Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction

Abstract

SUMMARYIn this paper a reaction-diffusion model describing two interacting pioneer and climax species is considered. The role of diffusivity and forcing (stocking or harvesting of the species) on the nonlinear stability of a coexistence equilibrium is analysed. The study is performed in the context of a new approach to nonlinear L 2 -stability based on the analysis of stability of the zero solution of a suitable linear system of ordinary differential equations. Theorems concerning the effect of forcing and diffusivity on the dynamics are established and stability-instability thresholds for the system are obtained. An example to illustrate the practical use of the results is also provided.

Dynamics of a Discrete-time Pioneer–climax Model

“…After the many initial studies into the basic dynamics of pioneer-climax models and the potential for reversing bifurcations and restoring stability in those models (e.g., Kim and Marlin (1999); Selgrade and Namkoong (1990); Selgrade and Roberds (1997); Sumner (1996)), research on pioneerclimax systems has indeed made the shift to spatiotemporal models, but it has largely focused on reaction-diffusion models. Much of this research has focused on proving the existence of traveling waves (Brown et al, 2005;Buonomo and Rionero, 2009;Cao and Weng, 2017;Huang and Weng, 2013;Yu et al, 2013;Yuan and Zou, 2010). Other studies have looked at Turing and Hopf bifurcations in reactiondiffusion systems (Buchanan, 1999;Liu and Wei, 2011).…”

Dynamics of a discrete-time pioneer–climax model

“…After the many initial studies into the basic dynamics of pioneer-climax models and the potential for reversing bifurcations and restoring stability in those models (e.g., Kim and Marlin 1999;Selgrade and Namkoong 1990;Selgrade and Roberds 1997;Sumner 1996), research on pioneer-climax systems has indeed made the shift to spatiotemporal models, but it has largely focused on reaction-diffusion models. Much of this research has focused on proving the existence of traveling waves (Brown et al 2005;Buonomo and Rionero 2009;Cao and Weng 2017;Huang and Weng 2013;Yu et al 2013;Yuan and Zou 2010). Other studies have looked at Turing and Hopf bifurcations in reaction-diffusion systems (Buchanan 2005;Liu and Wei 2011).…”

Traveling Wavefronts of Competing Pioneer and Climax Model with Nonlocal Diffusion

“…There are some existing results about the stability and traveling wave solutions for (2) ( [4][5][6]). About traveling wave solutions, Brown et al [4] studied the traveling wave of (1) connecting two boundary equilibria by singular perturbation technique, and Yuan and Zou [6] obtained the existence of traveling wave solutions connecting a monoculture state and a coexistence state by upper-lower solution method combined with the Schauder fixed point theorem.…”

“…Selgrade and Roberds [9], Sumner [10] analyzed the Hopf bifurcation of (5), and Selgrade and Namkoong [11], Sumner [12] considered the stable periodic behavior of (5). Because of the existence of rich equilibria and the various ranges of parameters, the dynamics of ordinary differential system (5) are complex, and a detailed review of all equilibrium types can be found in Buchanan [13,14]. Although the Laplacian operator Δ := 2 / 2 is always used to model the diffusion of the species, it suggests that the population at the location can only be influenced by the variation of the population near .…”