Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Slides - University of Alberta
Document related concepts
Transcript
Predator-Prey Cycles and Effects of Climate Change Hao Wang Department of Mathematical and Statistical Sciences Centre for Mathematical Biology University of Alberta CO2 Temperature Global Climate Change Hypothesis Period Amplitude Incidence Predator-Prey Cycles Population Cycles Large groups Taxon Testing # Periodic # Fraction Bird 139 18 13% Mammals 328 109 33% Fish 129 56 43% Insects 79 13 16% Crustaceans 12 6 50% Gastropods 3 1 33% Bivalves 3 1 33% All 694 204 29% (Bruce Kendall, John Prendergast and Ottar Bjornstad 1998, Ecology Letters, 1: 160-164) Population Cycles in a Small Mammal - Lemmings brown lemming in Alaska collared lemming in Greenland Empirical Data lemming (prey) density stoat (predator) density (Olivier Gilg, Ilkka Hanski et al 2003, Science 302:866-868) Lemming Background The population numbers of lemming fluctuate greatly over a period of 3-4 years until now. Ecologists suspect that such oscillations are controlled by a trophic mechanism. Their principal summer foods are moss and vascular plants. Lemmings are eaten by many animals, like weasels, arctic fox and a bigger number dies when they try to cross rivers, lakes or seas. Trophic Mechanisms Brown lemmings in Alaska: bottom-up regulation (P.Turchin et al 2000, Nature). Collared lemmings in NE Greenland: topdown regulation (Olivier Gilg et al 2003, Science). Top-down Regulation The lemming cycle driven by predators For collared lemmings in NE Greenland, the major sources of mortality are predation by stoats, arctic foxes, snowy owls and long-tailed skuas. Only the stoat is a specialist. Lemming-Stoat Delayed Model lemming 2 dx x bx (1 (x/K ))/( 1 K ) y 2 2 Dx dt modified logistic lemming growth Holling Type III functional response stoat dy x2(t ) exp( dj) 2 2 y (t )d (x )y dt Dx(t ) lemming stoat growth hmaturation witdelay depende stoat death Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10 (2009) Functional Response Test (Olivier Gilg et al 2003, Science) Predation by stoat is modeled with Holling Type III functional response, which was used to incorporate a possible "refuge" for the lemming at very low densities. when lemmings are so dispersed, then they must become very hard to locate for the stoat. Lemming-Stoat Delayed Model lemming 2 dx x bx (1 (x/K ))/( 1 K ) y 2 2 Dx dt modified logistic lemming growth Holling Type III functional response stoat dy x2(t ) exp( dj) 2 2 y (t )d (x )y dt Dx(t ) lemming stoat growth hmaturation witdelay depende stoat death Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10 (2009) Stoat Maturation Delay The stoat maturation delay is about 3 months. The stoat juvenile/maturation death rate is chosen to be the maximum stoat death rate, 4/year. Lemming-Stoat Delayed Model lemming 2 dx x bx (1 (x/K ))/( 1 K ) y 2 2 Dx dt modified logistic lemming growth Holling Type III functional response stoat dy x2(t ) exp( dj) 2 2 y (t )d (x )y dt Dx(t ) lemming stoat growth hmaturation witdelay depende stoat death Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10 (2009) Prey Dependent Death Rate The stoat death rate depends on lemming density d ( x ) d ( d d ) ( 1 / 2 arcta b ( x N )) / ) h h l 1 c This is tested by Olivier Gilg from field. Lemming-Stoat Delayed Model lemming 2 dx x bx (1 (x/K ))/( 1 K ) y 2 2 Dx dt modified logistic lemming growth Holling Type III functional response stoat dy x2(t ) exp( dj) 2 2 y (t )d (x )y dt Dx(t ) lemming stoat growth hmaturation witdelay depende stoat death Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10 (2009) Modified Logistic Growth for the lemming Per capita growth rate 0.3 mammals Population density x Per capita growth rate b ( 1 ( x / K ) ) （ / 1 K ) (Richard M. Sibly et al and John D. Reynolds et al 2005, Science) Empirical Data Fitting Sensitivity Analysis Parameters dj,, , and are key factors for period. theIncreasing stoat maturatio death rate dj and delay ,or decreasing predation rate and conversion rate can enlarge the period. Compare the lemming cycle to the snowshoe hare cycle lynx hare Hare-Lynx Delayed System In general view, the snowshoe hare cycle is also controlled by predators like collared lemming in NE Greenland. Differences: (i) Holling Type II functional response; (ii) constant lynx death rate. dx x bx ( 1 ( x /K )) /( 1 K ) y dt D x dy x ( t ) exp( d ) y ( t ) dy j dt D x ( t ) 1 10-year period 1976 1996 Sensitivity Analysis Goal: 4<10 lemming Predation rate and hareconversion rate are comparable. lemming hare Lynx maturation death rate is less than that of stoat ( d ) ( d ) opposite directio lemming j hare whichjmakes the period of snowshoe hare cycle smaller than the period of lemming cycle. right hare Lynxlemming maturation delay isdirection 1.5 years, much larger than the stoat maturation delay, 3 months. This makes ‘4<10’ possible. Therefore the predator maturation delay is the key factor to generate different periods (4-year and 10year) of lemming and hare cycles. 38-year moose cycle for moose-wolf interaction Maturation Hypothesis Climate change leads to variation of predator maturation period. Who knows the evolution direction? Snow Melt Hypothesis Climate change results in more frequent freeze-thaw events leading to a more shallow and icy snow pack, which will probably act to expose small mammals to predators…… (Callaghan et al 2004, Ambio; Callaghan et al 1995, Phil. Trans. R. Soc. Lond. A) Lemming-Stoat Delayed Model predation rate increases with snow melt lemming 2 dx x bx (1 (x/K ))/( 1 K ) y 2 2 Dx dt modified logistic lemming growth Holling Type III functional response stoat dy x2(t ) exp( dj) 2 2 y (t )d (x )y dt Dx(t ) lemming stoat growth hmaturation witdelay depende stoat death Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10 (2009) Observation For collared lemmings in NE Greenland, period of the cycle is decreasing and amplitude is almost unchanged. Climate change has different effects on lemming cycles in different locations because of different mechanisms triggering them.
