Population Growth Models: Unlimited and Limited Growth Rates, Papers of Forestry

Download Population Growth Models: Unlimited and Limited Growth Rates and more Papers Forestry in PDF only on Docsity! Population Models • Introduction to population growth models – Unlimited resources, density independent growth • Deterministic models, simple exponential growth • Stochastic unlimited growth – Limited resources, density dependent growth • Hood River winter steelhead (Oregon) • Estimated relative fitness for hatchery- reared adults reproducing in wild over several generations using microsatellite pedigree analyses. Araki et al. (2007): Genetic effects of captive breeding Araki et al. (2007): Genetic effects of captive breeding Effect of domestication on reproductive success? How does hatchery rearing alter adult life history traits and behavior affecting fitness? Araki et al. (2007) ~40% in relative fitness / generation in captivity when returned to the wild! -Will this result be general for all animals? Limited growth • Last time we talked about unlimited growth where b and d did not vary or varied randomly around a constant value. • Populations don’t grow indefinitely Growth rates • In unlimited environments, birth rate is constant • death rate is constant • r = b - d • dN / dt = f(N) N • Exponential growth f(N) = r • dN / dt = rN Rates: unlimited growth t R at eN N Nt = Noert = Noe(b-d)t d b Limited growth • Hastings (1997) Figure 4.1, Dynamics of sheep numbers in Tasmania after introduction (1820-1940). • Hastings (1997) Figure 4.5 E. coli from experiment of McKendrick (1911). Limited growth • First, growth is exponential • Then growth rate slows down as population increases (grow rate deccelarates) • and then the population size fluctuates around a an apparent equilibrium population size – Why? Growth rates • birth rate declines • and/or death rate increases • One or more limiting resources at higher population size Limited growth • Rev. Thomas Malthus (1798), An Essay on the Principles of Population • Human population grows expoentially, doubling ~30 years • Food supply increases arthimetically Limited growth • Rev. Thomas Malthus (1798), An Essay on the Principles of Population • Human population grows expoentially, doubling ~30 years • Food supply increases arthimetically • Verhulst (1800): environment is limited so there is some maximum number of organisms that can be supported in a area • = K (the carrying capacity) Caughley and Sinclair 1994 • What biological mechanisms explain logistic growth? • Suggest logistic growth results from animals consuming a renewable resource where: – animals have no influence on rate of renewal – animals consume the “interest” (excess production) – don’t consume the “capital” (basis for production) Caughley and Sinclair 1994 • i = satiating intake day-1 • g = production of resource ha-1 day-1 • b = maintenance intake individual-1 day-1 • N = no. individuals / ha • proportion of resource channeled into maintenance and replacement = bN/g • leaving rest, 1-(bN/g), for population growth Caughley and Sinclair 1994 • when production is in excess, g/N > i, then • dN / dt = rm N • when g/N < i, that is production is less than intake, then dN / dt = rm N (1-bN/g) • If we set g/b (production/per captia maintenance )= K then we get : dN / dt = rm N (1-N/K) Estimating Parameters • Dennis and Taper (1994) suggested a better way to estimate: • rt = ln (λt) = ln (Nt / Nt-1) = a + bNt • Do a regression of ln λt on Nt to statistically test for evidence of density dependence Estimating Parameters Nt ln (Nt / Nt-1) rt b a Logistic Growth Assumptions • 1) The population starts with a stable age distribution • 2) Density is measured in appropriate units • 3) There is a real attribute of the population corresponding to r (or rmax) Assumptions: Age distribution F0 F1 F2 F3 F4 F5 P0 0 0 0 0 0 0 P1 0 0 0 0 0 0 P2 0 0 0 0 0 0 P3 0 0 0 0 0 0 P4 0 A =Nt= N0 N1 N2 N3 N4 N5 Where Fx = f (N) and / or Px = f (N) Assumptions • 4) Relationship between density and rate of increase per individual is linear – Fowler’s work suggested that a non-linear relationship is more appropriate for large mammals – Non-linear models are commonly used: • The Beverton-Holt and Ricker spawner-recruit models used in fisheries assume two different non- linear relationships between r and N Theta logistic model – A more general form of the logistic model is the Theta logistic model (Ayalla 1973) – Logistic model is a special case assuming a linear relationship between r and N – Non-linear relationships modeled by adding a parameter (theta) that describes the shape of the relationship. Assumptions • 5) No time lags • 6) K does not change through time • 7) Population is large and the environment is constant so that there are no stochastic or random effects (i.e., no demographic or environmental stochasticity) • 8) Population grows continuously with overlapping generations Time lags? • Robert May (1974), in a famous paper in Science (186:645-647) explored implications of a discrete time version: Nt+1 = Nt e[r(1-Nt /K)] New individuals don’t appear until next time step (i.e., annual breeding cycle) • Simply adding a time lag produced some very surprising results May (1974) • when r < 1.0, smooth monotonic (only increasing) logistic population growth t N t/ K 1.0 May (1974) • when 1.0 < r < 2.0, logistic growth with damped oscillations settling to K Nt / K May (1974) • when 2.0 < r < 2.69, population shows stable limit cycles Nt / K 4 point cycle 2 point cycle May (1974) • when r > 2.69, populations exhibit chaos May (1974) • when r > 2.69, populations exhibit chaos – Chaotic dynamics: predictable, deterministic, and repeatable, but dependent on initial conditions May (1974) • when r > 2.69, populations exhibit chaos – Chaotic dynamics: predictable, deterministic, and repeatable, but dependent on initial conditions Nt / K Nt / K = 0.075 Nt / K = 1.5 May (1974) • Deterministic processes producing patterns of population dynamics that look like random processes • While r > 2.69 (λ > 14.9) unlikely, chaos can occur at lower values of r when models assume time lags, non-linear dynamics etc., • May be very difficult to distinguish stable limit cycles, chaos, and stochastic processes in populations, but some evidence for all three.