Unlimited growth, carrying capacity, and limited growth

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Unlimited growth, carrying capacity, and limited
growth
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Models of population growth

• This is the simplest model of population growth
  for species with discrete breeding seasons.
• In this model, there is no competition, and
  population dynamics are governed solely by the
  net reproductive rate, R.
• If R gt 1, the population increases indefinitely
  and exponentially.

Nt1 NtR Or Nt N0Rt
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Models of population growth incorporating
competition

• Graphically, we can see that the population
  increases exponentially when Nt is very low.
• But the rate of increase declines as population
  size rises.
• At carrying capacity, the growth rate is zero.
• Above carrying capacity, the population will
  decline.
• K is therefore a stable equilibrium.

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St. Matthew Island, Alaska
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Reindeer on St. Matthew Island, Alaska

• In 1944, 29 reindeer introduced to St. Matthew
  Island (300 km2)
• Approximate initial density 0.1/km2
• 24 females, 5 males, all 2 years old

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Reindeer on St. Matthew Island, Alaska

• R. Rausch visited the island in 1954, and on the
  basis of counts, estimated the population size at
  400-500.
• C.J. Rhode visited the island in 1955, and
  estimated the population size at 700-900.

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Reindeer on St. Matthew Island, Alaska

• David Klein visited the island in 1957, and made
  a total count of 1,350 animals.
• This implies an average annual growth rate of 34
  percent.
• Klein assumed that the population growth rate
  earlier in the explosion must have been near the
  theoretical maximum for the species.

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Reindeer on St. Matthew Island, Alaska

• Population growth during this period looks like
  unlimited growth.
• Klein recognized the potential importance of this
  study during his 1957 visit.

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Natural mortality was assessed from skeletons
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Physical condition was assessed from animals shot
during fieldwork
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Physical condition was assessed from animals shot
during fieldwork
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Physical condition was assessed from animals shot
during fieldwork
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At a density of 4.5 inds./km2, the animals were
in excellent condition

• Noticeable, extensive fat deposition, especially
  on large males
• Weights of all reindeer collected exceeded the
  average weight range for other Alaskan reindeer
• No external parasites noted
• Very large and uniform antler growth on males and
  females

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What contributed to the unlimited growth and
excellent condition of reindeer on St. Matthew
Island?

• Abundant winter and summer forage
• No competitors
• No large predators
• No large herbivores had been there previously
• But Klein sensed there was trouble on the horizon

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Signs that limits to population growth were
imminent in 1957

• Lichen beds were showing signs of fracturing due
  to overgrazing and trampling (winter range)
• Prostrate willows were also showing signs of
  heavy browsing (summer range)
• Calf percentage of 26 was below the indicated
  level of previous years
• Klein concluded/warned that the population
  decline may be rapid after the peak is reached.
  
  
  
  
  
  
  
  
  

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What happened next

• Klein revisited the island in 1963 and surveyed
  it with 2 Coast Guard helicopters.
• As their boots hit the shore, they saw reindeer
  tracks, reindeer droppings, bent-over willows,
  and reindeer after reindeer. Ned Rozell,
  Alaska Science Forum
• The survey revealed the population had increased
  to 6000
• Calf percentage was lower than in 1957
• Recruitment was down from 29 in 1957 to 17 in
  1963
• There was ample evidence of overpopulation, and
  the stage was apparently set for wholesale
  die-off.

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What happened next

• May 1964 an aerial survey of the island located
  no reindeer. We were unaware, of course, that a
  die-off had already taken place.

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The introduction, increase, and crash of reindeer
on St. Matthew Island Klein, D.R. 1968. J.
Wildl. Manage. 32350-367.

• Upon returning in 1966, Klein found only 42
  reindeer
• Of these, 1 male the rest were females 2yrs old
  and older
• No calves or yearlings, indicating the crash
  die-off probably occurred in late winter 1964.

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What caused the crash die-off?

• Extremely high density (20/km2)
• Unusually harsh winter in 1963-64 (exceptionally
  cold, with unusually deep snow)
• Long bones of examined skeletons contained no
  marrow fat, indicating starvation
• Many skeletal remains were found in groups,
  suggesting the animals died over a very short
  period.
• By the mid 1980s, there were 0 reindeer on the
  island.

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Sex and age composition of the die-off
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Compare natural mortality (1957) with crash
die-off (1966)
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• Physical characteristics of the animals in 1957
  and 1963
• Avg body weight declined by 38 for adult females
  and by 43 for adult males
• Not only were they smaller just before the crash,
  regressions between body weight and skeletal
  parameters indicated growth rates were lower in
  1963
• Lichens had been completely eliminated as a
  significant component of the winter diet

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Carrying capacity

• Klein (1968) suggested that forage quantity
  primarily governs population size, while quality
  determines the size of the individual.
• The winter component governs the upper limit of
  the population, and the summer component
  determines the stature of the individual.

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Klein (1968) attributed the large-scale die-off
to the following factors

• Overgrazing of lichens, with no possibility of
  the reindeer expanding into alternative range
• Excessive density of reindeer competing for a
  very restricted winter resource
• Relatively poor condition of reindeer going into
  the winter of 1963, resulting from intense
  competition
• Extreme weather conditions, primarily deep snow,
  during the winter of 1963-64.

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Intraspecific competition and carrying capacity

• Competition may be defined as (Begon et al.
  1984)
• An interaction between individuals, brought about
  by a shared requirement for a resource in limited
  supply, and leading to a reduction in the
  survivorship, growth, and/or reproduction of the
  competing individuals.

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Effects of competition on individuals

• Increased energy expenditure (searching for the
  unexploited resource), increased risk of
  mortality, and decreased rate of food intake may
  all decrease individuals chances of survival

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Effects of competition on individuals

• Increased energy expenditure and decreased food
  intake may leave less energy available for
  development and less available for reproduction.
• Increases in density will therefore decrease the
  contribution made by each individual to the next
  generation.

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Common features of intraspecific competition

• The ultimate effect of competition is a decreased
  contribution to the next generation
• Intraspecific competition leads to decreased
  rates of resource intake per individual,
  decreased rates of individual growth or
  development, or to decreases in the amounts of
  stored reserves
• These may lead to decreases in survival and/or
  fecundity.
• Evidence from St. Matthew Island?

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Common features of intraspecific competition

• The resource for which individuals compete must
  be in limited supply
• Competing individuals might or might not interact
  directly
• Exploitation competition occurs when individuals
  remove an item needed by others
• Interference competition occurs when individuals
  interact directly and prevent others from
  occupying a portion of habitat and exploiting its
  resources
• Which type presumably occurred on St. Matthew
  Island?

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Common features of intraspecific competition

• The competing individuals are in essence
  equivalent, but in practice they are not
• One-sided reciprocity or Asymmetric
  competition
• The effects of competition are not the same on
  all individuals in the population
• Evidence of asymmetry on St. Matthew Island?

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Common features of intraspecific competition

• The likely effect of competition on any
  individual is greater the more competitors there
  are.
• The effects of intraspecific competition are thus
  said to be density dependent.