Did Something Change? Thresholds in Population Models

Abstract

Dedication: This paper is dedicated to our good friend Professor Willi Jäger on the occasion of his 60th birthday. The three of us have collaborated since a momentous year (for us) at the Courant Institute of Mathematical Sciences in 1969-70.Abstract: The goal of this article is to illustrate several interesting bifurcations that can arise in population biology. These are of interest since it is often through bifurcation phenomena that changes significant enough to be measured occur. For example, a minor change in some environmental parameter can cause a system to change from being at rest to oscillating. We illustrate here the role of several canonical types of bifurcations in population modeling. IntroductionThe subject of mathematical population biology is a vast one, much too large to be surveyed in a single article. Since the thrust of this volume is to present material in a way to make the subject attractive to students, we have chosen to focus on specific examples and to illustrate the phenomenon of change. This forms one of the connections between theories and experiments since experiments often focus on when a moderate change in a parameter produces a major change in the outcome. Such changes in the model's solutions can reflect the occurrence of a bifurcation of solutions and with it an exchange of stabilities. For example, a stable static state can bifurcate into a new stable state or a stable oscillation as model parameters change. We present here several examples of where bifurcation phenomena have been observed in population biology and how they have been used to explain, predict and control outcomes. A major proponent of this approach is Rene Thom [30] who used canonical forms of singularities to identify bifurcation phenomena in developmental biology. In [14] a systematic development of certain bifurcations that arise in neuroscience is presented, and there are many excellent texts that present bifurcation phenomena for mathematicians and engineers, for example [2, 8,

Penentuan Bifurkasi Hopf Pada Predator Prey

“…). Karena nilai eigen imajiner λ 1,2 = ±i 3 8 bagian real nol, berarti titik setimbang T c = ( 1 4 , 9 16 ) bersifat center. Dalam keadaan ini terdapat populasi prey dan juga populasi predator yang hidup secara berdampingan bersama-sama dalam satu lingkungan.…”

Divergence Criterion for Generic Planar Systems

“…In many instances, experiments can be designed to detect such thresholds to test a particular model and/or theory. We refer the reader to [11] for an expository article on bifurcations in mathematical biology. Many population models are described by planar dynamical systems, and simply detecting the existence of a Hopf bifurcation is not difficult.…”