부산대학교 도서관

This paper investigates the spatio-temporal patterns of a freshwater tussock sedge model with discrete time and space variables. The authors find, in the spatially homogeneous case, flip and Neĭmark-Sacker bifurcations, respectively. Adding spatial diffusion, they show that the obtained stable homogeneous solutions can experience Turing instability under certain conditions. Through careful numerical simulations, they also find periodic doubling cascade, periodic window, invariant cycles, chaotic behaviors (evidenced by orbit diagram and positive Lyapunov exponents), and some interesting spatial patterns, which are induced by four mechanisms: pure-Turing instability, flip-Turing instability, Neĭmark-Sacker-Turing instability, and chaos. The authors note that the discretized model presents a greater richness of dynamics and spatial patterns than the continuous system it derives from. \par The discrete model is based on the continuous reaction-diffusion system of PDEs: $$ \cases \displaystyle\frac{\partial P}{\partial t}=P(1-P)-sP-\theta PW+d_p\Delta P,\\ \displaystyle\frac{\partial W}{\partial t}=sP-bW+d_w\Delta W, \endcases $$ where the predator-prey type relationship is between the plant mass $P(x,y,t)$ and the wrack (dead biomass) $W(x,y,t)$. To obtain a discretized system in both time and space, the authors use a coupled map lattice (CML) of the following form. The model first diffuses according to $$ \cases \displaystyle P'(i,j,t)=P(i,j,t)+\frac\tau\delta^2d_P\nabla^2P(i,j,t),\\ \displaystyle W'(i,j,t)=W(i,j,t)+\frac\tau\delta^2d_W\nabla^2W(i,j,t), \endcases $$ where $\nabla^2$ is the usual discretized Laplacian. Then the model applies the reaction $$ \cases P(i,j,t+1)=f(P'(i,j,t),W'(i,j,t)),\\ W(i,j,t+1)=g(P'(i,j,t),W'(i,j,t)), \endcases $$ where $$ \align f(P,W)&=P+\tau\left(sP+\xi P^2-PW\right),\\ g(P,W)&=W+\tau(P-W). \endalign $$ Toroidal boundary conditions are used.