Study Questions for Final

An alternative formulation of the equations for competition among two species is as follows (from Renshaw 1991):

n₁[t+1] = a₁ n₁[t] / (1 + b₁ n₁[t] + c₁ n₂[t])

n₂[t+1] = a₂ n₂[t] / (1 + b₂ n₂[t] + c₂ n₁[t])

a_i is a measure of the growth rate of population i (equivalent to 1+r_i), b_i is a measure of the effects of intraspecific competition on growth rate, and c_i is a measure of the effects of interspecific competition on growth rate.

(a) Identify all the equilibrium states of the population.

(b) Three of these equilibria lack one or both species. Determine the conditions under which each of these three equilibria are stable.

(c) Gause (1932) studied competition in yeast (Saccharomyces cerevisiae and Schizosaccharomyces pombe) and estimated the following parameters: a₁ = 1.2439, a₂ = 1.0626, b₁ = 0.0188, b₂ = 0.0108, c₁ = 0.0591, c₂ = 0.0047. Using these numbers, determine whether any of the three equilibria studied in (b) is stable. If any equilibrium is stable, explain why the absent species does not spread in terms of the parameters of the model.