We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldings. Moreover, we show the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter. We also find the existence of Bautin bifurcation numerically, then simulate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.

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This paper is concerned with the effect of predator cannibalism in a delayed diffusive predator–prey system. We aim for the case where the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, leading to Hopf–Hopf bifurcation. An approach of center manifold reduction is applied to derive the normal form for such nonresonant Hopf–Hopf bifurcations. We find that the system exhibits very rich dynamics, including the coexistence of periodic and quasi-periodic oscillations. Numerically, we show that Hopf–Hopf bifurcation is induced if the strength of the predator cannibalism term belongs to an appropriate interval.

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.

In this paper, we study the stability and bifurcation behavior of a three-dimensional melanoma model with immune response. The system has at least one and at most three positive equilibria. It is proved that the system undergoes Hopf bifurcation and saddle-node bifurcation at the positive equilibrium. We investigate the direction of Hopf bifurcation and stability of the bifurcating periodic solution by center manifold theorem and normal form theory. Moreover, codimension two bifurcations of the system are analyzed. We demonstrate the existence of Bautin bifurcation and Bogdanov–Takens bifurcation of the system. The normal form of Bautin bifurcation and Bogdanov–Takens bifurcation are given. Finally, some numerical simulations are demonstrated to support our theoretical results, and the importance of some parameters of the system is discussed, in particular the activation rate of CD8[Formula: see text]T cells.

<abstract><p>A diffusive predator-prey system with advection and time delay is considered. Choosing the conversion delay $ \tau $ as a bifurcation parameter, we find that as $ \tau $ varies, the system will generate Hopf bifurcation. Then, for the reaction diffusion model proposed in this paper, we use an improved center manifold reduction method and normal form theory to derive an algorithm for determining the direction and stability of Hopf bifurcation. Finally, we provide simulations to illustrate the effects of time delay $ \tau $ and advection $ \alpha $ on system behaviors.</p></abstract>

<p style='text-indent:20px;'>We propose two compartment models to study the disease transmission dynamics, then apply the models to the current COVID-19 pandemic and to explore the potential impact of the interventions, and try to provide insights into the future health care demand. Starting with an SEAIQR model by combining the effect from exposure, asymptomatic and quarantine, then extending the model to the one with ages below and above 65 years old, and classify the infectious individuals according to their severity. We focus our analysis on each model with and without vital dynamics. In the models with vital dynamics, we study the dynamical properties including the global stability of the disease free equilibrium and the existence of endemic equilibrium, with respect to the basic reproduction number. Whereas in the models without vital dynamics, we address the final epidemic size rigorously, which is one of the common but difficult questions regarding an epidemic. Finally, we apply our models to estimate the basic reproduction number and the final epidemic size of disease by using the data of COVID-19 confirmed cases in Canada and Newfoundland &amp; Labrador province.</p>