Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain

The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

Our Shadow Immigration System

This chapter explains how legal and institutional developments in immigration enforcement coincided with the dramatic acceleration of illegal immigration during the final third of the twentieth century. Together, these legal and demographic phenomena gave rise to a massive shadow immigration system that today operates alongside the formal immigration regime. This shadow system has rendered Congress’s intricate, detailed code of immigration rules increasingly less central to defining the content and character of the immigrant population. Instead, the Executive’s enforcement judgments—decisions about whom to target from the pool of deportable immigrants—have taken center stage. Indeed, the rise of the shadow system has effectively delegated vast screening authority to the President and other executive branch officials—authority that has culminated in events as dramatic as President Barack Obama’s Deferred Action for Childhood Arrivals (DACA). The large number of unauthorized immigrants living in the United States today amplifies the role of enforcement discretion and further entrenches the shadow immigration system.

Executive Governance Through Enforcement

chapter grapples with the risks associated with executive governance through enforcement, tracing them to core and undisputed executive powers whose reach has been magnified in immigration law by the emergence of the shadow system. Two interrelated features of executive governance should prompt vigilance. First, when executive branch officials pursue a policy agenda through their management of the enforcement bureaucracy, discretionary decision-making drives their choices about how to threaten or wield force, as well as offer forbearance. It is natural to worry that this discretion will lead to the lawless, arbitrary exercise of power. Second, discretionary decision-making processes are often opaque. This feature exacerbates the possibility of abuse and makes it difficult to hold government power accountable. After defining these risks, the chapter then focuses on how the domain of de facto delegation can be structured to preserve the virtues of executive governance while promoting rule of law values. It offers a qualified defense of the centralized, political control of enforcement discretion as a means of disciplining the Executive’s awesome power.

Positive Solutions of a Predator–Prey Model with Additive Allee Effect

This paper deals with the positive solutions of a predator–prey model with additive Allee effect under Neumann boundary conditions. By applying the bifurcation theory, we provide a proof of the existence of local bifurcation solutions and describe the global behavior of these solutions. The result shows that the bifurcation curves can be extended infinitely along [Formula: see text] in the one-dimensional case. Moreover, the limiting behavior of the steady states is clarified using a shadow system approach. It appears that the shadow system exists with a positive solution and it can go into a terminal point when the parameter [Formula: see text] is sufficiently large.