The fibonacci sequence and the jacobian matrix in food web models

Abstract

Where t is generation class, produces a number sequence for an exponentially expanding population-1, 1, 2, 3, 5, 8, 13, ... , which projects, through time, mating pairs of rabbits and offspring over t monthly generations. Over time, the ratio between successive generations of Fibonacci’s rabbits ( 1 t t n n − ) converges to the golden ratio Φ (1.618...). The largest eigenvalue of this matrix equals Φ exactly, and is the exponentiated growth rate (r) of the population, that is, λ1=e r. The next largest eigenvalue equals φ (0.618...), where φ=1/Φ. We are concerned here with limits to growth for an ecological community that can be modelled electronically. The dynamics of n interacting species can also be described by Lotka-Volterra equations1 which can be applied to biosensors, but we focus here on a qualitative analysis with a quantitative outline. The purpose of this paper is to use matrix and graph theory to highlight the presence of the Fibonacci sequence in the adjoint of Jacobian matrices which can arise in simple food web models. We then utilize an application to define terms and show the place of the Fibonacci sequence in the development of the main ideas and more complex models.