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Population Cycling.docx

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Department
Biology
Course
BIOLOGY 3SS3
Professor
Ben Bolker
Semester
Winter

Description
March 7 , 2014 Biology 3SS3: Population Ecology Population Cycling Cycling Populations ­ Many populations seem to cycle ­ Population densities increase and decrease more or less regularly ­ Over multiple generations/years ­ Name an example of a population that cycles • Small mammals, especially temperate/boreal (lynx/hare) • Epidemic diseases • Insects especially forest Lepidoptera • Red grouse ­ Interesting (but maybe overreported) Population Regulation ­ Population regulation is a necessary condition for cycling ­ Things must get worse/per capita growth rates must decrease as population  increases ­ We will not count the special case of non­overlapping cohorts in structured  populations What We Have so Far ­ Unregulated unstructured models: exponential growth or declined ­ Unregulated structured models: exponential growth or decline in long­term  (averaged across cohorts) ­ Regulated, unstructured models in continuous time • R<1: stable equilibrium at zero • R>1, no Allele effects: stable positive equilibrium • R>1, allee effects: unstable and stable positive equilibria Crossing ­ IF two populations are following the same deterministic rules ­ E.g. dN/dt = Nr(N) ­ And are in the same state ­ Then they must go tot the same place next ­ Trajectories can’t cross ­ Cycles are impossible What can Allow Cycles? ­ Discrete time ­ Age structure ­ Delayed effects (e.g. childhood crowding lowers adult fecundity) ­ Seasonal variation ­ Interaction with other populations (prey/depleatable resources, predators) ­ Regulation is always necessary ­ Give an example of one of these effects in real populations ­ Endogenous cycles: population cycles that are driven by stuff happening in the  biology ­ Exogenous: being forced or driven from outside Conceptual Model ­ Discrete­time, deterministic, unstructured, regulated ­ λ(B)=p(N)+f (N) ­ For simplicity we’ll assume p(N) = 0 ­ f(N) must decline as N gets large Mathematical Model ­ N T+1=λ ( )T T ­ Equilibrium when” • Nt+1 = N(T) • Lambda (N) = 1 ­ We will assume  λN=f exp⁡( −N ) (ricker model) 0 N c ­ N C= characteris−Nt/Ncale ­ N T+1=f 0 eT ­ What is R(0) for this model? ­ R(0)=N(0)=f 0 Simple Case ­ f  = 1.5, N  = 1 0 c ­ f0>1, so the population should grow initially
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