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In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a(8,0)-mode Turing–Hopf bifurcationand unstable for a(3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a(0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.
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This content will become publicly available on January 1, 2026
Dynamics of a state-dependent delay-differential equation
We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.
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- Award ID(s):
- 1951510
- PAR ID:
- 10640635
- Publisher / Repository:
- Electronic Journal of Qualitative Theory of Differential Equations 2025, No. 37, 1–94; https://doi.org/10.14232/ejqtde.2025.1.37 www.math.u-szeged.hu/ejqtde/
- Date Published:
- Journal Name:
- Electronic Journal of Qualitative Theory of Differential Equations
- Issue:
- 37
- ISSN:
- 1417-3875
- Page Range / eLocation ID:
- 1 to 94
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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