Population modeling of marine mammal populations Lecture 1: simple models of population counts Eli Holmes National Marine Fisheries Service
nmfs.noaa.gov
Today s lecture topics: Density-independent growth Exponential growth Density-dependent growth Hockey Stick Logistic Growth Optimal sustainable population Depensation (Allee effects)
Some real population counts for long-lived species 15 x 104 Northern fur seals 10 5 Real data 4 x 104 California sea lions 3 2 λ > 1 0 0 10 20 30 40 50 Northern Resident Killer Whales up to 2000 250 1 0 5 10 15 20 25 African Elephants 300 200 200 150 100 0 5 10 15 20 25 30 Whooping Cranes 150 100 Exponential growth 100 1500 0 0 20 40 60 80 Wandering Albatross 50 1000 λ < 1 0 0 10 20 30 40 50 500 0 5 10 15 20 25 30
Classical models for population counts Numbers this year = Numbers last year + births - deaths Models differ in how these depend on population density DENSITY-DEPENDENCE
Classical models for population counts Exponential growth (or decline) Rate of population growth is not influenced by the number or density of individuals in the population. Density-independent Numbers this year = Numbers last year + births - deaths Births = b x Numbers last year Deaths = d x Numbers last year exponential increase For example, b=0.1 (10%) and d=0.2 (20%) Year 2010 2011 2012 2013 2014. Count 100 90 81 72 65. to 0 exponential decline
Classical models for population counts Exponential growth (or decline) Rate of population growth is not influenced by the number or density of individuals in the population. Density-independent Numbers this year = Numbers last year + births - deaths N t+1 = N t + (b-d)n t = 1+(b-d)N t = N t + rn t r is the intrinsic rate of increase r>0 the population is increasing r<0 the population is decreasing (1+r) = λ λ >0 the population is increasing λ <0 the population is decreasing exponential increase exponential decline
Classical models for population counts Hockey Stick Rate of population growth is not influenced by the density of individuals in the population until it hits some threshold. Density-dependent N t+1 = N t + rn t N t+1 = N t + r exponential linear K carrying capacity Until N reaches K then linear N t+1 = K exponential
Classical models for population counts Logistic Growth Model Rate of population growth is influenced by the density of individuals in the population at all densities. Densitydependent N t+1 = N t + rn t (1-[N t /K]) K carrying capacity Compare to exponential sigmoidal N t+1 = N t + rn t exponential
Exponential versus logistic growth Numbers Time
6 Effects of growth rate ( r ) on the shape of the logistic curve big r Small r S. Holt
5 Comparing exponential and logistic curves Number The per capita rate of increase is lower here The change in N is highest here This is where the net productivity is highest The per capita rate of increase is highest here
Optimum sustainable population (OSP) OSP: Key management element of the Marine Mammal Protection Act of 1972: the number of animals which will result in the maximum productivity of the population or the species, keeping in mind the carrying capacity of the habitat and the health of the ecosystem
Optimum sustainable population* Working definition developed by the National Marine Fisheries Service: Population size between MNPL and K, where MNPL is the population size that produces the maximum net productivity level K MNPL OSP *This is a concept from the field of Fisheries. In non-fisheries fields (Conservation Biology, Ecology), this concept is not taught nor used. A K/2 decline is reason to consider a species for the IUCN Red List!
Optimum sustainable population (OSP) depends on the shape of densitydependence 10 N t+1 = N t + N t r(1-[n t /K] z ) z<1, r big = high growth rate at low population size, but higher density-dependence at high population size z>1, r small = lower growth rate at low population size, but less density-dependence at high population size Numbers OSP z=0.5 r=0.32 z=1 r=.2 z=2 r=0.15 OSP Different shapes of densitydependence lead to significant changes OSP! Time (yrs)
Optimum sustainable population (OSP) depends on the shape of densitydependence 10 N t+1 = N t + N t r(1-[n t /K] z ) z<1, r big = high growth rate at low population size, but higher density-dependence at high population size z>1, r small = lower growth rate at low population size, but less density-dependence at high population size Numbers OSP z=0.5 r=0.32 z=1 r=.2 z=2 r=0.15 OSP Marine mammals densitydependence tends to look like the red curve. Time (yrs)
11 Depensation (Allee effects) = reduced per capita production at small population size Per capita production (N t+1 N t )/N logistic Numbers threshold Pop. Growth rate exponential
11 Depensation (Allee effects) = reduced per capita production at small population size Potential causes include inbreeding depression difficulty finding a mate intensive predation (same # of predators but fewer prey) lack of social cohesion (esp. for social hunters) or just chance due to small numbers (demographic stochasticity) Numbers threshold
13 Severe reduction in population size can lead to the extinction vortex Where low densities produce a significant risk of extinction if the population becomes depleted There is much current research trying to determine how to define these extinction vortex thresholds. Per capita growth Extinction vortex threshold N J.D. Flores
Monday s lecture topic: Age-structures models of marine mammals How we figure out what s going wrong with a population and how best to reverse declines
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