ecology

ecology Data Analysis Module DA  Figure  1  illustrates  stochastic  variation  in  births and deaths in a population of killer whales (orcas) ( Orci-nus  orca)  that  reside  off  the  coast  of  British  Columbia, Canada. These data are reproduced in the Excel spread-sheet that accompanies this module as well. You can see that although the number of reproductive females in the population  varied  between  16  and  28  individuals,  the number of young born each year varied between 0 and 8. Annual deaths varied between 0 and 7. Step 1: Calculate the average rate of population change and its standard deviation. Using  the  numbers  of  births  and  deaths  and  the  initial population  size  of  73  individuals  in  1974,  calculate  the changes  in  population  size  in  the  Excel  spreadsheet through 2001. The average exponential rate of population growth (r) during year  i is calculated from the number of individuals at the beginning and at the end of the year as r i  ln   N N i i 1   Remember that the exponential growth rate (r) is equal to the natural logarithm of the geometric growth rate  (see page 225). •  Did  the  population  increase  or  decrease  over  this period? Was the average exponential growth rate positive or negative? To  explore  how  stochastic  environmental  variation affects the size of a population and the probability of its extinction, we need to develop a model based on random changes in the factors that determine changes in popula -tion size. Such models often include an upper limit to the size of the population to incorporate the effect of density dependence.  However,  in  the  simplest  case,  the  expo-nential rate of increase of a population is independent of its size (density-independent) and has a mean of  r and a standard deviation of S. Thus, on average, the population grows at an exponential rate of  r, but during some peri -ods the growth rate is above this level, and during other periods it is below this level. If a population experiences many  periods  of  below-average  growth,  it  runs  the  risk of  extinction.  The  critical  parameter  that  influences  the average time to extinction under random environmental variation is the variance in r. The variance is the square of the standard deviation, or S 2 . In the special case in which population size is, on average, balanced ( r  0; that is, births equal deaths under Stochastic Extinction with Variable Population Growth Rates The random nature of births, the number and sex of off-spring,  and  particularly  deaths  can  lead  to  variation  in population  size  even  in  a  constant  environment.  Such stochastic  variation  generally  is  not  a  problem  for  large populations  because  these  chance  events  average  out over  many  individuals.  However,  small  populations  can suffer from random variations in births and deaths, which can lead to random variation in population size and even to extinction. The  kakapo  ( Strigops  habroptilus )  is  a  large,  flightless parrot  found  only  in  New  Zealand.  Because  it  is  flight-less,  it  is  vulnerable  to  introduced  predators  such  as cats,  opossums,  and  weasels.  By  1976,  only  14  kakapos were known to be alive on New Zealand’s South Island; sadly, all of them were males. If males and females were hatched  with  equal  frequency  on  average,  what  is  the probability that 14 individuals would include no females? If each individual is considered a trial, and being male is considered a success with probability (p)   0.5, then the probability that  n trials will all be successes is ( p n ). If you think this is unlikely, why might the New Zealand Wildlife Service have failed to locate any female kakapos? Fortu-nately, additional kakapos were later discovered on Stew-art Island, at the southern tip of South Island, and kakapos were then introduced onto two small islands from which all  predators  had  been  removed.  (For  more  on  this  fas-cinating  bird,  see  http://en.wikipedia.org/wiki/Kakapo; http://animaldiversity.ummz.umich.edu/site/accounts/ information/Strigops_habroptila.html.) When the environment varies, all individuals in a pop -ulation  can  be  affected  in  the  same  way,  and  dramatic changes, even in large populations, can result. All members of a population feel a prolonged drought or a cold snap. Conversely, a period of exceptionally favorable conditions may  increase  the  fecundity  of  all  individuals  or  increase their  probability  of  survival.  The  kakapo,  for  example, breeds primarily in years when the rimu tree, an endemic conifer of the podocarp group, produces good fruit crops. Variations  in  environmental  conditions  may  be  random and essentially unpredictable —what is referred to as sto-chastic  environmental  variation—or  they  may  occur  with some  regularity.  Understanding  the  connection  between changes  in  the  environment  and  changes  in  population size can suggest interventions, such as supplemental feed -ing during critical periods of limited food supply, that can reduce the chance of population decline and extinction. Data a nalysis MoD ule page 2 Data Analysis Module average conditions), the average time to extinction ( T ) of a population of size N is T(N )   S 2 2 ln(1  S 2 N )  1 Step 2: Estimate the average time until extinction for a population of killer whales. Assume that for our killer whale population, the average growth rate (r) is 0. •  Starting with the population size in 2001, what is your estimate of the time to extinction? •  How does T(N ) change with the size of the initial popu -lation and with the variance in the rate of change in popu -lation size? •  Fill in the expected times to extinction for the range of population sizes (N ) and standard deviations of the popu -lation growth rate ( S ) in DA Table 1. Assuming  that  the  time  units  are  years,  these  values suggest that small populations in particular have relatively short life expectancies. •  What would T(N ) be for the killer whale population at its largest and smallest sizes? If a population grows just by chance,  does  this  mean  that  its  prospects  for  long-term persistence  improve?  Assume  that  the  sample  standard deviation of r in the spreadsheet accurately estimates the underlying value of S. The  implication  of  this  model  for  conservation  is clearly that populations should be managed to maintain as large a population size as possible and to prevent strong depression in the population growth rate during periods of poor environmental conditions. The latter strategy might Age and sex structure of population Births and deaths 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2 000 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2 000 2001 10 15 20 25 30 35 40 45 Deaths Number Births (a) (b) KEY Calves and juveniles Adult males Reproductive females Births Postreproductive females DA Figure 1  Stochastic variation in births and deaths of a population of killer whales. Data from Taylor and Plater, 2001. Initial population size S 10 100 1,000 10,000 0.05 0.1 0.2 0.5 Da table 1   Calculating time to extinction page 3 Data Analysis Module involve supplemental feeding or predator and pathogen control programs at critical times. The model described here lacks density dependence. Normally, ecologists believe that the growth potential of populations reduced to low numbers is greatly increased during  normal  conditions,  which  should  allow  them  to increase and draw back from the brink of extinction. This is a fundamental message of the logistic equation and one of the most basic foundations of ecology. •  If this were always the case, why should we be worried about  small  populations?  Under  what  conditions  might you expect a population not to increase when reduced to low numbers? This certainly has been the case for many endangered species that have become extinct or now tee-ter on the brink of extinction. Do some populations simply not “have what it takes” to maintain healthy numbers? Let’s  consider  the  effects  of  adding  normal  density dependence  to  models  incorporating  stochastic  environ-mental variation the intrinsic exponential growth rate, r 0 . According to the logistic equation, the average growth rate of populations reduced below their usual carrying capac-ity always exceeds r   0, and such populations tend to recover quickly. But a long series of unfavorable periods might still be enough to drive population size below 1 indi-vidual and cause extinction. The important parameters for predicting the average time to extinction in models with density dependence are, first, the product of the average value of  r 0 and K, and second, the ratio of the standard deviation of r 0 to its mean; in other words,  S/r 0 . The equa -tions for time to extinction under density dependence are messy, but as you would expect, the addition of negative density  dependence  greatly  increases  the  expected  time to extinction. For example, when N   100,  r 0  0.1, and S    0.22;  T(100)  is  equal  to  nearly  26,000  time  units, rather than the value of about 81 (see DA Table 1) in the absence of density dependence. Literature Cited Taylor, M., and B. Plater. 2001. Population viability analysis for the southern  resident  population  of  the  killer  whale  ( Orcinus  orca). The  Center  for  Biological  Diversity,  Tucson,  Arizona  (http:// www.biologicaldiversity.org/swcbd/species/orca/pva.pdf). PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)