induce Hopf bifurcation, destabilizing the steady state.

The first Hopf bifurcation value approaches that of the averaged DDE model as diffusion increases.

Abstract

In this paper, we consider the diffusive Nicholson's blowflies model in spatially heterogeneous environment when the diffusion rate is large. We show that the ratio of the average of the maximum per capita egg production rate to that of the death rate affects the dynamics of the model. The unique positive steady state is locally asymptotically stable if the ratio is less than a critical value. However, when the ratio is greater than the critical value, large time delay can make the unique positive steady state unstable through Hopf bifurcation. Especially, the first Hopf bifurcation value tends to that of the ''average'' DDE model when the diffusion rate tends to infinity. Moreover, we show that the direction of the Hopf bifurcation is forward, and the bifurcating periodic solution from the first Hopf bifurcation value is orbitally asymptotically stable, which improves the earlier result by Wei and Li (Nonlinear. Anal., 60: 1351-1367, 2005).