A short proof of the Cushing‐Henson conjecture

Abstract

We give a short proof of the Cushing-Henson conjecture concerning Beverton-Holt difference equation, which is important in theoretical ecology. The main result shows that a periodic environment is always deleterious for populations modeled by the Beverton-Holt difference equation.

Global behavior of solutions of a periodically forced Sigmoid Beverton–Holt model

“…The dynamics and properties of both equations have been studied in great detail in recent literature (see, for example [3][4][5][6][9][10][11][12][13][16][17][18][19][20][21][22][23][24][25][26]30,36,39])…”

Periodicity of a second-order switched difference system over integers

“…The first and second equalities of (13) show that the trace of the product is the product of the traces, thus it follows by (12) and (13) that the leading coefficient of trðMÞ is Q k j¼1 ðF n j þ1 À F n j Þ ¼ F n 1 À1 F n 2 À1 . .…”

“…Especially, the periodic property of difference equations or systems had been widely investigated, such as [10][11][12][13][14][15][16][17]. Mathematically, the subsystems of a switched system are often depicted by a collection of indexed differential or difference equations.…”

On the stability of some systems of exponential difference equations

“…A large number of papers dealing with such or related equations have been published (see, e.g, [6,16,[22][23][24]). In [34], the following model incorporating litter inhibition is discussed:…”