arXiv subjects

Tonghua Zhang

Publications and source records attributed to Tonghua Zhang.

5 recordsLinked to original sources

arXiv2026

Less Is More in Chemotherapy of Breast Cancer

This study presents a mathematical model that captures the interactions among tumor cells, healthy cells, and immune cells in a tumor-bearing host, with a specific focus on breast cancer. Incorporating the concept of delay, the model consists of four differential equations to analyze these cellular dynamics. The findings demonstrate the superior efficacy of metronomic chemotherapy compared to the maximum tolerated dose (MTD) method and underscore the necessity of adjunct therapies. Oscillatory tumor cell dynamics revealed by the model highlight the challenges of achieving complete tumor elimination through chemotherapy alone. Sensitivity analysis confirms the robustness of the model, particularly under metronomic treatment protocols, aligning with experimental observations regarding metronomic-to-MTD dosage ratios. Furthermore, the results emphasize the importance of synergistic effects from combination therapies. This biologically consistent framework provides valuable insights into tumor-immune interactions and offers a foundation for optimizing therapeutic strategies in cancer treatment.

q-bio.QM↗

arXiv2024

Dynamics of a memory-based diffusion model with spatial heterogeneity and nonlinear boundary condition

In this work, we study the dynamics of a spatially heterogeneous single population model with the memory effect and nonlinear boundary condition. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. By using bifurcation analysis, the Hopf bifurcation associated with one spatially nonconstant positive steady state is found to occur. The results complement the existing ones. Specifically, it is found that with the interaction of spatial heterogeneity and nonlinear boundary condition, when the memory term is stronger than the interaction of the interior reaction term and the boundary one, the memory-based diffusive model has a single stability switch from stability to instability, with the increase of the delayed memory value. Therefore, the memory delay will lead to a single stability switch of such memory-based diffusive model and consequently the Hopf bifurcation will happen in the model.

math.DS↗

arXiv2022

Traveling waves of a generalized Rotation-Camassa-Holm equation with the Coriolis effect

In this paper, we analyze the dynamics of a generalized Rotation-Camassa-Holm equation, which is the $θ$-equation augmented with the Coriolis effect, induced by the earth rotation. The generalized Rotation-Camassa-Holm equation (named as Rotation-$θ$ equation) is a generalization of a family of models (including the Rotation-Camassa-Holm equation for $θ=\frac{1}{3}$, asymptotic Rotation-Camassa-Holm equation for $θ=1$ and the Rotation-Degasperis-Procesi (DP) equation for $θ=\frac{1}{4}$). Our study is conducted via the bifurcation method and qualitative theory of dynamical systems. The existence of not only smooth solitary wave solutions, periodic wave solutions, but also peakons and periodic peakon solutions is shown. The chosen values $θ$, allow us to assess the difference in behavior between the classical $θ$-equation and the Rotation-$θ$ equation. We conclude that the Coriolis effect does affect the traveling wave solutions. We summarize the bifurcations and explicit expressions of waves solutions in three theorems in Section 4. A conclusion ends the paper.

math-ph↗

arXiv2022

Double Hopf bifurcation analysis in the memory-based diffusion system

In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.

math.DS↗

arXiv2021

The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system

The memory-based diffusion systems have wide applications in practice. Hopf bifurcations are observed from such systems. To meet the demand for computing the normal forms of the Hopf bifurcations of such systems, we develop an effective new algorithm where the memory delay is treated as the perturbation parameter. To illustrate the effectiveness of the algorithm, we consider a diffusive predator-prey system with memory-based diffusion and Holling type-II functional response. By employing this newly developed procedure, we investigate the direction and stability of the delay-induced mode-1 and mode-2 Hopf bifurcations. Numerical simulations confirm our theoretical findings, that is the existence of stable spatially inhomogeneous periodic solutions with mode-1 and mode-2 spatial patterns, and the transition from the unstable mode-2 spatially inhomogeneous periodic solution to the stable mode-1 spatially inhomogeneous periodic solution.

math.DS↗