Abstract
In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., digestion, gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in the proposed system. The developed algorithm for calculating the normal form of Hopf bifurcation can be used to investigate the direction and stability of Hopf bifurcation. As a real application, we consider a diffusive predator-prey model with ratio-dependent Holling type-III functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. By using the developed algorithm for calculating the normal form of Hopf bifurcation, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.
Highlights
4 on the Fick’s law, that is the movement flux is in the direction of negative gradient of the density distribution function [4]
In this paper, we have developed an algorithm for calculating the normal form of Hopf bifurcation in 520 a diffusive system with memory and general delays
By considering that apart from the memory delay appears in the diffusion term, the general delay occurs in the reaction term, the traditional algorithm
Summary
4 on the Fick’s law, that is the movement flux is in the direction of negative gradient of the density distribution function [4]. 14 at the present time, there is a directed movement toward the negative or positive gradient of the density distribution function at past time, and they proposed the following diffusive model with spatial memory. Song et al [26] developed an algorithm for calculating the normal form of Hopf bifurcation in the system (1.2), and they studied the direction. 64 for calculating the normal form of Hopf bifurcation induced jointly by memory and general delays. 4, we consider a diffusive predator-prey model with ratio-dependent Holling type-III functional response, 68 which includes with memory and gestation delays. Algorithm for calculating the normal form of Hopf bifurcation induced jointly by memory and general delays.
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