Research On Bifurcation Dynamics And Pattern Formation Of Two Discrete Food Webs

The food web is the most basic component of the ecosystem,and its research plays a key role in revealing the basic laws of the ecosystem and guiding the restoration of the natural ecology.This paper analyzes the bifurcation dynamics and self-organized pattern of the spatiotemporal discrete food web systems(chain food web system is also referred to as food chain system in this paper).The main findings are as follows:(1).The bifurcation dynamics and pattern formation of a three-species disceret-time food chain system with Beddington-DeAngelis functional response were studied.On the one hand,the time step is used as the bifurcation parameter,and the stability of the fixed point of the systel is analyzed.The occurrence conditions of period-doubling bifurcation and Neimark-Sacker bifurcation are deterlined by the central manifold theorem and the bifurcation theorem.Numerical simulations show the dynamic behaviors leading to chaos,including period-doubling bifurcations,invariant curve,chaotic attractors.sub-Neimark-Sacker bifurcations,sub-period-doubling cascades,chaotic internal crises,sub-period window and various periodic behaviors such as period-2,3,4.6,8.9.12,16,18,19,20,23,24,38?45,64,72 orbit.On the other hand,the Turing instability condition was analyzed.The numerical simulation showed the time evolution paltcrn of the food chain system,and found various types of patterns such as stripe,labyrinth,gap,spiral wave and fragmentation.(2)The bifurcation dynamics and pattern formation of a thrcc-species disceret-time food web system with mixed functional response were studied.On the one hand,the intrinsic growth rate of common prey is used as the bifurcation parameter,and the stability of the fixed point of the system is analyzed asc well as the occurrence condition of period-doubling bifurcation and Neimark-Sacker bifurcation is determined by the central manifold theorem and the bifurcation theorem.Numerical simulations show dynamic behaviors leading to chaotic paths,including period-doubling bifurcations,inverse Neimark-Sacker bifurcations,sub-period-doubling cascades.sub-Neimark-Sacker bifurcations,paroxysmal chaos,periods-2,3.4,5,6.8,10,13.23 and other periodic orbits.Oin the other hand,the Turing instability condition was analyzed.The numerical simulation showed the time evolution pattern of the food web system.In addition to the common and transitional spiral patterns,many rare pattern types were found.For example,shark-tooth shape,wave-band shape,symmetrical pattern,and the like.