Lecture%2015%20first%2012%20slides1

Lecture%2015%20first%2012%20slides1 - N 1 N 2 K 2 K 2...

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Interspecific competition: theory Competitive exclusion (Gause): Two species cannot coexist if their niches are too similar
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Competing species can coexist if their niches are different enough
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" N t = rN ( K # N K ) Logistic growth equation dN dt = 1.0 N Exponential growth Logistic growth dN dt = 1.0 N 1,500 N 1,500 K = 1,500 0 5 10 15 0 500 1,000 1,500 2,000 Number of generations Population size ( N )
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" N 1 t = rN 1 ( K 1 # N 1 K 1 ) N 1 t = r 1 N 1 ( K 1 # N 1 #$ 12 N 2 K 1 ) N 2 t = r 2 N 2 ( K 2 # N 2 #$ 21 N 1 K 2 ) Single species logistic population growth Now add a competitor (species 2) that lowers population growth Where α 12 is the effect of an individual of species 2 on species 1
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N = population size t = time K = carrying capacity r = intrinsic rate of increase α = competition coefficient " N 1 t = 0 When r 1 = 0, or N 1 = 0 N 1 t = r 1 N 1 ( K 1 # N 1 #$ 12 N 2 K 1 ) N 1 = "# 12 N 2 + K 1 When does ? K 1 " N 1 "# 12 N 2 = 0 Or when: Which can be rearranged as:
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N 1 N 2 N 1 = "# 12 N 2 + K 1 K 1 K 1 " 12 ( y = mx + b ) Zero growth isocline, species 1
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N 1 N 2 N 2 = "# 21 N 1 + K 2 K 2 K 2 " 21 Zero growth isocline, sp. 2
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Unformatted text preview: N 1 N 2 K 2 K 2 " 21 K 1 K 1 12 Scenario 1: species one goes to carrying capacity, competitively excludes species two. N 1 N 2 K 2 K 2 " 21 K 1 K 1 12 Scenario 2: species two goes to carrying capacity, competitively excludes species one. N 1 N 2 K 2 K 2 " 21 K 1 K 1 12 Scenario 3: stable coexistence of both species N 1 N 2 K 2 K 2 " 21 K 1 K 1 12 Scenario 4: Unstable coexistence. Any perturbation from the equilibrium point leads to competitive exclusion of one species or the other. K 1 > " 12 K 2 and K 2 > 21 K 1 Conditions for stable coexistence K 2 21 > K 1 and K 1 12 > K 2 N 1 N 2 K 2 K 2 21 K 1 K 1 12...
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Lecture%2015%20first%2012%20slides1 - N 1 N 2 K 2 K 2...

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