RelativeResoufrceManager

RelativeResoufrceManager - Theoretical Ecology BIOL 434...

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1 Theoretical Ecology – BIOL 434 Dynamics of predator– prey interactions Readings: Case, chapters 12-13 The Lotka–Volterra model (Lotka 1925; Volterra 1926) exponential growth dN dt = rN dP dt = mP # $ % % & % % N = prey population size P = predator population size r = intrinsic rate of natural increase of prey m = density-independent mortality rate of predator mortality (exponential decline)
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2 The Lotka–Volterra model (Lotka 1925; Volterra 1926) exponential growth predation dN dt = rN α NP dP dt = β NP mP % & ' ' ( ' ' N = prey population size P = predator population size r = intrinsic rate of natural increase of prey m = density-independent mortality rate of predator = prey consumption rate of predator = predator production rate per prey consumed mortality (exponential decline) predation = εα where ε = conversion factor Lotka–Volterra model: Isocline analysis 1. Null isoclines of prey dN dt = N ( r P ) = 0 (1) N = 0 P = r (2) P N r/ 0 0
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3 Lotka–Volterra model: Isocline analysis 2. Null isoclines of predator dP dt = P ( β N m ) = 0 (1) (2) P = 0 N = m P N m/ 0 0 Lotka–Volterra model: Isocline analysis P N r/ α 0 0 m/ P N The predator–prey interaction acts as a form of indirect, delayed density dependence, which tends to generate population cycles
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4 Experiments of Gause (1934) Cyclic prey–predator fluctuations in nature
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5 Local stability analysis 1. Linearisation of the system around the equilibrium dX i dt = F i X 1 , X 2 ,..., X n ( ) i = 1,. .., n ( ) 0 ,..., * * 1 * = = n i i X X F F Equilibrium: Linear approximation around the equilibrium by Taylor expansion: dX i dt = F i * + F i X 1 * X 1 X 1 * ( ) + + F i X n * X n X n * ( ) = 0 a i 1 x 1 = perturbation from equilibrium Local stability analysis 1. Linearisation of the system around the equilibrium dx i dt = a i 1 x 1 + + a in x n d x dt = Ax A = a ij ( ) , x = x i ( ) A = Jacobian matrix = community matrix * * j i j i ij X dt dX X F a = = Effect of species j on the growth rate of species i Note : Case uses a different definition of “community matrix”
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