lecture+13+competition-prednew

# Species from using resource modeling competition

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species from using resource)

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Modeling Competition Start with the logistic equation for a single species: dN/dt = r 0 N(K-N/K) Where r 0 is the rate of increase when population size is small (and resources aren’t limiting), N is the population size, and K is the equilibrium population size. This equation has an equilibrium population (dN/dt = 0) when N = K. This equation is readily modified to account for the effect of a competitor: dN 1 /dt = r 1 N 1 (K 1 -N 1 - a N 2 )/K 1
Competition (continued) dN 1 /dt = r 1 N 1 (K 1 -N 1 - a N 2 )/K 1 Where the subscript makes clear which of the two competing populations is being modeled, N 1 is the population size of the species we’re focusing upon, N 2 is the population size of species 2 (the competitor), a is a coefficient that expresses the effect of an individual of species 2 on the growth of species 1. For example, if a=2, then an additional individual of species 2 has twice the effect on the growth of species 1’s population as does an additional individual of species 1.

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dN 1 /dt = 0 where K 1 = N 1 , or where K 1 = a N 2 , or at any combination of N 1 and N 2 that’s equivalent to K 1
The population growth of species 2, the other competitor, can be described by: dN 2 /dt = r 2 N 2 (K 2 -N 2 - b N 1 )/K 2 Which is equivalent to the previous equation, substituting b (the effect of species 1 on species 2) for a . Can calculate the equilibria for species 2 similarly to those for species 1; dN 2 /dt = 0 where K 2 = N 2 , or where K 2 = b N 1 , or at any combination of N 1 and N 2 that’s equivalent to K 2 .

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