In this talk, I will give two examples to show how time delays affect the dynamics in biological problems.

The first part is about the population in ecology. The oscillation of populations is a fascinating phenomenon in ecology models. It is important to understand what

mechanisms generate such oscillations. In this talk, I will introduce a two-patch model for a single species with dispersal and time delay. For this model, we found that there exists a critical value τc for the time delay τ such that the unique positive equilibrium of the model is locally asymptotically stable for τ ∈ [0, τc) and unstable for τ > τc. Moreover, the system generates periodic solution when τ is closed to τc.

The second part is about the somitogenesis. During somite formation, there are three important dynamics of the gene expression which are the synchronous oscillation in the tail bud, traveling wave pattern, and the oscillation-arrested in the anterior region in the embryo. For biologists, they are interested in what mechanism generates these pattern and can mathematical models generate all these phenomena. For this part, we considered kinetic models for coupled cells to investigate these phenomena. We first performed the sequential-contracting technique to analyze the global behavior of this system to find some conditions such that every solution converges to a equilibrium. This dynamics corresponds to no oscillation in the anterior region. We also used delay Hopf bifurcation theory to analyze how time delays generate the synchronous periodic solutions. This result corresponds to the dynamics in the tail bud. Based on this analytical result and numerical simulation, we further constructed a lattice system for coupled N cells to generate these three phenomena in somitogenesis.