[70 points) The principle of competitive exclusion in biology states that it is unlikely for two or more similar species to coexist in a common...

Unsure how to work with the equilibria found when n=3 and generalize to any n.

2. [70 points) The principle of competitive exclusion in biology states that it is unlikely fortwo or more similar species to coexist in a common environment. A version of the principle ofcompetitive exclusion can be demonstrated mathematically by developing a model of interactionbetween species. Let us denote the population of the various species by .121, :52, ..., at“. Each of thespecies imposes a somewhat different burden on the environment. We assume that the aggregateburden is a linear combination where on 3:» i] for all 1?. Then? the population growth is governed by the following set of equations: :i:1[i’-l = [#91 — ’i’i Flffllllmillli‘zill = [#32 — ”T2Flfl3lll3r‘2ill in“) : llgn _ ”inF'lflfilllfi-‘inill where .53.: > U and "It > U for all i. The iii’s represent the natural growth rates in the absence ofcrowding effects. The 'rfis represent the sensitivities of the growth rates to the aggregate growingeffect. In addition1 we will assume that 31' P33" 'l'i ”l'jfor all 1' .75 j. Find the equilibria of the system and discuss their stability. Explain why the system exhibits com-petitive exclusion. llint: Try woking through a = 3 or 4 first and then try to generalize.