부산대학교 도서관

The authors study a type of Gierer-Meinhardt system of reaction-diffusion equations with time delay which, after non-dimensionalization, takes the form $$ \alignat2 &\frac{\partial u(x, t)}{\partial t}= \epsilon D \frac{\partial^2 u(x, t)}{\partial x^2} +\gamma\left(p-qu(x, t)+\frac{u^2(x, t-\tau)}{v(x, t-\tau)}\right),& x \in (0, \pi),\ t>0, \\ &\frac{\partial v(x, t)}{\partial t}= D \frac{\partial^2 v(x, t)}{\partial x^2} +\gamma\left(u^2(x, t-\tau)-v(x, t)\right), & x \in (0, \pi),\ t>0, \\ &u_x(0, t) = u_x(\pi, t) = v_x(0, t) = v_x(\pi, t)=0 ,& x \in (0, \pi),\ t>0, \\ &u(x, t) \geq 0,\ v(x, t) \geq 0, &(x, t) \in [0, \pi] \times [-\tau, 0]. \endalignat $$ \par By analyzing the characteristic equations at the positive equilibrium, the authors establish the conditions for Turing, Hopf and Turing-Hopf bifurcations. They then compute a normal form to order 3 of the delayed systems in a neighborhood of a Turing-Hopf bifurcation. They use this normal form to show that the system exhibits various interesting spatial, temporal and spatiotemporal patterns. Numerical simulations are shown to illustrate the theoretical results.