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This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter k measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when k is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as k is not large. [ABSTRACT FROM AUTHOR]

A reaction–diffusion Gierer–Meinhardt model of morphogenesis subject to Dirichlet fixed boundary condition in the one-dimensional spatial domain is considered. We perform a detailed Hopf bifurcation analysis and steady state bifurcation analysis to the system. Our results suggest the existence of spatially nonhomogenous periodic orbits and nonconstant positive steady state solutions, which imply the possibility of complex spatiotemporal patterns of the system. Numerical simulations are carried out to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family: z → f(C,A)(z, ...) ≡ z + z[sup 2] + C + A... where z is a complex dynamic (phase) variable, ... its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varied. For A = 0, the resulting two-real-parameter family is a familiar onecomplex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A ≠ 0, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of R[sup 2]. Saddle-node and period-doubling loci evolve from points on the main cardioid for A = 0 into closed curves for A ≠ 0. As A is varied further from 0 in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well. [ABSTRACT FROM AUTHOR]