For delayed reaction-diffusion Schnakenberg systems with Neumann boundary conditions, critical conditions for Turing instability are derived, which are necessary and sufficient. And existence conditions for Turing, Hopf and Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at Turing-Hopf singularity of codimension 2, are derived. By investigating Turing-Hopf bifurcation, the parameter regions for the stability of a periodic solution, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic solutions, are derived in (τ, ε) parameter plane (τ for time delay, ε for diffusion rate). It is revealed that joint effects of diffusion and delay can lead to the occurrence of mixed spatial and temporal patterns. Moreover, it is also demonstrated that various spatially inhomogeneous patterns with different spatial frequencies can be achieved via changing the diffusion rate. And, the phenomenon that time delay may induce a failure of Turing instability observed by Gaffney and Monk (2006) are theoretically explained.
The effects of predator-taxis and conversion time delay on formations of
spatiotemporal patterns in a predator-prey model are explored. Firstly,
the well-posedness, which implies global existence of classical
solutions, is proved. Then, we establish critical conditions for the
destabilization of coexistence equilibrium through Turing/Turing-Turing
bifurcations via describing the first Turing bifurcation curve, and
theoretically predict possible bi-stable/multi-stable spatially
heterogeneous patterns. Next, we demonstrate that coexistence
equilibrium can also be destabilized through Hopf, Hopf-Hopf,
Turing-Hopf bifurcations, and possible stable/bi-stable spatially
inhomogeneous staggered periodic patterns, bi-stable spatially
inhomogeneous synchronous periodic patterns, are theoretically
predicted. Finally, numerical experiments also support theoretical
predictions and partially extend them. In a word, theoretical analyses
indicate that, on the one hand, large predator-taxis can eliminate
spatial patterns caused by self-diffusion; on the other hand, the joint
effects of predator-taxis and conversion time delay can induce complex
survival patterns, e.g., bi-stable spatially heterogeneous
staggered/synchronous periodic patterns, thus diversify populations’
survival patterns.
<p style='text-indent:20px;'>A diffusive Rosenzweig-MacArthur model involving nonlocal prey competition is studied. Via considering joint effects of prey's carrying capacity and predator's diffusion rate, the first Turing (Hopf) bifurcation curve is precisely described, which can help to determine the parameter region where coexistence equilibrium is stable. Particularly, coexistence equilibrium can lose its stability through not only codimension one Turing (Hopf) bifurcation, but also codimension two Bogdanov-Takens, Turing-Hopf and Hopf-Hopf bifurcations, even codimension three Bogdanov-Takens-Hopf bifurcation, etc., thus the concept of Turing (Hopf) instability is extended to high codimension bifurcation instability, such as Bogdanov-Takens instability. To meticulously describe spatiotemporal patterns resulting from <inline-formula><tex-math id="M2">\begin{document}$ Z_2 $\end{document}</tex-math></inline-formula> symmetric Bogdanov-Takens bifurcation, the corresponding third-order normal form for partial functional differential equations (PFDEs) involving nonlocal interactions is derived, which is expressed concisely by original PFDEs' parameters, making it convenient to analyze effects of original parameters on dynamics and also to calculate normal form on computer. With the aid of these formulas, complex spatiotemporal patterns are theoretically predicted and numerically shown, including tri-stable nonuniform patterns with the shape of <inline-formula><tex-math id="M3">\begin{document}$ \cos \omega t\cos \frac{x}{l}- $\end{document}</tex-math></inline-formula>like or <inline-formula><tex-math id="M4">\begin{document}$ \cos \frac{x}{l}- $\end{document}</tex-math></inline-formula>like, which reflects the effects of nonlocal interactions, such as stabilizing spatiotemporal nonuniform patterns.</p>
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